 everybody to this second week of the school. Thank you very much for being here. So today we are going to have a different format compared to the ones we are used to from last week. So today is my pleasure to introduce the lecturer of today who is Jordy Vasconte. Jordy is a professor of ecology and evolutionary biology at the University of Zurich. in Switzerland and his research is mostly focused on ecological network and in particular on mutualistic networks. So as you know, Jordy Vasconte has uploaded two pre-recorded lectures that were available on the website starting last week and this session of today is a Q&A session. So it's an opportunity to discuss with Jordy the material presented in the lectures and I think it goes also a little bit beyond that. So thank you very much, Jordy, for recording the lectures and for being with us. Thanks to you, Ciacopo. It's really a pleasure and welcome everybody. And so now we are open to questions. So if you have any question on either of the two lectures, please use the raise and button of Zoom or write it in the chat or if you're following from YouTube, you can write it in the chat of YouTube. So we have a hand raised by Sylvia. So Sylvia, if you want to ask the question, please unmute and ask. Yeah, can you hear me? Yes. So I have a question on the first lecture concerning how the nested structure emerges. In fact, like in general, one could hypothesize that the structure could emerge either from the ecological dynamics or from the evolutionary dynamics, right? And on Friday, we heard a lecture by Stefano Alesina that was talking about community assembly and we learned that there will be a structure in the interactions only if there was a structure in the original pool of species. So this would suggest that the nested structure can only emerge from the evolutionary dynamics. So could you comment on this? And is there any of your works where you dealt with how the nested structure can emerge? Yes, that's a great question. Thanks for posing that. I think that as we've learned more about these ecological networks and the suite of mechanisms that are compatible with the structure, we tend to move from an initial focus where people were picking up a particular mechanism to considering like simultaneous mechanisms that can potentially be at war here. So early on, right after the first wave of a study describing the structure of these networks, people start focusing on particular mechanisms. For example, one of those is species abundance. So people realize that if most abundant species tend to be more available by chance, it's more likely that species would tend to interact with these more abundant than less abundant species. So this kind of neutral approach become one important mechanism in generating networks. At the same time, other people were focusing on other aspects. For example, a phylogenetic signal is well known that there is a phylogenetic signal, meaning that species close in the phylogeny tend to play similar roles in the network of interactions. What this is suggesting is that past evolutionary history may be important in understanding contemporary patterns of network built up. So for a while, as it tends to happen in science, it seemed like there was a little bit of a competition between specific mechanisms. I think where we are standing today is in a place where we recognize that there's a suite of mechanisms. There's not only one mechanism. So ecology certainly plays a role because species abundance, it's important in order to explain these patterns, but most likely it's not enough. And it's true that this coexist with this evolutionary signal with trade matching, for example, which is another set of explanations that emphasize that trades are the currency that explains interactions. And therefore, for example, one trade would be the length of pollinator stone, another trade could be the length of plants corolla, flowers corolla, and therefore whether or not there's a matching between trades may be key in order to explain those interactions. So what I think it's now the state where we are is in trying to understand, not proving that this is the mechanism or that's the mechanism, but in trying to weigh the relative importance of a series of mechanisms, always taking into account the phylogenetic structure. Because in these comparative studies, one cannot forget that species are not independent units, but they form part of this process of related species coming from an ancestor origin. So that's where I think we are right now. I think that evolutionary mechanisms certainly are important, what you were emphasizing, but there are other set of mechanisms like ecological ones. And I don't think there's nothing wrong on that. I think that as with many other things in science, a pluralistic view is most likely going to be at worst. So yes, I would highlight evolutionary mechanisms, trade mechanisms, and neutral or species abundance mechanisms. If I can ask just a little verification on this, like when you mentioned the importance of abundance, so are the species that are most generalist, typically the more abundant ones? Yes, that's something that people realize from an early stage. And if something, the debate was, I mean, are species more abundant because they are more generalist or the other way around? So that's really a bit where the discussion was. But from very much the early stage of these studies on ecological networks, and in particular these mutualistic networks, it was clear, and particularly, I'm thinking about the Great War by Diego Pazquez in Argentina and colleagues, that abundance was really important. The thing though is that, for example, going back to our own work, we had a paper laid by a former student, Abe Krishna, that proved with a simple model that while abundance is important, when you combine abundance with trade matching, the fit to the data is even higher. Great. So Silvia, you wanted to follow up? No, I'm fine. Thank you very much for the answer. So there is a question from Washington, Taylor. Yeah, hi, thanks. I thought your lecture was very interesting. You found some, there's some nice clear and simple ways of starting to address some really interesting and deep questions. I had actually two