Tuesday, April 17, 2012

Community ecology is complicated: revisiting Robert May’s weak interactions

When it comes to explaining species diversity, Stefano Allesina differs from the traditional approach. Community ecology has long focused on the role of two species interactions in determining coexistence (Lotka-Volterra models, etc), particularly in theory. The question then is whether two species interactions are representative of the interactions that are maintaining the millions of species in the world, and Allesina strongly feels that they are not.

In the paper “Stability criteria for complex ecosystems”, Stephano Allesina and Si Tang revisit and expand on an idea proposed by Robert May in 1972. In his paper “Will a large complex system be stable?” Robert May showed analytically that the probability a large system of interacting species is stable – i.e. will return to equilibrium following perturbation – is a function of the number of species and their average interactions strength. Systems with many species are more likely to be stable when the interactions among species are weak.

May’s paper was necessarily limited by the available mathematics of the time. His approach examined a large community matrix, with a large number of interacting species. The sign and strength of the interactions among species were chosen at random. Stability then could be assessed based on the sign of the eigenvalues of the matrix – if the eigenvalues of the matrix are all negative the system is likely to be stable. Solving for the distribution of the eigenvalues of such a large system relied on the semi-circle law for random matrices, and looking at more realistic matrices, such as those representing predator-prey, mutualistic, or competitive interactions, was not possible in 1972. However, more modern theorems for the distribution of eigenvalues from large matrices allowed Allesina and Tang to reevaluate May’s conclusions and expand them to examine how specific types of interactions affect the stability of complex systems.

Allesina and Tang examined matrices where the interactions among species (sign and strength) were randomly selected, similar to those May analyzed. They also looked at more realistic community matrices, for example matrices in which pairs of species have opposite-signed interactions (+ & -) representing predator prey systems (since the effect of a prey species is positive on its predator, but that predator has a negative effect on its prey). A matrix could also contain pairs of species with interactions of the same sign, creating a system with both competition (- & -) and mutualism (+ & +). When these different types of matrices were analyzed for stability, Allesina and Tang found that there was a hierarchy in which mixed competition/mutualism matrices were the least likely to be stable, random matrices (similar to those May used) are intermediate, and predator–prey matrices were the most likely to be stable (figure below).

When the authors looked at more realistic situations where the mean interaction strength for the matrix wasn’t zero (e.g. so a system could have all competitive or all mutualistic interactions), they found such systems were much less likely to be stable. Similarly, realistic structures based on accepted food web models (cascade or niche type) also resulted in less stable systems.

The authors reexamined May’s results that showed that weak interactions made large systems more likely to be stable. In particular they examined how the distribution of interactions strengths, rather than the mean value alone, affected system stability. In contrast to accepted ideas, they found that when there were many weak interactions, predator-prey systems tended to become less stable, suggesting that weak interactions destabilize predator-prey systems. In contrast, weak interactions tended to stabilize competitive and mutualistic systems. The authors concluded, “Our analysis shows that, all other things being equal, weak interactions can be either stabilizing or destabilizing depending on the type of interactions between species.” 

Approaching diversity and coexistence from the idea of large systems and many weak interactions  flies in the face of how much community ecology is practiced today. For that reason, it wouldn't be surprising if this paper has little influence. Allesina suggests that focusing on two species interactions is ultimately misleading, since if species experience a wide range of interactions that vary in strength and direction, sampling only a single interaction will likely misrepresent the overall distribution of interactions. Even when researchers do carry out experiments with multiple species, finding a result of very weak interactions between species is often interpreted as a failure to elucidate the processes maintaining diversity in the system. That said, Allesina’s work (which is worth reading, few people explain complex ideas so clearly) doesn’t necessarily make itself amenable to being tested or applied to concrete questions. Still, there’s unexplored space between traditional, two-species interactions and systems of weak interactions among many species, and exploring this space could be very fruitful. 

Monday, April 9, 2012

Disagreeing about ecology: how debate advances science







