Okay welcome everybody to our Friday session of the winter school. My name is Antoni Solani from my STP and I will be chairing this short session of today. As always the rules are that you are invited to type your questions in the chat and at the end of the talk there will be a discussion session which your questions will be answered. For the viewers who are following us from the youtube live stream you can write down your questions in the chat and I will read them out for you in the end. Okay so with that I will leave the floor to Andrei Rinaldo from the eCorpor technique for the other sun. We will deliver the first of his three lectures. Thank you Andrei. Thank you so much Antonio and we're like welcome everybody. Let me share my screen I'll be mostly counting on that and I may be a little bit abrupt in stopping at the 45th minute but I'm confident that because I have three lectures I can compensate the material should I be somewhat get carried away in the presentation because I kind of like the subject quite a bit. So let me show you what's a rationale. The network says ecological corridors and the subtitle that reads species populations and pathogens and in fact I think the boy from Bangladesh about who might be speculating later on in the third lecture in particular was trying to convince me that the mighty waters of a magna river couldn't carry a pathogen that could infect him from cholera and given that this happened like a couple hundred meters downstream of the largest derail hospital in the world. One wonders whether our limited knowledge in fact is a permanent liability on our ability to put price tag on ecosystem services. So here I'd be advocating for a particular robustness of our capabilities of predicting what happens in a case of a particular substrate for ecological interactions which is through the channel network and I am told that this is not a seminar supposed to be a lecture at times I'll try to be somewhat technical and what I can say is that whatever I'll be doing and I think and I gather that what Marino Gato will be doing as well in his lectures is taken from the book we published it came out like this month or maybe the end of November and if anybody's interested of course details and much details but maybe too much details contained therein and it's not by chance the picture which I took in Bangladesh doing feed work in fact is a very good example. So river networks as ecological corridors this picture is taking mine in my beloved hometown it's surrounding the city of Venice these are tidal networks not river networks just for those of you more versed in geomorphology would have spotted that immediately but in any way the coexistence of the built in the natural environment suggests that this is a this is a long-standing issue that has been exploited with a limited understanding for a long long time and in fact my pitch would be that essentially the intrinsic substrates for ecological interaction in particular that of river networks of course of which this is a particular if you please geomorphological relevant exception but it's so beautiful the picture that couldn't resist in fact even showing those wide spots which are nets fishermen's net nets in fact but anyways my pitch is for the intrinsic structures for ecological interaction that bear rather fundamental consequences on the number of processes in particular patterns of biodiversity but much more in my view to control the spread of say a number of things the species okay populations like dynamics of populations into the fluvial connections and throw their into models of infections like COVID-19 as Marina would be doing so the plan of my first 45 minutes is why in fact I believe that we can move from abstract models of dendritic ecosystem yet with some important constraints and why they're important in the quantitative valuation of ecosystem services in particular I'd be trying to show you progressively through abstract theoretical work more refined theoretical work and laboratory and fieldwork we carried out in my lab in Lausanne that directional dispersal and the spatial ecology kind of concert the spatial situation for on an essential bearing on form and function of a river network that is fluvial ecosystems in a sense but as we'll try to show much more than that but the issue would be again the dynamics of fuel ecosystem services not today biological invasions not today populations migrations not today and then transmission of waterborne disease today we set the premise and for these to be verified on practical cases and again if I may choose a punchline of a hydrologist at heart and so I I guess we're inching towards a fair distribution of water pillars of hydrology are floods droughts and the fair distribution of water and this I think would be an important step in that direction so let me first start chatting a bit about something of which in particular Ignacio Rodriguez and myself have been working for like 35 years at this point we studied at length the substrate for ecological interactions but through a landscape because one of the issues that we keep having in mind and echoing one of the organizers of these of these winter school Simon Levine who's the importance of whose work can never be overestimated in ecology and beyond is that although natural ecosystems are characterized by striking diversity of form and function uh at very many very quite a few times these often they exhibit they exhibit the deep similarity these structural similarities and that the times they emerge across scales of space time and ecological complexity so we wonder where the certain universal features that appear in the entity structures like the ones that I'll be pitching in for the fact that are statistically the same regardless of climate vegetation exposed lithology you name it there is a self-organizing process beyond this which is super strong and self-organizing the sense that it will generate realizations that statistically identical regardless of there's no fine there's no celestial tuning or parameters that allow them to happen it happens nonetheless so it's not the critical phenomenon but it's self-organized critical phenomena now these are those of you that being at ICTP for a long time know how important it is being the cultural activities there so let me talk to you about the river landscape and why in fact this is going to be brief but heartfelt that we have quite a few tools this is taken from again google maps and what you take a picture what you can do nowadays you can filter and realize what vegetation is and you can actually expose the terrain to the point that you can have lighter or leader whatever you want to call it maps in which essentially you have the size of the stamp really centimeters at this point once you have filtered the vegetation you have the surface of the terrain in an exquisite detail which you see in these cases exactly the same bend of the amazon and the accuracy of the vertical direction is now becoming more than enough to generate description of details in geomorphology I really changed the goal again forever and these so this is another picture at the same time once you filter the vegetation the detail of geomorphology that you're having there objectively manipulated and and automatically retrieved and remotely retrieved it's really phenomenal over a range of scale which is unheard of so essentially what you have I'm still attached to the old picture the first one we took out in 1989 or what in 90 whatever it was etc this is a small catchment in which the restitution was of your few 100 square kilometers and this was given the pixel size which you have by general tools it will be like of the order of this kind of 30 by 30 meters and the accuracy in the third dimension was of the order like half a meter so how this is that essentially you have a surface remotely acquired and objectively manipulated that tells you how you can calculate gradients and gradients means steepest descent directions you have a field z sub i which is a scalar right and you can calculate a vector which is the gradient which is the steepest descent direction which by all means in hydrology and geomorphology makes sense to assume that the strongest force is gravity by order of magnitude so steepest descent direction is also the direction of the flow direction so aggregation pass can be devised directly the shape of this kind I mean I'm essentially engrossing a particular of that so you can delineate a line of this kind and we have a trick which is a useful trick in fact of using a scaling relation for tracing the black line that shows the mainstream in here which is a well-known leopoldian rule for the scaling of a width of a river channel this is a trick for showing how the system behave but certainly it has a major implication you can calculate elevations you can calculate laplacians like let's say essentially the average over the nearest neighbors in a place you can easily decide what's the sign of the system that is whether the surface is concave down or concave up you have some sort of tricks to the point that we assume early on and essentially because of a statistically statistical well-worn argument essentially our theorem is that square root of two is equal to one that is big as a gradient the steepest descent direction in those exceptions can be can be generated so essentially you extract a very even network from data over the scales around the order of a meter even less than that now to the order of thousands of kilometers still we are talking about the so-called runoff generating part of a landscape where in fact you do have aggregation you don't have a transportational zoning we have no significant injection but still you're talking about six clean orders of magnitude that can be ground truth quite easily and quite a few at the time we used to go at the physics conference in particular in Trieste and it seems that the data set over which we started learning how nature works in fact hardly find a match than in the case of a fluvial basis so it's something we know fairly well so essentially we defined the master variable for this which is total contributing area at the site like in this case you have 12 connected pixels upstairs that is you essentially have and it's a typical equation that you had in the aggregation process if you assume one is the size of the pixel whatever square meters you have you have a connectivity matrix in the system and I'm getting to the point that now a system of size this is a well I skipped an important issue or which I shall return I'm assuming this is a tree that is in every node you have a unique path leading to that node by different directions you don't have loops a section and I'll get back to that well it's also historically quite important because that's the assumption that was made in the first manipulation of the of the system as we see it and yet it proved a lucky shot because we didn't search for like optimal configurational some kind by searching loopy structures and it turned out to be a fluke but anyways what you have is that wji means whether it's it's something which you have a one if this j is connected to i i'm sorry if i is connected to j and zero vice versa that is in this case here you're bringing a guy who carries a weight one a guy who carries a gate a weight one and a guy who carries a weight one plus one that makes four if you move here you add one plus four plus three plus three plus one and you get 12 pixels a second and this means that the this is a a quantity which introduces and the statistical physicist among you has spotted immediately this is a non-local interaction which is applying locally and this has plenty of consequences of course now what is interesting that is this is a tree this connectivity matrix has all zero eigenvalues so you know that immediately so if by any chance you thought that you tried to perturb this configuration generating loops you would know immediately with elementary numerical checks a source has area equal to one and the like so one is the basic scale which is delta square the size of this guy here i'm sorry i trivialized it a little bit but it shows how we can in fact show a remarkable capability we have to remotely acquire that it is actually manipulate the scripture is a super accurate of natural landforms over i would say up to six order of magnitude it was interesting here i removed the scale bar and one of the main tenets and that's the book where Ignacio and I wrote years back which was uh were received in fact is still used uh in quite a few even graduate classes or for it's well cited like so what i'm saying is that if you remove a scale bar you really don't know whether this is a large or this is a small catchment this could be the amazon that could be small creek in the dolomites nearby because nature tends to produce those shapes uh in a remarkably similar shape regardless again of climate regardless of vegetation cover exposure lithology you name it or any kind of perturbation of the system how do i know that well um actually we can specialize a little bit the uh a little bit more morphology geomorphological extraction of the geomorphological observation how to extract the proper channel part of a landscape for instance this is a the the network that connects all the concave sides the one for which nabla square of z i is larger than zero concave up so eliminate the dots here that have the concave sides which has a well defined geomorphological reason to happen the concave sides are called colluvium in geology geomorphology or you can take a subset of that which is further