somewhat unrelated questions. Maybe I'll throw both of them at you and you can choose which one or address them whichever you want. So one is, you focused on climate as a driver of extinction events and as kind of a primary thing throughout your lecture. But as I'm sure you're very well aware, most of the things currently driving species extinction are other things like habitat loss, human use of the ecosystem, pollution, invasive species and things like that. So the first question is whether those would have a similar impact or whether there are some climate specific signatures in what you were describing. And then the second question is, when you describe these essentially two level networks, I mean, of course, in a real ecosystem, there's all kinds of very subtle other species playing niche roles that are mediating interactions and maybe playing key roles as, you know, in all kinds of places in the system. So I'm wondering whether that you've explored whether there's any sense in which these networks you have are robust against intermediate species that are involved in the system going extinct or, you know, how that plays in the missing pieces that you don't have in the network. So those are sort of two questions. Yes, these are very great questions. Going back to the first one, you're totally right. I mean, climate change is just one driver of global environmental change, the other being habitat fragmentation or nitrogen deposition and so on. While our focus was here, probably the reason we're focusing on climate change is for historical reasons, because from early on there was this attempt of bridging between these somehow distant approaches, one the network approach and the other the climate change ecology, right? So to some extent, focusing on climate was a consequence of these two different approaches. But there's been a previous work by other people who focus on habitat transformation. Some of these work is theoretical using models of habitat loss, habitat fragmentation and looking at, what's the rate and shape of network collapse. And I would say, I would expect finding the same type of qualitative results. I don't think those results were quite specific about climate in terms of the rate and shape of collapse, in terms of like finding this phylogenetic signal or having a moment where the rules of the game, so to speak, change and then we're focusing different species from the phylogenetic tree. So I would say those are quite general results, although to be totally honest, only a subset of these questions have been addressed using other drivers of climate change. So here I'm kind of telling a little bit my gut feeling. For example, the one is for sure similar is this kind of abrupt collapse, this idea that the consequence, as we are moving through this axis of global environmental change, the thing that early on, nothing seems to change too much up to a point where suddenly there's a collapse. That's the kind of result that different people have seen when looking at different drivers. The other ones, I'm not so well aware of the studies, the ones looking, for example, at how functional diversity is eroded or evolutionary history is eroded. But my gut feeling would be that these are quite, have to be quite general consequences. So I would expect a similar kind of signal. In relation to the other question, that's very interesting. It's true that one should expect having this kind of, if you want, gist on a species, species that play a major role in bringing the network together. And that's a piece of research that has focus on that. But in my view, it's more a static one. For example, research looking at the modularity or compartmentalization of these networks. This research tends to look not only at this tendency to be organized in modules, but also at the role that different species play. And in particular, this role by a few species in bridging across different modules in being the sort of the glue that keeps this module together. Now, if I understood your question correctly, you were asking whether some studies have focused on what happens when these species disappear. And I can think of a study that was looking at a genetic diversity of some of these species that emphasized this thing that once you remove one of these species, you can have a major change because suddenly you can have a network that previously was more or less cohesive. And now you have like a collection of different networks. Great. Thanks. Those are great answers to both questions. And on the second question, I guess one, if I could just go a little further on that. So I guess one of the things I was also asking about there was you have this database of interactions between different species in these different habitats. And I guess I'm wondering, you know, probably there are important species that were missed in each of those databases. So for instance, you may have 73 species in the given area, but there may be another 30 species that really play a keystone role. So part of the question is, even if you missed some of those key species, and those were somehow just encoded secretly in the interactions, are the results that you're getting robust against, say, replacing the network with one where you imagine that you only know about a subset of the species and then test the same hypothesis? Yes, you're totally right. And that's the kind of, I would say that encapsulates a series of criticism that was about using these large database studies where, you know, I mean, it was a nice attempt in the sense of looking at generality, but there are trade-offs and part of the consequence of that. And I think it's a fair point, part of the consequence of people who were more critical about our work and the work by many others for the same sake, was that each of these networks has been compiled by a different author or a different team spending different time or using slightly different methodologies. And therefore, in the same way that when looking at studies on species diversity, it's very clear now, pretty much everybody knows and understands that we should use rarefaction cubes, we were not at this stage. And that arise some questions about the value of this generality. Now there's been, as they feel mature, and