 n, and I hit the subscript runs from one to s, the number of species in the pool. And okay, these are just the usual lookable terodynamics. I'm sure you've seen them before with migration from a regional species pool. And of course, even in the simplest of models, still the matrix alpha here, the network of interactions alpha has tons of numbers in it. You can imagine if there's 50 species, then there's more than 200, sorry, 2,000 numbers parameterizing this matrix. And so there's a lot going on, even in this simple framework. And okay, this is it. Later, I'll touch briefly on a few other things on meta-communities, on resource competition, on functional groups, on demographic noise. Of course, very briefly, you're welcome to be happy to discuss more. I won't have any evolution, I mean, there will be an evolution in this talk. And I'll only focus on theory today. Now, I should say at this point, I welcome, please interrupt, please ask questions at any time. I can't see the chat, so Yacopo, if there's anything, feel free to stop me at any time. Okay, any questions so far, anything? Okay. Right. So the basic idea we're working with is that, perhaps, when there are many combined things happening in my system, any interaction mechanisms, many things happening, you know, a cow goes to eat one kind of grass and it steps on another kind of grass and so on, then maybe the interaction matrix of, you know, all species in the pool would look, they won't see any conspicuous order and maybe we can take it as random. This is by now a rather standard step, but it is by no means a trivial one. I do not want to say that this is something that you should believe in just like that. On the other hand, if there's, you know, one dominant interaction mechanism, then you can see this matrix here on the right, it is very order. Okay, so at least there's this very strong contrast between these two cases. I want to emphasize that what I mean when I say this very, this math only random matrix, and this means that all the numbers are independent random numbers, this is for the pool. So once we assemble the community, structure is formed in the assembled, the network, the assembled network, and we will discuss this in some detail. And so one way to think of what we're doing here is that this maximum random interactions in the pool is a null model to understand the filtering that the assembly process applies to your network. Is this a good starting point? At the end of the day, you just, you know, generate predictions and you check and the proof is in the pudding. And I'll say a few words about how to add additional orders such as functional groups, you know, trophic levels and so on to this simplest of models. All right, Yapoko, anything so far? No questions. Okay, go ahead. Great. Or actually, who am I talking to? Just so I... I'm Matel Marsili, I'm chatting this session. Yeah, okay. Sorry, Matel, yeah. Because I was seeing Yapoko in the video, that's why I was, but I heard the voice, it wasn't his. Excellent. Okay, right. So if you run this one simple model and it looks, you know, again, you put very little into it, you already see quite a rich phenomenology of things that can happen. So the first thing is you just run the model and what you see here on the top left is you see an assembly process. So you start with all the species in the system. You put them each line here and with the color is one of the species. The x-axis is time and you can see them going up and down and reach a fixed point. Some of the species, some fraction of the species survive and others go locally extinct. So that's assembly. Now, if you start the same, you run the same equations with different initial conditions, you might reach the same fixed point again and again for the same interaction matrix or you might reach different fixed points and that represents multiple alternative stable states and historical contingency that could come with something like that. For example, dependence on the order in which you inject the species if you do it sequentially. There are other cases, other parameters in the model where you don't see the system reaching any fixed point. Instead, there's dynamical resistance. In fact, it's a very high-dimensional chaos and there's even another behavior which I won't talk about much where you see actually succession patterns, true directionality. But again, I won't say much about that. The question is how is it that you see all these very different things in one very simple model and how do you order and organize things? If you're a physicist, this reminds you immediately of this idea that ice and water, if you've never seen ice turning melt into water, you swear that these are completely different materials, but really they're just the same thing once you change some parameter by a bit parameter called temperature. What you're starting to think is that maybe this is what is happening in these models. The phase transition between a phase of ice and a phase of water would have similar things in community ecology. The phases would be, for example, the single equilibrium behavior to multiple alternative stable states or from a single equilibrium to chaos to this successional behavior. These are just examples of what might happen. All right, so over the years we've done quite a bit of work on this very basic model. We know quite a bit, I should say analytically, properties of the equilibrium, total diversity, biomass, fluctuations. We see that these things are said deterministically when the system size, but sorry when the diversity is large, all kinds of things about the dynamics that I will be touching up a bit. But what is the first important thing that I want to say is that what we find in all these calculations is that the distribution itself of these numbers alpha i j doesn't enter. So it can be a Gaussian, it can be uniform, it can be anything as long as it has a