 Thank you, thank you, Mercedes, and thank you, all of you, all the organizers, Jacopo, Mateo, Silvia, for this nice and interesting workshop, because all the, all the talks are going, are being excellent. And we are learning a lot of limits of diversity in ecological communities. It's a pleasure for me being invited for you. Today I'm going to talk about how can theory explain- Jose, sorry to interrupt you. We are seeing the presenter view. Can you put it into the- Jose, how do you call it? Yeah, I know, I know. So it's kind of, sorry. Maybe if you just share your screen and then you just put it in the mode, the presentation mode. That's better. Yeah, that's better, sorry. Okay, fantastic. Sorry. But I was saying, yeah, the purpose of my talk is going to be kind of investigating a little bit of how can theory tell something about the limits of diversity in ecological communities. We have some theoretical predictions and a simplified assumption. This is kind of related to the ideas that guy put forward in his talk before. Can we make some simplifications in our models to get some hint of how diversity can be predicted under some certain settings? Well, this is kind of the first slide I used to introduce the topic as species diversity is huge. There are a lot of species reported around the world and it is assumed that 80% of the species are still undiscovered. So this huge amount of diversity needs a well-grounded theoretical explanation. This is puzzling because what theory tells us about diversity is just the opposite. All of you know this complexity stability trade off which was put forward by Robert Mayne in 1972 when he challenged the view that ecological communities are diverse, which is puzzling because we observe that ecological communities are really diverse. We have two Jacobian metrics of a locable terradynamics distributed with random interactions with a given variance and some sort of connections which tells you the linkage density and showed that using elementary random matrix theory that there is an upper bound on the diversity that the ecological community can accommodate. Our intention in doing the work that I'm going to present is try to find an explanation of why do we see this overwhelming diversity in ecological communities and what are the limits that these theoretical models can set to the number of species that we observe. Until like 10 years or 10 years before, the idea of studying stability in ecological communities was just considering the whole set of species as coexisting together. And there were a lot of studies studying the stability and the feasibility of the equilibrium points of a given dynamics, but considering the whole set of species as the assembled community. As you can imagine, the probability that this whole set of species, the pool for example, is feasible, is really negligible. So in this talk, I'm trying to change the perspective which is kind of something that has been commented in the talks before. Here I'm going to change the perspective in the sense that the species that survive that we observe in local communities are basically a selected portion of the species pool, which actually has been proved by a certain population dynamics. So the question that I'm going to try to answer in my talk is whether we can predict the distribution of the number of species that survive when we assemble the community from a pool of species that in principle interact randomly. I'm not going to make any assumption of how the interactions are, whether they are safe, whether they show patterns, we take this simplified and new model perspective that guy was commented in his talk. And yeah, this is, I'm interested in predicting the distribution of the number of species in the in the minimal setting that one can imagine. So we are going to talk about top down species assembly here, I'm going to put those pieces together and let the dynamics unfold. And I will analyze, I will discuss here to two examples. One, when the interactions are basically random, and then when you consider phylogenies to generate the interactions. So if you have questions up to this point, I cannot see the chat at this moment. Sorry. No questions for now I will let you know Jose. Fantastic method. Thank you. The first part of the of the talk is what we call the random soup problem, which is work done in collaboration with Carlos, Sir van. And Stefano Alessina from the University of Chicago. Also, Jacopo really is was a key part of this of this collaboration in when I when I was visiting the University of Chicago in 2017. Also Ken Morrison collaborated. I give you here the reference if you are interested. So the way we do we call this the random shoe problem. Well, consider the following thought experiment. Consider that you can you take a large shoe. And at some point you open all the cages. If we were to come back after like 10 years, 20 years, can we predict how many species will still coexist in the soup. In other way, this will be kind of the outcome of when at some point Noah opens his arc after after what you know the you know the story right so this is the idea. And we are considering we put this metaphor in this time because we are considered and basically random interactions. So the practical setting of the problem is that we take a basically lockable terraria dynamics with variability in principle in growth rates. Also by variability in random interest and I don't see, I don't know whether you can see my pointer. Yeah. Fantastic. And this will be the general setting. So how can we compute the average number of species that one expect to observe after the community has reached to an equilibrium. Well, we for for doing that. We have to make some assumptions. Our strong assumption is this Lyapunov diagonal stability. We assume a piece it has some of this kind of related to to the talk of the dog by guy because well we are here we're considering that the dynamics has only just one single attractor. This can be achieved by assuming that our interaction matrix. The definition of interaction of the lockable terraria dynamics is such that there exists a diagonal matrix D does such that this combination is definite negative definite right in this case. And this is not so strong because you can always satisfy this condition by letting this self regulation terms being large enough, right. Okay, if this condition is satisfied is a theorem by our own segment that it guarantees you that for every vector of rates there is a single globally stable stable equilibrium point of the dynamics. So we are in a setting that that we can characterize if we study the properties of this single equilibrium point we can characterize the the ecological community. That is totally right. So, this, this is the characteristics of the of this equilibrium point that this is, it is not invisible by the remaining in minus kind of species, right. I mean if you if you if only K species survive in the community the other species in the pool cannot invade the the equilibrium this these two conditions the feasibility of the equilibrium and then no non impossibility of the equilibrium to understand the, the size of the attractor and the distribution of the pieces that survive and which is basically what I'm interested in. So, here I mentioned this this paper by by Carlos and Stefano, where I, although I'm interested in top down assembly process by just putting all the pieces together. In this case, it is easy to see that the sequential assemble community will be would be the same. But there are many conditions in this paper I recommended you to look at to look at it that give you necessary and sufficient conditions in which top down and bottom up assembly are the same. So, our setting is interaction matrix are basically a diagonal metric matrix mean term and random part right which has kind of mean zero and vector rates are have this mean and kind of a variability. So, to characterize this saturated fixes fixed point, we just need to impose the two conditions that define the, the, the saturated point which is basically the feasibility and the non impossibility feasibility is basically that the, if we order the the the species. The one that survived here in the, in the, in the first place, the feasibility condition is basically that this sub matrix times this vector plus the vectors of rates is equal to zero so they satisfy the equilibrium condition for the non impossibility the species rates has shouldn't grow when rare when rare. So this translates in this sub matrix times this green vector plus the remaining part of the vector of rates is equal to a vector that is negative so species. The remaining species in the pool cannot invade. I'm going to speak about two cases. The first case with this within this random two problem is the case of mean zero. In this case we can calculate with very simple symmetry arguments, the distribution of the pieces that that survive. We can consider arbitrary run interactions with mean zero and arbitrary growth rates also with with means you in this case, the distribution of the species that survive is basically a binomial with probability one half. What is this? Well, first you can analyze feasibility, the feasibility properties kind of intuitive right because which is the probability that out of then you can get in a species surviving right. This is basically one over two to the end. This is kind of intuitive which is the probability that all species relay in this in this in this quadrant if I if I take this one randomly well you have basically to solve a system of two equations a linear system of two equations with two unknowns and the probability of laying in these in this quadrant is kind of intuitive to be one half or one over four right. So you can you can prove that the probability of feasibility is one over two to the end. In this case of mean zero. Similarly, by a symmetry argument you can also calculate that the probability of this equilibrium point to be is not is feasible. What is feasible is basically this, this, this probability, the probability that n minus case species cannot invade condition that case survive times the probability that these cave species are feasible. This conditional probability is basically equal to one over two to the n minus k because any sign pattern for this vector is equally likely. It is simple to see that. So multiplying this this result to the with the times the the the other the other result you get that the probability is one over two to the n times the combinatorial factor because species can can be arranged can be sorted. In this number of combinations, right. So basically we get the binomial the binomial distribution in the case of mean zero, which is kind of unrealistic from the ecological point of view. Can we say something about the case of non zero mean. Well, in this case we we just choose. We just chose the, the simplest scenario where we were able to do the calculation up to the end, which was basically considering that species rates are normally distributed. We include a randomness in growth rates, right, but we switch off the random part in the interactions, right, so we cannot consider random interactions and allow variability for growth rates. I'm not going to give to give all the all the details but we can see as in the means in our case we can calculate here the distribution of species that that survive. We have to give a hint of how the calculation proceeds basically you just take the probability distribution of the of the rates which is basically a product of product of gaze and gaussians. See, you just only have to perform this change of variables, right, change the airs into the X and into the C into the vector C, right. If you do this change of variable which is very simple then you only have to compute the probability that that this vector X has all is components positive, and this vector C has all is terms. Which is basically this, this, this thing, right. Yeah. I think we have some raised hand. Can you take the question. I was fixated on a little detail might have missed the part of what you were saying but are you a priori assuming global stability and just imposing some kind of sign structure. No, no, no, no, no, no, no, I am assuming a global stability, right. This is this is something I'm assuming in the in the dynamics, I'm assuming that matrix a satisfied this condition the X is this diagonal matrix that this combination is definitely negative, but negative definite but what I was saying about this design combination is that all the sign combinations for this vector for the C vector and the X vector are equally likely. Right, you can if you think a little bit about it, about that you do realize that all the sign combinations are equally likely. Right. So this gives you a one half times a probability of one half raised to the, to the number of elements that you have or them to the number of entries that your vector has. Okay. So then we don't necessarily know what the interaction structure looks like we're just keeping it arbitrary or just do you mean any distribution only with the with the restriction that the mini zero, basically, right. And that, and that, I mean, but how do we know that implies the global stability, other than I mean of course that condition the diagonality. You, you can, you can do some to some extent in incorporate diagonal stability by just setting the self regulation large enough in order to get this, this, this, the stability, right. Okay, so you can you can send some of the plug in into the into the dynamics. Okay. Thank you. So, yeah. So after doing this calculation which is one kind of conversion calculation, you can get an analytical prediction of the mode of the distribution. Okay. So the, which is basically related this by this by this equation to model parameters this Q star would be the, the, the fraction of species that survive. I mean the, the, the number of species that survive divided by the size of the pool. And then this is an analytical prediction that can serve as a new model to test something about coexistence in problems that can be related to to do this to the system. I show some results about the fully connected network, because we can, we can, we can kind of, once we have the prediction of the expected diversity, we can kind of complicate the model or add some ingredients to the to the new approach by, for example, considering a network structure. In the case of a window not consider network network structure, we have a very nice prediction of the number of species that survive in this case the distribution is no longer binomial, and then you can have regions where the expected number of species is very large and regions where the expected species is really, really low. Okay, but the key point here, or the idea I'm trying to transmit here is that we can find an analytical predictions that an analytical prediction that can serve for further model testing or to compare with the system. In the case that you impose some network pattern, I mean this is basically introduced introducing a connected structure and a pattern of zero entries of the of the matrix in the in the model right and we can we can do it by introducing another range of distribution, power load distribution or some bipartite or modular structure. And if we do this and compare with our analytical prediction, we have that the network has really a moderate effect in the prediction for the diversity, right. Because the full line is basically the the analytical prediction for the for the fully connected network. Okay, so maybe I can stop here because sorry. Okay, because I'm going to talk about the effect of phylogeny when I don't know if some questions on someone has something to ask. I don't see questions in the chat. If someone has a question you can raise hand or unmute yourself before we go into this next section. Jose, I think you can continue for now. Thank you. Thank you. In this in the second part I'm going to do the same stuff, something, something similar, but I'm not going to consider that interactions are fixed as in the as in the previous in the previous random soup problem. So I'm going to consider the effect of phylogeny, which will be a way to introduce variability in interactions in the model, right. So I work down in collaboration also with with Stefano and Carlos also with with Zach from all of them from from the university of Chicago and I refer you to do this, to this to this reference where you can, we can see the details of the things I'm going to summarize here. Well, the idea here is just can we say something about the limits that are imposed to diversity when you impose a phylogenetic structure in in the community or when you when interactions are inherited by kind of phylogenetic structure. So for that purpose we start with phylogenetic tree, assuming that you have a pool of five species here and, for example, you have this perfectly hierarchical tree. This gives you a correlation between the, between the species, even by the basically by the covariance matrix of the of the tree, and we formulate the interactions in terms of trades. Okay, we can see that at the, at the, at this point of the tree, each trade, and in order to calculate the tree, the trade of each species is basically a random walk on the, on the, on the leaves. This basically gives you that the vector of trades is a multivariate normal distribution whose correlation matrix of his or whose covariance matrix is basically the covariance matrix of of the tree. Okay, so if we consider L trades. We have this this G matrix, which was also mentioned by by guy in the in his talk. And we just take as a measure of the interaction between species they share overlap between the between the vector of trades. And these, these sample covariance metrics a equal to one over L G G just post as the way we are going to introduce the, the, the interactions in in our model. If the vector of trace is basically multivariate normal then the sample covariance method follows the wizard distribution, right, depending on the parameters and which is the size of the pool, L is the number of trades and Sigma is the correlation of the tree. Now, we basically do the same thing as before we do top down assembly. In this case, things are kind of simple because a is symmetric and positive definite. So this means that all the again values are positive, right. This immediately gives you a diagonal stability, we have not. We don't have to impose diagonal stability on in this case. So you can see that local terraria dynamics in this setting we are assuming that fitness difference are absent. We assume that all all species grow at the same rate in terms of this Chessonian paradigm. And we are interested in considering like competitive ability differences, right in the industries, we are a kind of switching off fitness difference in order to get the picture more tractable. So in principle, you can consider a swimming and then you have this local terraria dynamics as in the as in the case before before notice the difference in design here because this is not important. But so, since we have a global stability, we can characterize the number of existing species by feasibility and noninvasibility. We can see simply that these these mean interaction does not affect species identity identities and also basically the vector of rates rescales biomass is so we can remove them from the dynamics and then we have to down assembly and we are reaching a unique endpoint. Okay. What are my biodiversity my diversity prediction with this model. Well, first, I'm going to consider the deterministic limits. I also want to mention that this model can be also recast in terms of MacArthur consumer resource model so I'll be con other things I'm going to say here about the, the number of trades else. It can be translated in immediately immediately on on a number of resources, which could be L as well in MacArthur model. Okay. So in the deterministic in deterministic limit, we consider that the number of trades is very large compared to the size of the pool. In this case, the sample covariance tends to the covariance of the tree. It is simple to show in this case that every covariance made in this induced by a tree yields full feasibility we cannot worry about in this limit about feasibility because it is guaranteed. The, the idea is simple as I'm trying to sketch by this. Basically, if you have a tree with this with this shape, you can show that the covariance matrix of the full tree is basically this block matrix, which this Sigma one is basically the covariance matrix of the subtree and so on, plus this constant 10, which is related to the time to the split of the tree. So if this is blocks, each block is feasible then by induction you get simply that the full team would be with the feasibility. This is a very strong result. Right. So we don't, you don't have to worry about feasibility. There are also some predictions about a total biomass and the abundance distribution in order to get to get things tractable we considered the limit of a perfectly hierarchical tree and perfectly balanced tree, right. So we have predictions of the total biomass and the, and the number of the abundance distribution. For example, in the case of the total biomass, you can see that the biomass, the biomass scales basically with the square root of the number of the number of species that you can have in the tree. In order to compare with a with an empirical phylogeny, we consider here the Senna phylogenetic tree, and we see that the we basically took some samples of the tree in order to get the this increasing number and we basically observed that the, the, the empirical trees basically kind of in the middle of this perfectly hierarchical and the, and the perfectly balanced, right. But in the deterministic limit, most of the things are easily predictable. We also can ask ourselves, what is the prediction for a general result. I know, of course, going to it in to enter into into a full detail bit, but using the properties of the wizard distribution you can calculate basically expressions for the probability of feasibility on the probability of noninvasibility. And these, these are the the expression which are basically the an integral over the, the key square distribution times the probability of a multivariate distribution over a certain condition. And the probability of noninvasibility is the same and this is the probability that the given multivariate normal distribution is containing a negative orphaned. So in principle, this, this, this result give you the exact probability of feasibility and noninvasibility and in principle this can be computed numerically, right, to a certain accuracy. Also, you can show that these two probabilities are independent of each other. So the probability of being an attractor factorizes and you have the probability that the attack those has size M. Okay, so again, we can have this theoretical prediction if we consider a simple case, which is the case of a star phylogeny, the case of a star phylogeny is basically that you have a point that every species is split and no more. In this case, we can carry out all these integrations and all these stuff, and we can have a prediction for the number of species that that survive in average. This is, this is the, the relation this reminds the experimental result that Martina showed in the in the first start on Tuesday, because these gamma is basically the number of traits, or the number of resources if you think of this in in a MacArthur risk setting. And we observe that the number of resource as the number of resources increase the the average fraction of species that you observe is is an increasing function, right. Those are simulations and lines are our analytical analytical prediction. This is interesting interesting because the prediction of the competitive exclusion would be here in gamma at gamma equal equal to one when the number of resources is equal to the number of species. And our theory predicts that the expected number of species is well beyond to what classical theoretical competitive exclusion predicts. Okay. So we have fully characterized the diversity as in the other case we have regions where the diversity increases to a high level. Basically, the idea is if you increase the number of traits in the limit of infinite number of trades we see basically the deterministic limit you get full coexistence. So, so if you increase correlation between between trades, then this is opposite to to coexistence. Also, we have predictions. Maybe I don't know because we started a little bit late but if I'm correct, you don't have a lot of time left are you do you have a lot to show. I have two slides just two slides. Fantastic. Yes, I just wanted to show how the total biomass depends on the on the on the number of resources on the number of of today's we find a good agreement with simulations. And it's a decreasing function. It is because kind of the biomass splits into into it kind of divides into the set of traits that you have into the number of traits. So this is it. I hope I convinced you that under some simplifying assumptions, you can do some interesting predictions on the diversity, even in the in related to the distribution of the number of coexistence pieces, assuming top down assembly in these two meaningful situations. So these, I have studied these, these two models where I will have analyzed the variability in rates and also the variability in interaction so in this case these two approaches are kind of complimentary. We don't have a new expectation of how many species should coexist and we can use it to probe the effect of other mechanisms in coexistence so this is a basically the idea home on carrying out these all these ugly calculations. We can we can introduce, for example, in networks, patterns, for example, modularity. Not yet in the position of the non zero elements of the networks, but also introducing patterns, patterns in the in the strengths, which have in the past have been shown to strongly stabilizers or destabilize the system. And to the to the talk by by guy. It would be interesting to relax all the assumption that we impose on diagonal stability and consider multiple multiple attractors, attractors. And basically this is this is this is all I wanted to say, I want to thank all my collaborators, most of them are in this in this picture also is done here. And Jacob is not in this picture because he was precisely taking taking the photo so thank you very much for your attention I'm happy to answer any question you may have.