these kind of studies become a little bit more mature, there's been already a subset of those that started using similar methodologies, and in particular use these rarefaction cubes. And this allows to think, first it allows to focus on the smaller subset of networks that have been sampled enough, so have a similar level of sampling, and then focusing on those, but also like looking at how these properties may change across a gradient in sampling, and there are properties that may vary quite a lot, but some of these properties do not vary that much, like in particular that nested structure I was emphasizing during that particular first talk, each one that is a little bit like the different builds of an onion, right? Essentially you have this, this core of generally species and then these few generalists, the thing is that if you sample more, you start having a longer tail of a species, and normally those species tend to be specialist and to be less common species, but also they tend to attach to the most generality, so depending on the type of dimension of a structure if you want or perspective, these may not depend that much on on the level of sampling, but in general I think that's a poor answer, and I think that now what we should try to do is every new study try to have a sort of rarefaction cubes, and people can do that in the same way that we look at how many different species we have when we sample 100, 500, 1000 individuals, we can do the same for example with a number of interactions, and oftentimes we have enough of a sample that people consider a little bit of an assing to it. Great, thanks a lot. My pleasure. Great, so the next one in line is Alfonso, so Alfonso please unmute yourself. Hello everybody, thank you for your great lecture, and my first question is, can the mutualistic networks account for weighted interaction between plants and insects, and in that case how much the distribution of these weights might change the results of the number of species, the maximum number of species supported by the network, or the change of extinction in case that, in the case of non-random extinction, like the extinction that you talked about in the lecture, and my second question, I just come up with that question recently, is that, is there a relation of in like generalist species tend to be more abundant? My question is related with the species abundance distribution, if there are a relation between the species abundance distribution of a community and the network structure. Very good, very good set of questions. Let me start by the second one. The answer is yes, and some of these approaches we were referring to a few minutes ago in terms of like checking the relative role of a different mechanisms in explaining network structure, one of those, this kind of a neutral approach, was using these observed skew species abundance relationships, and then for the plants on one hand, for the animals on the other, and then assuming that the probability of drawing an interaction is going to be proportional to the to the product of these species abundance, so it's kind of simultaneously taking into account both the skew distribution for both sets, right? And when one does something like that, you come up with a network of interactions that tends to be similar than the one we observe in nature. So people would tend to think, okay, that's that kind of supports the idea that neutral processes, species abundance is certainly important. Again, as I said, when you have a model that takes into account these and other factors, you can get an even better fit, which tends to support the idea that there's not only one mechanism, but most likely a suite of mechanisms. So you are totally right, species abundance and the particular empirical distributions is something that may be very important and has been empirically used in order to come up with the expectation of these network of interactions and then matching that expectation with them, with the observed one. I don't know if that answered your question. Yes, I think that answered my question. Lovely. Then in regards to the first one, you are totally right. I think my talk, you probably realize that I mean, when I give this talk on mutualistic networks, now it's a talk that spans now 20 years. So early on, when we were mainly looking at the structure, I was going through different levels of the structure and focusing on interaction strength. As more results were packed, I tend to reduce the focus on structure and just focus on that particular dimension, the nested one, but you are totally right. And although early on, the first set of papers were looking at binary data, and that refers to, for example, this nested pattern I was talking about, but also these connectivity distributions, whether they are, they follow overlaw or a truncated power law or an exponential that was like the kind of things people in natural research were doing at the same, at the time. I mean, a few people again were critical about that and they would say that, okay, all these results may be meaningful without like considering embracing the fact that there may be a huge variability in the wave in the strength of those interactions. And actually, there was a few studies that were looking at the structure, but using weighted data. In that big repository, I mentioned during my talk, right now there's almost half of these networks that contain information not only on who interacts with whom, but on the relative ways of this interaction. Oftentimes, this is the surrogate of frequency of interactions is used or number of fruits removed, for example, things among those lines. So there's this kind of information and some of these studies describing network structure were focusing on that component. For example, they were looking at the dependence of an animal in a plant. And they were focusing or they were highlighting this idea of asymmetry in the interaction, which can be also observed in binary data. One of the results of an instead pattern is this asymmetry in the sense that specialists tend to interact with the most generalist. Now, when you look at weighted network, you can see that this also happens in a pairwise scale. So like, for example, a plant that depends very much