refined by some sort of a criterion with the fashion that you're having there you can read in the signatures for instance of past climates as seen in a minute so essentially if i enlarge the thing you have the channeled pixels which are the proper parts which are um a part of the domain of a fluid domain of river methods as ecological corridors as we said so again this is what we have seen and the key result is that you take this as a master variable that is they take you consider that the total contributing area at any pixel at any site is a random variable whose probability distribution in this case by no chance we use a probability of exceedance to avoid a promise of binning as you may know or which we may return in case is essentially proportional to this value a to a power the infinitely popular terms like power laws which is essentially dictating what Simon Olivier was saying about the deep structural similarities that emerge across scales of space in this case not in time but equally well if you have activity on the network so essentially oops sorry go in the wrong direction so this is the what happens of course if you take um subcatchments you sub sample within a basin you had that the maximum area would be limited to the maximum size of the of the of the sub basin you have but what you see that you take nested uh sub basis bigger and bigger you see that the argument you're having the system in reality your base an argument is called the finite size scaling which is something which is well known to a statistical physicist in which you see that essentially you can have these curves collapsing directly um onto one another showing that in fact the deep and remarkable um a similarity emerging across scales it's a well established fact over which there is a significant literature consequences that the system so it's a well known fractal if you have signatures etc um the geometrical language as Benoit Mendelbrot whom um who we miss a lot and uh to whom we pay respects all the time for these visionary ideas uh it's a common name for that so it's the language that nature speaks and what you have is that if you take for instance gross features of a domain uh for interactions like uh uh transverse and and horizontal length you have relationships like the variety distributions uh are with a remarkably consistent coefficient of less than 0.5 hiding on trivial or the lengths to the source of any point are well codified and well known and linked scaling exponents fully characterize the network forms in fact so these are you have you have there's a zoo of cases that have been investigated in the early 90s and this is a remarkably uh uh robust result that we show now the finite size argument also allows you to get interesting features for instance if a finite size argument applies the ratio of consecutive moments of a of a finite size scaling uh uh a finite size scaling um the distribution produces a particular relationship which shows that regardless of a value of the subsequent moments that you're using um you should have a different uh the same exponent essentially so this is a particular kind of a fractal machine which allows us to calculate the elongation of a catchment which is something which uh without math but that Mandelbrot spotted on early and these are the data it's for you to show what happens etc so it's actually a consistent across scale it can be done because you can use these to generate for instance variations in time of the features that generate the network but what we really know at this point is that essentially the spatial imprinting then generates to the contributing area which rightly called the master variable for the uh for the system is what generates something like this is what happened to Mount St Helens after a few minutes out of the first rainfall in fact so the imprinting um it's been like Kenny on like but he will stay forever the planner in printing of the system that settled the issue or um if you take for instance Martian landforms if you apply to the same things etc what you seem to be or what you see in fact as something fluid has to be um operating in eroding those surfaces but obviously I should not insist even for saying that we have tools for and then I'm back to the not really fluvial but um tidal networks it can tools in which by studying the landscape you can start lengths of landforms of a very fine scale indeed and so these are many other things that we've been studying for years or for instance if you take like this looks like a photograph but it's a digital terrain map of an accurate money which you can actually start devising or whether you can discuss where these are natural or artificial forms in fact so that's what we have again this is a digital terrain map or our they treasure our data set that you can in fact show how nature works from again 10 centimeters to easily in the through your landscape to uh thousands of kilometers and um uh very that opens the the thing to I still allow myself to have like five minutes of networks and then I move on to the first exploitation of that it would be if whether all trees are equal and the idea is that whether the loopless properties particularly distinctly is something also important and we say why comparing networks is important for instance I'm claiming that these networks not this one but the three of them for instance had the same topological properties and what distinguishes them are metric properties of a different thing but it doesn't take I mean a scientist to see that the piano network which I'm showing here is different from this one but they are topologically identical indistinguishable so this deserves some uh some extra thinking this is piano network which is something on which we worked a lot and it was essentially devised by Mandelbrot is an exact fractal you have a hell of a lot of properties which are hydrological and geomorphological irrelevant they can be solved exactly for this construct essentially because they map uh multi fractality for instance exactly it's a binomial multiplicative process in this case it's a characterizing something which is very important for hydrology is a benchmark but again the topology is the same over the networks so it allows you to get exact solution for properties that topologically dominated and we'll see next class so there are a few when I'm spending some time on optimal channel networks which is something which we invented in Nationalite and this is a tool through which you essentially calculate a spanning tree a tree has to be in fact um over a given domain in this case square but it could be anything it could be with boundary conditions which are periodic boundary conditions whatever you want to have it and so the idea is the following there is an exact statement taken from the general landscape evolution equation which is nothing but the vast balance of the of the elevation field z in which you have that in steady state condition and the small gradient approximation in the case of reparameterization in variance in fact you get that there exists a Hamiltonian of the configuration of the system which you're seeing here which minimizes energy dissipation which is essentially related to the master variable the non-local variable at any place raised to a power gamma now on this I shall not spend time because this is that well digested if you if you want you can find the exact solution in a certain places etc but what I'm saying is that what is relevant for us is that there is a zoo of possible cases that we know of for instance if gamma is equal to two these are called random resistor networks that been studied by engineers in particular for a long long time s is the set of see the configuration of the system is essentially the number of l square sides the total contributing area which is in begging the connectivity structure and the aggregation level now in the case of gamma equal to one this is a very interesting thing these are called the well mapping the so-called abelian sand piles in self-organization that is you essentially have a network which minimizes the mean length of the outlet and what is interesting is that all directed networks in fact have the same energy dissipation of gamma is equal to one but mostly interesting is that if you assume that this is a problem a problem which you multiply flow time a gradient like you have normally the power dissipation the energy dissipation as you want to call it um then a is proportional to the landscape forming this charge it's a well known result in ideology and the slope is proportional to area to a power which is less than one so essentially you have a classical of host optimal channel networks that obey this property instead of gamma equal to zero you have spanning networks and it's a different ball game in its own all i'm saying is they keep going in the system what you have in fact in some case an issue is that from any initial condition you can essentially rearrange the system by disconnecting a place getting a new tree going to have a direction and checking whether they well you can actually find whether you may accept changes in this case which is the basis from this procedure or you can put some probability à la Gibbs like a simulated annealing and you can actually generate these figures here now an interesting point and i won't follow up on that is you you can calculate in the case of trees the true thermodynamic entropy that is you can count the number log of a number of states that have the same energy we scale with a certain condition and total energy dissipation scales with a with a different exponent which is larger than one so in the thermodynamic limit if the size of the network is big enough energy dissipation is also minimizing the thermodynamic entropy so these are the figures that we can generate and in fact you can show that if you start making good comparisons you can calculate fairly well how these exponents are perfectly matched in fact so match scaling exponents are the true tool for comparing different trees that's it also interesting point that by another lecture that you have i must find out was the lead the lead author and a former icdp professor of them of a ground state the scaling property of a ground state for this case and what is interesting in fact the scaling exponents for the ground states are not the ones we observe in nature nor the ones that we get in the case of so-called optimal feasible optimality that is optimality which is readily that is optimality which is dynamically accessible randomness is not a thing so essentially this is an even growth which is entirely random dominated choice it should make a good comparison of these two networks look at the different boundary conditions that look similar but if you look carefully they are not like your eyes telling you and these are the two different conditions these two scale differently they are both network and this the chance that the system has for each ground state is nil in practice and this can be seen progressively by relaxing boundary conditions size and the likes so i'm these are the typical networks over which i shall be talking about and i will jump now to the i'm showing i'm i'm skipping details essentially on the fact on why in fact networks are i may get back to that and later on in one of the following classes i think it's more important that i move on but the idea is that i hope i convey that can give you the material on that is that it's a coherent thinking on how in fact we have similarities structural similarities emerging in the substrate for ecological interaction emerging across scales of space so the first thing that we had in mind is using on a subset of this kind the neutral theory of biodiversity why neutral theory of biodiversity you may know that neutral theory was originally proposed in complete analogy with the neutral theory of molecular evolution which assumes that gene mutations are selectively neutral that these new genes are demographically equivalent to the old genes and they do not give any advantage in terms of decreased mortality and or increased fertility and it's a paradigm that dominated the molecular processes and still does for a long time the main advocate in fact for a neutral theory of biodiversity was a colleague from Princeton now at UCLA Hubble who wrote the fundamental book about the neutral theory of biodiversity it is he was asking at whether the idea starting from finding from works done on tropical forests in which essentially he proposed to have mutations replaced by the occurrence of new species in a landscape so the idea is that it was a revolutionary one because essentially what you do you assume that all species are considered equivalent to per capita level and on which i shall build more in lecture two but what i'm showing you and i hopefully i will convey the my main idea is that the if you have a neutral process over a particular topology of a substrate this has consequences so the conditions for a species to occupy a site for instance and maintain a population are the dispersal ability of a species the habitat suitability and the susceptibility to any kind of biotic killing of any kind etc but we want to see okay now of course we have conditions through the to grow and maintain a viable population in the place but what happens that was the key questions knowing that we know a lot now about the recurrent characters of fluvial landscapes in fact we should have a spanning tree over which interactions occur what would be the consequence of the extension of the neutral theory biodiversity to space explicit ecological setting