mean and variance. And so what doesn't enter in all the calculations again and again are three parameters, rescaled mean of the interaction. See here's the number of non-zero links. If you like, it's all the species that you interact with. It can be any number as long as it's much larger than one. And then there's the variability in the interactions. And then there's the symmetry of the interactions. How correlated is the interaction of i on j with the interaction of j on i? And so these are the parameters that drive everything. And system-wide properties again, like diversity, fluctuations are fixed deterministically. They're what are called self-averaging quantities by these three parameters. So we have the variables. We have what are the axes that we should start thinking about. And now... So if I may, why don't you consider lambda as a relevant parameter? So the immigration. Right. Sorry. That's an excellent question. So we're looking at very small numbers. So the strength of migration is much smaller than all the other things. It's true that in the chaotic regime, it can enter in some very subtle way. But for fixed points, there's a well-defined, at least for fixed points, there's a well-defined limit of lambda going to zero, but larger than zero, which we can take. But it's true that if it was order one quantity, then it would affect everything. Any other questions in the chat or something? Yes. Mercedes asks whether you can clarify what C is. Yeah. So if we're looking at matrices or interaction networks where some of the interactions are zero, then this counts the number of interactions. We're assuming anyway here that the number of interactions is much larger than one. And so we're also going to assume that the distribution, because it's a large number and we're going to assume the distribution of this number is not very broad, so you can just take one number for that. If you like for simplicity for everything that follows, you can take C to be equal to S to all the, everyone interacting with everyone else. And then it's just a fixed number. It won't change anything in what follows. Okay. Sorry. So in that sense, it's like the degree, right, in the network? Absolutely. Absolutely. Because otherwise I confuse it with the density or the connectance, but it is the degree. It's, yeah. Sorry. That was my questions. Thank you. Absolutely. Okay. It's the number that when you're interacting with everyone, you get S. That's, by the way, it's not the assembled degree. It's the degree in the network before assembly. Yeah. Okay. Anything else? Yeah. And also as a question. Yes. Okay. So I can see it now. Yes. So as I said, you need the mean and the variance to be finite, right? So it's not heavy-tailed in that sense. Yes. Okay. So, right. So you could ask, okay, this is an extremely non-biological model. So my colleagues in Michelot's group, Mathieu Balbié and Jeff Almondie have looked at many other properties of networks that people are, have been using and looking at how much these three numbers, if you just use them blindly, how much you get system-wise properties correctly. And it turns out that many structures that people do use in the literature do not affect the system-wide properties very much. However, there are things that matter. So if you have, you know, plants and pollinators, then this bipartite structure is very important. And then what you need to do is to extend this very simple model to the next simplest model with a few more parameters. So that's basically the idea. I'm not going to go any more into that, but yeah. So that's it. Okay. A question on Monday asked whether the distribution is in the mean or the interaction strength? The distribution is in the interaction strength. So you have a matrix, you have, you sample numbers in it, and they're all sampled from the same distribution, and you fix that matrix and then you run the dynamics. That's how it works. Okay. Right. So the puzzle, so we have the axis for the puzzle. And now here comes the pieces of the puzzle with these axes. So again, so let's put on the x-axis here, the mean of the interaction strength. So more competitive is more to the right. And on the y-axis, this variability that I mentioned before that I call sigma. And we see these different, what we cannot call phases. And I want to start focusing on this unique equilibrium phase where whatever the initial conditions you start with, as long as all the species are there initially, you end up with the same fixed point. Okay. And yeah, so that's a very simple dynamics. But now I want to talk a bit about the assembly process in this case. All right. So let's see what happens to the network during the assembly in this case. And so right. So you have, sorry, the notation here is slightly different. I'm sorry for this. So you have, before the assembly, you have what I call here S pool, S sub pool species. Okay. And here for it, I slides for something from somewhere else. Here it's S survivors. Okay. And the S species in the assembled coexisting community. So maybe you could call it S star or something. And so for all these species, the important thing is that for all the species that have managed to coexist, for them to coexist in the fixed point, you need, first of all, the final abundances to be positive. That's, that's, otherwise, they're not really coexisting. And then the fixed point has to be dynamically stable. If you push away a bit, like some press perturbation, small press perturbation, for example, then you need to be able to return from that press perturbation to your fixed point. And then there's also the issue that you have to be uninvatable from the outside, but I don't want to talk about that much now because let's, let's just focus on the coexisting species and not on the properties of those that cannot. Okay. And now the question is this, if we look at the matrix or the network of the assembled