on an animal for its pollination. Normally, you encounter that the animal depends very little on that particular plant. So some of these results you can still see when you move from binary to weighted networks. And other results only make sense or only kind of tools or approaches when you have a weighted network. So overall, yes, you have this kind of information, you can address new questions that you could not with a binary data. And a few questions you can check with both. And I would say you tend to find similar patterns. Great, Alfonso, you have a follow up? No, I think that that answers were really nice. So thank you very much. Thanks to you for the question. Great. So next in line is Violeta. Yes. Hi. Hello. Thank you, Jordi. Yes, my question is about structural stability. More trade off there. Okay. For me, crystal clear, how do you firm the nested net of the network? You just measure it with some algorithm. But what about the mutualistic trade off? I mean, how have you inferred this mutualistic trade off? Because then if you see the equations that generalize a couple of the right questions there, the parameter space is multidimensional and very, very wide. So I suppose that you just chose certain parameters. But can we be sure that about the general generality of the results when doing that? Yes. In terms of how one does this, on one hand, you have for these weighted networks, you have empirical information. Essentially, that's information that relates the degree, how many species, a focal species interact with, that may be two, five or ten species, but you also have information on the weight of each one of these interactions. So this is information that you have empirically from the network. So that means that you can determine where the point is in that figure. In terms of how to explore, it's very much similar than with nestedness. I mean, with nestedness, you take the network as it is, and then you randomize that with a series of assumptions. Normally, you tend to preserve total number of plants, the number of animals, total number of interactions, and approximately how many interactions each species has. With the trade, obviously the same, you could have like a randomization where you maintain the degree of each species. You maintain like the number of interactions they have, but you can shuffle the weights of each one of those interactions. So that would allow you to explore the taxes in parameter space. But anyway, that axis is also related with the intrinsic weight of the species and also the competition parameters. So all of those. That axis, it's not related to anything else, because it's the way you define it. You define an axis like being only a value of nestedness, and then it's only structural, and you can change it. What it's related to these other parameters, and in particular to growth rates, which this was the variable we're looking at, is in terms of the measure of structural stability. So on one hand, you have parameters that define the structure of the network, and then you use growth rates as a way to quantify how much variability in those growth rates the model can cope with before one or more species is driven extinct. There's a little bit of an uncoupling. Some of these variables of network structure are just used to show how much variability you could have. And then for each level of variability, the demographic parameters are used in order to explain the range in growth rates in this case in particular. So you choose like the center of this domain, the center where this domain of feasibility, where all the species can coexist, there's ways by which you can focus on that center, and then you start perturbing growth rates, changing growth rates, increasing, increasing, increasing up to the point where one or more species disappears. So there's a little bit of a decoupling on how you treat these different different parameters. Okay, sorry, last question. When you are varying the range of the intrinsic growth rate during your equations, you must fix the other betas and gammas. So maybe it happens that for those 13 parameters of gamma and betas, your intrinsic growth rate behaves in some way, but maybe there's a tiny region where they don't. So is that actually important or not? Yes, you are totally right. And to make a long story short, essentially that's for a specific values of the other parameters. What one can do is then repeat the analysis for another value for each one of these parameters. So this gives you an idea of how robust results are for variation in the other parameters. But to be honest with you, essentially that's more like, okay, you fix the other parameters and then you focus on variability on growth rates. And yes, you cannot rule out the possibility that there may be some combination of parameters where now you could have a slightly larger or a smaller range of conditions compatible with feasibility. You are right. It's a very complex problem just because of the dimensionality in parameters. So to some extent, you fix some of the parameters and then you focus on growth rates. So if you want, it's a partial account of the much more complex variability in parameter space. Okay. No, but thank you. Thank you very much. Very interesting. Great. Thanks a lot for the questions. So the next in line is Martina. Hi. Hello. And thank you for your lectures. So I have two questions. One is a clarification. The other one is more general. So maybe I'll start with clarification. So when you were talking about extinctions that were driven by climate compared to the secondary extinctions that are driven by interactions, you had those phylogenetic trees and with the circles. And you were saying that you can predict the direct extinctions with the geographic location, whether you predict the others with the traits. And I was wondering, maybe it's completely wrong. Sorry. Just to be precise, the single most important variable in explaining extinction is what we call network ID. So a property of the network. Yes. Okay. And I was wondering whether, okay, if you have interactions, you're supposed to, you have them in the same location. So I was wondering, how can you get these, say, switching the first predictor? Essentially, the way is by using these generalized linear models where you can have different factors. One is geographic location. The other, it's a factor called network ID. So a property of the network, which it's unique and it's not affected by the others. And then through this kind of a statistical approach, you can wave the relative contribution of one of these variables accounting for the other ones. So it's a way by which you can focus on the relative role of geographic location while keeping into account, if you want, keeping fixed the role of network ID. That's one component. And the other one is, or the complement to that, it's by using what's called a Caikis information criteria because other things being equal, I mean, the more variables you have in the model, the better it gets. But you have to penalize it. To some extent, it's an artifact or just a corollary of having more variables. So the idea is to really focus on the variables that explain the variability in a model, taking into account the number of variables or the dimensionality of the model itself. So that's a kind of statistical approach to come up with this kind of result. So essentially, it's statistically, you can do that. Even when you have several variables, these kind of models allow you to focus on one variable at the time and kind of give you the relative relevance of that variable while keeping the others constant. And also the interaction because sometimes the interaction between factors or variables. Oftentimes, it's not just that variable x or variable y are important, but sometimes you can also find a significant interaction between these two variables. And so given these models, so the network ID is in the fixed part or in the random part? That's a good point. To be honest, now I don't remember if we treat it as a random factor. This I cannot remember now. It's fine. I can read the paper, probably. Yes. I mean, I'm sure the details are there. Because that's a very interesting thing, and it's not trivial. The way people who are really knowledgeable about these models, the way they treat these factors as either a random or a fix, that depends very much on the structure of the question, but also a little bit the constraints or the limitations of the data. So again, if you are interested, details are there. I just do not remember now. Okay. And the other one is more general. So you find these very nice relationships between nestedness and biodiversity. And do you know what happens if you have modularity on the x-axis? That's a very good point. We've not really checked that in the context of that framework. So I cannot give you a solid answer to that question just because we did not check. Obviously, one could think about these two dimensions of network structure not being totally independent, but somehow related. And although it's not perfect relationship, people tend to assume that the higher the nestedness, the lower modularity, that's not necessarily the case because that depends on a level of connectivity. So below that threshold of connectivity for low-connectivity, you actually find a positive relationship. You find the networks which are more nested. They are also more modular. But when the network is well connected, you find this kind of in their relationship. So one could conclude that because of that, you would find the opposite sort of trend. Okay. Thank you. My pleasure. Okay. So we can move on with Sree Rama. Thank you for your talk. Am I audible? Hello. Hi. Yes, I couldn't hear you very well. Sorry, am I audible? Okay. So actually, I have a general question. Suppose if you take a general predator or anything, they have some referential structure. So there's something like optimal forages. So what happens is every time the network structure changes because they can prefer one prey or another prey, so the network becomes a dynamics. So how this dynamic network can be modelled generally? That's an excellent point. You are totally right. And one of the limitations of the last part of my talk, talking about these models of climate change, is that we were not taking into account that. So essentially in this kind of approach and also the approach by many others, the thing is that once a species runs out of resources or the fraction of resources disappear, those species have a higher probability of being driven coextine. And we know that things are a little bit more complicated because as you already point out, there is a trophic flexibility, the fact that some species, whenever their favourite prey items are not abandoned enough, they can shift to another one. So I mean, to my knowledge, the very first person who brought this forward was Mikio Kondo, a theoretical ecologist based in Japan that in a paper in science, that's probably now about 15 years ago, proved that this flexibility can certainly shift. He was focusing on the relationship between stability and complexity. It can be shifted. So when you have trophic flexibility, you can see that more complex food webs tend to be more stable. Also in the context of mutualistic neighbors, people in Fernanda Valdobinos, it's a good example, have shown that this relationship between stability and complexity can largely change through that trophic flexibility. So I would say that the sort of models that we and others have used, ignoring trophic flexibility would be a sort of worst-case scenario. Whenever you have a trophic flexibility, things become a little bit better. But I would say that some of the qualitative results, for example, the existence of these tipping points are still there, only that you can shift the tipping point to higher values of habitat laws or species extinctions and things like that. I think now the question, though, is how to really bring a biological informed model of network rewiring. This is what we're trying to do now in the lab with Marilia Gallarsa. And I think that, again, that phylogenetic signal can be key here. The fact that, okay, although there's a potential for rewiring, oftentimes this is not going to be random. So any species will most likely not have the possibility to shift to any other item. But we think that it may be a good starting point to consider phylogenetic signal, meaning that if one species runs out of resources, most likely may shift towards resources that a species close in the phylogeny depends on. So I think that would be a good way to start introducing these rewiring a little bit in a more biological informed way than just assuming every species has a probability to rewire with any other species randomly picking the community. Can I add one thing? Generally, most of this optimum phrasing is something, suppose they are not able to see that they are not referring, they are not, the feeding may not be giving the required growth or anything, they will divert to another species. So that means something like, suppose if they are seeing that we are having a lot of energy we are spending in time of predation, but we are not getting required energy to our growth. Then we have to shift for another species. So how can you do this? That is what my question, I mean, I don't know, I'm able to properly convey that or not because I'm not a biologist, I'm a mathematician, I'm working in this area. So yeah, I think it's a very good point. I mean, to be honest, we know very little about that. And the reason is again, because these kind of studies have gone quite independent from each other. A little bit I was emphasizing that independence between network research and climate change research, and that was a little bit of the rationale for us trying to bridge them. Another big gap exists between these models or this approach of ecological networks that tend to be quite static and this optimal foraging, which obviously emphasized a dynamic component. I think it would be a very interesting direction to try to bridge those. And for example, try to see what would be the predictions, what would come up out of these basic ideas of optimal foraging. So for example, allowing like a few species of animals to forage in a given landscape and then trying to see how out of these basic rules of optimal foraging, what kind of network structure would arise. I think there's a little bit of that. I seem to recall a paper probably by Morales and perhaps Diego Vázquez as well. I may be wrong, but I think these are the authors who tried to do that. And that was certainly very, very interesting. But I think there's lots of room to kind of expand this kind of bridging between optimal foraging and network structure. My pleasure. Great. So we are going at very high pace here. So there is another question by Ankit. Hi. Hello. Thank you for this very broad survey of mutualistic networks. So I had a question regarding like which is somewhat related to May's result of stability in large complex ecosystems. So there obviously he looks at like entirely random networks. But let's say if you talk about large mutualistic networks, where you have some asymmetry between the number of plants and the animals, like let's say there are very few plants, but many animals which like depend on these plants. So in that case like is there a theoretical result of like stability in the same sense as May or like yeah, or like is it difficult to like define that because I guess in such a setting you would have a lot of negative interactions instead of like just totally random interactions. What do you mean by negative interactions? Non random? Just non random? Yeah, random negative interactions, but they don't sum up to zero. I mean like the interactions of the entire matrix. Yes, that's a very good point. You are totally right. I mean, Bob's May great paper has to be seen as like a baseline expectation. So some of the criticism the paper had actually did not arise because of the paper itself, but our faulty way of interpreting that paper. So we could not interpret that May said that most complex communities have to be unstable. Rather, what he said is that our baseline expectation, if communities were randomly organized is that there's a limit to complexity. And therefore, I think that paper was extremely influential in shaping the field in many directions. One of those was just asking ourselves what may be those mechanisms or these dimensions of structure that can help reconcile being complex and being stable. So yeah, our original work on trying to bridge between this network structure and stability that was our science paper in 2006. We tried to do that in following a very similar approach than Bob May with lots of limitations. I mean, that was our first attempt and therefore we had to simplify a lot of things. But what I want to emphasize in the context of your question is that yes, you come up with an equation for the linear stability and feasibility condition for both being feasible and linearly stable that very much resembles Bob May. Essentially, you have in Bob May, you have that the average strength of interactions has to be lower than an amount, amount that involves number of species and connectivity. So what you have here is a similar thing, but instead of having like the average strength of interactions, you have the average product of the strength of interaction of the animal and the strength of interaction of the plan. And this is what has to be less than an amount. And that allow us to predict that what's relevant in these motoristic networks is that either you have species that depend very little on others or when one species depends a lot on a second, that second depends very little in the one. So even when one term is large, if the other is very, very small, the product still remains small. So that would be an example of a very similar kind of criteria than Bob May, but it had some interesting variability and that variability allow us to start thinking about how these dependencies of a plan and an animal and the animal on that plan have to be rearranged to keep stable communities. Yeah, thanks. I'll also look at your paper. Thank you. Great. So is there any other question? I don't see anyone with the raise hand in the participant list. No question in the chat, but we had a 50 minute, very intense question session. So I think that if no one has a question, we can move forward. Great. Well, I think this was a sort of experiment for us to have this pre-record sessions plus question, but I personally think it worked very, very well. And I'd like to thank Jordy for being with us and for answering all the questions, as well as pre-recording the lectures. