Cyclic dominance in evolutionary games: A review
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paper_title: Resolving social dilemmas on evolving random networks

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We show that strategy-independent adaptations of random interaction networks can induce powerful mechanisms, ranging from the Red Queen to group selection, which promote cooperation in evolutionary social dilemmas. These two mechanisms emerge spontaneously as dynamical processes due to deletions and additions of links, which are performed whenever players adopt new strategies and after a certain number of game iterations, respectively. The potency of cooperation promotion, as well as the mechanism responsible for it, can thereby be tuned via a single parameter determining the frequency of link additions. We thus demonstrate that coevolving random networks may evoke an appropriate mechanism for each social dilemma, such that cooperation prevails even in highly unfavorable conditions.
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paper_title: Rock–scissors–paper and the survival of the weakest

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In the children9s game of rock–scissors–paper, players each choose one of three strategies. A rock beats a pair of scissors, scissors beat a sheet of paper and paper beats a rock, so the strategies form a competitive cycle. Although cycles in competitive ability appear to be reasonably rare among terrestrial plants, they are common among marine sessile organisms and have been reported in other contexts. Here we consider a system with three species in a competitive loop and show that this simple ecology exhibits two counter–intuitive phenomena. First, the species that is least competitive is expected to have the largest population and, where there are oscillations in a finite population, to be the least likely to die out. As a consequence an apparent weakening of a species leads to an increase in its population. Second, evolution favours the most competitive individuals within a species, which leads to a decline in its population. This is analogous to the tragedy of the commons, but here, rather than leading to a collapse, the ‘tragedy’ acts to maintain diversity.
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paper_title: Chemical Warfare Among Invaders: A Detoxification Interaction Facilitates an Ant Invasion

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As tawny crazy ants ( Nylanderia fulva ) invade the southern USA, they often displace imported fire ants ( Solenopsis invicta ). Following exposure to S. invicta venom, N. fulva applies abdominal exocrine gland secretions to its cuticle. Bioassays reveal that these secretions detoxify S. invicta venom. Further, formic acid, from N. fulva venom, is the detoxifying agent. N. fulva exhibits this detoxification behavior after conflict with a variety of ant species; however, it expresses it most intensely after interactions with S. invicta . This behavior may have evolved in their shared South American native range. The unique capacity to detoxify a major competitor’s venom likely contributes substantially to its ability to displace S. invicta populations, making this behavior a causative agent in the ecological transformation of regional arthropod assemblages.
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paper_title: Evolutionary conservation of species' roles in food webs.

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Studies of ecological networks (the web of interactions between species in a community) demonstrate an intricate link between a community's structure and its long-term viability. It remains unclear, however, how much a community's persistence depends on the identities of the species present, or how much the role played by each species varies as a function of the community in which it is found. We measured species' roles by studying how species are embedded within the overall network and the subsequent dynamic implications. Using data from 32 empirical food webs, we find that species' roles and dynamic importance are inherent species attributes and can be extrapolated across communities on the basis of taxonomic classification alone. Our results illustrate the variability of roles across species and communities and the relative importance of distinct species groups when attempting to conserve ecological communities.
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paper_title: A synthetic oscillatory network of transcriptional regulators

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Networks of interacting biomolecules carry out many essential functions in living cells, but the 'design principles' underlying the functioning of such intracellular networks remain poorly understood, despite intensive efforts including quantitative analysis of relatively simple systems. Here we present a complementary approach to this problem: the design and construction of a synthetic network to implement a particular function. We used three transcriptional repressor systems that are not part of any natural biological clock to build an oscillating network, termed the repressilator, in Escherichia coli. The network periodically induces the synthesis of green fluorescent protein as a readout of its state in individual cells. The resulting oscillations, with typical periods of hours, are slower than the cell-division cycle, so the state of the oscillator has to be transmitted from generation to generation. This artificial clock displays noisy behaviour, possibly because of stochastic fluctuations of its components. Such 'rational network design may lead both to the engineering of new cellular behaviours and to an improved understanding of naturally occurring networks.
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paper_title: Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

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The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.
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paper_title: Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

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Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.
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paper_title: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

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Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.
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paper_title: Zero-one survival behavior of cyclically competing species

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The coexistence of competing species is, due to unavoidable fluctuations, always transient. In this Letter, we investigate the ultimate survival probabilities characterizing different species in cyclic competition. We show that they often obey a surprisingly simple, though nontrivial behavior. Within a model where coexistence is neutrally stable, we demonstrate a robust zero-one law: When the interactions between the three species are (generically) asymmetric, the "weakest" species survives at a probability that tends to one for large population sizes, while the other two are guaranteed to extinction. We rationalize our findings from stochastic simulations by an analytic approach.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

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Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
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paper_title: The rock–paper–scissors game and the evolution of alternative male strategies

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MANY species exhibit colour polymorphisms associated with alternative male reproductive strategies, including territorial males and 'sneaker males' that behave and look like females1–3. The prevalence of multiple morphs is a challenge to evolutionary theory because a single strategy should prevail unless morphs have exactly equal fitness4,5 or a fitness advantage when rare6,7. We report here the application of an evolutionary stable strategy model to a three-morph mating system in the side-blotched lizard. Using parameter estimates from field data, the model predicted oscillations in morph frequency, and the frequencies of the three male morphs were found to oscillate over a six-year period in the field. The fitnesses of each morph relative to other morphs were non-transitive in that each morph could invade another morph when rare, but was itself invadable by another morph when common. Concordance between frequency-dependent selection and the among-year changes in morph fitnesses suggest that male interactions drive a dynamic 'rock–paper–scissors' game7.
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paper_title: Costs for switching partners reduce network dynamics but not cooperative behaviour

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Social networks represent the structuring of interactions between group members. Above all, many interactions are profoundly cooperative in humans and other animals. In accordance with this natural observation, theoretical work demonstrates that certain network structures favour the evolution of cooperation. Yet, recent experimental evidence suggests that static networks do not enhance cooperative behaviour in humans. By contrast, dynamic networks do foster cooperation. However, costs associated with dynamism such as time or resource investments in finding and establishing new partnerships have been neglected so far. Here, we show that human participants are much less likely to break links when costs arise for building new links. Especially, when costs were high, the network was nearly static. Surprisingly, cooperation levels in Prisoner's Dilemma games were not affected by reduced dynamism in social networks. We conclude that the mere potential to quit collaborations is sufficient in humans to reach high levels of cooperative behaviour. Effects of self-structuring processes or assortment on the network played a minor role: participants simply adjusted their cooperative behaviour in response to the threats of losing a partner or of being expelled.
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paper_title: Complex Competitive Relationships Among Genotypes of Three Perennial Grasses: Implications for Species Coexistence

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Competitive relationships were studied among genotypes of Agropyron repens, Poa pratensis, and Phleum pratense collected from a grassland community in southeastern Ontario. Field surveys revealed no significant correlations of abundance among the three species within randomly placed survey plots. The greater performance of target plants in removal plots versus control plots for all three species in a vegetation-removal experiment, however, suggests that each of these three species was suppressed by competitive interactions in the field. One clone (genotype) of each species was collected from each of 10 sites (7-cm2 neighborhoods) within the community. Each genotype was propagated vegetatively and grown in the greenhouse in monocultures and in pairwise mixtures with all other genotypes collected from the community. Differences in relative competitive ability among species or genotypes were measured as significant differences between the yield-suppression coefficients of the two components of a mixture. A s...
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paper_title: Competing associations in six-species predator-prey models

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We study a set of six-species ecological models where each species has two predators and two prey. On a square lattice the time evolution is governed by iterated invasions between the neighbouring predator–prey pairs chosen at random and by a site exchange with a probability Xs between the neutral pairs. These models involve the possibility of spontaneous formation of different defensive alliances whose members protect each other from the external invaders. The Monte Carlo simulations show a surprisingly rich variety of the stable spatial distributions of species and subsequent phase transitions when tuning the control parameter Xs. These very simple models are able to demonstrate that the competition between these associations influences their composition. Sometimes the dominant association is developed via a domain growth. In other cases larger and larger invasion processes precede the prevalence of one of the stable associations. Under some conditions the survival of all the species can be maintained by the cyclic dominance occurring between these associations.
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paper_title: Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors

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One of the central aims of ecology is to identify mechanisms that maintain biodiversity. Numerous theoretical models have shown that competing species can coexist if ecological processes such as dispersal, movement, and interaction occur over small spatial scales. In particular, this may be the case for non-transitive communities, that is, those without strict competitive hierarchies. The classic non-transitive system involves a community of three competing species satisfying a relationship similar to the children's game rock-paper-scissors, where rock crushes scissors, scissors cuts paper, and paper covers rock. Such relationships have been demonstrated in several natural systems. Some models predict that local interaction and dispersal are sufficient to ensure coexistence of all three species in such a community, whereas diversity is lost when ecological processes occur over larger scales. Here, we test these predictions empirically using a non-transitive model community containing three populations of Escherichia coli. We find that diversity is rapidly lost in our experimental community when dispersal and interaction occur over relatively large spatial scales, whereas all populations coexist when ecological processes are localized.
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paper_title: Volunteering leads to rock–paper–scissors dynamics in a public goods game

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Collective efforts are a trademark of both insect and human societies1. They are achieved through relatedness in the former2 and unknown mechanisms in the latter. The problem of achieving cooperation among non-kin has been described as the ‘tragedy of the commons’, prophesying the inescapable collapse of many human enterprises3,4. In public goods experiments, initial cooperation usually drops quickly to almost zero5. It can be maintained by the opportunity to punish defectors6 or the need to maintain good reputation7. Both schemes require that defectors are identified. Theorists propose that a simple but effective mechanism operates under full anonymity. With optional participation in the public goods game, ‘loners’ (players who do not join the group), defectors and cooperators will coexist through rock–paper–scissors dynamics8,9. Here we show experimentally that volunteering generates these dynamics in public goods games and that manipulating initial conditions can produce each predicted direction. If, by manipulating displayed decisions, it is pretended that defectors have the highest frequency, loners soon become most frequent, as do cooperators after loners and defectors after cooperators. On average, cooperation is perpetuated at a substantial level.
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paper_title: Spatial organization in cyclic Lotka-Volterra systems

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We study the evolution of a system of $N$ interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5 species, spatial inhomogeneities develop spontaneously in initially homogeneous systems. The arising spatial patterns form a mosaic of single-species domains with algebraically growing size, $\ell(t)\sim t^\alpha$, where $\alpha=3/4$ (1/2) and 1/3 for N=3 with sequential (parallel) dynamics and N=4, respectively. The domain distribution also exhibits a self-similar spatial structure which is characterized by an additional length scale, ${\cal L}(t)\sim t^\beta$, with $\beta=1$ and 2/3 for N=3 and 4, respectively. For $N\geq 5$, the system quickly reaches a frozen state with non interacting neighboring species. We investigate the time distribution of the number of mutations of a site using scaling arguments as well as an exact solution for N=3. Some possible extensions of the system are analyzed.
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paper_title: Fixation in a cyclic Lotka-Volterra model

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We study a cyclic Lotka-Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species N_c(d) above which the system fixates is determined analytically. We find N_c=5,14,23 in dimensions d=1,2,3, in remarkably good agreement with simulation results in two dimensions.
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paper_title: Effects of punishment in a mobile population playing the prisoner's dilemma game

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We deal with a system of prisoner's dilemma players undergoing continuous motion in a two-dimensional plane. In contrast to previous work, we introduce altruistic punishment after the game. We find punishing only a few of the cooperator-defector interactions is enough to lead the system to a cooperative state in environments where otherwise defection would take over the population. This happens even with soft nonsocial punishment (where both cooperators and defectors punish other players, a behavior observed in many human populations). For high enough mobilities or temptations to defect, low rates of social punishment can no longer avoid the breakdown of cooperation.
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paper_title: Dynamically generated cyclic dominance in spatial prisoner's dilemma games

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We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a step-like function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time-dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.
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paper_title: Testosterone, Endurance, and Darwinian Fitness: Natural and Sexual Selection on the Physiological Bases of Alternative Male Behaviors in Side-Blotched Lizards

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The mechanistic bases of natural and sexual selection on physiological and behavioral traits were examined in male morphs of three colors of the side-blotched lizard, Uta stansburiana. Orange-throated males are aggressive and defend large territories with many females. Blue-throated males defend smaller territories with fewer females; however, blue-throated males assiduously mate guard females on their territory. Yellow-throated males do not defend a territory, but patrol a large home range. They obtain secretive copulations from females on the territories of dominant males. Males with bright orange throats had higher levels of plasma testosterone (T), endurance, activity, and home range size and concomitantly gained greater control over female home ranges than blue- or yellow-throated males. Experimentally elevating plasma T in yellow- and blue-throated males increased their endurance, activity, home range size, and control over female territories to levels that were seen in unmanipulated orange-throated males that had naturally high plasma T. However, the enhanced performance of orange-throated males is not without costs. Orange-throated males had low survival compared to the other morphs. Finally, some yellow-throated males transformed to a partial blue morphology late in the season and the endurance of these transforming yellow-throated males increased from early to late in the season. In addition, yellow-throated males that transformed to blue also had significantly higher plasma T late in the season compared to the plasma T earlier in the season. T appears to play an important role in the physiological changes that all three color morphs undergo during the process of maturation. In some yellow males, T plays an additional role in plastic changes in behavior and physiology late in the reproductive season. We discuss natural and sexual selection on physiological and behavioral traits that leads to the evolution of steroid regulation in the context of alternative male strategies.
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paper_title: Reward and cooperation in the spatial public goods game

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The promise of punishment and reward in promoting public cooperation is debatable. While punishment is traditionally considered more successful than reward, the fact that the cost of punishment frequently fails to offset gains from enhanced cooperation has lead some to reconsider reward as the main catalyst behind collaborative efforts. Here we elaborate on the"stick versus carrot"dilemma by studying the evolution of cooperation in the spatial public goods game, where besides the traditional cooperators and defectors, rewarding cooperators supplement the array of possible strategies. The latter are willing to reward cooperative actions at a personal cost, thus effectively downgrading pure cooperators to second-order free-riders due to their unwillingness to bear these additional costs. Consequently, we find that defection remains viable, especially if the rewarding is costly. Rewards, however, can promote cooperation, especially if the synergetic effects of cooperation are low. Surprisingly, moderate rewards may promote cooperation better than high rewards, which is due to the spontaneous emergence of cyclic dominance between the three strategies.
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paper_title: Evolutionary establishment of moral and double moral standards through spatial interactions

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Situations where individuals have to contribute to joint efforts or share scarce resources are ubiquitous. Yet, without proper mechanisms to ensure cooperation, the evolutionary pressure to maximize individual success tends to create a tragedy of the commons (such as over-fishing or the destruction of our environment). This contribution addresses a number of related puzzles of human behavior with an evolutionary game theoretical approach as it has been successfully used to explain the behavior of other biological species many times, from bacteria to vertebrates. Our agent-based model distinguishes individuals applying four different behavioral strategies: non-cooperative individuals ("defectors"), cooperative individuals abstaining from punishment efforts (called "cooperators" or "second-order free-riders"), cooperators who punish non-cooperative behavior ("moralists"), and defectors, who punish other defectors despite being non-cooperative themselves ("immoralists"). By considering spatial interactions with neighboring individuals, our model reveals several interesting effects: First, moralists can fully eliminate cooperators. This spreading of punishing behavior requires a segregation of behavioral strategies and solves the "second-order free-rider problem". Second, the system behavior changes its character significantly even after very long times ("who laughs last laughs best effect"). Third, the presence of a number of defectors can largely accelerate the victory of moralists over non-punishing cooperators. Fourth, in order to succeed, moralists may profit from immoralists in a way that appears like an "unholy collaboration". Our findings suggest that the consideration of punishment strategies allows one to understand the establishment and spreading of "moral behavior" by means of game-theoretical concepts. This demonstrates that quantitative biological modeling approaches are powerful even in domains that have been addressed with non-mathematical concepts so far. The complex dynamics of certain social behaviors become understandable as the result of an evolutionary competition between different behavioral strategies.
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paper_title: When does cyclic dominance lead to stable spiral waves?

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Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
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paper_title: Cellular Automaton Models of Interspecific Competition for Space--The Effect of Pattern on Process

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Species in plant communities generally shown an aggregated distribution at one or more spatial scales. This, and the fact that competition between sessile organisms occurs chiefly between neighbours, suggests that the spatial configuration of plants should affect the process and outcome of interspecific competition. Cellular automaton models were constructed to simulate the competitive interaction of five grass species, Agrostis stolonifera, Holcus lanatus, Cynosurus cristatus, Poa trivialis and Lolium perenne, based on experimentally determined rates of invasion. A model with a random initial starting arrangement showed a very rapid loss of species compared to initial arrangements in which species occurred in monospecific bands (...)
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paper_title: Antibiotic-mediated antagonism leads to a bacterial game of rock–paper–scissors in vivo

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Colicins are narrow-spectrum antibiotics produced by and active against Escherichia coli and its close relatives. Colicin-producing strains cannot coexist with sensitive or resistant strains in a well-mixed culture, yet all three phenotypes are recovered in natural populations1. Recent in vitro results conclude that strain diversity can be promoted by colicin production in a spatially structured, non-transitive interaction2, as in the classic non-transitive model rock–paper–scissors (RPS). In the colicin version of the RPS model, strains that produce colicins (C) kill sensitive (S) strains, which outcompete resistant (R) strains, which outcompete C strains. Pairwise in vitro competitions between these three strains are resolved in a predictable order (C beats S, S beats R, and R beats C), but the complete system of three strains presents the opportunity for dynamic equilibrium2. Here we provide conclusive evidence of an in vivo antagonistic role for colicins and show that colicins (and potentially other bacteriocins) may promote, rather than eliminate, microbial diversity in the environment.
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paper_title: Reward and punishment

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Minigames capturing the essence of Public Goods experiments show that even in the absence of rationality assumptions, both punishment and reward will fail to bring about prosocial behavior. This result holds in particular for the well-known Ultimatum Game, which emerges as a special case. But reputation can induce fairness and cooperation in populations adapting through learning or imitation. Indeed, the inclusion of reputation effects in the corresponding dynamical models leads to the evolution of economically productive behavior, with agents contributing to the public good and either punishing those who do not or rewarding those who do. Reward and punishment correspond to two types of bifurcation with intriguing complementarity. The analysis suggests that reputation is essential for fostering social behavior among selfish agents, and that it is considerably more effective with punishment than with reward.
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paper_title: Phase diagrams for the spatial public goods game with pool-punishment

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The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways.
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paper_title: Coevolutionary Dynamics: From Finite to Infinite Populations

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Traditionally, frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes. Only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations. However, the relationship between deterministic and stochastic approaches remained unclear. Here we solve this problem by explicitly considering large populations. In particular, we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics. Moreover, differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process.
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paper_title: Cyclic dominance and biodiversity in well-mixed populations

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Coevolutionary dynamics is investigated in chemical catalysis, biological evolution, social and economic systems. The dynamics of these systems can be analyzed within the unifying framework of evolutionary game theory. In this Letter, we show that even in well-mixed finite populations, where the dynamics is inherently stochastic, biodiversity is possible with three cyclic-dominant strategies. We show how the interplay of evolutionary dynamics, discreteness of the population, and the nature of the interactions influences the coexistence of strategies. We calculate a critical population size above which coexistence is likely.
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paper_title: Interfaces with internal structures in generalized rock-paper-scissors models

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In this work we investigate the development of stable dynamical structures along interfaces separating domains belonging to enemy partnerships, in the context of cyclic predator-prey models with an even number of species $N \ge 8$. We use both stochastic and field theory simulations in one and two spatial dimensions, as well as analytical arguments, to describe the association at the interfaces of mutually neutral individuals belonging to enemy partnerships and to probe their role in the development of the dynamical structures at the interfaces. We identify an interesting behaviour associated to the symmetric or asymmetric evolution of the interface profiles depending on whether $N/2$ is odd or even, respectively. We also show that the macroscopic evolution of the interface network is not very sensitive internal structure of the interfaces. Although this work focus on cyclic predator prey-models with an even number of species, we argue that the results are expected to be quite generic in the context of spatial stochastic May-Leonard models.
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paper_title: Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model

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Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan Nature 418, 171 (2002)] and B. Kirkup and M. A. Riley Nature 428, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely, the "rock-paper-scissors" or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting in regular (yet neutrally stable) oscillations of the population densities, we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems.
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paper_title: Coaction versus reciprocity in continuous-time models of cooperation.