network, how is it that all these species coexist? So one thing you can do is you can keep reducing the diversity of the assembled community. I mean, if you get to one species, certainly they all coexist, it's fine. But another thing you can do is you can add structure. The assembly itself can generate a matrix which is no longer completely independent random numbers. Okay. And the question is, first of all, does this happen? Does this structure exist? And if it exists, which of these requirements that we've discussed before, these two things having positive abundances and stability, which of these things is affected by the structure and how? Okay. And so it turns out that the assembled matrix is in fact not maximally random. There is structure. For example, you can show that there are correlations along rows and columns. And we can calculate them analytically from some pattern I'll show you next. And so these correlations did not exist in the pool. So assembly does generate structure in the interactions. So that's one thing. The second thing is that these correlations are there, but so... Okay. Let me finish the sentence. The second point is that these correlations are not sufficient to explain coexistence. So if I just have these correlations, or in fact, any enrichment of some local interactions, I do not generate the full coexistence. So I need to... It's just not a full explanation of what's going on. Okay. Let me look at if I can see the chat. Do you also have a stable limit cycle? So we do not see stable limit cycles and the limit of large S, of larger, of highly diverse systems and this asset product limit that doesn't happen. In intermediate values, there is in the transition from a single fixed point to chaos, you do see a limit cycle. Are we taking... Yeah. We're only talking right now about... I'm still in the single equilibrium single fixed point phase and I'm looking at assembly, the effective network structure on assembly on network structure in that case. I'm not even touching dynamics just... Okay. Okay. So those are the... So this is what happens. Now... Okay. So I said that there are these correlations, but they don't explain the coexistence. But okay. First of all, there is structure in the assembly level. Okay. That's one. What is this structure? And importantly, how can we cease to understand what is the structure that is in fact relevant to the coexistence? So for me, the functionality in this case, what this system has to do is to coexist. Okay. So I need to understand what are the... What is the structure that is sufficient for this coexistence? And it turns out that the one way to describe it is the following, that if you look at the assembled matrix or network, it looks very much like a random network as before, but there are subtle biases in the strength of the underlying interactions. In particular... Sorry. In particular, for some reason I can't see. Sorry. Can you tell me what the time is and when did we start? Just so I know how I'm doing? We started five minutes late. So you have 20 minutes more. 20 minutes to the 35... Okay. So it turns out that the structure is... The following looks very random. It's hard to see by eye what's going on, but there's an underlying, as I was saying, bias in the same interactions. So it's my mistake. So it's 12 minutes. 12 minutes. That's also fine. Yeah. I can do that. So this underlying pattern, this bias of the interaction strength can be understood as follows. You organize your species in the assembled network from the least successful one to the most successful one, where by success we mean the ratio of its abundance in the community to its carrying capacity, to its monoculture abundance. And it turns out that the abundant species do not... are sort of specialized. They do not interact... Do not compete as strongly with other abundant species or with other successful species. So it's sort of this mafia or this click type of thing, whereas the unsuccessful species just compete with everyone at the same strength. So that's what you see. It's some form of specialization of the successful species. That's how I think about it. Okay. I don't want to go very much into the details, but I will say that this structure can be proven to guarantee the positive abundances that the interaction network has once assembled. And one last thing I want to say about this is that we've actually derived this same structure for a completely different framework, and we've actually validated, we've seen it in... We've shown it's there in plant competition experiments and biodiversity experiments. Again, this takes us into a different topic, so I won't go into it, but it was just published now. You can take a look if you're interested. All right. So what did we get here? So before we've already seen that there is structured in the assembled network. And so the next question is which of the requirements that we've seen before for an assembled network, namely having positive abundances. Being stable, which of these requirements is affected by this new structure? So we said that abundance has become positive once you use the structure. Stability, it turns out, is unaffected by the assembly process. Okay. And this is an important point. It's a subtle point, but it can be important. I don't want to get into the technicalities, but maybe for the experts, just in one word, you know that there's a spectrum of the matrix which affects its stability. You cannot squeeze this spectrum just by the assembly process. There's a simple reason I'm happy to discuss this later with those interested. Okay. So what the assembly does is it affects feasibility and the abundances, but it cannot push the stability at least for large systems. Okay. So that's all I wanted to say about the unique fixed point phase and how the assembly affects the structure of the network. So anything so far, any questions? There's no question in the