but in the particular case of a river network so essentially the experiment i'll be showing you now to introduce the following discussions is one in which you have essentially or suppose you have a lattice okay which in a hydrologic term called savannas because essentially what you have is that every single node is allowed to interact with the nearest neighbors whereas in any chunk of the river network you can have two sides that are neighbors nearest neighbors but they are completely connectivity wise they take they're a very long distance from one another so essentially introduced because of a directional dispersal which is embedded in the network structure with or without drift because that's that's the point and you have a system in which the connectivity matrix is a completely different picture so we can apply the spatial ecology same to like till manna the famous work with caravan 1997 is extended in the model which i'll be discussing now extended to embed the topology of the substrate so this is what happens in setting the simplest possible node so you have a assume a distribution of colors and assume the color is a species this is what in social sizes statistical physics was called the boat or model in the beginning so assume that at random you kill any existing color at any place and then you have two options with probability me you replace it by one of the colors that is not existing in the particular place and probably one minus me and this is a very small number you assume that in this case the color that will be taken out of this would be the one in which or nearest neighbor in the most abundant of the colors in your nearest neighborhood now this is interesting because you assume that there's no stronger species than another right so you assume that if this is a political opinion there's no political opinion will be stronger than the others and you'll be simply the what you actually have as the it's the wisdom of the of the numbers of the large number of the system so in these every now and then like either 10 to the 5 uh uh trials etc the the point you choose at random and you kill is just parachuted from uh from uh outside the domains mimicking either speciation or migration or immigration from different places so the only difference within the two is essentially the topology of a connection and when we saw this we we got kind of excited because they said look what happens is that the only ingredient you have changed is the a matrix is the neighborhood the definition of neighborhood in one case the one you see uh on the left was open and thereby more open to interactions and the second one was more protected in the sense it was due to the presence of a network itself now what is interesting that if you just make it quantitative that you have it here you essentially have not only you have the different degrees of irregularity the boundaries which is but in this case which is the network system here the neutral system here what is clearly that any statistical measure which you have of the distribution of the colors is essentially uh completely changed and the rank abundance curve it shows that the biodiversity of a network system um is much bigger and of course first thing you say okay is this a realistic model uh the neutral model uh of a of a fluvial ecosystem of a fluvial uh stream uh microbial ecosystem uh distribution or not well it so happens that um um patterns um uh may not be neutral patterns do not require a neutral process perhaps or equifinality issues can pitch in but the fact is this was an intriguing suggestion early on it happened in 2009 before we embarked in this study etc but for us it was worth uh checking why this is interesting because this is a result that it's absolutely robust with respect to a number of generalizations namely this is one point one species but this could be evolving into a meta-population model it's one node because a local community with a certain number of species the same rules of engagement for the interactions and you can also have this that you can relax the possibility that it's only the nearest neighbors giving you by sheer majority the new colonizer in a place in which you don't parachute the new species and um and the result is the same so a a dendritic uh substrate for interactions changes completely the ball game regardless of a number of assumptions still in the obvious assumption that you're assuming that all colors are equally valid and you're assuming that all species are equivalent to the per capital level but we thought it was interesting and the first thing that we did we applied the same rule individual based but or meta-community based but with a kernel that is the radius of influence of the area over which you average progressively to get the majority replacing the place could be immediate neighborhood or mean field that is the one you essentially go and check what happens in the system you can do that on a 2d lattice you can do it on a network you can do it on a 3d lattice on a 3d network and much to our surprise what have we found okay one good measure to see what happens in this case uh would be um essentially uh related to how long would a certain color last from onset when it's created or in a particular place or um it has a local extinction and um this is called the persistence time of the color or the species into a landscape only in this case affected by technicalities like the structure of a network and much to our surprise the distribution whatever you want to call it certainly there is a huge there's a huge range of uh uh scales that covered by by power law in this case show there is a distinct and unmistakable effect of the topology of the substrate for topological interaction so um in our case um we took on to a first empirical check because it's interesting thing we did too in fact so we started the this the we had like 41 daily values 41 year of observation of breeding birds in North America into a place and you see you see the what is this um the uh persistence time in this case meaning suppose that you are in this area here that you have yeah here it is I should add in this area here you count in those observational sites you count uh how many times here you spotted for the first time a particular bird and here for a number of years you haven't so these are independent persistent signs and you can study the value that you have a different level of aggregation for instance or you can do the uh cancer spray herbus herbaceous uh uh plants etc so you can characterize it it's long story short what I'm saying is that you can isolate the effect of a finiteness of a sample which you have an atom or probability if you have a species that's been continuously observed for instance and you can characterize this interesting feature so long story short what you have is that you confirm what you have seen topology means a lot and topology of the substrate for interactions has an effect and to the point that you start for instance aggregating a different spaces you can transform this scaling effect exponent of a lifetime into a distribution of biodiversity so essentially by taking coarse grained versions of the same space you can find the probability distribution of the survival how it behaves on the coarse grain and thereby generating a true species area relationship and the next step which I want to show you and on this we are relatively fast it's only the last five minutes of a class we said okay say look of course and you talk to an ecologist you say hey but this is bullshit I mean species are not equal at per capita level how you can do that etc and yet we thought earlier there was something um uh two I mean manifest to be completely false a topological effect of the substrate on the nature the recursive nature of a form and function which is embedded in the structure of the networks regardless of whatever it is truly a universal feature so we decided to um decide to test on in my lab these swift living communities in which there's nothing neutral there is absolutely nothing neutral so essentially what we do is just we choose for instance you see how rudimentary the city is we should have had the money to buy a robot and the in the absence of robot to have so this is essentially what we have in the in the uh uh lattice in which nearest neighbors are the four nearest neighbors I mean the in which nearest neighbors are the topological ones the adjacent things and the structure in which these two guys are not nearest neighbors and we did that brute force this is Francesco Carrara now it is Z and this is Enrico Bertuzzo now a professor in the University of Venice now long story short this is what we got replicas were as many as needed and this is the experiment that you have in the random network system that you have or in the thing that you have and the color essentially is a measure of species richness so in these the theoretical idea was telling us look guys there's something which pertains to the fact that you have a certain topology of the connections it's regardless of everything else it's but the prediction came directly from a neutral model um if you have like a lattice of interactions this doesn't happen experimentally and this again uh this is the living communities I'm talking about in this case the the uh the I'm giving you the details that took a note someplace here about what is the experimental community I'm talking about 21 uh protozoan species um uh in which we had I'm sorry nine protozoan species one rotifer which is a multicellular one and a set of freshwater bacteria as food resource in the place and the dispersal was done manually but it was done essentially according to the things and I can give you the details if you care for an interesting tool was also published in another set of painfully uh uh conducted experiments in which essentially get okay now suppose that we take into account the fact that habitat capacity in a river is a hierarchical okay so essentially you assume that the medium in any single well depends on the number of upstream sides so you have a hierarchical structure or you get system in which the same total amount of medium is randomly distributed or in which you take a uniform distribution of the system believe it or not um the hierarchical system is a completely different structure so um and that's the take for the next classes I'm saying that uh whether neutral models or models of anything as modeling or riparian vegetation modeling or species diversity etc we're talking about the landscape um quite a bit when you have an environmental matrix which is made by nodes and connections um the substrate matters and I'll show you how this in fact has a say on the oriented graph is something which we give you don't assign a network interaction this is given by the by the geometry of the system and um and in good timing in fact I'm showing you the conclusions are general conclusions but I'm essentially what I have shown is that eco-hydrological footprints of rivers as ecological corridors were suspected from early abstract theoretical work and confirmed by empirical analysis uh on a particular set of persistence times measured for breeding birds or plants in the cancer spraying or experimental work done on living communities um in this place so I'm ready for your questions of course if you're following classes what I'll be doing I'm talking about species about the biological invasions I'll be showing you why biological invasions slowed uh by by forketing structure of the system and then I'm talking about disease which is an interesting point disease is tackled in this manner I have special expertise in that okay let me check where I have chats okay uh hi everyone I'm having serious problem with electricity supplies uh oh no I'm sorry that's that's a question about the electricity supply it cannot help you I'm sorry so we are ready to take questions from the audience if you want to speak just raise your hand in the place of participants and I can give you the floor okay uh we have a question from Miguel Rodriguez Miguel hello uh Miguel that that was a that was a fantastic lecture thank you for this last part uh where you do this experiments with microorganisms moving from well to well um would that would that pattern be robust to different degrees of base diversity would that be true for much larger uh values of richness for example in those okay it's a super question super question now uh they they uh uh well we have experimented with a varying number of protists in fact because you needed to have a size of thing and and I have to say the protists have been a good model organism in fact so these kind of studies for a long long time and if you look at what there's a number holly oak uh florian alternat which was actually working with us on these etc they work a lot on so-called dendritic substrates for interaction but what the head the dendrites were just a pipe and a few bifurcations that's not the river network that's why I keep saying and I think I convinced Florian who followed us etc to the point that in fact that he put on uh you'll find an old he put an open source code for having his own networks in the place because you have to have the number of sides the number of sides that you have are constrained by the total area it's not that you can assign them independently the connectivity which is suited to a fluid ecosystem uh is a specific one highly constrained one a highly recurrent one it's not one you just had like a pipe and a few uh branching patterns in there and so the question is that we have experimented with different number of processes