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Cooperating animals frequently show closely coordinated behaviours organized by a continuous flow of information between interacting partners. Such real-time coaction is not captured by the iterated prisoner's dilemma and other discrete-time reciprocal cooperation games, which inherently feature a delay in information exchange. Here, we study the evolution of cooperation when individuals can dynamically respond to each other's actions. We develop continuous-time analogues of iterated-game models and describe their dynamics in terms of two variables, the propensity of individuals to initiate cooperation (altruism) and their tendency to mirror their partner's actions (coordination). These components of cooperation stabilize at an evolutionary equilibrium or show oscillations, depending on the chosen payoff parameters. Unlike reciprocal altruism, cooperation by coaction does not require that those willing to initiate cooperation pay in advance for uncertain future benefits. Correspondingly, we show that introducing a delay to information transfer between players is equivalent to increasing the cost of cooperation. Cooperative coaction can therefore evolve much more easily than reciprocal cooperation. When delays entirely prevent coordination, we recover results from the discrete-time alternating prisoner's dilemma, indicating that coaction and reciprocity are connected by a continuum of opportunities for real-time information exchange.
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paper_title: Nonlinear Aspects of Competition Between Three Species

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It is shown that for three competitors, the classic Gause–Lotka–Volterra equations possess a special class of periodic limit cycle solutions, and a general class of solutions in which the system exhibits nonperiodic population oscillations of bounded amplitude but ever increasing cycle time. Biologically, the result is interesting as a caricature of the complexities that nonlinearities can introduce even into the simplest equations of population biology ; mathematically, the model illustrates some novel tactical tricks and dynamical peculiarities for 3-dimensional nonlinear systems.
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paper_title: Spatial Rock-Paper-Scissors Models with Inhomogeneous Reaction Rates

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We study several variants of the stochastic four-state rock-paper-scissors game or, equivalently, cyclic three-species predator-prey models with conserved total particle density, by means of Monte Carlo simulations on one- and two-dimensional lattices. Specifically, we investigate the influence of spatial variability of the reaction rates and site occupancy restrictions on the transient oscillations of the species densities and on spatial correlation functions in the quasistationary coexistence state. For small systems, we also numerically determine the dependence of typical extinction times on the number of lattice sites. In stark contrast with two-species stochastic Lotka-Volterra systems, we find that for our three-species models with cyclic competition quenched disorder in the reaction rates has very little effect on the dynamics and the long-time properties of the coexistence state. Similarly, we observe that site restriction only has a minor influence on the system's dynamical properties. Our results therefore demonstrate that the features of the spatial rock-paper-scissors system are remarkably robust with respect to model variations, and stochastic fluctuations as well as spatial correlations play a comparatively minor role.
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paper_title: Evolutionary games on graphs

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Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
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paper_title: Interaction strengths in food webs : issues and opportunities

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Summary 1. Recent efforts to understand how the patterning of interaction strength affects both structure and dynamics in food webs have highlighted several obstacles to productive synthesis. Issues arise with respect to goals and driving questions, methods and approaches, and placing results in the context of broader ecological theory. 2. Much confusion stems from lack of clarity about whether the questions posed relate to community-level patterns or to species dynamics, and to what authors actually mean by the term ‘interaction strength’. Here, we describe the various ways in which this term has been applied and discuss the implications of loose terminology and definition for the development of this field. 3. Of particular concern is the clear gap between theoretical and empirical investigations of interaction strengths and food web dynamics. The ecological community urgently needs to explore new ways to estimate biologically reasonable model coefficients from empirical data, such as foraging rates, body size, metabolic rate, biomass distribution and other species traits. 4. Combining numerical and analytical modelling approaches should allow exploration of the conditions under which different interaction strengths metrics are interchangeable with regard to relative magnitude, system responses, and species identity. 5. Finally, the prime focus on predator‐prey links in much of the research to date on interaction strengths in food webs has meant that the potential significance of nontrophic interactions, such as competition, facilitation and biotic disturbance, has been largely ignored by the food web community. Such interactions may be important dynamically and should be routinely included in future food web research programmes.
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paper_title: Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

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The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes. Strong emphasis is placed on systematic approximation methods for solving problems. Stochastic adiabatic elimination is newly formulated. The book contains the "folklore" of stochastic methods in systematic form and is suitable for use as a reference work.
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paper_title: Meet the New Boss – Same as the Old Boss

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A key paradox of subalternity for the subject throwing off the colonial yoke is the degree to which the collective emergence from the nation-state is to be in the image of the colonizer; that is, as a modern state, notionally on a par with the mother/father country. With a view to understanding the meaning of such symbolic investments, this paper surveys the lyrics of national anthems from a wide range of postcolonial countries. ::: ::: Despite the avowals one regularly finds in the lyrics of postcolonial anthems, and despite the expression of sometimes rote resistance to a putative colonial oppressor, those singing songs which imitate European anthems, and the feeling such anthems inspire, invest identity in the mimetic, rather than the unique. On the basis of a range of observations of anthems and their circumstances, this paper dares to ask finally whether the singing of anthems makes for better worlds.
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paper_title: Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations

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We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude.
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paper_title: Evolutionary games on minimally structured populations

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Population structure induced by both spatial embedding and more general networks of interaction, such as model social networks, have been shown to have a fundamental effect on the dynamics and outcome of evolutionary games. These effects have, however, proved to be sensitive to the details of the underlying topology and dynamics. Here we introduce a minimal population structure that is described by two distinct hierarchical levels of interaction, similar to the structured metapopulation concept of ecology and island models in population genetics. We believe this model is able to identify effects of spatial structure that do not depend on the details of the topology. While effects depending on such details clearly lie outside the scope of our approach, we expect that those we are able to reproduce should be generally applicable to a wide range of models. We derive the dynamics governing the evolution of a system starting from fundamental individual level stochastic processes through two successive mean-field approximations. In our model of population structure the topology of interactions is described by only two parameters: the effective population size at the local scale and the relative strength of local dynamics to global mixing. We demonstrate, for example, the existence of a continuous transition leading to the dominance of cooperation in populations with hierarchical levels of unstructured mixing as the benefit to cost ratio becomes smaller then the local population size. Applying our model of spatial structure to the repeated prisoner's dilemma we uncover a counterintuitive mechanism by which the constant influx of defectors sustains cooperation. Further exploring the phase space of the repeated prisoner's dilemma and also of the ``rock-paper-scissor'' game we find indications of rich structure and are able to reproduce several effects observed in other models with explicit spatial embedding, such as the maintenance of biodiversity and the emergence of global oscillations.
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paper_title: Imitation, internal absorption and the reversal of local drift in stochastic evolutionary games

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Evolutionary game dynamics in finite populations is typically subject to noise, inducing effects which are not present in deterministic systems, including fixation and extinction. In the first part of this paper we investigate the phenomenon of drift reversal in finite populations, taking into account that drift is a local quantity in strategy space. Secondly, we study a simple imitation dynamics, and show that it can lead to fixation at internal mixed-strategy fixed points even in finite populations. Imitation in infinite populations is adequately described by conventional replicator dynamics, and these equations are known to have internal fixed points. Internal absorption in finite populations on the other hand is a novel dynamic phenomenon. Due to an outward drift in finite populations this type of dynamic arrest is not found in other commonly studied microscopic dynamics, not even in those with the same deterministic replicator limit as imitation.
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paper_title: Noise-guided evolution within cyclical interactions

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We study a stochastic predator–prey model on a square lattice, where each of the six species has two superior and two inferior partners. The invasion probabilities between species depend on the predator–prey pair and are supplemented by Gaussian noise. Conditions are identified that warrant the largest impact of noise on the evolutionary process, and the results of Monte Carlo simulations are qualitatively reproduced by a four-point cluster dynamical mean-field approximation. The observed noise-guided evolution is deeply routed in short-range spatial correlations, which is supported by simulations on other host lattice topologies. Our findings are conceptually related to the coherence resonance phenomenon in dynamical systems via the mechanism of threshold duality. We also show that the introduced concept of noise-guided evolution via the exploitation of threshold duality is not limited to predator–prey cyclical interactions, but may apply to models of evolutionary game theory as well, thus indicating its applicability in several different fields of research.
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paper_title: Extinction in neutrally stable stochastic Lotka-Volterra models

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Populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions. The processes leading to extinction of species, whose understanding is a key component in the study of evolution and biodiversity, are influenced by both of these factors. Here, we investigate a class of stochastic population dynamics models based on generalized Lotka-Volterra systems. In the case of neutral stability of the underlying deterministic model, the impact of intrinsic noise on the survival of species is dramatic: It destroys coexistence of interacting species on a time scale proportional to the population size. We introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower-dimensional effective dynamics. This is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations. The extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations.
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paper_title: Replicator dynamics of reward & reputation in public goods games.

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Public goods games have become the mathematical metaphor for game theoretical investigations of cooperative behavior in groups of interacting individuals. Cooperation is a conundrum because cooperators make a sacrifice to benefit others at some cost to themselves. Exploiters or defectors reap the benefits and forgo costs. Despite the fact that groups of cooperators outperform groups of defectors, Darwinian selection or utilitarian principles based on rational choice should favor defectors. In order to overcome this social dilemma, much effort has been expended for investigations pertaining to punishment and sanctioning measures against defectors. Interestingly, the complementary approach to create positive incentives and to reward cooperation has received considerably less attention-despite being heavily advocated in education and social sciences for increasing productivity or preventing conflicts. Here we show that rewards can indeed stimulate cooperation in interaction groups of arbitrary size but, in contrast to punishment, fail to stabilize it. In both cases, however, reputation is essential. The combination of reward and reputation result in complex dynamics dominated by unpredictable oscillations.
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paper_title: Spatial aspects of interspecific competition.

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Using several variants of a stochastic spatial model introduced by Silvertown et al., we investigate the effect of spatial distribution of individuals on the outcome of competition. First, we prove rigorously that if one species has a competitive advantage over each of the others, then eventually it takes over all the sites in the system. Second, we examine tradeoffs between competition and dispersal distance in a two-species system. Third, we consider a cyclic competitive relationship between three types. In this case, a nonspatial treatment leads to densities that follow neutrally stable cycles or even unstable spiral solutions, while a spatial model yields a stationary distribution with an interesting spatial structure.
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paper_title: Alleopathy and spatial competition among coral reef invertebrates.

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Species of ectoprocts and solitary encrusting animals were subjected in aquaria to homogenates of 11 sympatric species of sponges and colonial ascidians. Five of the nine sponge species and one of the two ascidian species exhibited species-specific allelochemical effects. Evidence suggests that alleochemical provide a wide-spread, specific, and complex mechanism for interference competition for space among natural populations of coral reef organisms. The existence of such species-specific mechanisms may provide a basis for maintenance of diversity in space-limited systems in the absence of high levels of predation and physical disturbance.
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paper_title: Social learning promotes institutions for governing the commons

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Cooperation in evolutionary games can be stabilized through punishment of non-cooperators, at a cost to those who do the punishing. Punishment can take different forms, in particular peer-punishment, in which individuals punish free-riders after the event, and pool-punishment, in which a fund for sanctioning is set up beforehand. These authors show that pool-punishment is superior to peer-punishment in dealing with second-order free-riders, who cooperate in the main game but refuse to contribute to punishment.
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paper_title: Modelling patch dynamics on rocky shores using deterministic cellular automata

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Information on biodiversity and community structure is vital for monitoring the effects of climate change and other anthropogenic impacts. Benthic ecosystems of 5 sites off Viti Levu (Fiji),comprising 50 stations were sampled quantitatively revealing 13 128 individuals of 230 species at a mean density of 273.5 ind. m–2. Common taxa included polychaetes (89 species), crustaceans (84 species), molluscs (50 species) and echinoderms (7 species). No species occurred in all 50 stations; the maximum distribution range was 45 stations occupied by the polychaete Aglaophamus sp. A total of 81 species (35.2%) were restricted to single sites (‘uniques’), highlighting spot endemism. Species richness and rarefaction curves provided high estimates of diversity. Multivariate analyses incorporating biological abundances and environmental factors showed 3 distinct clusters among sites characterising differences in benthic community structure. Strongest determinants of faunal distribution were depth, distance from reef and river, and sand content. The presence of heterogeneous faunal assemblages suggests the interplay of these factors at each site. Fauna in Nadi Bay (Shannon-Weiner diversity index H’: 3.26), Suva Harbour (H’: 3.19) and Laucala Bay Lagoon (H’: 3.06) had high diversity indicative of biologically accommodated communities. Rewa River Estuary (H’: 2.42) and Nukubuco Reef drop-off (H’: 2.48) had low diversities, typical of habitats subjected to fluctuating environmental conditions. Benthic community structure in the lagoons around Viti Levu was rich and diverse. Biodiversity was greater than previously recorded from the Great Astrolabe Reef, Fiji (207 ::: to 211 species) and Australia’s Great Barrier Reef (154 species), but lower than in New Caledonia (311 species) and Tahiti (315 species).
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paper_title: Mutual Feedbacks Maintain Both Genetic and Species Diversity in a Plant Community

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The forces that maintain genetic diversity among individuals and diversity among species are usually studied separately. Nevertheless, diversity at one of these levels may depend on the diversity at the other. We have combined observations of natural populations, quantitative genetics, and field experiments to show that genetic variation in the concentration of an allelopathic secondary compound in Brassica nigra is necessary for the coexistence of B. nigra and its competitor species. In addition, the diversity of competing species was required for the maintenance of genetic variation in the trait within B. nigra. Thus, conservation of species diversity may also necessitate maintenance of the processes that sustain the genetic diversity of each individual species.
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paper_title: Local migration promotes competitive restraint in a host–pathogen 'tragedy of the commons'

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These T4 page and their E. coli hosts are the model for a typical 'victim-exploiter' interaction in a study of the role of migration patterns in a 'tragedy of the commons' competition for limited resources within fragmented communities. In this host-pathogen system, growing in 96-well microtitre plates, coexistence, stability and evolution within the separated communities depend critically on migration: restricted migration can promote restraint in the use of the common resource. In this experiment and in theory, highly connected social networks favour virulence. Fragmented populations possess an intriguing duplicity: even if subpopulations are reliably extinction-prone, asynchrony in local extinctions and recolonizations makes global persistence possible1,2,3,4,5,6,7,8. Migration is a double-edged sword in such cases: too little migration prevents recolonization of extinct patches, whereas too much synchronizes subpopulations, raising the likelihood of global extinction. Both edges of this proverbial sword have been explored by manipulating the rate of migration within experimental populations1,3,4,5,6,8. However, few experiments have examined how the evolutionary ecology of fragmented populations depends on the pattern of migration5. Here, we show that the migration pattern affects both coexistence and evolution within a community of bacterial hosts (Escherichia coli) and viral pathogens (T4 coliphage) distributed across a large network of subpopulations. In particular, different patterns of migration select for distinct pathogen strategies, which we term 'rapacious' and 'prudent'. These strategies define a 'tragedy of the commons'9: rapacious phage displace prudent variants for shared host resources, but prudent phage are more productive when alone. We find that prudent phage dominate when migration is spatially restricted, while rapacious phage evolve under unrestricted migration. Thus, migration pattern alone can determine whether a de novo tragedy of the commons is resolved in favour of restraint.
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paper_title: Evolution and the Theory of Games

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In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.
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paper_title: Evolution of restraint in a structured rock-paper-scissors community.

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It is not immediately clear how costly behavior that benefits others evolves by natural selection. By saving on inherent costs, individuals that do not contribute socially have a selective advantage over altruists if both types receive equal benefits. Restrained consumption of a common resource is a form of altruism. The cost of this kind of prudent behavior is that restrained individuals give up resources to less-restrained individuals. The benefit of restraint is that better resource management may prolong the persistence of the group. One way to dodge the problem of defection is for altruists to interact disproportionately with other altruists. With limited dispersal, restrained individuals persist because of interaction with like types, whereas it is the unrestrained individuals that must face the negative long-term consequences of their rapacity. Here, we study the evolution of restraint in a community of three competitors exhibiting a nontransitive (rock–paper–scissors) relationship. The nontransitivity ensures a form of negative feedback, whereby improvement in growth of one competitor has the counterintuitive consequence of lowering the density of that improved player. This negative feedback generates detrimental long-term consequences for unrestrained growth. Using both computer simulations and evolution experiments with a nontransitive community of Escherichia coli, we find that restrained growth can evolve under conditions of limited dispersal in which negative feedback is present. This research, thus, highlights a set of ecological conditions sufficient for the evolution of one form of altruism.
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paper_title: Competing associations in bacterial warfare with two toxins

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Simple combinations of common competitive mechanisms can easily result in cyclic competitive dominance relationships between species. The topological features of such competitive networks allow for complex spatial coexistence patterns. We investigate self-organization and coexistence in a lattice model, describing the spatial population dynamics of competing bacterial strains. With increasing diffusion rate the community of the nine possible toxicity/resistance types undergoes two phase transitions. Below a critical level of diffusion, the system exhibits expanding domains of three different defensive alliances, each consisting of three cyclically dominant species. Due to the neutral relationship between these alliances and the finite system size effect, ultimately only one of them remains. At large diffusion rates the system admits three coexisting domains, each containing mutually neutral species. Because of the cyclical dominance between these domains, a long term stable coexistence of all species is ensured. In the third phase at intermediate diffusion the spatial structure becomes even more complicated with domains of mutually neutral species persisting along the borders of defensive alliances. The study reveals that cyclic competitive relationships may produce a large variety of complex coexistence patterns, exhibiting common features of natural ecosystems, like hierarchical organization, phase transitions and sudden, large-scale fluctuations.
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paper_title: Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

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Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.
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paper_title: Chemical warfare between microbes promotes biodiversity

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Evolutionary processes generating biodiversity and ecological mechanisms maintaining biodiversity seem to be diverse themselves. Conventional explanations of biodiversity such as niche differentiation, density-dependent predation pressure, or habitat heterogeneity seem satisfactory to explain diversity in communities of macrobial organisms such as higher plants and animals. For a long time the often high diversity among microscopic organisms in seemingly uniform environments, the famous “paradox of the plankton,” has been difficult to understand. The biodiversity in bacterial communities has been shown to be sometimes orders of magnitudes higher than the diversity of known macrobial systems. Based on a spatially explicit game theoretical model with multiply cyclic dominance structures, we suggest that antibiotic interactions within microbial communities may be very effective in maintaining diversity.
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paper_title: Stability and robustness analysis of cooperation cycles driven by destructive agents in finite populations

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The emergence and promotion of cooperation are two of the main issues in evolutionary game theory, as cooperation is amenable to exploitation by defectors, which take advantage of cooperative individuals at no cost, dooming them to extinction. It has been recently shown that the existence of purely destructive agents (termed jokers) acting on the common enterprises (public goods games) can induce stable limit cycles among cooperation, defection, and destruction when infinite populations are considered. These cycles allow for time lapses in which cooperators represent a relevant fraction of the population, providing a mechanism for the emergence of cooperative states in nature and human societies. Here we study analytically and through agent-based simulations the dynamics generated by jokers in finite populations for several selection rules. Cycles appear in all cases studied, thus showing that the joker dynamics generically yields a robust cyclic behavior not restricted to infinite populations. We also compute the average time in which the population consists mostly of just one strategy and compare the results with numerical simulations.
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paper_title: Evolutionary classification of toxin mediated interactions in microorganisms

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A trade-off between the parameters of Lotka-Volterra systems is used to give verifications of relations between intrinsic growth rate and limiting capacity and the stability type of the resulting dynamical system. The well known rock-paper-scissors game serves as a template for toxin mediated interactions, which is best represented by the bacteriocin producing Escherichia coli bacteria. There, we have three strains of the same species. The producer produces a toxin lethal to the sensitive, while the resistant is able to protect itself from that toxin. Due to the fact that there are costs for production and for resistance, a dynamics similar to the rock-paper-scissors game results. By using an adaptive dynamics approach for competitive Lotka-Volterra systems and assuming an inverse relation (trade-off) between intrinsic growth rate (IGR) and limiting capacity (LC) we obtain evolutionary and convergence stable relations between the IGR's and the LC's. Furthermore this evolutionary process leads to a phase topology of the population dynamics with a globally stable interior fixed point by leaving the interaction parameters constant. While the inverse trade-off stabilizes coexistence and does not allow branching, toxicity itself can promote diversification. The results are discussed in view of several biological examples indicating that the above results are structurally valid.
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paper_title: Chemical warfare and survival strategies in bacterial range expansions

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Dispersal of species is a fundamental ecological process in the evolution and maintenance of biodiversity. Limited control over ecological parameters has hindered progress in understanding of what enables species to colonize new areas, as well as the importance of interspecies interactions. Such control is necessary to construct reliable mathematical models of ecosystems. In our work, we studied dispersal in the context of bacterial range expansions and identified the major determinants of species coexistence for a bacterial model system of three Escherichia coli strains (toxin-producing, sensitive and resistant). Genetic engineering allowed us to tune strain growth rates and to design different ecological scenarios (cyclic and hierarchical). We found that coexistence of all strains depended on three strongly interdependent factors: composition of inoculum, relative strain growth rates and effective toxin range. Robust agreement between our experiments and a thoroughly calibrated computational model enabled us to extrapolate these intricate interdependencies in terms of phenomenological biodiversity laws. Our mathematical analysis also suggested that cyclic dominance between strains is not a prerequisite for coexistence in competitive range expansions. Instead, robust three-strain coexistence required a balance between growth rates and either a reduced initial ratio of the toxin-producing strain, or a sufficiently short toxin range.
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paper_title: Evolutionary Games and Population Dynamics

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Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms. ::: ::: Comprehensive treatment of ecological and game theoretic dynamics ::: Invasion dynamics and permanence as key concepts ::: Explanation in terms of games of things like competition between species
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paper_title: Coevolutionary games - a mini review

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Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.
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paper_title: A three-species model explaining cyclic dominance of pacific salmon

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The four-year oscillations of the number of spawning sockeye salmon (Oncorhynchus nerka) that return to their native stream within the Fraser River basin in Canada are a striking example of population oscillations. The period of the oscillation corresponds to the dominant generation time of these fish. Various - not fully convincing - explanations for these oscillations have been proposed, including stochastic influences, depensatory fishing, or genetic effects. Here, we show that the oscillations can be explained as a stable dynamical attractor of the population dynamics, resulting from a strong resonance near a Neimark Sacker bifurcation. This explains not only the long-term persistence of these oscillations, but also reproduces correctly the empirical sequence of salmon abundance within one period of the oscillations. Furthermore, it explains the observation that these oscillations occur only in sockeye stocks originating from large oligotrophic lakes, and that they are usually not observed in salmon species that have a longer generation time.
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paper_title: Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities

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Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. ::: ::: A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics.
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paper_title: Labyrinthine clustering in a spatial rock-paper-scissors ecosystem

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The spatial rock-paper-scissors ecosystem, where three species interact cyclically, is a model example of how spatial structure can maintain biodiversity. We here consider such a system for a broad range of interaction rates. When one species grows very slowly, this species and its prey dominate the system by self-organizing into a labyrinthine configuration in which the third species propagates. The cluster size distributions of the two dominating species have heavy tails and the configuration is stabilized through a complex spatial feedback loop. We introduce a statistical measure that quantifies the amount of clustering in the spatial system by comparison with its mean-field approximation. Hereby, we are able to quantitatively explain how the labyrinthine configuration slows down the dynamics and stabilizes the system.
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paper_title: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

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Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.
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paper_title: Three- and four-state rock-paper-scissors games with diffusion

paper_content:

Cyclic dominance of three species is a commonly occurring interaction dynamics, often denoted the rock-paper-scissors (RPS) game. Such a type of interactions is known to promote species coexistence. Here, we generalize recent results of Reichenbach [Nature (London) 448, 1046 (2007)] of a four-state variant of the RPS game. We show that spiral formation takes place only without a conservation law for the total density. Nevertheless, in general, fast diffusion can destroy species coexistence. We also generalize the four-state model to slightly varying reaction rates. This is shown both analytically and numerically not to change pattern formation, or the effective wavelength of the spirals, and therefore not to alter the qualitative properties of the crossover to extinction.
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paper_title: When does cyclic dominance lead to stable spiral waves?

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Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
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paper_title: Co-existence in the two-dimensional May-Leonard model with random rates

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We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-) steady state in two-dimensional stochastic May-Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May-Leonard system (for small system sizes): (1) as the mobility rate exceeds a threshold that separates a species coexistence (quasi-) steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ∼ e cN /N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.
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paper_title: Spontaneous formation of dynamical patterns with fractal fronts in the cyclic lattice Lotka-Volterra model.

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Dynamical patterns, in the form of consecutive moving stripes or rings, are shown to develop spontaneously in the cyclic lattice Lotka-Volterra model, when realized on square lattice, at the reaction limited regime. Each stripe consists of different particles (species) and the borderlines between consecutive stripes are fractal. The interface width w between the different species scales as w(L,t) approximately L(alpha)f(t/L(z)), where L is the linear size of the interface, t is the time, and alpha and z are the static and dynamical critical exponents, respectively. The critical exponents were computed as alpha=0.49+/-0.03 and z=1.53+/-0.13 and the propagating fronts show dynamical characteristics similar to those of the Eden growth models.
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paper_title: Instability of spatial patterns and its ambiguous impact on species diversity

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Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable "blurred" patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals' mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.
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paper_title: Rock-scissors-paper game in a chaotic flow: the effect of dispersion on the cyclic competition of microorganisms.

paper_content:

Laboratory experiments and numerical simulations have shown that the outcome of cyclic competition is significantly affected by the spatial distribution of the competitors. Short-range interaction and limited dispersion allows for coexistence of competing species that cannot coexist in a well-mixed environment. In order to elucidate the mechanisms that destroy species diversity we study the intermediate situation of imperfect mixing, typical in aquatic media, in a model of cyclic competition between toxin producing, sensitive and resistant phenotypes. It is found, that chaotic mixing, by changing the character of the spatial distribution, induces coherent oscillations in the populations. The magnitude of the oscillations increases with the strength of mixing, leading to the extinction of some species beyond a critical mixing rate. When mixing is non-uniform in space, coexistence can be sustained at much stronger mixing by the formation of partially isolated regions, that prevent global extinction. The heterogeneity of mixing may enable toxin producing and sensitive strains to coexist for very long time at strong mixing.
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paper_title: Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model

paper_content:

Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan Nature 418, 171 (2002)] and B. Kirkup and M. A. Riley Nature 428, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely, the "rock-paper-scissors" or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting in regular (yet neutrally stable) oscillations of the population densities, we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems.
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paper_title: Chaotic Red Queen coevolution in three-species food chains

paper_content:

Coevolution between two antagonistic species follows the so-called ‘Red Queen dynamics’ when reciprocal selection results in an endless series of adaptation by one species and counteradaptation by the other. Red Queen dynamics are ‘genetically driven’ when selective sweeps involving new beneficial mutations result in perpetual oscillations of the coevolving traits on the slow evolutionary time scale. Mathematical models have shown that a prey and a predator can coevolve along a genetically driven Red Queen cycle. We found that embedding the prey–predator interaction into a three-species food chain that includes a coevolving superpredator often turns the genetically driven Red Queen cycle into chaos. A key condition is that the prey evolves fast enough. Red Queen chaos implies that the direction and strength of selection are intrinsically unpredictable beyond a short evolutionary time, with greatest evolutionary unpredictability in the superpredator. We hypothesize that genetically driven Red Queen chaos could explain why many natural populations are poised at the edge of ecological chaos. Over space, genetically driven chaos is expected to cause the evolutionary divergence of local populations, even under homogenizing environmental fluctuations, and thus to promote genetic diversity among ecological communities over long evolutionary time.
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paper_title: Continuous model for the rock-scissors-paper game between bacteriocin producing bacteria.

paper_content:

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock–scissors–paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May–Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka–Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.
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paper_title: How community size affects survival chances in cyclic competition games that microorganisms play

paper_content:

Cyclic competition is a mechanism underlying biodiversity in nature and the competition between large numbers of interacting individuals under multifaceted environmental conditions. It is commonly modeled with the popular children’s rock-paper-scissors game. Here we probe cyclic competition systematically in a community of three strains of bacteria Escherichia coli. Recent experiments and simulations indicated the resistant strain of E. coli to win the competition. Other data, however, predicted the sensitive strain to be the final winner. We find a generic feature of cyclic competition that solves this puzzle: community size plays a decisive role in selecting the surviving competitor. Size-dependent effects arise from an easily detectable “period of quasiextinction” and may be tested in experiments. We briefly indicate how.
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paper_title: Three-fold way to extinction in populations of cyclically competing species

paper_content:

Species extinction occurs regularly and unavoidably in ecological systems. The time scales for extinction can vary broadly and provide information on the ecosystem's stability. We study the spatio-temporal extinction dynamics of a paradigmatic population model where three species exhibit cyclic competition. The cyclic dynamics reflects the non-equilibrium nature of the species interactions. While previous work focuses on the coarsening process as a mechanism that drives the system to extinction, we found that unexpectedly the dynamics in going to extinction is much richer. We observed dynamics of three different types. In addition to coarsening, in the evolutionarily relevant limit of large times, oscillating traveling waves and heteroclinic orbits play a dominant role. The weights of the different processes depend on the degree of mixing and the system size. By means of analytical arguments and extensive numerical simulations we provide the full characteristics of scenarios leading to extinction in one of the most surprising models of ecology.
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paper_title: Noise and Correlations in a Spatial Population Model with Cyclic Competition

paper_content:

Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influence on the coevolution of populations with cyclic interspecies competition have been demonstrated in recent experiments e.g., B. Kerr , Nature (London) 418, 171 (2002)]. To reach a better theoretical understanding of these phenomena, we consider a paradigmatic spatial model where three species exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial differential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and spatial diffusion at different levels and show how fascinating patterns of entangled spirals emerge. We rationalize our analysis by computing the spatiotemporal correlation functions and provide analytical expressions for the front velocity and the wavelength of the propagating spiral waves.
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paper_title: Vortex dynamics in a three-state model under cyclic dominance.

paper_content:

The evolution of domain structure is investigated in a two-dimensional voter model with three states under cyclic dominance. The study focus on the dynamics of vortices, defined by the points where the three states (domains) meet. We can distinguish vortices and antivortices which walk randomly and annihilate each other. The domain wall motion can create vortex-antivortex pairs at a rate that is increased by the spiral formation due to cyclic dominance. This mechanism is contrasted with a branching annihilating random walk (BARW) in a particle-antiparticle system with density-dependent pair creation rate. Numerical estimates for the critical indices of the vortex density [beta=0.29(4)] and of its fluctuation [gamma=0.34(6)] improve an earlier Monte Carlo study [K. Tainaka and Y. Itoh, Europhys. Lett. 15, 399 (1991)] of the three-state cyclic model in two dimensions.
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paper_title: Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics

paper_content:

Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the so-called replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of non-mean-field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the non-trivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.
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paper_title: Globally synchronized oscillations in complex cyclic games

paper_content:

The rock-paper-scissors game and its generalizations with S>3 species are well-studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one-predator, one-prey symmetry. Lütz et al. [J. Theor. Biol. 317, 286 (2013)] have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators, but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food-web subloops (if any). Here we show that upon replacing a minimum fraction Q of the short-range interactions by long-range ones, there is a Hopf bifurcation, and global oscillations become stable. Surprisingly, to build such long-distance, global synchronization, the four-species coexistence phase requires fewer long-range interactions than the three-species phase, while one would naively expect the opposite to be true. Moreover, deviations from highly homogeneous conditions (χ=0 or 1) increase Qc, and the more heterogeneous is the food web, the harder the synchronization is. By further increasing Q, while the three-species phase remains stable, the four-species one has a transition to an absorbing, single-species state. The existence of a phase with global oscillations for S>3, when the interaction graph has multiple subloops and several possible local cycles, leads to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

paper_content:

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
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paper_title: Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors

paper_content:

One of the central aims of ecology is to identify mechanisms that maintain biodiversity. Numerous theoretical models have shown that competing species can coexist if ecological processes such as dispersal, movement, and interaction occur over small spatial scales. In particular, this may be the case for non-transitive communities, that is, those without strict competitive hierarchies. The classic non-transitive system involves a community of three competing species satisfying a relationship similar to the children's game rock-paper-scissors, where rock crushes scissors, scissors cuts paper, and paper covers rock. Such relationships have been demonstrated in several natural systems. Some models predict that local interaction and dispersal are sufficient to ensure coexistence of all three species in such a community, whereas diversity is lost when ecological processes occur over larger scales. Here, we test these predictions empirically using a non-transitive model community containing three populations of Escherichia coli. We find that diversity is rapidly lost in our experimental community when dispersal and interaction occur over relatively large spatial scales, whereas all populations coexist when ecological processes are localized.
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paper_title: Statistical mechanics of complex networks

paper_content:

The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. ::: Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. ::: The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. ::: The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.
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paper_title: Evolutionary games on graphs

paper_content:

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
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paper_title: Phase transitions for rock-scissors-paper game on different networks.

paper_content:

Monte Carlo simulations and dynamical mean-field approximations are performed to study the phase transitions in the rock-scissors-paper game on different host networks. These graphs are originated from lattices by introducing quenched and annealed randomness simultaneously. In the resulting phase diagrams three different stationary states are identified for all structures. The comparison of results on different networks suggests that the value of the clustering coefficient plays an irrelevant role in the emergence of a global oscillating phase. The critical behavior of phase transitions seems to be universal and can be described by the same exponents.
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paper_title: Rock-scissors-paper game on regular small-world networks

paper_content:

The spatial rock-scissors-paper game (or cyclic Lotka–Volterra system) is extended to study how the spatiotemporal patterns are affected by the rewired host lattice providing uniform number of neighbours (degree) at each site. On the square lattice this system exhibits a self-organizing pattern with equal concentration of the competing strategies (species). If the quenched background is constructed by substituting random links for the nearest-neighbour bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value. This transition can also be observed if the standard link is replaced temporarily by a random one with a probability P at each step of iteration. Above a second threshold value of P the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous (absorbing) states. In this case the results of Monte Carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one-, two-, four- and six-site clusters.
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paper_title: Networks with dispersed degrees save stable coexistence of species in cyclic competition

paper_content:

Coexistence of individuals with different species or phenotypes is often found in nature in spite of competition between them. Stable coexistence of multiple types of individuals have implications for maintenance of ecological biodiversity and emergence of altruism in society, to name a few. Various mechanisms of coexistence including spatial structure of populations, heterogeneous individuals, and heterogeneous environments, have been proposed. In reality, individuals disperse and interact on complex networks. We examine how heterogeneous degree distributions of networks influence coexistence, focusing on models of cyclically competing species. We show analytically and numerically that heterogeneity in degree distributions promotes stable coexistence.
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paper_title: Defined spatial structure stabilizes a synthetic multispecies bacterial community

paper_content:

This paper shows that for microbial communities, “fences make good neighbors.” Communities of soil microorganisms perform critical functions: controlling climate, enhancing crop production, and remediation of environmental contamination. Microbial communities in the oral cavity and the gut are of high biomedical interest. Understanding and harnessing the function of these communities is difficult: artificial microbial communities in the laboratory become unstable because of “winner-takes-all” competition among species. We constructed a community of three different species of wild-type soil bacteria with syntrophic interactions using a microfluidic device to control spatial structure and chemical communication. We found that defined microscale spatial structure is both necessary and sufficient for the stable coexistence of interacting bacterial species in the synthetic community. A mathematical model describes how spatial structure can balance the competition and positive interactions within the community, even when the rates of production and consumption of nutrients by species are mismatched, by exploiting nonlinearities of these processes. These findings provide experimental and modeling evidence for a class of communities that require microscale spatial structure for stability, and these results predict that controlling spatial structure may enable harnessing the function of natural and synthetic multispecies communities in the laboratory.
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paper_title: THE EVOLUTION OF RESTRAINT IN BACTERIAL BIOFILMS UNDER NONTRANSITIVE COMPETITION

paper_content:

Abstract Theoretical and empirical evidence indicates that competing species can coexist if dispersal, migration, and competitive interactions occur over relatively small spatial scales. In particular, spatial structure appears to be critical to certain communities with nontransitive competition. A typical nontransitive system involves three competing species that satisfy a relationship similar to the children's game of rock–paper–scissors. Although the ecological dynamics of nontransitive systems in spatially structured communities have received some attention, fewer studies have incorporated evolutionary change. Here we investigate evolution within toxic bacterial biofilms using an agent-based simulation that represents a nontransitive community containing three populations of Escherichia coli. In structured, nontransitive communities, strains evolve that do not maximize their competitive ability: They do not reduce their probability of death to a minimum or increase their toxicity to a maximum. That is...
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paper_title: Complex networks: Structure and dynamics

paper_content:

Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering. © 2005 Elsevier B.V. All rights reserved.
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paper_title: Evolution of restraint in a structured rock-paper-scissors community.

paper_content:

It is not immediately clear how costly behavior that benefits others evolves by natural selection. By saving on inherent costs, individuals that do not contribute socially have a selective advantage over altruists if both types receive equal benefits. Restrained consumption of a common resource is a form of altruism. The cost of this kind of prudent behavior is that restrained individuals give up resources to less-restrained individuals. The benefit of restraint is that better resource management may prolong the persistence of the group. One way to dodge the problem of defection is for altruists to interact disproportionately with other altruists. With limited dispersal, restrained individuals persist because of interaction with like types, whereas it is the unrestrained individuals that must face the negative long-term consequences of their rapacity. Here, we study the evolution of restraint in a community of three competitors exhibiting a nontransitive (rock–paper–scissors) relationship. The nontransitivity ensures a form of negative feedback, whereby improvement in growth of one competitor has the counterintuitive consequence of lowering the density of that improved player. This negative feedback generates detrimental long-term consequences for unrestrained growth. Using both computer simulations and evolution experiments with a nontransitive community of Escherichia coli, we find that restrained growth can evolve under conditions of limited dispersal in which negative feedback is present. This research, thus, highlights a set of ecological conditions sufficient for the evolution of one form of altruism.
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paper_title: Coevolutionary games - a mini review

paper_content:

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.
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paper_title: Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities

paper_content:

Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. ::: ::: A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics.
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paper_title: Resolving social dilemmas on evolving random networks

paper_content:

We show that strategy-independent adaptations of random interaction networks can induce powerful mechanisms, ranging from the Red Queen to group selection, which promote cooperation in evolutionary social dilemmas. These two mechanisms emerge spontaneously as dynamical processes due to deletions and additions of links, which are performed whenever players adopt new strategies and after a certain number of game iterations, respectively. The potency of cooperation promotion, as well as the mechanism responsible for it, can thereby be tuned via a single parameter determining the frequency of link additions. We thus demonstrate that coevolving random networks may evoke an appropriate mechanism for each social dilemma, such that cooperation prevails even in highly unfavorable conditions.
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paper_title: Population interaction structure and the coexistence of bacterial strains playing ‘rock-paper-scissors’

paper_content:

The simplest example of non-transitive competition is the game rock–paper–scissors (RPS), which exhibits characteristic cyclic strategy replacement: paper beats rock, which in turn beats scissors, which in turn beats paper. In addition to its familiar use in understanding human decision-making, rock–paper–scissors is also played in many biological systems. Among other reasons, this is important because it potentially provides a mechanism whereby species- or strain coexistence can occur in the face of intense competition. Kerr et al. (2002, Nature 418: 171–174) use complementary experiments and simulations to show that RPS-playing toxic, resistant, and susceptible E. coli bacteria can coexist when interactions between the strains are spatially explicit. This raises the question of whether limited interactions associated with space are sufficient to allow strain coexistence, or whether space per se is crucial. I approach this question by extending the Kerr et al. model to include different (aspatial) population network structures with the same degree distributions as corresponding spatial lattice models. I show that the coexistence that occurs for some parameter combinations when simulated bacterial strains compete on lattices is absent when they compete on random regular graphs. Further, considering small-world networks of intermediate ‘quenched randomness’ between lattices and random regular graphs, I show that only small deviations from pure spatial interactions are sufficient to prevent strain coexistence. These results emphasize the explicit role of space, rather than merely limited interactions, as being decisive in allowing the coexistence of toxic, resistant, and susceptible strains in this model system.
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paper_title: Phase transitions for a rock–scissors–paper model with long-range-directed interactions

paper_content:

We have investigated a rock–scissors–paper model with long-range-directed interactions in two dimensions where every site has four outgoing links but a fraction q of the outgoing links to the nearest neighbour sites are rewired to other long-distance sites chosen randomly and the lattice structure is replaced again after a Monte Carlo step. It is found that, with q increasing, the system changes from a three species coexistence self-organizing state to a global oscillation state and then to one of the homogeneous states. However when q exceeds a third threshold value, the system returns to a self-organizing state. When we restrict the maximum number of ingoing links of a site to four, the last self-organizing state disappears, the system stays in the homogeneous state forever after q exceeds the second threshold value. And when we restrict the maximum number of ingoing links of a site to five or six, the system exhibits a transition from the homogeneous state to a global oscillation state again and then to the last self-organizing state with q increasing. The comparison of results on different networks suggests that the sites with zero ingoing links should play a significant role in the emergences of the later self-organizing state and the subsequent global oscillation.
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paper_title: Cyclic dominance in adaptive networks

paper_content:

The Rock-Paper-Scissors (RPS) game is a paradigmatic model for cyclic dominance in biological systems. Here we consider this game in the social context of competition between opinions in a networked society. In our model, every agent has an opinion which is drawn from the three choices: rock, paper or scissors. In every timestep a link is selected randomly and the game is played between the nodes connected by the link. The loser either adopts the opinion of the winner or rewires the link. These rules define an adaptive network on which the agents’ opinions coevolve with the network topology of social contacts. We show analytically and numerically that nonequilibrium phase transitions occur as a function of the rewiring strength. The transitions separate four distinct phases which differ in the observed dynamics of opinions and topology. In particular, there is one phase where the population settles to an arbitrary consensus opinion. We present a detailed analysis of the corresponding transitions revealing an apparently paradoxical behavior. The system approaches consensus states where they are unstable, whereas other dynamics prevail when the consensus states are stable.
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paper_title: Coevolutionary cycling of host sociality and pathogen virulence in contact networks.

paper_content:

Infectious diseases may place strong selection on the social organization of animals. Conversely, the structure of social systems can influence the evolutionary trajectories of pathogens. While much attention has focused on the evolution of host sociality or pathogen virulence separately, few studies have looked at their coevolution. Here we use an agent-based simulation to explore host-pathogen coevolution in social contact networks. Our results indicate that under certain conditions, both host sociality and pathogen virulence exhibit continuous cycling. The way pathogens move through the network (e.g., their interhost transmission and probability of superinfection) and the structure of the network can influence the existence and form of cycling.
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paper_title: Evolutionary games on graphs

paper_content:

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
---
paper_title: Phase transitions for rock-scissors-paper game on different networks.

paper_content:

Monte Carlo simulations and dynamical mean-field approximations are performed to study the phase transitions in the rock-scissors-paper game on different host networks. These graphs are originated from lattices by introducing quenched and annealed randomness simultaneously. In the resulting phase diagrams three different stationary states are identified for all structures. The comparison of results on different networks suggests that the value of the clustering coefficient plays an irrelevant role in the emergence of a global oscillating phase. The critical behavior of phase transitions seems to be universal and can be described by the same exponents.
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paper_title: Emergence of multilevel selection in the prisoner's dilemma game on coevolving random networks

paper_content:

We study the evolution of cooperation in the prisoner's dilemma game, whereby a coevolutionary rule is introduced that molds the random topology of the interaction network in two ways. First, existing links are deleted whenever a player adopts a new strategy or its degree exceeds a threshold value; second, new links are added randomly after a given number of game iterations. These coevolutionary processes correspond to the generic formation of new links and deletion of existing links that, especially in human societies, appear frequently as a consequence of ongoing socialization, change of lifestyle or death. Due to the counteraction of deletions and additions of links the initial heterogeneity of the interaction network is qualitatively preserved, and thus cannot be held responsible for the observed promotion of cooperation. Indeed, the coevolutionary rule evokes the spontaneous emergence of a powerful multilevel selection mechanism, which despite the sustained random topology of the evolving network, maintains cooperation across the whole span of defection temptation values.
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paper_title: Coevolutionary games - a mini review

paper_content:

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.
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paper_title: Mobility-Dependent Selection of Competing Strategy Associations

paper_content:

Standard models of population dynamics focus on the interaction, survival, and extinction of the competing species individually. Real ecological systems, however, are characterized by an abundance of species (or strategies, in the terminology of evolutionary-game theory) that form intricate, complex interaction networks. The description of the ensuing dynamics may be aided by studying associations of certain strategies rather than individual ones. Here we show how such a higher-level description can bear fruitful insight. Motivated from different strains of colicinogenic Escherichia coli bacteria, we investigate a four-strategy system which contains a three-strategy cycle and a neutral alliance of two strategies. We find that the stochastic, spatial model exhibits a mobility-dependent selection of either the three-strategy cycle or of the neutral pair. We analyze this intriguing phenomenon numerically and analytically.
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paper_title: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

paper_content:

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.
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paper_title: Discriminating the effects of spatial extent and population size in cyclic competition among species

paper_content:

We introduce a population model for species under cyclic competition. This model allows individuals to coexist and interact on single cells while migration takes place between adjacent cells. In contrast to the model introduced by Reichenbach, Mobilia, and Frey [Reichenbach, Mobilia, and Frey, Nature (London) 448, 1046 (2007)], we find that the emergence of spirals results in an ambiguous behavior regarding the stability of coexistence. The typical time until extinction exhibits, however, a qualitatively opposite dependence on the newly introduced nonunit carrying capacity in the spiraling and the nonspiraling regimes. This allows us to determine a critical mobility that marks the onset of this spiraling state sharply. In contrast, we demonstrate that the conventional finite size stability analysis with respect to spatial size is of limited use for identifying the onset of the spiraling regime.
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paper_title: Basins of attraction for species extinction and coexistence in spatial rock-paper-scissors games

paper_content:

We study the collective dynamics of mobile species under cyclic competition by breaking the symmetry in the initial populations and examining the basins of the two distinct asymptotic states: extinction and coexistence, the latter maintaining biodiversity. We find a rich dependence of dynamical properties on initial conditions. In particular, for high mobility, only extinction basins exist and they are spirally entangled, but a basin of coexistence emerges when the mobility parameter is decreased through a critical value, whose area increases monotonically as the parameter is further decreased. The structure of extinction basins for high mobility can be predicted by a mean-field theory. These results provide a more comprehensive picture for the fundamental issue of species coexistence than previously achieved.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

paper_content:

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
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paper_title: Three- and four-state rock-paper-scissors games with diffusion

paper_content:

Cyclic dominance of three species is a commonly occurring interaction dynamics, often denoted the rock-paper-scissors (RPS) game. Such a type of interactions is known to promote species coexistence. Here, we generalize recent results of Reichenbach [Nature (London) 448, 1046 (2007)] of a four-state variant of the RPS game. We show that spiral formation takes place only without a conservation law for the total density. Nevertheless, in general, fast diffusion can destroy species coexistence. We also generalize the four-state model to slightly varying reaction rates. This is shown both analytically and numerically not to change pattern formation, or the effective wavelength of the spirals, and therefore not to alter the qualitative properties of the crossover to extinction.
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paper_title: Vertex dynamics during domain growth in three-state models

paper_content:

Topological aspects of interfaces are studied by comparing quantitatively the evolving three-color patterns in three different models, such as the three-state voter, Potts, and extended voter models. The statistical analysis of some geometrical features allows us to explore the role of different elementary processes during distinct coarsening phenomena in the above models.
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paper_title: When does cyclic dominance lead to stable spiral waves?

paper_content:

Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
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paper_title: Instability of spatial patterns and its ambiguous impact on species diversity

paper_content:

Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable "blurred" patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals' mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.
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paper_title: Spatial social dilemmas: dilution, mobility and grouping effects with imitation dynamics

paper_content:

We present an extensive, systematic study of the Prisoner's Dilemma and Snowdrift games on a square lattice under a synchronous, noiseless imitation dynamics. We show that for both the occupancy of the network and the (random) mobility of the agents there are intermediate values that may increase the amount of cooperators in the system and new phases appear. We analytically determine the transition lines between these phases and compare with the mean field prediction and the observed behavior on a square lattice. We point out which are the more relevant microscopic processes that entitle cooperators to invade a population of defectors in the presence of mobility and discuss the universality of these results.
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paper_title: Evolutionary games on graphs

paper_content:

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
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paper_title: Random mobility and spatial structure often enhance cooperation

paper_content:

The effects of an unconditional move rule in the spatial Prisoner's Dilemma, Snowdrift and Stag Hunt games are studied. Spatial structure by itself is known to modify the outcome of many games when compared with a randomly mixed population, sometimes promoting, sometimes inhibiting cooperation. Here we show that random dilution and mobility may suppress the inhibiting factors of the spatial structure in the Snowdrift game, while enhancing the already larger cooperation found in the Prisoner's dilemma and Stag Hunt games.
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paper_title: Coevolutionary dynamics in structured populations ofthree species

paper_content:

Inspired by the experiments with the three strains of E. coli bacteria as well as the three morphs of Uta stansburiana lizards, a model of cyclic dominance was proposed to investigate the mechanisms facilitating the maintenance of biodiversity in spatially structured populations. Subsequent studies enriched the original model with various biologically motivated extension repeating the proposed mathematical ::: analysis and computer simulations. ::: The research presented in this thesis unifies and generalises these models by combining the birth, selection-removal, selection-replacement and mutation processes as well as two forms of mobility into a generic metapopulation model. Instead of the standard mathematical treatment, more controlled analysis with inverse system size and multiscale asymptotic expansions is presented to derive an approximation of the system dynamics in terms of a well-known pattern forming equation. The novel analysis, capable of increased accuracy, is evaluated with improved numerical experiments performed with bespoke software developed for simulating the stochastic and deterministic descriptions of the generic metapopulation model. ::: The emergence of spiral waves facilitating the long term biodiversity is confirmed in the computer simulations as predicted by the theory. The derived conditions on the stability of spiral patterns for different values of the biological parameters are studied resulting in discoveries of interesting phenomena such as spiral annihilation or instabilities caused by nonlinear diffusive terms.
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paper_title: Velocity-enhanced Cooperation of Moving Agents playing Public Goods Games

paper_content:

In this Brief Report we study the evolutionary dynamics of the Public Goods Game in a population of mobile agents embedded in a 2-dimensional space. In this framework, the backbone of interactions between agents changes in time, allowing us to study the impact that mobility has on the emergence of cooperation in structured populations. We compare our results with a static case in which agents interact on top of a Random Geometric Graph. Our results point out that a low degree of mobility enhances the onset of cooperation in the system while a moderate velocity favors the fixation of the full-cooperative state.
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paper_title: Spiral Waves Emergence in a Cyclic Predator-Prey Model

paper_content:

Based on a cyclic predator-prey model of three species, spiral waves on global level of the system are obtained. It is found that the predation intensity greatly affects on the behaviors of spiral waves. The wavelength of spiral waves alter with the mobility in the form of λ~D θ . Values of θ are determined by predation rates between species. It indicates the behaviors of spiral waves varying with mobility are universal at the same predation rate which reveals competition of resources among species.
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paper_title: Effect of epidemic spreading on species coexistence in spatial rock-paper-scissors games

paper_content:

A fundamental question in nonlinear science and evolutionary biology is how epidemic spreading may affect coexistence. We address this question in the framework of mobile species under cyclic competitions by investigating the roles of both intra- and interspecies spreading. A surprising finding is that intraspecies infection can strongly promote coexistence while interspecies spreading cannot. These results are quantified and a theoretical paradigm based on nonlinear partial differential equations is derived to explain the numerical results.
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paper_title: Pattern formation, synchronization, and outbreak of biodiversity in cyclically competing games

paper_content:

Species in nature are typically mobile over diverse distance scales, examples of which range from bacteria run to long-distance animal migrations. These behaviors can have a significant impact on biodiversity. Addressing the role of migration in biodiversity microscopically is fundamental but remains a challenging problem in interdisciplinary science. We incorporate both intra- and inter-patch migrations in stochastic games of cyclic competitions and find that the interplay between the migrations at the local and global scales can lead to robust species coexistence characterized dynamically by the occurrence of remarkable target-wave patterns in the absence of any external control. The waves can emerge from either mixed populations or isolated species in different patches, regardless of the size and the location of the migration target. We also find that, even in a single-species system, target waves can arise from rare mutations, leading to an outbreak of biodiversity. A surprising phenomenon is that target waves in different patches can exhibit synchronization and time-delayed synchronization, where the latter potentially enables the prediction of future evolutionary dynamics. We provide a physical theory based on the spatiotemporal organization of the target waves to explain the synchronization phenomena. We also investigate the basins of coexistence and extinction to establish the robustness of biodiversity through migrations. Our results are relevant to issues of general and broader interest such as pattern formation, control in excitable systems, and the origin of order arising from self-organization in social and natural systems.
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paper_title: Cyclic competition of mobile species on continuous space: pattern formation and coexistence.

paper_content:

We propose a model for cyclically competing species on continuous space and investigate the effect of the interplay between the interaction range and mobility on coexistence. A transition from coexistence to extinction is uncovered with a strikingly nonmonotonic behavior in the coexistence probability. About the minimum in the probability, switches between spiral and plane-wave patterns arise. A strong mobility can either promote or hamper coexistence, depending on the radius of the interaction range. These phenomena are absent in any lattice-based model, and we demonstrate that they can be explained using nonlinear partial differential equations. Our continuous-space model is more physical and we expect the findings to generate experimental interest.
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paper_title: Effects of competition on pattern formation in the rock-paper-scissors game

paper_content:

We investigate the impact of cyclic competition on pattern formation in the rock-paper-scissors game. By separately considering random and prepared initial conditions, we observe a critical influence of the competition rate p on the stability of spiral waves and on the emergence of biodiversity. In particular, while increasing values of p promote biodiversity, they may act detrimentally on spatial pattern formation. For random initial conditions, we observe a phase transition from biodiversity to an absorbing phase, whereby the critical value of mobility grows linearly with increasing values of p on a log-log scale but then saturates as p becomes large. For prepared initial conditions, we observe the formation of single-armed spirals, but only for values of p that are below a critical value. Once above that value, the spirals break up and form disordered spatial structures, mainly because of the percolation of vacant sites. Thus there exists a critical value of the competition rates p(c) for stable single-armed spirals in finite populations. Importantly though, p(c) increases with increasing system size because noise reinforces the disintegration of ordered patterns. In addition, we also find that p(c) increases with the mobility. These phenomena are reproduced by a deterministic model that is based on nonlinear partial differential equations. Our findings indicate that competition is vital for the sustenance of biodiversity and the emergence of pattern formation in ecosystems governed by cyclical interactions.
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paper_title: Self-Organization of Mobile Populations in Cyclic Competition

paper_content:

The formation of out-of-equilibrium patterns is a characteristic feature of spatially extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the 'rock-paper-scissors' game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals' mobility on the spatial structures emerging in rock-paper-scissors games. We devise a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior-in the proper continuum limit-is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals' wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system. These findings and our methods have important applications for understanding the formation of noisy patterns, e.g. in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)-chemical reactions.
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paper_title: Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species

paper_content:

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.
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paper_title: Does mobility decrease cooperation?

paper_content:

We explore the minimal conditions for sustainable cooperation on a spatially distributed population of memoryless, unconditional strategies (cooperators and defectors) in presence of unbiased, non-contingent mobility in the context of the Prisoner's Dilemma game. We find that cooperative behavior is not only possible but may even be enhanced by such an "always-move" rule, when compared with the strongly viscous ("never-move") case. In addition, mobility also increases the capability of cooperation to emerge and invade a population of defectors, what may have a fundamental role in the problem of the onset of cooperation.
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paper_title: Effects of mobility in a population of Prisoner's Dilemma players

paper_content:

We address the problem of how the survival of cooperation in a social system depends on the motion of the individuals. Specifically, we study a model in which prisoner's dilemma players are allowed to move in a two-dimensional plane. Our results show that cooperation can survive in such a system provided that both the temptation to defect and the velocity at which agents move are not too high. Moreover, we show that when these conditions are fulfilled, the only asymptotic state of the system is that in which all players are cooperators. Our results might have implications for the design of cooperative strategies in motion coordination and other applications including wireless networks.
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paper_title: Noise and Correlations in a Spatial Population Model with Cyclic Competition

paper_content:

Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influence on the coevolution of populations with cyclic interspecies competition have been demonstrated in recent experiments e.g., B. Kerr , Nature (London) 418, 171 (2002)]. To reach a better theoretical understanding of these phenomena, we consider a paradigmatic spatial model where three species exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial differential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and spatial diffusion at different levels and show how fascinating patterns of entangled spirals emerge. We rationalize our analysis by computing the spatiotemporal correlation functions and provide analytical expressions for the front velocity and the wavelength of the propagating spiral waves.
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paper_title: Vortex dynamics in a three-state model under cyclic dominance.

paper_content:

The evolution of domain structure is investigated in a two-dimensional voter model with three states under cyclic dominance. The study focus on the dynamics of vortices, defined by the points where the three states (domains) meet. We can distinguish vortices and antivortices which walk randomly and annihilate each other. The domain wall motion can create vortex-antivortex pairs at a rate that is increased by the spiral formation due to cyclic dominance. This mechanism is contrasted with a branching annihilating random walk (BARW) in a particle-antiparticle system with density-dependent pair creation rate. Numerical estimates for the critical indices of the vortex density [beta=0.29(4)] and of its fluctuation [gamma=0.34(6)] improve an earlier Monte Carlo study [K. Tainaka and Y. Itoh, Europhys. Lett. 15, 399 (1991)] of the three-state cyclic model in two dimensions.
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paper_title: Emergence of target waves in paced populations of cyclically competing species

paper_content:

We investigate the emergence of target waves in a cyclic predator-prey model incorporating a periodic current of the three competing species in a small area situated at the center of the square lattice. The periodic current acts as a pacemaker, trying to impose its rhythm on the overall spatiotemporal evolution of the three species. We show that the pacemaker is able to nucleate target waves that eventually spread across the whole population, whereby three routes leading to this phenomenon can be distinguished depending on the mobility of the three species and the oscillation period of the localized current. First, target waves can emerge due to the synchronization between the periodic current and oscillations of the density of the three species on the spatial grid. The second route is similar to the first, the difference being that the synchronization sets in only intermittently. Finally, the third route towards target waves is realized when the frequency of the pacemaker is much higher than that characterizing the oscillations of the overall density of the three species. By considering mobility and the frequency of the current as variable parameters, we thus provide insights into the mechanisms of pattern formation resulting from the interplay between local and global dynamics in systems governed by cyclically competing species.
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paper_title: Discriminating the effects of spatial extent and population size in cyclic competition among species

paper_content:

We introduce a population model for species under cyclic competition. This model allows individuals to coexist and interact on single cells while migration takes place between adjacent cells. In contrast to the model introduced by Reichenbach, Mobilia, and Frey [Reichenbach, Mobilia, and Frey, Nature (London) 448, 1046 (2007)], we find that the emergence of spirals results in an ambiguous behavior regarding the stability of coexistence. The typical time until extinction exhibits, however, a qualitatively opposite dependence on the newly introduced nonunit carrying capacity in the spiraling and the nonspiraling regimes. This allows us to determine a critical mobility that marks the onset of this spiraling state sharply. In contrast, we demonstrate that the conventional finite size stability analysis with respect to spatial size is of limited use for identifying the onset of the spiraling regime.
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paper_title: Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors

paper_content:

One of the central aims of ecology is to identify mechanisms that maintain biodiversity. Numerous theoretical models have shown that competing species can coexist if ecological processes such as dispersal, movement, and interaction occur over small spatial scales. In particular, this may be the case for non-transitive communities, that is, those without strict competitive hierarchies. The classic non-transitive system involves a community of three competing species satisfying a relationship similar to the children's game rock-paper-scissors, where rock crushes scissors, scissors cuts paper, and paper covers rock. Such relationships have been demonstrated in several natural systems. Some models predict that local interaction and dispersal are sufficient to ensure coexistence of all three species in such a community, whereas diversity is lost when ecological processes occur over larger scales. Here, we test these predictions empirically using a non-transitive model community containing three populations of Escherichia coli. We find that diversity is rapidly lost in our experimental community when dispersal and interaction occur over relatively large spatial scales, whereas all populations coexist when ecological processes are localized.
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paper_title: When does cyclic dominance lead to stable spiral waves?

paper_content:

Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
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paper_title: Co-existence in the two-dimensional May-Leonard model with random rates

paper_content:

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-) steady state in two-dimensional stochastic May-Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May-Leonard system (for small system sizes): (1) as the mobility rate exceeds a threshold that separates a species coexistence (quasi-) steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ∼ e cN /N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.
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paper_title: Spatial Rock-Paper-Scissors Models with Inhomogeneous Reaction Rates

paper_content:

We study several variants of the stochastic four-state rock-paper-scissors game or, equivalently, cyclic three-species predator-prey models with conserved total particle density, by means of Monte Carlo simulations on one- and two-dimensional lattices. Specifically, we investigate the influence of spatial variability of the reaction rates and site occupancy restrictions on the transient oscillations of the species densities and on spatial correlation functions in the quasistationary coexistence state. For small systems, we also numerically determine the dependence of typical extinction times on the number of lattice sites. In stark contrast with two-species stochastic Lotka-Volterra systems, we find that for our three-species models with cyclic competition quenched disorder in the reaction rates has very little effect on the dynamics and the long-time properties of the coexistence state. Similarly, we observe that site restriction only has a minor influence on the system's dynamical properties. Our results therefore demonstrate that the features of the spatial rock-paper-scissors system are remarkably robust with respect to model variations, and stochastic fluctuations as well as spatial correlations play a comparatively minor role.
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paper_title: Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

paper_content:

The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes. Strong emphasis is placed on systematic approximation methods for solving problems. Stochastic adiabatic elimination is newly formulated. The book contains the "folklore" of stochastic methods in systematic form and is suitable for use as a reference work.
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paper_title: Coevolutionary dynamics in structured populations ofthree species

paper_content:

Inspired by the experiments with the three strains of E. coli bacteria as well as the three morphs of Uta stansburiana lizards, a model of cyclic dominance was proposed to investigate the mechanisms facilitating the maintenance of biodiversity in spatially structured populations. Subsequent studies enriched the original model with various biologically motivated extension repeating the proposed mathematical ::: analysis and computer simulations. ::: The research presented in this thesis unifies and generalises these models by combining the birth, selection-removal, selection-replacement and mutation processes as well as two forms of mobility into a generic metapopulation model. Instead of the standard mathematical treatment, more controlled analysis with inverse system size and multiscale asymptotic expansions is presented to derive an approximation of the system dynamics in terms of a well-known pattern forming equation. The novel analysis, capable of increased accuracy, is evaluated with improved numerical experiments performed with bespoke software developed for simulating the stochastic and deterministic descriptions of the generic metapopulation model. ::: The emergence of spiral waves facilitating the long term biodiversity is confirmed in the computer simulations as predicted by the theory. The derived conditions on the stability of spiral patterns for different values of the biological parameters are studied resulting in discoveries of interesting phenomena such as spiral annihilation or instabilities caused by nonlinear diffusive terms.
---
paper_title: Global attractors and extinction dynamics of cyclically competing species

paper_content:

Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.
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paper_title: Local migration promotes competitive restraint in a host–pathogen 'tragedy of the commons'

paper_content:

These T4 page and their E. coli hosts are the model for a typical 'victim-exploiter' interaction in a study of the role of migration patterns in a 'tragedy of the commons' competition for limited resources within fragmented communities. In this host-pathogen system, growing in 96-well microtitre plates, coexistence, stability and evolution within the separated communities depend critically on migration: restricted migration can promote restraint in the use of the common resource. In this experiment and in theory, highly connected social networks favour virulence. Fragmented populations possess an intriguing duplicity: even if subpopulations are reliably extinction-prone, asynchrony in local extinctions and recolonizations makes global persistence possible1,2,3,4,5,6,7,8. Migration is a double-edged sword in such cases: too little migration prevents recolonization of extinct patches, whereas too much synchronizes subpopulations, raising the likelihood of global extinction. Both edges of this proverbial sword have been explored by manipulating the rate of migration within experimental populations1,3,4,5,6,8. However, few experiments have examined how the evolutionary ecology of fragmented populations depends on the pattern of migration5. Here, we show that the migration pattern affects both coexistence and evolution within a community of bacterial hosts (Escherichia coli) and viral pathogens (T4 coliphage) distributed across a large network of subpopulations. In particular, different patterns of migration select for distinct pathogen strategies, which we term 'rapacious' and 'prudent'. These strategies define a 'tragedy of the commons'9: rapacious phage displace prudent variants for shared host resources, but prudent phage are more productive when alone. We find that prudent phage dominate when migration is spatially restricted, while rapacious phage evolve under unrestricted migration. Thus, migration pattern alone can determine whether a de novo tragedy of the commons is resolved in favour of restraint.
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paper_title: Evolution of restraint in a structured rock-paper-scissors community.

paper_content:

It is not immediately clear how costly behavior that benefits others evolves by natural selection. By saving on inherent costs, individuals that do not contribute socially have a selective advantage over altruists if both types receive equal benefits. Restrained consumption of a common resource is a form of altruism. The cost of this kind of prudent behavior is that restrained individuals give up resources to less-restrained individuals. The benefit of restraint is that better resource management may prolong the persistence of the group. One way to dodge the problem of defection is for altruists to interact disproportionately with other altruists. With limited dispersal, restrained individuals persist because of interaction with like types, whereas it is the unrestrained individuals that must face the negative long-term consequences of their rapacity. Here, we study the evolution of restraint in a community of three competitors exhibiting a nontransitive (rock–paper–scissors) relationship. The nontransitivity ensures a form of negative feedback, whereby improvement in growth of one competitor has the counterintuitive consequence of lowering the density of that improved player. This negative feedback generates detrimental long-term consequences for unrestrained growth. Using both computer simulations and evolution experiments with a nontransitive community of Escherichia coli, we find that restrained growth can evolve under conditions of limited dispersal in which negative feedback is present. This research, thus, highlights a set of ecological conditions sufficient for the evolution of one form of altruism.
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paper_title: Characterization of spiraling patterns in spatial rock-paper-scissors games

paper_content:

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising “rock-paper-scissors” interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial “rock-paper-scissors” competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.
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paper_title: Spirals and coarsening patterns in the competition of many species: A complex Ginzburg-Landau approach

paper_content:

In order to model real ecological systems one has to consider many species that interact in complex ways. However, most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions. The few studies dealing with larger number of species and/or more complex interaction schemes are mostly restricted to numerical explorations. In this paper we determine, starting from the deterministic mean-field rate equations, for large classes of systems the space of coexistence fixed points at which biodiversity is maximal. For systems with a single coexistence fixed point we derive complex Ginzburg–Landau equations that allow to describe space-time pattern realized in two space dimensions. For selected cases we compare the theoretical predictions with the pattern observed in numerical simulations.
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paper_title: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

paper_content:

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.
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paper_title: Phase diagrams for three-strategy evolutionary prisoner's dilemma games on regular graphs

paper_content:

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.
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paper_title: Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

paper_content:

An important clement in the long-time dynamics of pattern forming systems is a class of solutions we will call "coherent structures". These are states that are either themselves localized, or that consist of domains of regular patterns connected by localized defects or interfaces. This paper summarizes and extends recent work on such coherent structures in the one-dimensional complex Ginzburg-Landau equation and its generalizations, for which rather complete information can be obtained on the existence and competition of fronts, pulses, sources and sinks. For the special subclass of uniformly translating structures, the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three-dimensional phase space. Fixed points of the flow correspond to the two basic building blocks of coherent structures, uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero. A study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures. The dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts, and is expressed in terms of a set of conjectures for front propagation that generalize the "marginal stability" and "pinch-point" approaches of earlier authors. These rules, together with an exact front solution whose dynamics plays an important role in the selection of patterns, yield an analytic expression for the upper limit of the range of existence of pulse solutions, as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist. Extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses. In the parameter ranges where no uniformly translating fronts can exist, examples are shown of irregularly spreading fronts that generate strongly chaotic regions, as well as nonuniformly translating fronts that lead to uniform amplitude states. Recent perturbative treatments based on expansions about the nonlinear Schr6dinger equation are generalized to perturbations of the cubic-quintic and derivative Schr~idinger equations, for which both pulses and fronts exist in the unperturbed system. Comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions. Nevertheless, those that are obtained are found to be consistent with the general conjectures, and in particular they provide an analytic demonstration of front/pulse competition. While the discussion of the competition between fronts and pulses focusses on the complex Ginzburg-Landau equation with quintic terms and a subcritical bifurcation, a number of results are also presented for the cubic equation. In particular, the existence of a family of moving source solutions derived by Bekki and Nozaki for this equation contradicts the naive counting arguments. We attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits.
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paper_title: When does cyclic dominance lead to stable spiral waves?

paper_content:

Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
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paper_title: Theory of interaction and bound states of spiral waves in oscillatory media

paper_content:

We present an alternative method for the calculation of the interaction between spirals in oscillatory media. This method is based on a rigorous evaluation of the perturbation of an isolated spiral resulting from neighboring spirals in a linear approximation. For the complex Ginzburg-Landau equation, the existence of bound states is identified with the parameter range where the perturbations behave in an oscillatory manner. The results for the equilibrium distance for two spirals in the bound state and also the dependence of the velocity of the spiral on the distance are in good agreement with numerical simulations. In the equally charged case, we find multiple bound states which may be interpreted as multiply armed spirals. Outside the oscillatory range, well-separated spirals appear to repel each other regardless of topological charge.
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paper_title: Instability of spatial patterns and its ambiguous impact on species diversity

paper_content:

Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable "blurred" patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals' mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.
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paper_title: The world of the complex Ginzburg-Landau equation

paper_content:

The cubic complex Ginzburg-Landau equation is one of the most-studied nonlinear equations in the physics community. It describes a vast variety of phenomena from nonlinear waves to second-order phase transitions, from superconductivity, superfluidity, and Bose-Einstein condensation to liquid crystals and strings in field theory. The authors give an overview of various phenomena described by the complex Ginzburg-Landau equation in one, two, and three dimensions from the point of view of condensed-matter physicists. Their aim is to study the relevant solutions in order to gain insight into nonequilibrium phenomena in spatially extended systems.
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paper_title: Nonlinear Aspects of Competition Between Three Species

paper_content:

It is shown that for three competitors, the classic Gause–Lotka–Volterra equations possess a special class of periodic limit cycle solutions, and a general class of solutions in which the system exhibits nonperiodic population oscillations of bounded amplitude but ever increasing cycle time. Biologically, the result is interesting as a caricature of the complexities that nonlinearities can introduce even into the simplest equations of population biology ; mathematically, the model illustrates some novel tactical tricks and dynamical peculiarities for 3-dimensional nonlinear systems.
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paper_title: Coevolutionary dynamics in structured populations ofthree species

paper_content:

Inspired by the experiments with the three strains of E. coli bacteria as well as the three morphs of Uta stansburiana lizards, a model of cyclic dominance was proposed to investigate the mechanisms facilitating the maintenance of biodiversity in spatially structured populations. Subsequent studies enriched the original model with various biologically motivated extension repeating the proposed mathematical ::: analysis and computer simulations. ::: The research presented in this thesis unifies and generalises these models by combining the birth, selection-removal, selection-replacement and mutation processes as well as two forms of mobility into a generic metapopulation model. Instead of the standard mathematical treatment, more controlled analysis with inverse system size and multiscale asymptotic expansions is presented to derive an approximation of the system dynamics in terms of a well-known pattern forming equation. The novel analysis, capable of increased accuracy, is evaluated with improved numerical experiments performed with bespoke software developed for simulating the stochastic and deterministic descriptions of the generic metapopulation model. ::: The emergence of spiral waves facilitating the long term biodiversity is confirmed in the computer simulations as predicted by the theory. The derived conditions on the stability of spiral patterns for different values of the biological parameters are studied resulting in discoveries of interesting phenomena such as spiral annihilation or instabilities caused by nonlinear diffusive terms.
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paper_title: A non-linear instability theory for a wave system in plane Poiseuille flow

paper_content:

The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are e t and e ½ ( x + a 1 r t ), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (− a 1 r ) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.
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paper_title: Non-linear wave-number interaction in near-critical two-dimensional flows

paper_content:

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Benard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.
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paper_title: Global attractors and extinction dynamics of cyclically competing species

paper_content:

Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases.
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paper_title: Self-Organization of Mobile Populations in Cyclic Competition

paper_content:

The formation of out-of-equilibrium patterns is a characteristic feature of spatially extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the 'rock-paper-scissors' game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals' mobility on the spatial structures emerging in rock-paper-scissors games. We devise a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior-in the proper continuum limit-is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals' wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system. These findings and our methods have important applications for understanding the formation of noisy patterns, e.g. in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)-chemical reactions.
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paper_title: Mutual Feedbacks Maintain Both Genetic and Species Diversity in a Plant Community

paper_content:

The forces that maintain genetic diversity among individuals and diversity among species are usually studied separately. Nevertheless, diversity at one of these levels may depend on the diversity at the other. We have combined observations of natural populations, quantitative genetics, and field experiments to show that genetic variation in the concentration of an allelopathic secondary compound in Brassica nigra is necessary for the coexistence of B. nigra and its competitor species. In addition, the diversity of competing species was required for the maintenance of genetic variation in the trait within B. nigra. Thus, conservation of species diversity may also necessitate maintenance of the processes that sustain the genetic diversity of each individual species.
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paper_title: Noise and Correlations in a Spatial Population Model with Cyclic Competition

paper_content:

Noise and spatial degrees of freedom characterize most ecosystems. Some aspects of their influence on the coevolution of populations with cyclic interspecies competition have been demonstrated in recent experiments e.g., B. Kerr , Nature (London) 418, 171 (2002)]. To reach a better theoretical understanding of these phenomena, we consider a paradigmatic spatial model where three species exhibit cyclic dominance. Using an individual-based description, as well as stochastic partial differential and deterministic reaction-diffusion equations, we account for stochastic fluctuations and spatial diffusion at different levels and show how fascinating patterns of entangled spirals emerge. We rationalize our analysis by computing the spatiotemporal correlation functions and provide analytical expressions for the front velocity and the wavelength of the propagating spiral waves.
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paper_title: A time dependent Ginzburg-Landau equation and its application to the problem of resistivity in the mixed state

paper_content:

A time dependent modification of the Ginzburg-Landau equation is given which is based on the assumption that the functional derivative of the Ginzburg-Landau free energy expression with respect to the wave function is a generalized force in the sense of irreversible thermodynamics acting on the wave function. This equation implies an energy theorem, according to which the energy can be dissipated by i) production of Joule heat; ii) irreversible variation of the wave function. The theory is a limiting case of the BCS theory, and hence, it contains no adjustable parameters. The application of the modified equation to the problem of resistivity in the mixed state reveals satisfactory agreement between experiment and theory for reduced temperatures higher than 0.6.
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paper_title: Characterization of spiraling patterns in spatial rock-paper-scissors games

paper_content:

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising “rock-paper-scissors” interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial “rock-paper-scissors” competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.
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paper_title: Evolutionary Games and Population Dynamics

paper_content:

Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms. ::: ::: Comprehensive treatment of ecological and game theoretic dynamics ::: Invasion dynamics and permanence as key concepts ::: Explanation in terms of games of things like competition between species
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paper_title: Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities

paper_content:

Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the self-organization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures the viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social” behavior. ::: ::: A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems. Intrinsic fluctuations, stemming from the discreteness of individuals, are ubiquitous, and can have an important impact on the stability of ecosystems. In the absence of speciation, extinction of species is unavoidable. It may, however, take very long times. We provide a general concept for defining survival and extinction on ecological time-scales. Spatial degrees of freedom come with a certain mobility of individuals. When the latter is sufficiently high, bacterial community structures can be understood through mapping individual-based models, in a continuum approach, onto stochastic partial differential equations. These allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds. We conclude with a perspective on the current challenges in quantifying bacterial pattern formation, and how this might have an impact on fundamental research in non-equilibrium physics.
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paper_title: Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies

paper_content:

Economic experiments reveal that humans value cooperation and fairness. Punishing unfair behavior is therefore common, and according to the theory of strong reciprocity, it is also directly related to rewarding cooperative behavior. However, empirical data fail to confirm that positive and negative reciprocity are correlated. Inspired by this disagreement, we determine whether the combined application of reward and punishment is evolutionary advantageous. We study a spatial public goods game, where in addition to the three elementary strategies of defection, rewarding and punishment, a fourth strategy combining the later two competes for space. We find rich dynamical behavior that gives rise to intricate phase diagrams where continuous and discontinuous phase transitions occur in succession. Indirect territorial competition, spontaneous emergence of cyclic dominance, as well as divergent fluctuations of oscillations that terminate in an absorbing phase are observed. Yet despite the high complexity of solutions, the combined strategy can survive only in very narrow and unrealistic parameter regions. Elementary strategies, either in pure or mixed phases, are much more common and likely to prevail. Our results highlight the importance of patterns and structure in human cooperation, which should be considered in future experiments.
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paper_title: Punishing and abstaining for public goods.

paper_content:

The evolution of cooperation within sizable groups of nonrelated humans offers many challenges for our understanding. Current research has highlighted two factors boosting cooperation in public goods interactions, namely, costly punishment of defectors and the option to abstain from the joint enterprise. A recent modeling approach has suggested that the autarkic option acts as a catalyzer for the ultimate fixation of altruistic punishment. We present an alternative, more microeconomically based model that yields a bistable outcome instead. Evolutionary dynamics can lead either to a Nash equilibrium of punishing and nonpunishing cooperators or to an oscillating state without punishers.
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paper_title: Self-organization of punishment in structured populations

paper_content:

Cooperation is crucial for the remarkable evolutionary success of the human species. Not surprisingly, some individuals are willing to bear additional costs in order to punish defectors. Current models assume that, once set, the fine and cost of punishment do not change over time. Here we show that relaxing this assumption by allowing players to adapt their sanctioning efforts in dependence on the success of cooperation can explain both the spontaneous emergence of punishment and its ability to deter defectors and those unwilling to punish them with globally negligible investments. By means of phase diagrams and the analysis of emerging spatial patterns, we demonstrate that adaptive punishment promotes public cooperation through the invigoration of spatial reciprocity, the prevention of the emergence of cyclic dominance, or the provision of competitive advantages to those that sanction antisocial behavior. The results presented indicate that the process of self-organization significantly elevates the effectiveness of punishment, and they reveal new mechanisms by means of which this fascinating and widespread social behavior could have evolved.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

paper_content:

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
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paper_title: Phase diagrams for three-strategy evolutionary prisoner's dilemma games on regular graphs

paper_content:

Evolutionary prisoner's dilemma games are studied with players located on square lattice and random regular graph defining four neighbors for each one. The players follow one of the three strategies: tit-for-tat, unconditional cooperation, and defection. The simplified payoff matrix is characterized by two parameters: the temptation b to choose defection and the cost c of inspection reducing the income of tit-for-tat. The strategy imitation from one of the neighbors is controlled by pairwise comparison at a fixed level of noise. Using Monte Carlo simulations and the extended versions of pair approximation we have evaluated the b-c phase diagrams indicating a rich plethora of phase transitions between stationary coexistence, absorbing, and oscillatory states, including continuous and discontinuous phase transitions. By reasonable costs the tit-for-tat strategy prevents extinction of cooperators across the whole span of b determining the prisoner's dilemma game, irrespective of the connectivity structure. We also demonstrate that the system can exhibit a repetitive succession of oscillatory and stationary states upon changing a single payoff value, which highlights the remarkable sensitivity of cyclical interactions on the parameters that define the strength of dominance.
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paper_title: Incentives and opportunism: from the carrot to the stick

paper_content:

Cooperation in public good games is greatly promoted by positive and negative incentives. In this paper, we use evolutionary game dynamics to study the evolution of opportunism (the readiness to be swayed by incentives) and the evolution of trust (the propensity to cooperate in the absence of information on the co-players). If both positive and negative incentives are available, evolution leads to a population where defectors are punished and players cooperate, except when they can get away with defection. Rewarding behaviour does not become fixed, but can play an essential role in catalysing the emergence of cooperation, especially if the information level is low.
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paper_title: The evolution of sanctioning institutions: an experimental approach to the social contract

paper_content:

A vast amount of empirical and theoretical research on public good games indicates that the threat of punishment can curb free-riding in human groups engaged in joint enterprises. Since punishment is often costly, however, this raises an issue of second-order free-riding: indeed, the sanctioning system itself is a common good which can be exploited. Most investigations, so far, considered peer punishment: players could impose fines on those who exploited them, at a cost to themselves. Only a minority considered so-called pool punishment. In this scenario, players contribute to a punishment pool before engaging in the joint enterprise, and without knowing who the free-riders will be. Theoretical investigations (Sigmund et al., Nature 466:861–863, 2010) have shown that peer punishment is more efficient, but pool punishment more stable. Social learning, i.e., the preferential imitation of successful strategies, should lead to pool punishment if sanctions are also imposed on second-order free-riders, but to peer punishment if they are not. Here we describe an economic experiment (the Mutual Aid game) which tests this prediction. We find that pool punishment only emerges if second-order free riders are punished, but that peer punishment is more stable than expected. Basically, our experiment shows that social learning can lead to a spontaneously emerging social contract, based on a sanctioning institution to overcome the free rider problem.
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paper_title: Volunteering leads to rock–paper–scissors dynamics in a public goods game

paper_content:

Collective efforts are a trademark of both insect and human societies1. They are achieved through relatedness in the former2 and unknown mechanisms in the latter. The problem of achieving cooperation among non-kin has been described as the ‘tragedy of the commons’, prophesying the inescapable collapse of many human enterprises3,4. In public goods experiments, initial cooperation usually drops quickly to almost zero5. It can be maintained by the opportunity to punish defectors6 or the need to maintain good reputation7. Both schemes require that defectors are identified. Theorists propose that a simple but effective mechanism operates under full anonymity. With optional participation in the public goods game, ‘loners’ (players who do not join the group), defectors and cooperators will coexist through rock–paper–scissors dynamics8,9. Here we show experimentally that volunteering generates these dynamics in public goods games and that manipulating initial conditions can produce each predicted direction. If, by manipulating displayed decisions, it is pretended that defectors have the highest frequency, loners soon become most frequent, as do cooperators after loners and defectors after cooperators. On average, cooperation is perpetuated at a substantial level.
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paper_title: Effects of punishment in a mobile population playing the prisoner's dilemma game

paper_content:

We deal with a system of prisoner's dilemma players undergoing continuous motion in a two-dimensional plane. In contrast to previous work, we introduce altruistic punishment after the game. We find punishing only a few of the cooperator-defector interactions is enough to lead the system to a cooperative state in environments where otherwise defection would take over the population. This happens even with soft nonsocial punishment (where both cooperators and defectors punish other players, a behavior observed in many human populations). For high enough mobilities or temptations to defect, low rates of social punishment can no longer avoid the breakdown of cooperation.
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paper_title: Evolutionary advantages of adaptive rewarding

paper_content:

Our well-being depends on both our personal success and the success of our society. The realization of this fact makes cooperation an essential trait. Experiments have shown that rewards can elevate our readiness to cooperate, but since giving a reward inevitably entails paying a cost for it, the emergence and stability of such behavior remains elusive. Here we show that allowing for the act of rewarding to self-organize in dependence on the success of cooperation creates several evolutionary advantages that instill new ways through which collaborative efforts are promoted. Ranging from indirect territorial battle to the spontaneous emergence and destruction of coexistence, phase diagrams and the underlying spatial patterns reveal fascinatingly rich social dynamics that explain why this costly behavior has evolved and persevered. Comparisons with adaptive punishment, however, uncover an Achilles heel of adaptive rewarding, coming from over-aggression, which in turn hinders optimal utilization of network reciprocity. This may explain why, despite its success, rewarding is not as firmly embedded into our societal organization as punishment.
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paper_title: Dynamically generated cyclic dominance in spatial prisoner's dilemma games

paper_content:

We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a step-like function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time-dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.
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paper_title: Reward and cooperation in the spatial public goods game

paper_content:

The promise of punishment and reward in promoting public cooperation is debatable. While punishment is traditionally considered more successful than reward, the fact that the cost of punishment frequently fails to offset gains from enhanced cooperation has lead some to reconsider reward as the main catalyst behind collaborative efforts. Here we elaborate on the"stick versus carrot"dilemma by studying the evolution of cooperation in the spatial public goods game, where besides the traditional cooperators and defectors, rewarding cooperators supplement the array of possible strategies. The latter are willing to reward cooperative actions at a personal cost, thus effectively downgrading pure cooperators to second-order free-riders due to their unwillingness to bear these additional costs. Consequently, we find that defection remains viable, especially if the rewarding is costly. Rewards, however, can promote cooperation, especially if the synergetic effects of cooperation are low. Surprisingly, moderate rewards may promote cooperation better than high rewards, which is due to the spontaneous emergence of cyclic dominance between the three strategies.
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paper_title: Evolutionary establishment of moral and double moral standards through spatial interactions

paper_content:

Situations where individuals have to contribute to joint efforts or share scarce resources are ubiquitous. Yet, without proper mechanisms to ensure cooperation, the evolutionary pressure to maximize individual success tends to create a tragedy of the commons (such as over-fishing or the destruction of our environment). This contribution addresses a number of related puzzles of human behavior with an evolutionary game theoretical approach as it has been successfully used to explain the behavior of other biological species many times, from bacteria to vertebrates. Our agent-based model distinguishes individuals applying four different behavioral strategies: non-cooperative individuals ("defectors"), cooperative individuals abstaining from punishment efforts (called "cooperators" or "second-order free-riders"), cooperators who punish non-cooperative behavior ("moralists"), and defectors, who punish other defectors despite being non-cooperative themselves ("immoralists"). By considering spatial interactions with neighboring individuals, our model reveals several interesting effects: First, moralists can fully eliminate cooperators. This spreading of punishing behavior requires a segregation of behavioral strategies and solves the "second-order free-rider problem". Second, the system behavior changes its character significantly even after very long times ("who laughs last laughs best effect"). Third, the presence of a number of defectors can largely accelerate the victory of moralists over non-punishing cooperators. Fourth, in order to succeed, moralists may profit from immoralists in a way that appears like an "unholy collaboration". Our findings suggest that the consideration of punishment strategies allows one to understand the establishment and spreading of "moral behavior" by means of game-theoretical concepts. This demonstrates that quantitative biological modeling approaches are powerful even in domains that have been addressed with non-mathematical concepts so far. The complex dynamics of certain social behaviors become understandable as the result of an evolutionary competition between different behavioral strategies.
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paper_title: Indirect reciprocity can stabilize cooperation without the second-order free rider problem

paper_content:

Models of large-scale human cooperation take two forms. ‘Indirect reciprocity’1 occurs when individuals help others in order to uphold a reputation and so be included in future cooperation. In ‘collective action’2, individuals engage in costly behaviour that benefits the group as a whole. Although the evolution of indirect reciprocity is theoretically plausible3,4,5,6, there is no consensus about how collective action evolves. Evidence suggests that punishing free riders can maintain cooperation7,8,9, but why individuals should engage in costly punishment is unclear. Solutions to this ‘second-order free rider problem’ include meta-punishment10, mutation11, conformism12, signalling13,14,15 and group-selection16,17,18. The threat of exclusion from indirect reciprocity can sustain collective action in the laboratory19. Here, we show that such exclusion is evolutionarily stable, providing an incentive to engage in costly cooperation, while avoiding the second-order free rider problem because punishers can withhold help from free riders without damaging their reputations. However, we also show that such a strategy cannot invade a population in which indirect reciprocity is not linked to collective action, thus leaving unexplained how collective action arises.
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paper_title: The evolution of altruistic punishment

paper_content:

Both laboratory and field data suggest that people punish noncooperators even in one-shot interactions. Although such “altruistic punishment” may explain the high levels of cooperation in human societies, it creates an evolutionary puzzle: existing models suggest that altruistic cooperation among nonrelatives is evolutionarily stable only in small groups. Thus, applying such models to the evolution of altruistic punishment leads to the prediction that people will not incur costs to punish others to provide benefits to large groups of nonrelatives. However, here we show that an important asymmetry between altruistic cooperation and altruistic punishment allows altruistic punishment to evolve in populations engaged in one-time, anonymous interactions. This process allows both altruistic punishment and altruistic cooperation to be maintained even when groups are large and other parameter values approximate conditions that characterize cultural evolution in the small-scale societies in which humans lived for most of our prehistory.
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paper_title: Reward and punishment

paper_content:

Minigames capturing the essence of Public Goods experiments show that even in the absence of rationality assumptions, both punishment and reward will fail to bring about prosocial behavior. This result holds in particular for the well-known Ultimatum Game, which emerges as a special case. But reputation can induce fairness and cooperation in populations adapting through learning or imitation. Indeed, the inclusion of reputation effects in the corresponding dynamical models leads to the evolution of economically productive behavior, with agents contributing to the public good and either punishing those who do not or rewarding those who do. Reward and punishment correspond to two types of bifurcation with intriguing complementarity. The analysis suggests that reputation is essential for fostering social behavior among selfish agents, and that it is considerably more effective with punishment than with reward.
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paper_title: Phase diagrams for the spatial public goods game with pool-punishment

paper_content:

The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways.
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paper_title: Rare but severe concerted punishment that favors cooperation.

paper_content:

As one of the mechanisms that are supposed to explain the evolution of cooperation among unrelated individuals, costly punishment, in which altruistic individuals privately bear the cost to punish defection, suffers from such drawbacks as decreasing individuals' welfare, inducing second-order free riding, the difficulty of catching defection, and the possibility of triggering retaliation. To improve this promising mechanism, here we propose an extended Public Goods game with rare but severe concerted punishment, in which once a defector is caught punishment is triggered and the cost of punishment is equally shared among the remainder of the group. Analytical results show that, when the probability for concerted punishment is above a threshold, cooperating is, while defecting is not, an evolutionarily stable strategy in finite populations, and that this way of punishment can considerably decrease the total cost of inhibiting defection, especially in large populations.
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paper_title: Public goods games with reward in finite populations

paper_content:

Public goods games paraphrase the problem of cooperation in game theoretical terms. Cooperators contribute to a public good and thereby increase the welfare of others at a cost to themselves. Defectors consume the public good but do not pay its cost and therefore outperform cooperators. Hence, according to genetic or cultural evolution, defectors should be favored and the public good disappear – despite the fact that groups of cooperators are better off than groups of defectors. The maximization of short term individual profits causes the demise of the common resource to the detriment of all. This outcome can be averted by introducing incentives to cooperate. Negative incentives based on the punishment of defectors efficiently stabilize cooperation once established but cannot initiate cooperation. Here we consider the complementary case of positive incentives created by allowing individuals to reward those that contribute to the public good. The finite-population stochastic dynamics of the public goods game with reward demonstrate that reward initiates cooperation by providing an escape hatch out of states of mutual defection. However, in contrast to punishment, reward is unable to stabilize cooperation but, instead, gives rise to a persistent minority of cooperators.
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paper_title: Cooperation for volunteering and partially random partnerships

paper_content:

Competition among cooperative, defective, and loner strategies is studied by considering an evolutionary prisoner's dilemma game for different partnerships. In this game each player can adopt one of its coplayer's strategy with a probability depending on the difference of payoffs coming from games with the corresponding coplayers. Our attention is focused on the effects of annealed and quenched randomness in the partnership for fixed number of coplayers. It is shown that only the loners survive if the four coplayers are chosen randomly (mean-field limit). On the contrary, on the square lattice all the three strategies are maintained by the cyclic invasions resulting in a self-organizing spatial pattern. If the fixed partnership is described by a regular small-world structure then a homogeneous oscillation occurs in the population dynamics when the measure of quenched randomness exceeds a threshold value. Similar behavior with higher sensitivity to the randomness is found if temporary partners are substituted for the standard ones with some probability at each step of iteration.
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paper_title: Emergence of synchronization induced by the interplay between two prisoner's dilemma games with volunteering in small-world networks

paper_content:

We studied synchronization between prisoner's dilemma games with voluntary participation in two Newman-Watts small-world networks. It was found that there are three kinds of synchronization: partial phase synchronization, total phase synchronization, and complete synchronization, for varied coupling factors. Besides, two games can reach complete synchronization for the large enough coupling factor. We also discussed the effect of the coupling factor on the amplitude of oscillation of cooperator density.
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paper_title: Defense mechanisms of empathetic players in the spatial ultimatum game

paper_content:

Experiments on the ultimatum game have revealed that humans are remarkably fond of fair play. When asked to share an amount of money, unfair offers are rare and their acceptance rate small. While empathy and spatiality may lead to the evolution of fairness, thus far considered continuous strategies have precluded the observation of solutions that would be driven by pattern formation. Here we introduce a spatial ultimatum game with discrete strategies, and we show that this simple alteration opens the gate to fascinatingly rich dynamical behavior. In addition to mixed stationary states, we report the occurrence of traveling waves and cyclic dominance, where one strategy in the cycle can be an alliance of two strategies. The highly webbed phase diagram, entailing continuous and discontinuous phase transitions, reveals hidden complexity in the pursuit of human fair play.
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paper_title: Phase transitions and volunteering in spatial public goods games.

paper_content:

We present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to "rock-scissors-paper"-type cyclic dominance of the three strategies, cooperate, defect, and loner. In spatial settings with players arranged on a regular lattice, this results in interesting dynamical properties and intriguing spatiotemporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two-, and three-strategy states which either are in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior.
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paper_title: Moral assessment in indirect reciprocity

paper_content:

Indirect reciprocity is one of the mechanisms for cooperation, and seems to be of particular interest for the evolution of human societies. A large part is based on assessing reputations and acting accordingly. This paper gives a brief overview of different assessment rules for indirect reciprocity, and studies them by using evolutionary game dynamics. Even the simplest binary assessment rules lead to complex outcomes and require considerable cognitive abilities.
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paper_title: The take-it-or-leave-it option allows small penalties to overcome social dilemmas

paper_content:

Self-interest frequently causes individuals engaged in joint enterprises to choose actions that are counterproductive. Free-riders can invade a society of cooperators, causing a tragedy of the commons. Such social dilemmas can be overcome by positive or negative incentives. Even though an incentive-providing institution may protect a cooperative society from invasion by free-riders, it cannot always convert a society of free-riders to cooperation. In the latter case, both norms, cooperation and defection, are stable: To avoid a collapse to full defection, cooperators must be sufficiently numerous initially. A society of free-riders is then caught in a social trap, and the institution is unable to provide an escape, except at a high, possibly prohibitive cost. Here, we analyze the interplay of (a) incentives provided by institutions and (b) the effects of voluntary participation. We show that this combination fundamentally improves the efficiency of incentives. In particular, optional participation allows institutions punishing free-riders to overcome the social dilemma at a much lower cost, and to promote a globally stable regime of cooperation. This removes the social trap and implies that whenever a society of cooperators cannot be invaded by free-riders, it will necessarily become established in the long run, through social learning, irrespective of the initial number of cooperators. We also demonstrate that punishing provides a “lighter touch” than rewarding, guaranteeing full cooperation at considerably lower cost.
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paper_title: Optional contributions have positive effects for volunteering public goods games

paper_content:

Public goods (PG) games with the volunteering mechanism are referred to as volunteering public goods (VPG) games, in which loners are introduced to the PG games, and a loner obtains a constant payoff but not participating the game. Considering that small contributions may have positive effects to encourage more players with bounded rationality to contribute, this paper introduces optional contributions (high value or low value) to these typical VPG games—a cooperator can contribute a high or low payoff to the public pools. With the low contribution, the logit dynamics show that cooperation can be promoted in a well mixed population comparing to the typical VPG games, furthermore, as the multiplication factor is greater than a threshold, the average payoff of the population is also enhanced. In spatial VPG games, we introduce a new adjusting mechanism that is an approximation to best response. Some results in agreement with the prediction of the logit dynamics are found. These simulation results reveal that for VPG games the option of low contributions may be a better method to stimulate the growth of cooperation frequency and the average payoff of the population.
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paper_title: Altruistic Punishment and the Origin of Cooperation

paper_content:

How did human cooperation evolve? Recent evidence shows that many people are willing to engage in altruistic punishment, voluntarily paying a cost to punish noncooperators. Although this behavior helps to explain how cooperation can persist, it creates an important puzzle. If altruistic punishment provides benefits to nonpunishers and is costly to punishers, then how could it evolve? Drawing on recent insights from voluntary public goods games, I present a simple evolutionary model in which altruistic punishers can enter and will always come to dominate a population of contributors, defectors, and nonparticipants. The model suggests that the cycle of strategies in voluntary public goods games does not persist in the presence of punishment strategies. It also suggests that punishment can only enforce payoff-improving strategies, contrary to a widely cited “folk theorem” result that suggests that punishment can allow the evolution of any strategy.
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paper_title: Fixation and escape times in stochastic game learning

paper_content:

Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2×2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulations. Simple cases are also amenable to analytical approaches and we provide estimates of the behaviour of so-called escape times as a function of the batch size. The differences and similarities with escape and fixation in evolutionary dynamics are discussed.
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paper_title: Dynamically generated cyclic dominance in spatial prisoner's dilemma games

paper_content:

We have studied the impact of time-dependent learning capacities of players in the framework of spatial prisoner's dilemma game. In our model, this capacity of players may decrease or increase in time after strategy adoption according to a step-like function. We investigated both possibilities separately and observed significantly different mechanisms that form the stationary pattern of the system. The time decreasing learning activity helps cooperator domains to recover the possible intrude of defectors hence supports cooperation. In the other case the temporary restrained learning activity generates a cyclic dominance between defector and cooperator strategies, which helps to maintain the diversity of strategies via propagating waves. The results are robust and remain valid by changing payoff values, interaction graphs or functions characterizing time-dependence of learning activity. Our observations suggest that dynamically generated mechanisms may offer alternative ways to keep cooperators alive even at very larger temptation to defect.
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paper_title: Impact of aging on the evolution of cooperation in the spatial prisoner's dilemma game

paper_content:

Aging is always present, tailoring our interactions with others, and postulating a finite lifespan during which we are able to exercise them. We consider the prisoner's dilemma game on a square lattice and examine how quenched age distributions and different aging protocols influence the evolution of cooperation when taking the life experience and knowledge accumulation into account as time passes. In agreement with previous studies, we find that a quenched assignment of age to players, introducing heterogeneity to the game, substantially promotes cooperative behavior. Introduction of aging and subsequent death as a coevolutionary process may act detrimental on cooperation but enhances it efficiently if the offspring of individuals that have successfully passed their strategy is considered newborn. We study resulting age distributions of players and show that the heterogeneity is vital-yet insufficient-for explaining the observed differences in cooperator abundance on the spatial grid. The unexpected increment of cooperation levels can be explained by a dynamical effect that has a highly selective impact on the propagation of cooperator and defector states.
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paper_title: Intrinsic noise in game dynamical learning

paper_content:

Demographic noise has profound effects on evolutionary and population dynamics, as well as on chemical reaction systems and models of epidemiology. Such noise is intrinsic and due to the discreteness of the dynamics in finite populations. We here show that similar noise-sustained trajectories arise in game dynamical learning, where the stochasticity has a different origin: agents sample a finite number of moves of their opponents in between adaptation events. The limit of infinite batches results in deterministic modified replicator equations, whereas finite sampling leads to a stochastic dynamics. The characteristics of these fluctuations can be computed analytically using methods from statistical physics, and such noise can affect the attractors significantly, leading to noise-sustained cycling or removing periodic orbits of the standard replicator dynamics.
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paper_title: Volunteering leads to rock–paper–scissors dynamics in a public goods game

paper_content:

Collective efforts are a trademark of both insect and human societies1. They are achieved through relatedness in the former2 and unknown mechanisms in the latter. The problem of achieving cooperation among non-kin has been described as the ‘tragedy of the commons’, prophesying the inescapable collapse of many human enterprises3,4. In public goods experiments, initial cooperation usually drops quickly to almost zero5. It can be maintained by the opportunity to punish defectors6 or the need to maintain good reputation7. Both schemes require that defectors are identified. Theorists propose that a simple but effective mechanism operates under full anonymity. With optional participation in the public goods game, ‘loners’ (players who do not join the group), defectors and cooperators will coexist through rock–paper–scissors dynamics8,9. Here we show experimentally that volunteering generates these dynamics in public goods games and that manipulating initial conditions can produce each predicted direction. If, by manipulating displayed decisions, it is pretended that defectors have the highest frequency, loners soon become most frequent, as do cooperators after loners and defectors after cooperators. On average, cooperation is perpetuated at a substantial level.
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paper_title: Spatial prisoner's dilemma game with volunteering in Newman-Watts small-world networks.

paper_content:

A modified spatial prisoner's dilemma game with voluntary participation in Newman-Watts small-world networks is studied. Some reasonable ingredients are introduced to the game evolutionary dynamics: each agent in the network is a pure strategist and can only take one of three strategies (cooperator, defector, and loner); its strategical transformation is associated with both the number of strategical states and the magnitude of average profits, which are adopted and acquired by its coplayers in the previous round of play; a stochastic strategy mutation is applied when it gets into the trouble of local commons that the agent and its neighbors are in the same state and get the same average payoffs. In the case of very low temptation to defect, it is found that agents are willing to participate in the game in typical small-world region and intensive collective oscillations arise in more random region.
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paper_title: The Joker effect: cooperation driven by destructive agents

paper_content:

Abstract Understanding the emergence of cooperation is a central issue in evolutionary game theory. The hardest setup for the attainment of cooperation in a population of individuals is the Public Goods game in which cooperative agents generate a common good at their own expenses, while defectors “free-ride” this good. Eventually this causes the exhaustion of the good, a situation which is bad for everybody. Previous results have shown that introducing reputation, allowing for volunteer participation, punishing defectors, rewarding cooperators or structuring agents, can enhance cooperation. Here we present a model which shows how the introduction of rare, malicious agents – that we term jokers – performing just destructive actions on the other agents induce bursts of cooperation. The appearance of jokers promotes a rock-paper-scissors dynamics, where jokers outbeat defectors and cooperators outperform jokers, which are subsequently invaded by defectors. Thus, paradoxically, the existence of destructive agents acting indiscriminately promotes cooperation.
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paper_title: Phase transitions for rock-scissors-paper game on different networks.

paper_content:

Monte Carlo simulations and dynamical mean-field approximations are performed to study the phase transitions in the rock-scissors-paper game on different host networks. These graphs are originated from lattices by introducing quenched and annealed randomness simultaneously. In the resulting phase diagrams three different stationary states are identified for all structures. The comparison of results on different networks suggests that the value of the clustering coefficient plays an irrelevant role in the emergence of a global oscillating phase. The critical behavior of phase transitions seems to be universal and can be described by the same exponents.
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paper_title: Rock-scissors-paper game on regular small-world networks

paper_content:

The spatial rock-scissors-paper game (or cyclic Lotka–Volterra system) is extended to study how the spatiotemporal patterns are affected by the rewired host lattice providing uniform number of neighbours (degree) at each site. On the square lattice this system exhibits a self-organizing pattern with equal concentration of the competing strategies (species). If the quenched background is constructed by substituting random links for the nearest-neighbour bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value. This transition can also be observed if the standard link is replaced temporarily by a random one with a probability P at each step of iteration. Above a second threshold value of P the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous (absorbing) states. In this case the results of Monte Carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one-, two-, four- and six-site clusters.
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paper_title: Cooperation for volunteering and partially random partnerships

paper_content:

Competition among cooperative, defective, and loner strategies is studied by considering an evolutionary prisoner's dilemma game for different partnerships. In this game each player can adopt one of its coplayer's strategy with a probability depending on the difference of payoffs coming from games with the corresponding coplayers. Our attention is focused on the effects of annealed and quenched randomness in the partnership for fixed number of coplayers. It is shown that only the loners survive if the four coplayers are chosen randomly (mean-field limit). On the contrary, on the square lattice all the three strategies are maintained by the cyclic invasions resulting in a self-organizing spatial pattern. If the fixed partnership is described by a regular small-world structure then a homogeneous oscillation occurs in the population dynamics when the measure of quenched randomness exceeds a threshold value. Similar behavior with higher sensitivity to the randomness is found if temporary partners are substituted for the standard ones with some probability at each step of iteration.
---
paper_title: Phase transitions and volunteering in spatial public goods games.

paper_content:

We present a simple yet effective mechanism promoting cooperation under full anonymity by allowing for voluntary participation in public goods games. This natural extension leads to "rock-scissors-paper"-type cyclic dominance of the three strategies, cooperate, defect, and loner. In spatial settings with players arranged on a regular lattice, this results in interesting dynamical properties and intriguing spatiotemporal patterns. In particular, variations of the value of the public good leads to transitions between one-, two-, and three-strategy states which either are in the class of directed percolation or show interesting analogies to Ising-type models. Although volunteering is incapable of stabilizing cooperation, it efficiently prevents successful spreading of selfish behavior.
---
paper_title: Stability and robustness analysis of cooperation cycles driven by destructive agents in finite populations

paper_content:

The emergence and promotion of cooperation are two of the main issues in evolutionary game theory, as cooperation is amenable to exploitation by defectors, which take advantage of cooperative individuals at no cost, dooming them to extinction. It has been recently shown that the existence of purely destructive agents (termed jokers) acting on the common enterprises (public goods games) can induce stable limit cycles among cooperation, defection, and destruction when infinite populations are considered. These cycles allow for time lapses in which cooperators represent a relevant fraction of the population, providing a mechanism for the emergence of cooperative states in nature and human societies. Here we study analytically and through agent-based simulations the dynamics generated by jokers in finite populations for several selection rules. Cycles appear in all cases studied, thus showing that the joker dynamics generically yields a robust cyclic behavior not restricted to infinite populations. We also compute the average time in which the population consists mostly of just one strategy and compare the results with numerical simulations.
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paper_title: Phase diagrams for the spatial public goods game with pool-punishment

paper_content:

The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool punishers as the three competing strategies. Fines and costs of pool punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects a payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single- and two-strategy stationary states, a rock-paper-scissors type of cyclic dominance can emerge in strikingly different ways.
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paper_title: The Economics of Fair Play.

paper_content:

Imagine that somebody offers you $100. All you have to do is agree with some other anonymous person on how to share the sum. The rules are strict. The two of you are in separate rooms and cannot exchange information. A coin toss decides which of you will propose how to share the money. Suppose that you are the proposer. You can make a single offer of how to split the sum, and the other person-the responder-can say yes or no. The responder also knows the rules and the total amount of money at stake. If her answer is yes, the deal goes ahead. If her answer is no, neither of you gets anything. In both cases, the game is over and will not be repeated. What will you do? ::: ::: Instinctively, many people feel they should offer 50 percent, because such a division is "fair" and therefore likely to be accepted. More daring people, however, think they might get away with offering somewhat less than half of the sum.
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paper_title: Defense mechanisms of empathetic players in the spatial ultimatum game

paper_content:

Experiments on the ultimatum game have revealed that humans are remarkably fond of fair play. When asked to share an amount of money, unfair offers are rare and their acceptance rate small. While empathy and spatiality may lead to the evolution of fairness, thus far considered continuous strategies have precluded the observation of solutions that would be driven by pattern formation. Here we introduce a spatial ultimatum game with discrete strategies, and we show that this simple alteration opens the gate to fascinatingly rich dynamical behavior. In addition to mixed stationary states, we report the occurrence of traveling waves and cyclic dominance, where one strategy in the cycle can be an alliance of two strategies. The highly webbed phase diagram, entailing continuous and discontinuous phase transitions, reveals hidden complexity in the pursuit of human fair play.
---
paper_title: Evolutionary conservation of species' roles in food webs.

paper_content:

Studies of ecological networks (the web of interactions between species in a community) demonstrate an intricate link between a community's structure and its long-term viability. It remains unclear, however, how much a community's persistence depends on the identities of the species present, or how much the role played by each species varies as a function of the community in which it is found. We measured species' roles by studying how species are embedded within the overall network and the subsequent dynamic implications. Using data from 32 empirical food webs, we find that species' roles and dynamic importance are inherent species attributes and can be extrapolated across communities on the basis of taxonomic classification alone. Our results illustrate the variability of roles across species and communities and the relative importance of distinct species groups when attempting to conserve ecological communities.
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paper_title: Spirals and coarsening patterns in the competition of many species: A complex Ginzburg-Landau approach

paper_content:

In order to model real ecological systems one has to consider many species that interact in complex ways. However, most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions. The few studies dealing with larger number of species and/or more complex interaction schemes are mostly restricted to numerical explorations. In this paper we determine, starting from the deterministic mean-field rate equations, for large classes of systems the space of coexistence fixed points at which biodiversity is maximal. For systems with a single coexistence fixed point we derive complex Ginzburg–Landau equations that allow to describe space-time pattern realized in two space dimensions. For selected cases we compare the theoretical predictions with the pattern observed in numerical simulations.
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paper_title: Globally synchronized oscillations in complex cyclic games

paper_content:

The rock-paper-scissors game and its generalizations with S>3 species are well-studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one-predator, one-prey symmetry. Lütz et al. [J. Theor. Biol. 317, 286 (2013)] have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators, but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food-web subloops (if any). Here we show that upon replacing a minimum fraction Q of the short-range interactions by long-range ones, there is a Hopf bifurcation, and global oscillations become stable. Surprisingly, to build such long-distance, global synchronization, the four-species coexistence phase requires fewer long-range interactions than the three-species phase, while one would naively expect the opposite to be true. Moreover, deviations from highly homogeneous conditions (χ=0 or 1) increase Qc, and the more heterogeneous is the food web, the harder the synchronization is. By further increasing Q, while the three-species phase remains stable, the four-species one has a transition to an absorbing, single-species state. The existence of a phase with global oscillations for S>3, when the interaction graph has multiple subloops and several possible local cycles, leads to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs.
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paper_title: Competing associations in six-species predator-prey models

paper_content:

We study a set of six-species ecological models where each species has two predators and two prey. On a square lattice the time evolution is governed by iterated invasions between the neighbouring predator–prey pairs chosen at random and by a site exchange with a probability Xs between the neutral pairs. These models involve the possibility of spontaneous formation of different defensive alliances whose members protect each other from the external invaders. The Monte Carlo simulations show a surprisingly rich variety of the stable spatial distributions of species and subsequent phase transitions when tuning the control parameter Xs. These very simple models are able to demonstrate that the competition between these associations influences their composition. Sometimes the dominant association is developed via a domain growth. In other cases larger and larger invasion processes precede the prevalence of one of the stable associations. Under some conditions the survival of all the species can be maintained by the cyclic dominance occurring between these associations.
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paper_title: Network structure, predator–prey modules, and stability in large food webs

paper_content:

Large, complex networks of ecological interactions with random structure tend invariably to instability. This mathematical relationship between complexity and local stability ignited a debate that has populated ecological literature for more than three decades. Here we show that, when species interact as predators and prey, systems as complex as the ones observed in nature can still be stable. Moreover, stability is highly robust to perturbations of interaction strength, and is largely a property of structure driven by predator–prey loops with the stability of these small modules cascading into that of the whole network. These results apply to empirical food webs and models that mimic the structure of natural systems as well. These findings are also robust to the inclusion of other types of ecological links, such as mutualism and interference competition, as long as consumer–resource interactions predominate. These considerations underscore the influence of food web structure on ecological dynamics and challenge the current view of interaction strength and long cycles as main drivers of stability in natural communities.
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paper_title: Fixation in a cyclic Lotka-Volterra model

paper_content:

We study a cyclic Lotka-Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species N_c(d) above which the system fixates is determined analytically. We find N_c=5,14,23 in dimensions d=1,2,3, in remarkably good agreement with simulation results in two dimensions.
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paper_title: Coexistence and Survival in Conservative Lotka-Volterra Networks

paper_content:

Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.
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paper_title: A competitive network theory of species diversity.

paper_content:

Nonhierarchical competition between species has been proposed as a potential mechanism for biodiversity maintenance, but theoretical and empirical research has thus far concentrated on systems composed of relatively few species. Here we develop a theory of biodiversity based on a network representation of competition for systems with large numbers of competitors. All species pairs are connected by an arrow from the inferior to the superior. Using game theory, we show how the equilibrium density of all species can be derived from the structure of the network. We show that when species are limited by multiple factors, the coexistence of a large number of species is the most probable outcome and that habitat heterogeneity interacts with network structure to favor diversity.
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paper_title: Interaction strengths in food webs : issues and opportunities

paper_content:

Summary 1. Recent efforts to understand how the patterning of interaction strength affects both structure and dynamics in food webs have highlighted several obstacles to productive synthesis. Issues arise with respect to goals and driving questions, methods and approaches, and placing results in the context of broader ecological theory. 2. Much confusion stems from lack of clarity about whether the questions posed relate to community-level patterns or to species dynamics, and to what authors actually mean by the term ‘interaction strength’. Here, we describe the various ways in which this term has been applied and discuss the implications of loose terminology and definition for the development of this field. 3. Of particular concern is the clear gap between theoretical and empirical investigations of interaction strengths and food web dynamics. The ecological community urgently needs to explore new ways to estimate biologically reasonable model coefficients from empirical data, such as foraging rates, body size, metabolic rate, biomass distribution and other species traits. 4. Combining numerical and analytical modelling approaches should allow exploration of the conditions under which different interaction strengths metrics are interchangeable with regard to relative magnitude, system responses, and species identity. 5. Finally, the prime focus on predator‐prey links in much of the research to date on interaction strengths in food webs has meant that the potential significance of nontrophic interactions, such as competition, facilitation and biotic disturbance, has been largely ignored by the food web community. Such interactions may be important dynamically and should be routinely included in future food web research programmes.
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paper_title: A General Model for Food Web Structure

paper_content:

A central problem in ecology is determining the processes that shape the complex networks known as food webs formed by species and their feeding relationships. The topology of these networks is a major determinant of ecosystems' dynamics and is ultimately responsible for their responses to human impacts. Several simple models have been proposed for the intricate food webs observed in nature. We show that the three main models proposed so far fail to fully replicate the empirical data, and we develop a likelihood-based approach for the direct comparison of alternative models based on the full structure of the network. Results drive a new model that is able to generate all the empirical data sets and to do so with the highest likelihood.
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paper_title: Defensive alliances in spatial models of cyclical population interactions

paper_content:

As a generalization of the three-strategy Rock-Scissors-Paper game dynamics in space, cyclical interaction models of six mutating species are studied on a square lattice, in which each species is supposed to have two dominant, two subordinated, and a neutral interacting partner. Depending on their interaction topologies, all imaginable systems can be classified into four (isomorphic) groups exhibiting significantly different behaviors as a function of mutation rate. In three out of four cases three (or four) species form defensive alliances that maintain themselves in a self-organizing polydomain structure via cyclic invasions. Varying the mutation rate, this mechanism results in an ordering phenomenon analogous to that of magnetic Ising systems. The model explains a very basic mechanism of community organization, which might gain important applications in biology, economics, and sociology.
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paper_title: Extinction in four species cyclic competition

paper_content:

When four species compete stochastically in a cyclic way, the formation of two teams of mutually neutral partners is observed. In this paper we study through numerical simulations the extinction processes that can take place in this system both in the well mixed case as well as on different types of lattices. The different routes to extinction are revealed by the probability distribution of the domination time, i.e.?the time needed for one team to fully occupy the system. If swapping is allowed between neutral partners, then the probability distribution is dominated by very long-lived states where a few very large domains persist, each domain being occupied by a mix of individuals from species that form one of the teams. Many aspects of the possible extinction scenarios are lost when only considering averaged quantities, such as for example the mean domination time.
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paper_title: Globally synchronized oscillations in complex cyclic games

paper_content:

The rock-paper-scissors game and its generalizations with S>3 species are well-studied models for cyclically interacting populations. Four is, however, the minimum number of species that, by allowing other interactions beyond the single, cyclic loop, breaks both the full intransitivity of the food graph and the one-predator, one-prey symmetry. Lütz et al. [J. Theor. Biol. 317, 286 (2013)] have shown the existence, on a square lattice, of two distinct phases, with either four or three coexisting species. In both phases, each agent is eventually replaced by one of its predators, but these strategy oscillations remain localized as long as the interactions are short ranged. Distant regions may be either out of phase or cycling through different food-web subloops (if any). Here we show that upon replacing a minimum fraction Q of the short-range interactions by long-range ones, there is a Hopf bifurcation, and global oscillations become stable. Surprisingly, to build such long-distance, global synchronization, the four-species coexistence phase requires fewer long-range interactions than the three-species phase, while one would naively expect the opposite to be true. Moreover, deviations from highly homogeneous conditions (χ=0 or 1) increase Qc, and the more heterogeneous is the food web, the harder the synchronization is. By further increasing Q, while the three-species phase remains stable, the four-species one has a transition to an absorbing, single-species state. The existence of a phase with global oscillations for S>3, when the interaction graph has multiple subloops and several possible local cycles, leads to the conjecture that global oscillations are a general characteristic, even for large, realistic food webs.
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paper_title: Three- and four-state rock-paper-scissors games with diffusion

paper_content:

Cyclic dominance of three species is a commonly occurring interaction dynamics, often denoted the rock-paper-scissors (RPS) game. Such a type of interactions is known to promote species coexistence. Here, we generalize recent results of Reichenbach [Nature (London) 448, 1046 (2007)] of a four-state variant of the RPS game. We show that spiral formation takes place only without a conservation law for the total density. Nevertheless, in general, fast diffusion can destroy species coexistence. We also generalize the four-state model to slightly varying reaction rates. This is shown both analytically and numerically not to change pattern formation, or the effective wavelength of the spirals, and therefore not to alter the qualitative properties of the crossover to extinction.
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paper_title: Cyclic competition of four species: mean field theory and stochastic evolution

paper_content:

Generalizing the cyclically competing three-species model (often referred to as the rock-paper-scissors game), we consider a simple system of population dynamics without spatial structures that involves four species. Unlike the previous model, the four form alliance pairs which resemble partnership in the game of Bridge. In a finite system with discrete stochastic dynamics, all but 4 of the absorbing states consist of coexistence of a partner-pair. From a master equation, we derive a set of mean field equations of evolution. This approach predicts complex time dependence of the system and that the surviving partner-pair is the one with the larger product of their strengths (rates of consumption). Simulations typically confirm these scenarios. Beyond that, much richer behavior is revealed, including complicated extinction probabilities and non-trivial distributions of the population ratio in the surviving pair. These discoveries naturally raise a number of intriguing questions, which in turn suggests a variety of future avenues of research, especially for more realistic models of multispecies competition in nature.
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paper_title: Cyclic competition of four species: domains and interfaces

paper_content:

We study numerically domain growth and interface fluctuations in one- and two-dimensional lattice systems composed of four species that interact in a cyclic way. Particle mobility is implemented through exchanges of particles located on neighboring lattice sites. For the chain we find that the details of the domain growth strongly depend on the mobility, with a higher mobility yielding a larger effective domain growth exponent. In two space dimensions, when also exchanges between mutually neutral particles are possible, both domain growth and interface fluctuations display universal regimes that are independent of the predation and exchange rates.
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paper_title: Coexistence and Survival in Conservative Lotka-Volterra Networks

paper_content:

Analyzing coexistence and survival scenarios of Lotka-Volterra (LV) networks in which the total biomass is conserved is of vital importance for the characterization of long-term dynamics of ecological communities. Here, we introduce a classification scheme for coexistence scenarios in these conservative LV models and quantify the extinction process by employing the Pfaffian of the network's interaction matrix. We illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time.
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paper_title: Theory of phase ordering kinetics

paper_content:

The theory of phase ordering kinetics is reviewed, and new results for systems with continuous symmetry presented. A generalisation of “Porod's law” for the tail of the structure factor, of the form S(k, t) ∼ k−(d+n)L(t)−n for kL(t) ≫ 1, where L(t) is the characteristic length scale at time t after the quench, is derived where, for a vector order parameter, n is simply the number of components of the vector. The power-law tail is shown to be associated with topological defects in the field, and its amplitude is calculated exactly in terms of the defect density. For a conserved vector order parameter the multiscaling form obtained for n = ∞ is argued to be special to this limit. Using an approximate theory due to Mazenko, it is shown that conventional scaling is recovered, for any finite n, when t→∞, with L(t)∼ (tln n)14 for n large.
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paper_title: Saddles, Arrows, and Spirals: Deterministic Trajectories in Cyclic Competition of Four Species

paper_content:

Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable that evolves simply as an exponential: Q ∝ e(λt), where λ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for t→-∞). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems.
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paper_title: Intransitivity and coexistence in four species cyclic games

paper_content:

Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species allowing other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the parameter setting the rate at which one species invades another changes, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity.
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paper_title: Phase transitions induced by variation of invasion rates in spatial cyclic predator-prey models with four or six species

paper_content:

Cyclic predator-prey models with four or six species are studied on a square lattice when the invasion rates are varied. It is found that the cyclic invasions maintain a self-organizing pattern as long as the deviation of the invasion rate(s) from a uniform value does not exceed a threshold value. For larger deviations the system exhibits a continuous phase transition into a frozen distribution of odd (or even) label species.
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paper_title: From pairwise to group interactions in games of cyclic dominance

paper_content:

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies, but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance too the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.
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paper_title: Mobility-Dependent Selection of Competing Strategy Associations

paper_content:

Standard models of population dynamics focus on the interaction, survival, and extinction of the competing species individually. Real ecological systems, however, are characterized by an abundance of species (or strategies, in the terminology of evolutionary-game theory) that form intricate, complex interaction networks. The description of the ensuing dynamics may be aided by studying associations of certain strategies rather than individual ones. Here we show how such a higher-level description can bear fruitful insight. Motivated from different strains of colicinogenic Escherichia coli bacteria, we investigate a four-strategy system which contains a three-strategy cycle and a neutral alliance of two strategies. We find that the stochastic, spatial model exhibits a mobility-dependent selection of either the three-strategy cycle or of the neutral pair. We analyze this intriguing phenomenon numerically and analytically.
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paper_title: Diverging fluctuations in a spatial five-species cyclic dominance game

paper_content:

A five-species predator-prey model is studied on a square lattice where each species has two prey and two predators on the analogy to the rock-paper-scissors-lizard-Spock game. The evolution of the spatial distribution of species is governed by site exchange and invasion between the neighboring predator-prey pairs, where the cyclic symmetry can be characterized by two different invasion rates. The mean-field analysis has indicated periodic oscillations in the species densities with a frequency becoming zero for a specific ratio of invasion rates. When varying the ratio of invasion rates, the appearance of this zero-eigenvalue mode is accompanied by neutrality between the species associations. Monte Carlo simulations of the spatial system reveal diverging fluctuations at a specific invasion rate, which can be related to the vanishing dominance between all pairs of species associations.
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paper_title: Spatial coherence resonance in excitable media

paper_content:

We study the phenomenon of spatial coherence resonance in a two-dimensional model of excitable media with FitzHugh-Nagumo local dynamics. In particular, we show that there exists an optimal level of additive noise for which an inherent spatial scale of the excitable media is best pronounced. We argue that the observed phenomenon occurs due to the existence of a noise robust excursion time that is characteristic for the local dynamics whereby the diffusion constant, representing the rate of diffusive spread, determines the actual resonant spatial frequency. Additionally, biological implications of presented results in the field of neuroscience are outlined.
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paper_title: Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games

paper_content:

Biodiversity is essential to the viability of ecological systems. Species diversity in ecosystems is promoted by cyclic, non-hierarchical interactions among competing populations. Such non-transitive relations lead to an evolution with central features represented by the `rock-paper-scissors' game, where rock crushes scissors, scissors cut paper, and paper wraps rock. In combination with spatial dispersal of static populations, this type of competition results in the stable coexistence of all species and the long-term maintenance of biodiversity. However, population mobility is a central feature of real ecosystems: animals migrate, bacteria run and tumble. Here, we observe a critical influence of mobility on species diversity. When mobility exceeds a certain value, biodiversity is jeopardized and lost. In contrast, below this critical threshold all subpopulations coexist and an entanglement of travelling spiral waves forms in the course of temporal evolution. We establish that this phenomenon is robust, it does not depend on the details of cyclic competition or spatial environment. These findings have important implications for maintenance and evolution of ecological systems and are relevant for the formation and propagation of patterns in excitable media, such as chemical kinetics or epidemic outbreaks.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

paper_content:

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
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paper_title: Competing associations in six-species predator-prey models

paper_content:

We study a set of six-species ecological models where each species has two predators and two prey. On a square lattice the time evolution is governed by iterated invasions between the neighbouring predator–prey pairs chosen at random and by a site exchange with a probability Xs between the neutral pairs. These models involve the possibility of spontaneous formation of different defensive alliances whose members protect each other from the external invaders. The Monte Carlo simulations show a surprisingly rich variety of the stable spatial distributions of species and subsequent phase transitions when tuning the control parameter Xs. These very simple models are able to demonstrate that the competition between these associations influences their composition. Sometimes the dominant association is developed via a domain growth. In other cases larger and larger invasion processes precede the prevalence of one of the stable associations. Under some conditions the survival of all the species can be maintained by the cyclic dominance occurring between these associations.
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paper_title: Critical coarsening without surface tension: the universality class of the voter model.

paper_content:

We show that the two-dimensional voter model, usually considered to be only a marginal coarsening system, represents a broad class of models for which phase ordering takes place without surface tension. We argue that voter-like growth is generically observed at order-disorder nonequilibrium transitions solely driven by interfacial noise between dynamically symmetric absorbing states.
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paper_title: Cyclical interactions with alliance-specific heterogeneous invasion rates

paper_content:

We study a six-species Lotka-Volterra-type system on different two-dimensional lattices when each species has two superior and two inferior partners. The invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator-prey pair, whereby cyclic symmetries within the two three-species defensive alliances are conserved. Monte Carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance-specific invasion rates. This behavior is qualitatively reproduced by a four-point mean-field approximation. The study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences.
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paper_title: Self-organizing patterns maintained by competing associations in a six-species predator-prey model

paper_content:

Formation and competition of associations are studied in a six-species ecological model where each species has two predators and two prey. Each site of a square lattice is occupied by an individual belonging to one of the six species. The evolution of the spatial distribution of species is governed by iterated invasions between the neighboring predator-prey pairs with species specific rates and by site exchange between the neutral pairs with a probability X . This dynamical rule yields the formation of five associations composed of two or three species with proper spatiotemporal patterns. For large X a cyclic dominance can occur between the three two-species associations whereas one of the two three-species associations prevails in the whole system for low values of X in the final state. Within an intermediate range of X all the five associations coexist due to the fact that cyclic invasions between the two-species associations reduce their resistance temporarily against the invasion of three-species associations.
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paper_title: Defensive alliances in spatial models of cyclical population interactions

paper_content:

As a generalization of the three-strategy Rock-Scissors-Paper game dynamics in space, cyclical interaction models of six mutating species are studied on a square lattice, in which each species is supposed to have two dominant, two subordinated, and a neutral interacting partner. Depending on their interaction topologies, all imaginable systems can be classified into four (isomorphic) groups exhibiting significantly different behaviors as a function of mutation rate. In three out of four cases three (or four) species form defensive alliances that maintain themselves in a self-organizing polydomain structure via cyclic invasions. Varying the mutation rate, this mechanism results in an ordering phenomenon analogous to that of magnetic Ising systems. The model explains a very basic mechanism of community organization, which might gain important applications in biology, economics, and sociology.
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paper_title: Noise-guided evolution within cyclical interactions

paper_content:

We study a stochastic predator–prey model on a square lattice, where each of the six species has two superior and two inferior partners. The invasion probabilities between species depend on the predator–prey pair and are supplemented by Gaussian noise. Conditions are identified that warrant the largest impact of noise on the evolutionary process, and the results of Monte Carlo simulations are qualitatively reproduced by a four-point cluster dynamical mean-field approximation. The observed noise-guided evolution is deeply routed in short-range spatial correlations, which is supported by simulations on other host lattice topologies. Our findings are conceptually related to the coherence resonance phenomenon in dynamical systems via the mechanism of threshold duality. We also show that the introduced concept of noise-guided evolution via the exploitation of threshold duality is not limited to predator–prey cyclical interactions, but may apply to models of evolutionary game theory as well, thus indicating its applicability in several different fields of research.
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paper_title: Competing associations in bacterial warfare with two toxins

paper_content:

Simple combinations of common competitive mechanisms can easily result in cyclic competitive dominance relationships between species. The topological features of such competitive networks allow for complex spatial coexistence patterns. We investigate self-organization and coexistence in a lattice model, describing the spatial population dynamics of competing bacterial strains. With increasing diffusion rate the community of the nine possible toxicity/resistance types undergoes two phase transitions. Below a critical level of diffusion, the system exhibits expanding domains of three different defensive alliances, each consisting of three cyclically dominant species. Due to the neutral relationship between these alliances and the finite system size effect, ultimately only one of them remains. At large diffusion rates the system admits three coexisting domains, each containing mutually neutral species. Because of the cyclical dominance between these domains, a long term stable coexistence of all species is ensured. In the third phase at intermediate diffusion the spatial structure becomes even more complicated with domains of mutually neutral species persisting along the borders of defensive alliances. The study reveals that cyclic competitive relationships may produce a large variety of complex coexistence patterns, exhibiting common features of natural ecosystems, like hierarchical organization, phase transitions and sudden, large-scale fluctuations.
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paper_title: Coherence resonance in a spatial prisoner's dilemma game

paper_content:

We study effects of additive spatiotemporal random variations, introduced to the payoffs of a spatial prisoner's dilemma game, on the evolution of cooperation. In the absence of explicit payoff variations the system exhibits a phase transition from a mixed state of cooperators and defectors to a homogenous state of defectors belonging to the directed percolation universality class. By introducing nonzero random variations to the payoffs, this phase transition can be reverted in a resonance-like manner depending on the variance of noise, thus marking coherence resonance in the system. We argue that explicit random payoff variations present a viable mechanism that promotes cooperation for defection temptation values substantially exceeding the one marking the transition point to homogeneity by deterministic payoffs.
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paper_title: A golden point rule in rock–paper–scissors–lizard–spock game

paper_content:

We study a novel five-species system on two-dimensional lattices when each species have two superior and two inferior partners. Here we simplify the huge parameter space of predation probability to only two parameters. Both of Monte Carlo simulation and Mean Field Theory reveal that two of strategies may die out when the ratio of the two parameters is close to the golden point 0.618, and the remaining three strategies are provided a cyclic dominance system.
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paper_title: Competition of individual and institutional punishments in spatial public goods games

paper_content:

We have studied the evolution of strategies in spatial public goods games where both individual (peer) and institutional (pool) punishments are present beside unconditional defector and cooperator strategies. The evolution of strategy distribution is governed by imitation based on random sequential comparison of neighbors' payoff for a fixed level of noise. Using numerical simulations we have evaluated the strategy frequencies and phase diagrams when varying the synergy factor, punishment cost, and fine. Our attention is focused on two extreme cases describing all the relevant behaviors in such a complex system. According to our numerical data peer punishers prevail and control the system behavior in a large segments of parameters while pool punishers can only survive in the limit of weak peer punishment when a rich variety of solutions is observed. Paradoxically, the two types of punishment may extinguish each other's impact resulting in the triumph of defectors. The technical difficulties and suggested methods are briefly discussed.
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paper_title: Does local competition increase the coexistence of species in intransitive networks?

paper_content:

Competitive intransitivity, a situation in which species' competitive ranks cannot be listed in a strict hierarchy, promotes species coexistence through "enemy's enemy indirect facilitation." Theory suggests that intransitivity-mediated coexistence is enhanced when competitive interactions occur at local spatial scales, although this hypothesis has not been thoroughly tested. Here, we use a lattice model to investigate the effect of local vs. global competition on intransitivity-mediated coexistence across a range of species richness values and levels of intransitivity. Our simulations show that local competition can enhance intransitivity-mediated coexistence in the short-term, yet hinder it in the long-term, when compared to global competition. This occurs because local competition slows species disaggregation, allowing weaker competitors to persist longer in the shifting spatial refuges of intransitive networks, enhancing short-term coexistence. Conversely, our simulations show that, in the long-term, local competition traps disaggregated species in unfavorable areas of the competitive arena, where they are excluded by superior competitors. As a result, in the long-term, global intransitive competition allows a greater number of species to coexist than local intransitive competition.
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paper_title: Chemical Warfare Among Invaders: A Detoxification Interaction Facilitates an Ant Invasion

paper_content:

As tawny crazy ants ( Nylanderia fulva ) invade the southern USA, they often displace imported fire ants ( Solenopsis invicta ). Following exposure to S. invicta venom, N. fulva applies abdominal exocrine gland secretions to its cuticle. Bioassays reveal that these secretions detoxify S. invicta venom. Further, formic acid, from N. fulva venom, is the detoxifying agent. N. fulva exhibits this detoxification behavior after conflict with a variety of ant species; however, it expresses it most intensely after interactions with S. invicta . This behavior may have evolved in their shared South American native range. The unique capacity to detoxify a major competitor’s venom likely contributes substantially to its ability to displace S. invicta populations, making this behavior a causative agent in the ecological transformation of regional arthropod assemblages.
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paper_title: Evolutionary dynamics of group interactions on structured populations: A review

paper_content:

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.
---
paper_title: The role of diversity in the evolution of cooperation

paper_content:

Understanding the evolutionary mechanisms that promote and maintain cooperative behavior is recognized as a major theoretical problem where the intricacy increases with the complexity of the participating individuals. This is epitomized by the diverse nature of Human interactions, contexts, preferences and social structures. Here we discuss how social diversity, in several of its flavors, catalyzes cooperative behavior. From the diversity in the number of interactions an individual is involved to differences in the choice of role models and contributions, diversity is shown to significantly increase the chances of cooperation. Individual diversity leads to an overall population dynamics in which the underlying dilemma of cooperation is changed, benefiting the society as whole. In addition, we show how diversity in social contexts can arise from the individual capacity for organizing their social ties. As such, Human diversity, on a grand scale, may be instrumental in shaping us as the most sophisticated cooperative entities on this planet.
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paper_title: Social diversity promotes the emergence of cooperation in public goods games

paper_content:

Although humans often cooperate with each other, the temptation to forego the public good mostly wins over collective cooperative action, leading to the so-called 'tragedy of the commons'. Many existing models treat individuals as equivalent, ignoring diversity and population structure. Santos et al. show theoretically that social diversity, introduced via heterogeneous graphs, promotes the emergence of cooperation in public goods games. Humans often cooperate with each other, but the temptation to forgo the public good mostly wins over collective cooperative action. Many existing models treat individuals as equivalent, ignoring diversity and population structure; however, here it's shown theoretically that social diversity, introduced via heterogeneous graphs, promotes the emergence of cooperation in public goods games. Humans often cooperate in public goods games1,2,3 and situations ranging from family issues to global warming4,5. However, evolutionary game theory predicts4,6 that the temptation to forgo the public good mostly wins over collective cooperative action, and this is often also seen in economic experiments7. Here we show how social diversity provides an escape from this apparent paradox. Up to now, individuals have been treated as equivalent in all respects4,8, in sharp contrast with real-life situations, where diversity is ubiquitous. We introduce social diversity by means of heterogeneous graphs and show that cooperation is promoted by the diversity associated with the number and size of the public goods game in which each individual participates and with the individual contribution to each such game. When social ties follow a scale-free distribution9, cooperation is enhanced whenever all individuals are expected to contribute a fixed amount irrespective of the plethora of public goods games in which they engage. Our results may help to explain the emergence of cooperation in the absence of mechanisms based on individual reputation and punishment10,11,12. Combining social diversity with reputation and punishment will provide instrumental clues on the self-organization of social communities and their economical implications.
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paper_title: Network structure, predator–prey modules, and stability in large food webs

paper_content:

Large, complex networks of ecological interactions with random structure tend invariably to instability. This mathematical relationship between complexity and local stability ignited a debate that has populated ecological literature for more than three decades. Here we show that, when species interact as predators and prey, systems as complex as the ones observed in nature can still be stable. Moreover, stability is highly robust to perturbations of interaction strength, and is largely a property of structure driven by predator–prey loops with the stability of these small modules cascading into that of the whole network. These results apply to empirical food webs and models that mimic the structure of natural systems as well. These findings are also robust to the inclusion of other types of ecological links, such as mutualism and interference competition, as long as consumer–resource interactions predominate. These considerations underscore the influence of food web structure on ecological dynamics and challenge the current view of interaction strength and long cycles as main drivers of stability in natural communities.
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paper_title: Competitive intransitivity promotes species coexistence.

paper_content:

Using a spatially explicit cellular automaton model with local competition, we investigate the potential for varied levels of competitive intransitivity (i.e., nonhierarchical competition) to promote species coexistence. As predicted, on average, increased levels of intransitivity result in more sustained coexistence within simulated communities, although the outcome of competition also becomes increasingly unpredictable. Interestingly, even a moderate degree of intransitivity within a community can promote coexistence, in terms of both the length of time until the first competitive exclusion and the number of species remaining in the community after 500 simulated generations. These results suggest that modest levels of intransitivity in nature, such as those that are thought to be characteristic of plant communities, can contribute to coexistence and, therefore, community-scale biodiversity. We explore a potential connection between competitive intransitivity and neutral theory, whereby competitive intransitivity may represent an important mechanism for "ecological equivalence."
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paper_title: A competitive network theory of species diversity.

paper_content:

Nonhierarchical competition between species has been proposed as a potential mechanism for biodiversity maintenance, but theoretical and empirical research has thus far concentrated on systems composed of relatively few species. Here we develop a theory of biodiversity based on a network representation of competition for systems with large numbers of competitors. All species pairs are connected by an arrow from the inferior to the superior. Using game theory, we show how the equilibrium density of all species can be derived from the structure of the network. We show that when species are limited by multiple factors, the coexistence of a large number of species is the most probable outcome and that habitat heterogeneity interacts with network structure to favor diversity.
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paper_title: Meet the New Boss – Same as the Old Boss

paper_content:

A key paradox of subalternity for the subject throwing off the colonial yoke is the degree to which the collective emergence from the nation-state is to be in the image of the colonizer; that is, as a modern state, notionally on a par with the mother/father country. With a view to understanding the meaning of such symbolic investments, this paper surveys the lyrics of national anthems from a wide range of postcolonial countries. ::: ::: Despite the avowals one regularly finds in the lyrics of postcolonial anthems, and despite the expression of sometimes rote resistance to a putative colonial oppressor, those singing songs which imitate European anthems, and the feeling such anthems inspire, invest identity in the mimetic, rather than the unique. On the basis of a range of observations of anthems and their circumstances, this paper dares to ask finally whether the singing of anthems makes for better worlds.
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paper_title: A General Model for Food Web Structure

paper_content:

A central problem in ecology is determining the processes that shape the complex networks known as food webs formed by species and their feeding relationships. The topology of these networks is a major determinant of ecosystems' dynamics and is ultimately responsible for their responses to human impacts. Several simple models have been proposed for the intricate food webs observed in nature. We show that the three main models proposed so far fail to fully replicate the empirical data, and we develop a likelihood-based approach for the direct comparison of alternative models based on the full structure of the network. Results drive a new model that is able to generate all the empirical data sets and to do so with the highest likelihood.
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paper_title: Dynamics of N-person snowdrift games in structured populations.

paper_content:

In many real-life situations, the completion of a task by a group toward achieving a common goal requires the cooperation of at least some of its members, who share the required workload. Such cases are conveniently modeled by the N-person snowdrift game, an example of a Public Goods Game. Here we study how an underlying network of contacts affects the evolutionary dynamics of collective action modeled in terms of such a Public Goods Game. We analyze the impact of different types of networks in the global, population-wide dynamics of cooperators and defectors. We show that homogeneous social structures enhance the chances of coordinating toward stable levels of cooperation, while heterogeneous network structures create multiple internal equilibria, departing significantly from the reference scenario of a well-mixed, structureless population.
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paper_title: Evolutionary dynamics of collective action in N-person stag hunt dilemmas

paper_content:

In the animal world, collective action to shelter, protect and nourish requires the cooperation of group members. Among humans, many situations require the cooperation of more than two individuals simultaneously. Most of the relevant literature has focused on an extreme case, the N-person Prisoner's Dilemma. Here we introduce a model in which a threshold less than the total group is required to produce benefits, with increasing participation leading to increasing productivity. This model constitutes a generalization of the two-person stag hunt game to an N-person game. Both finite and infinite population models are studied. In infinite populations this leads to a rich dynamics that admits multiple equilibria. Scenarios of defector dominance, pure coordination or coexistence may arise simultaneously. On the other hand, whenever one takes into account that populations are finite and when their size is of the same order of magnitude as the group size, the evolutionary dynamics is profoundly affected: it may ultimately invert the direction of natural selection, compared with the infinite population limit.
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paper_title: WHY DO POPULATIONS CYCLE? A SYNTHESIS OF STATISTICAL AND MECHANISTIC MODELING APPROACHES

paper_content:

Population cycles have long fascinated ecologists. Even in the most-studied populations, however, scientists continue to dispute the relative importance of various potential causes of the cycles. Over the past three decades, theoretical ecologists have cataloged a large number of mechanisms that are capable of generating cycles in population models. At the same time, statisticians have developed new techniques both for characterizing time series and for fitting population models to time-series data. Both disciplines are now sufficiently advanced that great gains in understanding can be made by synthesizing these complementary, and heretofore mostly independent, quantitative approaches. In this paper we demonstrate how to apply this synthesis to the problem of population cycles, using both long-term population time series and the often-rich observational and experimental data on the ecology of the species in question. We quantify hypotheses by writing mathematical models that embody the interactions and forces that might cause cycles. Some hypotheses can be rejected out of hand, as being unable to generate even qualitatively appropriate dynamics. We finish quantifying the remaining hypotheses by estimating parameters, both from independent experiments and from fitting the models to the time-series data using modern statistical techniques. Finally, we compare simulated time series generated by the models to the observed time series, using a variety of statistical descriptors, which we refer to collectively as “probes.” The model most similar to the data, as measured by these probes, is considered to be the most likely candidate to represent the mechanism underlying the population cycles. We illustrate this approach by analyzing one of Nicholson’s blowfly populations, in which we know the “true” governing mechanism. Our analysis, which uses only a subset of the information available about the population, uncovers the correct answer, suggesting that this synthetic approach might be successfully applied to field populations as well.
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paper_title: Coevolutionary games - a mini review

paper_content:

Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.
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paper_title: From pairwise to group interactions in games of cyclic dominance

paper_content:

We study the rock-paper-scissors game in structured populations, where the invasion rates determine individual payoffs that govern the process of strategy change. The traditional version of the game is recovered if the payoffs for each potential invasion stem from a single pairwise interaction. However, the transformation of invasion rates to payoffs also allows the usage of larger interaction ranges. In addition to the traditional pairwise interaction, we therefore consider simultaneous interactions with all nearest neighbors, as well as with all nearest and next-nearest neighbors, thus effectively going from single pair to group interactions in games of cyclic dominance. We show that differences in the interaction range affect not only the stationary fractions of strategies, but also their relations of dominance. The transition from pairwise to group interactions can thus decelerate and even revert the direction of the invasion between the competing strategies. Like in evolutionary social dilemmas, in games of cyclic dominance too the indirect multipoint interactions that are due to group interactions hence play a pivotal role. Our results indicate that, in addition to the invasion rates, the interaction range is at least as important for the maintenance of biodiversity among cyclically competing strategies.
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paper_title: Social diversity and promotion of cooperation in the spatial prisoner's dilemma game

paper_content:

The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determine the social diversity of players engaging in the prisoner's dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game payoffs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrinsic factors that determine the social diversity of players, and find that the power-law distribution enables the best promotion of cooperation. The facilitation of the cooperative strategy relies mostly on the inhomogeneous social state of players, resulting in the formation of cooperative clusters which are ruled by socially high-ranking players that are able to prevail against the defectors even when there is a large temptation to defect. To confirm this, we also study the impact of spatially correlated social diversity and find that cooperation deteriorates as the spatial correlation length increases. Our results suggest that the distribution of wealth and social status might have played a crucial role by the evolution of cooperation amongst egoistic individuals.
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Title: Cyclic dominance in evolutionary games: A review

Section 1: INTRODUCTION
Description 1: Introduce the concept of cyclic dominance in evolutionary games and its significance in explaining biodiversity and evolutionary dynamics. Highlight examples from nature and describe the role of spatial structure and mobility.

Section 2: ROCK-PAPER-SCISSORS GAME IN STRUCTURED POPULATIONS
Description 2: Discuss the dynamics of the rock-paper-scissors (RPS) game in structured populations, focusing on interaction networks and the effect of mobility on the outcome of these games.

Section 3: Interaction networks
Description 3: Explore how different types of interaction networks, such as lattices and small-world networks, impact the dynamics and outcome of cyclic dominance in evolutionary games.

Section 4: Mobility
Description 4: Review the impact of mobility on biodiversity and pattern formation in RPS games, including the emergence of spiral and target waves.

Section 5: Metapopulation and nonlinear mobility
Description 5: Introduce the metapopulation modeling approach and discuss nonlinear mobility's effects on the evolutionary dynamics of cyclic dominance games.

Section 6: Complex Ginzburg-Landau equation
Description 6: Explain the derivation and application of the Complex Ginzburg-Landau equation (CGLE) to predict spatio-temporal patterns in RPS games.

Section 7: EVOLUTIONARY GAMES WITH SPONTANEOUSLY EMERGING CYCLIC DOMINANCE
Description 7: Discuss instances where cyclic dominance emerges spontaneously in evolutionary games, covering topics such as time-dependent learning and voluntary participation.

Section 8: Time-dependent learning
Description 8: Analyze the role of time-dependent properties, like learning ability, in generating cyclic dominance in two-strategy and more complex games.

Section 9: Voluntary participation
Description 9: Review the effects of voluntary participation in public goods games, including the emergence and experimental validation of the "Red Queen" effect.

Section 10: When three competing strategies are more than three
Description 10: Examine cases where an alliance of strategies acts as an additional "strategy" in cyclic dominance, focusing on structured populations.

Section 11: CYCLIC DOMINANCE BETWEEN MORE THAN THREE STRATEGIES
Description 11: Investigate the complexity of evolutionary dynamics when the number of competing strategies exceeds three and discuss the emergence of defensive alliances and global oscillations.

Section 12: Alliances
Description 12: Detail the formation and impact of alliances within cyclic dominance games, highlighting examples and the conditions under which they form.

Section 13: CONCLUSIONS AND OUTLOOK
Description 13: Summarize the main findings and discuss unexplored problems related to cyclic dominance in evolutionary games, offering insights and directions for future research.