chat. Okay. Good. I'll give another second. Okay. Good. So now we step, we increase the variability of the interactions. In fact, we can increase diversity at fixed interaction parameters. It's the same thing here. And what we see is that we lose stability at this unique fixed point and instead we go into something new. Now what happens now depends on the symmetry of the interactions. If the interactions are symmetric, as is often used to model competition, what you get is many, many, many fixed points. You get exponentially many fixed points in the diversity of the system. So you can have, imagine 300 million alternative stable states, alternative equilibria here. This is very different from regime shifts and things like that where you imagine one or two. So it's very many sub-sensitive species that in fact form a stable equilibrium. So I see there's a question, the correlation gamma plays, yeah. So right now we're talking about this correlation gamma equal to one. And this is symmetric interactions. Okay. That's gamma equal to one. And that's what happens in this case. Okay. And right. And as advertised, this transition is indeed sharp, like a phase transition. Namely, you change the interaction parameters by a bit and you jump from one fixed point very, very, very many fixed points. And this, yeah. Okay. So it's as we thought it would be before a phase transition. All right. Okay. I don't have time for this. Right. So now I want to go into the case where the interactions are not symmetric. In fact, let's look at the case where gamma, the correlation is zero. So it's independent effect of i on j and j on i. And then what you get is a transition into a chaotic phase. Okay. Persistent dynamics, very high-dimensional persistent dynamics happens. It starts at a very well-defined strength, variability of interaction strength. But the size of these fluctuations grows continuously as you change the parameters. So at first, it looks almost like a fixed point with small fluctuations around it. As we increase the variability, you get more and more strong and stronger interaction, sorry, strong and stronger abundance fluctuations. Okay. So that's the, that's the phenomenology that we see in this case. All right. So one word about what the mechanism is that's going on here. Basically, the idea is that you have a species pool and invasions try to increase the diversity in the community. But then there's competition, which pushes some of the species out by because of the competition. They have negative growth rates. They leave the system. And so this alone balances, gives it some diversity that would balance these two processes. Now, at this diversity, the assembled community might or might not be stable. If it's stable, you get a single equilibrium. If it's not, then one of two things can happen. You can either say, okay, I cannot go beyond this stability diversity. I stick to the stability bound, and then I get multiple equilibria. Or you can say, I don't care. I'm just going to do dynamical fluctuations, not care about stability at all. And then you get chaos. And it's not an easy question to understand when and which of these scenarios will happen. There's still ongoing work on this question. Right. Okay. Mateo, how do I, with time? Because I can't see my phone. So four minutes. Okay. I'm wrapping up. Okay. So from this, we can generate specific predictions for what we expect from, for example, these chaotic fluctuations that are endogenously created by the system, as opposed to, for example, environmental fluctuations. For example, you expect the sideline fluctuations to grow continuously with diversity. The fluctuations look almost uncorrelated between different species when the diversity is high, as opposed to the things that you happen, you see in environmental fluctuations. And okay. And now the next aim, the next big aim is to look at experiments and data to start looking for these things. Okay. I don't have much time, and I'd rather leave time for questions. Let me just say that a similar, one thing I want to say is that a similar behavior happens if you have a meta-community that with patches connected via migration without external migration. So the system can hold this chaotic behavior without extinctions for a very long time. That's one result. Another thing I want to say is that people think that resource competition models are so very different that none of this will ever be seen there. That is not the case. So one of the issues is that there is this, many of you know, the MacArthur model is a model of resource competition. This is a very, very influential model, and I love it. But it's it's simplicity is sometimes misleading. If you change the model such that, for example, the resource uptake or the dependence of growth on resources is nonlinear, you can in fact see chaos, and you can go beyond the stability bound, which in this case is in fact the competitive exclusion limit. So you can have more species than the competitive exclusion limit using these persistent dynamics, and it's enough to in certain cases it's enough to introduce very little nonlinearity, and you're already there. So the MacArthur model is in a sense a bit single. All right, I won't talk about this directional phase. Let me just wrap up and open to questions. So what characterizes these high-dimensional, high-diversity assembly models and hopefully also systems out there? There's these very few relevant variables. Once you know that you know how to start seeing what start putting your puzzle together, there's these deterministic outcomes for things like dynamics and properties of the assembled community. There's these sharp transitions between very different qualitatively different behaviors. I talked about these interaction patterns. And again, there's extensions to various directions. I'm happy to discuss whatever is interesting in the questions. Yeah, thank you.