but we haven't explored it a hell of a lot all I'm saying is that as you've been seeing uh soon after these kind of confirmation we got more more enthusiastic I got an ERC grant and we started having lots of people working on different aspects of that about which I'll be talking the next two classes and I'll be showing how we did species we did uh uh population migration which is a fascinating subject I'll be talking to you because we start from migrations human migrations in the 19th century it's a long story and uh and then I'll be talking about disease believe it or not we have the same tools we started simulating spatial explicit models of cholera where water born of course it has to be right the pathogen and whatever has the pathogen living in it's uh or uh proliferative kidney disease in fish so anyways we haven't experimented a lot a lot more in the lab mind you at the time I didn't have the money for the robot so Francesco and Erika stayed up long nights in pipetting from one side to the other and that's what it's typical you know it's it's um uh uh you see typical of professors say what are these details it's not detailed it works a lot okay thanks we uh we had a question from Susie do you want to uh speak for yourself or do you wish us to read your question on the chat ah Susie I saw I see it I see it here want to repeat it Susie yourself otherwise I read it you can go ahead if you can understand the question okay the question is are there evidence for biodiversity reduction whenever rivers are diverted or dams or construction of course fantastic uh fantastic uh uh questions Susie yes there is in fact a group of fish biologists um in Spain uh it's Pepe Barquina I think is a guy in Santander that uh published it published quite extensively on the study of how biodiversity is impacted by it's um it's actually common sense it's common sense on number of counts for instance I could tell you that there are a number of studies especially the biofilm guys are very prominent on that if you change if you modulate the natural sequence of stream flows you change your ecosystem period that's a fact the color can be used for for example uh impact studies before prior to constructing dams to say that you know oh yeah oh yeah but that's I haven't done it but the Pepe Barquina and his group has done it quite a bit actually but I mean it's just but even a common sense I mean uh it's uh it's known by the what the dam does besides the interruption and the change in connectivity which is directly reproducible in the case of this because essentially you change at any downstream point the total contributing area right but the sequence of stream flows is not proportional to area anymore so essentially you change the structure of the engine for interactions and uh and I mean it's it's known people that study the biofilms know that which is the the basis the engine for the ecological interactions in the in the microbial ecosystems or fluvial microbial ecosystems and so there is evidence for that yes the answer is yes I haven't done it no I haven't done it you have a question from Victor Subrak okay within this model the river caves uh could uh we add the effects of factual changes in river curvature due to coastal erosion or even okay uh or rivers driven by deforestation or riverside vegetation okay the two different issues um first um the I'm referring okay it's two separate issues to the first one the answer is no why because the recurrent properties we see in rivers uh do not apply generally they apply the so-called runoff producing areas so the the river structure is essentially the runoff producing area where the landscape form and discharge is proportional to the total contributing area at the point the non-local quantity then for instance if you enter the desert like the colorado river does after uh uh glen canyon dam for instance when you have a place in which you have no injection of water so essentially the transportation thing is dominated by a geomorphological process and morphodynamics which is not related to the agrarian process at all and then you have the delta of the estuary which is the distributary system which is again governed by different rules what I'm looking here is that they're not producing areas remind you it's not the whole of the river basin but it is um something which can easily extend to scales of the order of thousands of kilometers inside if you're talking about deforestation or riverside vegetation of course yes and I work on that I'll be talking about briefly about models of vegetation uh over which we work a lot actually so the answer is yes to the second one is that okay vitor do you hear me and we cannot say presumed that vitor is satisfied yes yes we have uh any further questions in the audience can I ask one more one more question yes please go ahead uh in your in your uh persistent persistence time distributions patterns uh you you show that the the these networks these river networks sit somewhere between 1d and 2d fully connected uh uh grids uh can can will these be predicted by the dimensions in that fractal structure absolutely that's that's that's it's again a very good comment the what happens is that it's the topology of a substrate that matters in determining the persistence time so if you have you sit in the point you close your eyes close your ears and you measure for how long the color stays there okay you find that if you do that of course every side is a a different persistence time from the onset of a of a species we're not measuring abundance here so we don't measure the number of colors so color right in this place yes for how long it stays and you measure it so every particular from a onset to local extension is one value of a random variable persistence time local persistence time you do that for every single thing but what I'm saying is that the the game of simulating that in a couple of hours you can do that because it's absolutely trivial right if you give by hand for instance the connectivity which is slightly more complicated but okay but the essence is that one what happens is that what you see that depending on the topology of the system that you have you don't know anything of what's going on in the model etc you simply count and you see that those distributions once you start to make it big enough to have enough statistics etc behave completely different and that is the counterparting which is telling you look there may be something deep going on here because what we are saying is that depending on how I'm connected it's how I'm exposed to interactions and how in fact the the the distribution of my energy it completes completely different changes radically and what is super interesting is that this topological effect has been it's a single doubt exactly in fact but what is interesting if you look at data you know it's a tricky thing because if you look at the data for instance look at breeding birds in an area like of Georgia God knows what you know all the most society started like 50 years ago in collecting those like the network of the bird watchers it's a serious business in the in the north of the United States and the Mediterranean one what happens is that what you had said okay let's suppose the cardinal that was seen in this area now that he hasn't been seen on on like this year he's been spotted next year he hasn't so essentially you measure what happens locally and then what you do you can make it a bigger area so it changes because you may be local extension can be here and not be near us and so at the continental scale okay if you assume there's no migrations or carnals et cetera speciation that you're measuring is very is the the you can call it the speciation rate or a true speciation rate so what I'm saying is that you can empirically there is also an issue because if you have a finite sample in terms of length what happens is that you may ask yourself okay look if you see throughout always this you always spot the same bird in the same place you have an atom probability of one okay and that is very important because you see how this atom is reducing with the size of a sample it's telling you a hell of a lot of the underlying distribution and so being careful but you see I'm showing you in 45 minutes like four or five years of work and but again if you if you if you care for that I'm ask your library to buy the book because we spent a hell of a lot of time in putting it together thank you I think we have time for one couple of questions more then we'll close okay one to one is asking could you please remind me what is the difference in topological structure between the savannas and the ocm okay savannas means that in every side your nearest neighbors are the four dearest neighbors in the coordinate direction that is north south east and west that's all you check for okay so once you kill in in a place a system you replace the color you have in there by the most abundant color in the nearest neighbors and if all three if all four values are different you choose a random okay in the case of the ocm you have a directional disperser so you have a essentially drainage direction that I tell you who's connecting with whom so you may easily have that because of this directed structure nearest neighbors topographically may not be nearest neighbors in terms of interaction and that's a very nice direction which you get directly from the from the landscape elevation there's another one Zebsa Rabah always I think that there is a limit number of species because existing structure then all about special structure that's a good question too Zebsa what I'm saying is that they I'm talking about the abstract model now in which one side is one species okay but you can generalize this into a meta-community model which I'll be explaining next class in which for every node you have a meta-community a local community it is a community of communities so you can actually calculate carrying capacity if you're worried if you're worried about that you can calculate logistic curves etc you can put this into a into a spatial ecology framework like the spatial ecology framework that Tillman the mighty Tillman and Kareva put together only thing what we provide it's those recurrent features of channel network which is no better than anybody who worked for 20 years on those so the answer to your question is of course there's a carrying capacity on each node you bypass it but you see the pattern explained by the by the individual based say kernel one that is only nearest neighbors thing etc it's telling you a story which is conferred by making the kernel say mean field like the entire structure of the catchment or a number of species fixed or number of pieces variable or whatever the result that is the pattern is affected resists and is general thank you okay so maybe it's time for us to call an end to this first lecture and thank you Andrea thank you so much lecture we as usual we have the opportunity to be to participate into the breakout rooms where you will be sent automatically and otherwise we meet again at 345 for the next lecture of the school okay thank you ciao Antonio ciao ciao ciao okay so welcome back I think that everybody has joined again the main room and we are ready to start with our last lecture for today by Stefano Lezina from University of Chicago please Stefano good morning and good afternoon good evening everybody so this is the second lecture on the theory of ecological assemblies and just to recap what we've been doing like the first lecture we've been exploring a little bit of the history of the program of assembly in ecology and then we looked at some features of the model that we're going to use for our exploration which is the generalized lot of alter a model which is arguably the simplest model for population dynamics and just to give you a very brief recap of what we looked at so this is the form of the model here and it has just like the parameters are just a vector of growth or death rates and the metrics of interaction a and then we looked at the notion of an equilibrium which is like a point a certain density at which like dynamics are fixed and we saw that like this equilibrium if it exists it's unique provided that a is not singular that we call this equilibrium feasible if it is attainable by an ecological system meaning all the species have to have a positive density we saw that this equilibrium is the time average of the dynamics in the long run and then we saw also that the feasible equilibrium is necessary for quick existence is not sufficient to to get toward like sufficient conditions we looked at stability and in particular we looked at local stability meaning is this equilibrium gonna be you know like good trajectories go back to the equilibrium if we slightly perturb them away from the equilibrium that's like a local stability and we saw that this is very simple to do because all you have to do is to form this matrix which is like the diagonal matrix with the equilibrium on the diagonal times the matrix of interaction if this matrix and that is called in ecology the community matrix has only negative eigenvalues or better eigenvalues with negative real part then the equilibrium is locally asymptotically stable we also introduced a much stronger form of stability it's called global stability which simply says that if I start the system with species with all positive densities eventually the trajectories would converge to the equilibrium and to prove that what you have to do is to find a matrix C that is diagonal and positive meaning only positive coefficients of the diagonal such that these new metrics that is a symmetric matrix that is c times a plus the transpose of a times c is stable is negative definite right if you can find that then every a equilibrium is globally stable all right but we ever really got to the point of talking about assembly so that's what I want to do today and in particular what I would start with is like the reasons for which building a general theory of assembly is so difficult and for this I'm going to follow quite closely like an introduction that we have in this paper with my former student Carlos Cervantes that possibly will be published next year so we argue that there are three main issues with assembly in its general form right so imagine that this assembly is just a process by which some species enter a system they interact with the local community and then something happens for example the species is established send some other species extinct or whatnot the first problem that I would like to highlight is that of invasion rates right so if you think in this system we basically have two different timescales one timescale is the timescale at which new species join the system that's like the invasion rate and then we have the rate at which like dynamics evolve right so you can imagine like provided that when I put in these pieces something will go extinct how long does it take before they go extinct and the interplay with between these two timescales it's actually a complicated and the problem quite a bit and I think that the simplest case to study is what is one of like a rock paper scissor community right so imagine that we have three species of safe plants right such that each species like the species say rock can happily you know stay in its habitat by itself and I just like use this notation for that right and similarly the scissors plant or the paper plant they can happily colonize this habitat what is the problem that when I put two species together for example rock and paper what happens is that one of the two species will go extinct right paper when over rock and so so like if I wait for long enough I start with rock paper I end up with paper alone similarly rock scissor I end up with scissor alone no sorry this should be rock alone rock all right interestingly enough if I start the system with all three species together they can coexist they would cycle or do something like that right so now if we're thinking of the process of assembly imagine that we start with like the bare ground right there's nothing and then one species comes in for example rock and then after a while we go to these unstable configuration of say rock and paper right now if we wait for long enough we will end up with just paper but if scissor joins these configuration before rock goes extinct then we're in this state with rock paper scissor and therefore the system can assemble like to the full community so you can see that really what matters is the speed at which we have the dynamics versus the speed at which we have invasions right like so that's one first complication the second complication has to do with invasion size right so so what how many individuals of the new species come into the system at one point and here like the simplest example to think of is like two competitors in lotcavolterra where we have by stability meaning each species is happy by itself but then when i start with both species together i could end up with one stable state only species a or only species b depending on where i start right and so what happens is that if i'm sitting at the state with only species a and i invade with a little bit of b i go back to the state but if i invade with a large population of b then they can kind of switch like to the other half of the plane in which i projectors will need to be so the the outcome of the invasion depends on the size of the invasion the third problem that i see has really to do with the dynamics of the local community so imagine that we've seen like last time a set of species cycling right so imagine that you're a predator you have to come into the system and you want to have your prey available to establish if your prey are cycling maybe you could establish an invade when prey are high enough but not when they're low enough right so then the timing of invasion would determine the outcome right so these three main difficulties i think are the three main roadblocks to developing a full theory of assembly so in our paper and in these lectures today what i would like to do is to make three strong assumptions that get rid of these three problems the first we were saying like the invasion rate and so what we're going to post postulate is that invasion events are rare so rare in fact that the local community it's always at the state at its final state right like it's always at it's a tractor that would reach like it would be reached by the dynamics right so so what i mean is that if we have the rock paper scissor we will only see rock paper or scissors we will never see the three species together because that would basically require two invasions in a rapid sequence rapid enough to to allow the establishment of the full community before the second species goes extinct right this is a very strong requirement is in fact even stronger than it looks because like the the time of the dynamics could change depending on the composition these strong requirements and the strong assumption is what it's typically made in all studies of say population genetics or almost all studies of population genetics right where we always assume that there's a wild type and only one mutant there's no two mutants or three mutants at the time we do the same in ecology when we are performing invasion analysis right like we're saying these a community says it is resting at its state like China state stable state for example and now I introduce a my invader second to get rid of the problem of size what we're going to assume is that invaders arrive at very low abundance so low abundance that they don't feel self interactions for example right so self limitation or clouding with your own corn specifics they don't they don't feel that and the such they will only be influenced by like the state of the species that are in the local community we will see what I mean by that in a second and third to get rid of the problem of the invasion timing the simplest solution even though again it's a very strong solution is to say there's only equilibria right so so we can study systems where we know that all the dynamics are given by equilibria there's no cycling there's no chaos and as such the invasion timing problem is removed for example in generalized lot of altera if we had symmetric metrics of interaction we will only get a fixed point analysis all right throughout like these lectures I will distinguish between two different types of assembly I call them top down and bottom up and and they're quite different in the sense that top down assembly what I am thinking of is a massive invasion for all the species at the same time which is akin to saying what I start my system with all positive densities at arbitrary densities and I see what happens so in this case assembly is really like more of a disassembly in a sense right like so what will happen is that I put all my species in a system dynamics will do what they have to do they will probably send some of the species extinct so some sort of dynamical pruning and then I reach my finite configuration at the other extreme we have bottom up assembly where I start from the blank slate like I start from an empty environment and then species trickling one at a time and then I like the dynamics elapsed and then I add another species and then let the dynamics elapsed again and add another species and so on and so forth right so these you could think of like the two extremes of one invasion at a time bottom up assembly all invasions at one time the top down assembly all right now the last ingredient that we need to really like start studying assembly is to see what happens when a species invades a very low abundance a community that is resting at an equilibrium point so let's again go back to our generalized loss of altera and for example write the per capita growth rate right meaning like I take like the x i d t and I divide by x i of t right so so that would be per unit of individuals right and so you can see that the right hand side we have r i like the intrinsic growth rate of the species and then we have the self-effect right so because this is the first species that arrives in the system because it starts at very very very low abundance basically this term here it's negligible right it's basically zero and as such what we're left with is just a growth rate right so you can imagine that these species will start growing if it has a positive growth rate and then if it doesn't it will just immediately go extinct okay so so this is what happens when we have the first species coming in it better had a positive growth rate in that particular environment right and then what happens then if we have more than one species for example let's imagine that we have two species and so so in this case what we have is that there is one species that is resting at its equilibrium point right so this is what the species i here that is resting at its equilibrium point and now we add the second species and what we do is exactly the same right we compute the per capita growth rate that is here like by taking like the dxj dt and divided by xj of t and now we will have more than one term right we will have as we had before the growth rate of j but now we have all the influences of all the species which will be only like the species that was already there let's say i and the new species j so as before we assume that the effect of j on itself is basically zero and so this simplifies the problem a little bit and because xi is sitting at its equilibrium point then we have that this whole quantity has to be greater than zero for the species to start growing in the presence of the other species at equilibrium is this clear enough right so this is what we call an envision criterion and in fact we can extend it to multiple species and we will do that in a second just like to to make something even clearer what will happen when this condition is match right so imagine that these species too can really start growing well it could initially grow and then go extinct or it could displace the first species right displace species i and therefore we end up with an environment with only j or they could coexist together or in fact they could even like both go extinct that is unlikely with the parameters that we're going to all right now we can do the same for multiple species of course this will require a little more algebra right because because we have to divide like these interactions into the interactions of the species that are already there and the interactions of the species that are not there so what is really convenient to do is to divide my vector of the densities my growth rates and my metrics into blocks right and in particular you remember that we we put x star for a feasible equilibrium point now let's put this x bar for an equilibrium point that is composed of two pieces right the first piece is a bunch of species say k species that are resting at an equilibrium point and the other piece of this equilibrium is a vector of zero for all the species that are not there right so so we have split like the populations into two classes those that are already in the local environment and sitting in that equilibrium and then the remaining species that could potentially invade and we use the same order to write the growth rates so this would be the growth rates of the species that are already there these are the growth rates of the species that are not there yet then we have to do the same with the metrics right so now because for a vector we will have two blocks right but for a matrix we will have four blocks so a k k would be the interactions of the species that are already in the environment with themselves right with the other species already in the environment a n minus k m minus k would be the interactions between the species that are not there and the other species that are not there and then of course the off-diagonals are the interactions of the species that are not there with the ones that are there and vice versa right so now what we need to do to say this piece the system is at some equilibrium like of the kind that we said right like we first have to have that x of k must be feasible right and this is exactly in this condition right which only involves the first block on the diagonal here right the interactions of the species that are there with the other species that are there and then we can write a second condition which we call the non-invasibility condition which says in this point if I try to invade with any of the other species at low density they just die right so so that means that these a point where we are where there's k species that are coexisting in the other n minus k that are absent is somewhat a stable right like it's difficult to move away from this equilibrium in fact we will see that for lotcavolterra you know with certain condition it's actually impossible so so this will be the end state of the community and so we can write this condition for invisibility which looks exactly like the one for two species but now there's several species and now what we can say is that if we can find an equilibrium x bar such that the k part of the equilibrium is feasible and stable so stable when considering only the species that are already present and b the other species cannot invade this equilibrium these is called the feasible stable non-invisible solution or also like the saturated equilibrium right and like the book that I referenced like of power and sigma there's a very long discussion about this and in fact for this type of equilibrium you can show that again if our matrix interaction is a little special such that we can find a matrix c that makes like the symmetric part of a negative definite then there is always a saturated equilibrium in which like a bunch of components like a bunch of species are positive the rest are zero and this is going to be globally stable meaning I can start my top down assembly by starting the system at whatever positive condition for all the species I always end up with the saturated equilibrium and to prove this point you basically use the same type of algebra that we do we did for the Lyapunov function of the generalized linear model and just generalized Lyapunov model and yeah so I'm not going to go through the details of this because you can read in the north all right so this moves us to lecture two so maybe this is a good time to to look at the chat if there are any questions I don't see any questions all right so then we start exploring assembly with this top down assembly which means we take all the species and we put them in the environment at the same time and again we use generalized lot of altera we use a that is symmetric and stable right with random coefficients right so we just build like a random matrix make it like stable in this strong sense right that we have and then we choose like the growth rates at random again and then what we want to know is what happens in this case right so we want to draw some sort of very generic conclusions on generalized lot of altera with random coefficients even that the matrix of interaction is strongly stable as you make and this case was started by Carlos and some collaborators of mine you can find the paper here so let's start with the trivial case right then I think you know the answer already so imagine that we have a matrix of interaction that is diagonal so species only interact with themselves they don't interact with the other species at all right so each species is self-regulating to some extent so you can fit only so many individuals of any species in the environment but other than that species don't care about each other at this point what would be our criterion for invisibility of the species it has to have a positive growth rate in the environment right if it has a negative growth rate it will die if it has a positive growth rate it will go to its equilibrium because it doesn't the species does not care about the other species at all if we're saying we sample these growth rates at random from a distribution that it centered at zero right with the mean of zero then we would have 50 probability of getting a species to persist and therefore if I just like take a bunch of species say a hundred species how many species will persist on average 50 right in fact like we know the whole distribution it's the binomial distribution would give a will give me the number of species that persist right the binomial distribution with parameters and which is like the number of species are introducing the environment and one half where this one half is basically the probability of having a positive growth rate okay that was a bit trivial and so so let's start thinking about more complicated models where where species do interact with each other and just to start to get your sense of what we want to do let's start with some simulations right we will take like species that interact with each other and what we do is like we write a function that draws some sort of random metrics of interactions you can see here interactions are taken from a normal distribution makes this metric symmetric and makes it stable how do I make this metric stable I just look at the positive eigenvalues if there is any take the largest positive eigenvalues subtract this constant from the diagram by doing that I shift all of the eigenvalues of the matrix to be in the negative plane right like half plane and then we have a just like random growth rates that are taken from a normal distribution okay so we start with seven species we build our a stable symmetric matrix we draw random growth rate we started some random initial conditions and then we integrate our dynamics with these parameters and then this is what happens right so we started with seven species and you can see here the two species started immediately going extinct they never recovered and then they went extinct while the other species you can see that they go to their equilibrium which is a mark here as this dashed line right so so in the end we have five species coexisting the other two have gone extinct now you can see here that we reach some sort of equilibrium dynamics for this case and I told you that this is always the case when we choose this type of parameters and this is the what we would call this saturated equilibrium right like some species have positive components here and then two species have a component of zero right like these species are absent and they cannot invade when they're rare just to show you that this is in fact globally stable let's take the two species that go extinct and put in a very very high density and leave the other ones at low density at the beginning and what happens it's the same we see like they start very high but they fall down and eventually go extinct and the other species they go to the same equilibrium as before right so in the end we obtain exactly the same vector of density here so this is one very computationally expensive way to compute right like this equilibrium is to do what like we take the system we started in some initial condition and then we run the dynamics we get these saturated equilibrium at the end turns out that you can adapt some algorithm from that was originally developed in game theory it's called the lampehosen algorithm and you can use this algorithm to just very efficiently find this a saturated equilibrium these works when when we have this matrix A that is a strongly stable and symmetric but if you have that like a code that is up for you you can find it here then you can say get finite composition and what you get is exactly the same vector here you can see like 0.57 right but but it would take a much much shorter computation time and especially when you have many species this thing can you know it can take a while all right so in the case above we saw that we started with seven species and we ended up with five right can we say something more general about this process and the way in my laboratory always like jokes about this issue we call it the problem of the random zoo and the idea of the random zoo is that you go into a big city that has a big zoo and then you fence the area of the zoo off then you go inside the zoo and you open all the cages of all the animals and just let them free and then come back after 50 years right and the question is how many species do we have after 50 years and like the zoo is nice because like these species were not chosen to coexist with each other they're random to some extent species there are species that we like to see in a zoo and therefore like they haven't really established these interactions before right like we will have a lion that is looking at the penguin thinking what is this thing they've never seen this before right so so that's the idea of the random zoo and with these kind of a code that hand we can actually simulate in the random zoo and start with maybe several observations so let's say we have five species and we repeat these random zoo experiment with these five species two thousand times right just to see how many times do we get all the five species how many times do we get only one species and whatnot and so what we can do is just like for each one of the simulations build a matrix that is stable draw random growth rates find the final composition using these lampe-hosen algorithm and then count how many components of x star are positive that's the number of existing species right and then we can take this data and plot it so this is the the result of our simulations and you actually have code you can do the simulations with larger number of species or whatever you want but what I just want to point out is that this is a distribution with what with a very strong central tendency the probability of having two or three species is the same one and four is the same zero and five is the same which is what the binomial distribution again in fact we can overlay the binomial distribution here and you can see that we're for if we had chosen like two million simulations we would converge perfectly to the binomial distribution so this is an interesting result in the sense that a neutral to some extent model right like a model where species do not interact with each other gave us the binomial distribution a model with random interactions right strong but random interaction gives the exactly the same result and this is it goes back to a discussion we were having in the in the first a class right so how do you go about proving a result like this that that is an interesting question and I just want to outline this proof that is like something that Carlos did which I find it very very beautiful and so like this proof has two pieces right like one you have to find what is the size of the number of species that coexist and then what is the probability that this would be known in baseball let's just look at the first part of this so let's say I want to show you that the probability of having all species coexisting right so so that would be the five here is exactly one over two to the end right so in this case because the matrix is stable so all we care about is feasibility so so to have full coexistence what we need is to have a feasible equilibrium point which you can write as like the negative a product of like the inverse of a times r has to be all positive so to do this Carlos concocted this trick that I like a lot so imagine that you have a certain matrix which is exactly like the identity matrix so let me maybe draw it here the d of k is like a matrix that has zero everywhere but in the diagonal and then in the diagonal it has all ones besides in position k where it has minus one okay so this is my dk and now we have to think of what happens when I take this matrix dk I use it to multiply right left and right the matrix of interactions and I use it also to multiply the vector of growth rate so what happens when I do this operation is that I flip the k sign of the equilibrium x star right so so imagine that I have all positive equilibrium already I choose one of these numbers to be minus one I say the first one what and then I apply this transformation and what happens is that now x star of the first case is negative while it was positive before so I can use these to basically change my parameters to make an equilibrium a feasible like let me just show you this in practice how does it work so for example here we build a certain matrix that is stable we build random growth rates and then we solve for the equilibrium and you can see here that the first component is negative right so this is not a feasible equilibrium but now if I choose then dk with like the first element of the diagonal to be a negative let me just clear my drawings right now what do I have that when I solve these new system of equations right like that is like the modified matrix A and the modified vector of growth rates are now this equilibrium is positive okay so then we can do this mental experiment of thinking a I have my matrix and then they multiply these metrics using these left and right multiplication and infinite number of times like as many times as I want or let's say n times so if I choose appropriately where to put these numbers like or the order in which I do these operations then I can always recover a feasible equilibrium right so so recovering a feasible equilibrium is akin to having chosen the right sequence of matrices to use to multiply these equations right in fact these transformations because they gain values of k are the numbers on the diagonal of dk and it inverse it's the same as dk it's a similarity to transformation right so so what it means that when I do this operation I change the equilibrium but I do not change the stability of matrix A right so if I take my matrix A before the transformation you can see that this was a stable matrix right it has only real eigenvalues that are negative if I compute the eigenvalues for my modified matrix right where I multiply left and right by this matrix d1 I get exactly the same eigenvalues right so so so if the probability of getting a certain equilibrium to be feasible is just like the same as having chosen the right sequence of these matrices dk then you can see that this has to be one over two to the end right just because you know I had to choose which one should they flip to minus one or one and in fact like you can think that the matrix A and these metrics d1 A d1 are actually sampled from the same exact ensemble as before right so these are still symmetric matrices they are stable with the parameters chosen in exactly the same distribution right using the same argument and you can see like the paper for a foolproof but using the same argument you can prove that under this parametrization the probability that these you know k species coexist and n minus k's are not able to invade is exactly like the binomial. All right you can complicate this model as much as you want like I would not suggest doing that because then the math gets a little difficult but for example in the paper we have what if now my matrix A is fixed and I choose random r's and the matrix does not have a mean of zero right like or something like that and what you find in general is the same that you start with a bunch of species you end up with a much smaller typically a number of species but the proportion is fixed by the parametrization so if I choose like my r's from a certain distribution my a from a certain distribution I have a typical proportion of species that persists right so so what you would think at the point is that the proportion if I were to plot initial number of species versus final number of species what I would find is that there should be some sort of a line right like it's directly proportional to each other now you might think that this random zoo is a bit like a tried example right very artificial but I would like to argue that this type of experiment is actually performed by scientists all the time and just to prove my point I just like want to discuss briefly this paper by Lenora Brithelstone and collaborators in like the lab of autocornero at MIT and what they did in this paper that is referenced here is to sample picture plants right these are like carnival plants so so they have this picture and inside the picture there's like this liquid that they used to digest insects and this has a very rich bacteria community so so so what they did is they took the bacteria community from 10 different picture plants and then cultured this community in some synthetic medium that they they they concocted right which is like normal medium but I think they put some sort of plant extract as well and what they did is that then they serially cultured this first for I think about two months right so so this is somewhat a random zoo right we pick a community that would have its own interactions and now we put it in a different environment where we might think that these species would establish some sort of random relationship with each other and what would happen then that like these species could have a positive growth rate in the new environment or not if they don't they should go extinct almost immediately if they do then they could potentially grow grow but because we started with like a little bit of these community and now it's growing in time like they will feel these interactions stronger and stronger so eventually like interactions between the species we kick in and then we should be in the random zoo experiment again and so I want to like this is I'm reporting like the figure two of the paper and so here you can see like on the x axis is there the number of ASV or imagine of species to some extent that they found that day three right so once like all the species that are negative growth rate died and then on the y axis you find the number of species that were there after two months and what you can see is that there is a striking linear relationship between these two quantities which is exactly what we would expect under some sort of random zoo all right now in the run let me just go over the chat to see whether there are questions that that are important oh yes I forgot to post the lecture notes again but somebody beat me to the to this thank you very much Alfonso for doing that and then in the random zoo model are also the interaction coefficient chosen randomly yes they're there are samples randomly from a normal distribution and you can see in the code above so they're just like a random symmetric matrix and then all all we're doing is to change the diagonal a little bit to make it stable right so so these interactions are sampled from a distribution centered at zero and it's a symmetric matrix and you find the code here here right so this says sample them randomly from a normal distribution with mean zero and variance one Martina asks can we really assume that the interactions amongst species sample from a picture plants at random absolutely not in fact like we think that like they already assembled inside the picture plant and they are able to some extent coexist if you're able to sample them because of that what I'm saying is if we take this community and then we put it in a completely different environment these interactions may change right because these interactions we're thinking are mostly like due to the resource utilization or things like that so so if you change the resources you're changing the interactions so just to be sure it's not it's not abiding by all the rules of the random zoo is more of like the random zoo is a very simplistic metaphor of what we're trying to do and can we give species a functional role I'm sure you can I don't think they went into there were a very many species like you can see here so I don't think they characterize fully all of them right but they like they start with like more than 120 species sometimes right so it's a bit difficult to see what each species it's doing they would require a lot of expert all right but just to say that even like this very theoretical abstract a model of a random zoo can have some sort of at least helps us forming some sort of new expectation of what we should be doing and in fact like I can tell you that the day like Lenora came to Chicago to present this talk and I jumped on my chair and I was looking at Carlos saying have you seen that you know when she presented and then I was talking with Otto actually at another like meeting organized by ICTP and we had a lot of fun thinking about how do these two experiments really all right now I just want to give you a little taste for for these one at a time assembly like this bottom up assembly and so let's try to take the same matrices the same type of parameterization but then at each step we introduce a species at low abundance and we start from blank slate from an empty community we compute the new configuration using this lemke-hosen algorithm and then we introduce another species and then we repeat this process a bunch of times right so so we can see like the assembly of the community right so so this would be called to assemble one step right so we have our metrics well what we assume is that a certain number of speed a certain proportion of like these x would be zeros those are the species that are absent another proportion would be the ones that are present in the local community we pick a random species to be one that is already there or not and then we introduce it at very low abundance get the new composition and then repeat right so we start with 10 species we build a matrix that is stable we build random coefficients exactly as we need for the random zoom and then we start with like x being empty there's no species and then we just do 40 invasions right and then we track these invasions in time and see what happens whenever we add another species right so so just to say for example here you know we start with the we with the invading by species one species one happens to have a negative growth rate it does then you know we have a like species two that can establish and then right and then we introduce now species one when species two is present species one can invade and in fact like it resists like the invasions by some other species if we plot this which I think it's a little simpler you can see here we are at the blank slate a species tries to invade phase then we get species two then we get one and two then at some point species nine comes in kicks out species one and now we have two and nine ten attaches to the community eight attaches to the community six comes in but kicks out species ten etc etc etc and at the end we arrive with these one two four five six seven eight right so so this would be like the assembly if we were to track the number of species you can see that we go up but sometimes we also go down right like here we had eight species at some point the time step say 30 but at time step 40 we have only seven species all right contrary to what you can do in the real world in a computer you can go back in time and do this experiment again with the same exact parameters so we now go to a parallel universe where we can assemble the system again all right let's let's roll back history and assemble again and if we look at this graph now we have a slightly different picture right so we start with pieces four it resists a bunch of invasions and then finally species ten comes in attaches to the community three comes in etc etc etc we get to eight species and then we fall back to seven species one two four five six seven eight which is exactly what we had before turns out that exactly because of the saturated equilibrium we can do this experiment as many kinds as we want and we will always find that at the end you know really wait for long enough right such that all possible invasions have happened we would end up always with the seven species if it's sitting together in fact these are the species that I would expect if I were to throw all the species together right so in this case of these very stable matrices the assembly is not super interesting right saying if I wait for long enough I always go to the same place okay and next lecture we will actually try to have a pictorial representation of this type of phenomenon this is not the case when when the matrix is not symmetric right for example and in fact we I mean there is like an interesting observation that you can find in the supplement of this paper that the probability of finding a system whose final composition cannot be assembled one species at a time so imagine that we could have like some sort of system where we can get one species at a time up to a certain level but then to reach like another composition we have to invade with two species at a time three species at a time four species at a time and again it's something we will discuss in the next lecture but our feeling at least for random coefficients is that even when this matrix is not stable the probability of finding such a system goes down quite quickly with the number of species now another thing that I just want to raise as the last point of today just because it's fun and just because it is like something that leads to a lot of misunderstandings in the field right so so we go in the field and as Martina was saying before right like these communities that we sample are there right persisting happily in the such I don't think they should be random whatever random means right and in fact I can find a some sort of like a yes in the literature saying they must have been some somewhat like selected in a very known adaptive selection type of way to persist right like that if I start with something there's this sweep like that is like this selective process like imagine that the bottom in the top down assembly would be a true troll the species together dynamics removes all the species that cannot fit in and now I end up with a final configuration that is different from my initial configuration and we saw that there's a different number of species but did like these pruning select for certain patterns of interactions or not right that's the question so let's let's do these experiments we start with a much larger sample right say imagine 200 species random coefficients and then we get the final composition and then we contrast our initial matrix a with our prune matrix right like the a of only the coexisting part of the community and similarly we contrast our initial growth rates with the growth rates in the community right so so and now we can ask are these two matrices similar right that is not an easy question like to to to answer because what does it mean for two matrices to be similar one thing that we could think is like let's look at these metrics as a graph right like imagine that we take all the coefficients if they're big we draw an edge between i and j if they're small we leave it alone right so so we we form a graph of strong interactions between these species right so these would be the spaghetti ball that is the first matrix so you can see it's like a very dense random looking graph and then this is like the prune matrix right which will be fewer number of species and fewer connections as such it looks also quite random so this is not the greatest way to visualize whether these two matrices are similar you know being who I am I like a spectrum graph theory a spectrum like a method so so what we could do is to take these two matrices and plot the distribution of their eigenvalues these two matrices are symmetric so they will only have real eigenvalues and we can look at what is the shape of the distribution of the eigenvalues here we'll help by what that we know that if we take a random symmetric matrix with like a coefficients the sample for a certain distribution you can actually determine analytically what the spectrum should look like why what this distribution of eigenvalues should look like and in fact is called beginner semicircle law and so we plot the eigenvalues of the initial matrix and sure enough you see they fall in this kind of shape which is like beginner semicircle law and now we can do the same on the matrix that was pruned right like the selected by the dynamics and what we find is exactly the same right so these are like two histograms that look exactly like you would if you were to just sample pieces at random right so it seems that these selective like a pressure imposed by the environment does not really leave much of a signature in the structure of this matrix let's look at the growth rate now here we expect something to happen why because you have a negative growth rate in the environment you're going to die right if you have a positive growth rate you might survive but you only might right because you're interacting with all the other species and in fact like the strongest the positive growth rate the more likely to some extent you will be to survive and in fact you can see that these is like the red is the initial distribution of growth rates and in blue we have the final distribution of growth rates and you can see that this has been shipped by the dynamics I have no way right now to calculate the distribution the expected distribution again so if somebody is really good with these type of problems go ahead this is a problem to work with because no one has the answer all right and so this is like well we took random species did like our dynamics we end up with basically no like a random growth rate but you know from a different distribution and the interactions look exactly as they did before now let's try to do a different exploration that maybe it's closer to what happens like in nature right like instead of say I just randomly pick species from the whole world and throw them all together I could say there is one species and then the new species that comes in is actually a modification of the previous species right there's some sort of mutation and re-invasion like I don't know two populations like split geographically they evolve independently and now they try to re-invade right so this would be a different type of assembly and this would boil down to what like the parameters of like the k species and the k plus one species should be somewhat similar right so so what I'm doing here is like I take some matrix I start with just one species that interacts with itself with the coefficient of minus one that's the only coefficient in the matrix and now I start building these metrics by saying species two will have very strong interactions with species one and with itself maybe a little weaker with species one than they do with itself and then I build these metrics and then I make it stable exactly as I did before right so these metrics now you can see that this is not a random random matrix in the sense that there is this very strong you know diagonal and then you're very interacting very strongly with the species around you right in fact this you can actually see if you do again as we did before plot this graph right like you plot a graph of interactions and you can see that there's some sort of like gradient right like the species that were maybe early on in history species that were very late in history and then in between so this kind of axis maps it to basically like the time right and then when I take these and I prune it of course now I get many fewer species right why is that because they interact very very strongly with each other and they're very similar with each other but you can see that the structure is basically the same right and so again like we look at the eigenvalues these should follow some other distribution which I'm not going to talk about but then you plot the before pruning and after pruning and again you find basically the same distribution so so what this means is if we throw in like all the species and these species have a matrix of interaction that is structured what we get out is a structured matrix of interaction if what we put in is a random matrix of interactions what we get out is a random matrix of interactions which then tells us what that when we go in a system and we're saying oh look these interactions have a structure they this structure must be there for some reason the reason might be that this structure was already present in the pool right like this structure is due to this way with which like we explored the space of possible species rather than having some sort of adaptive right and this actually is in connection with like the work of gold and the one thing in 1979 famous paper on the spandrels of san marco and in fact even more and that's what inspired me to to write this paper the work by ricard soler and sergi valverde in 2006 where they were making this argument for network spandrels right so this is like a paper that we published with dan maynard and carlo serban in 2018 if you want to read it i think it's a lot of fun so that's what i wanted to say for for for today so right now i'm just going to go in the chat and and see whether i can find some interesting questions and and tony if you have other questions please go ahead thank you thank you stephan let's open the floor to the audience if they want to raise their hands or five further questions yeah there is a technical question by kiseok Lee is saying what happens when we subtract these diagonal from like the matrix to make it stable to the value is not very much right all the value which we just shifted of a certain quantity these will however influence like the the probability of finding like a feasible random growth rates sure and it's in a non-trivial way so so that again we don't really have full control over this problem but certainly we'll do something and the saturated equilibrium as i said is like this solution where we go to where a certain number of species are present the other ones are absent cannot invade and we have a question from the youtube chat i can read it out for you is there a way to distinguish if two different collections of experimental observations only problem this is we're assembled using different metrics and samples the answer unfortunately not that i know of right maybe you you can look into that what i'm just saying is like let's not take for granted that if i sample the community i measure all the interaction strengths with which there are metals to do like don't it is actually perfect and i find that this is a highly structured matrix then i'm saying this must be the by it must be the product of some sort of pressure that led to a very interesting configuration right like what i'm saying is that we should always allow for for the hypothesis that these structures just a byproduct of the way in which we chose the species to put in the first place right because the ecosystems are not assembled by taking all the you know 8.9 million species on earth and just throwing them all in the same environment to see what happens right like we can only invade with species that are geographically close evolutionarily like not to be distant etc etc yes so is the symmetry in a assuming that competition mutualism are the only relevant interactions to some extent right like you could do the same like with only negative numbers say only competition then you add the constant it complicates the calculation slightly it's not hugely important if it's not symmetric the problem there is not really the known symmetry of this thing right you could do this for a non-symmetric matrix that abides by the same rule that only gives me equilibrium point dynamics so it would that like to be a very special non-symmetric matrix the problem is that when a is not symmetric i cannot even speak of this problem to some extent because i could have like chaotic attractors cycles and whatnot which makes it like very difficult that was problem number three that i tried to remove right so that's that is why i'm not talking a lot about symmetry in the paper with Carlos we actually look at non-symmetric matrices that again abide by these three rules that it set out it's a slightly more involved formulation so i did that use it for the lectures yes so i did ask what happens in bottom up assembly when the new species are mutants of existing species right with very correlated interactions what will happen is that they're very similar i would expect that it's very unlikely that they both will eventually be in the mix at the end of the interaction at the end of the dynamics and so what what happens in fact like if you look in the paper on the natural spenders is that it takes a lot longer to you know build a system of 100 species maybe you know in the random case you need i don't know 300 invasions there you would need maybe 15,000 invasions to get to 100 species coexistence and there are some raised ends thank you Flavia but i don't know if i can see that i'm Keith as a resident but i think yeah hi like can i ask my question yes please okay yeah so like you have this result about getting a binomial distribution for the number of surviving species where you had a fixed mean of zero and some random interactions for the off diagonal so like is the variance of the interaction matrix fixed in that case or like how does it work yeah like you can show that this is going to be the same if you choose any distribution for the growth rates and the interactions that is centered at zero it doesn't matter the values okay because i've seen in some cases where like the number of surviving species like it changes with the variance and oh yeah like you also lose stability after a certain point or is it because of the symmetric nature of the matrix i i actually i agree 100 percent i wrote several papers on the effect of the variance and the correlation in these in these matrices and how does it affect stability in this case it's like a little special because these things are centered at zero okay and i think that that is a and also like yes these are symmetric stable matrices so we don't have to worry about stability to some extent because they the problem of the variance is really like that it makes the distribution of eigenvalues they like more spread right uh Geordi also has raised that yeah um can you hear me yes yeah first thanks for this awesome set of lectures uh Stefano and my question is maybe a little bit more speculative and it's regarding the spandrels idea do you think that there would be instances in maybe some other ecological um ensembles for example in micro in microecology where um the precursor set of structures are not so stable so to speak because maybe evolution is happening in situ or at the same time as you could see the assembly going on and therefore we will not be able to see these stable structures or maybe that's not a very good assessment thank you yes like yeah like of course these are like cartoons right of what's going on of course species are adapting to each other co-evolving or like trying to find like solutions to this problem of surviving you know in this in this environment so that becomes a very complicated i would say process but but it's interesting here it's really that this is going to be true for species right invading each other it's also going to be true for like say gene regulation network and in fact like if you look at this paper from from solem but they they are thinking more about gene regulation network and say do we observe like gene regulation networks they have a certain pattern of motifs because these motifs are useful to some extent right they provide some sort of structure like the design of stable or like robust or something like that or is it just because like the process of gene duplication is very much like this second example that we were saying right that you duplicate a gene now you have two copies of the gene you can change one basically freely because you still have the other copy but their function would be somewhat correlated right like so that would give you only a certain way of exploring this space which would be reflected then in the structure of this network thank you yeah so if there are at the race end otherwise we have some questions in the chat oh but it's also 945 so i don't know if i should stop but i don't know what do you say if you wish to ask a few more questions i think that's not a problem sure so somebody asked what would be the biological meaning of having identical similar eigenvalue distribution does this provide for similar stability yes like you can imagine that like any class of say random matrices will produce if it's large enough like about the same like a distribution of eigenvalues and they would have basically the same properties yes and then could we see the top-down assembly dynamics as an approximation for any strong disturbance in which new niches are created that is an interesting question like i'm thinking more of like you know some sort of like event in which you know we're thinking of invasion always says like one little thing coming in but many cases you know maybe you get like a whole bunch of species because it's brought out like by these i don't know animal brings all these other parasites and other things with them yeah but when new niches are created i'm not sure what exactly it means but then i would have to think about sorry i cannot respond on what is the definition of a structure matrix that's an excellent question no one really has the answer right all the matrices have some structure to some extent what i'm saying is like if i draw for example this graph right this graph does not look like a spaghetti bowl it looks like some class of graph these i would say it has very structured matrix while if i look at something it looks really like a random erdo charny say random graph then these i would argue it has basically no structure of course it has some structure but it's not like prominent all right anyway if you have other questions we have another session so so think of your questions before if you want to email me like with some questions i'm happy to answer and i that said i would just say have a good weekend and i see you on monday thank you thank you stephan oh nice to see you as well thank you bye everybody bye bye thank you very much thank you sir thank you so have a nice weekend thank you bye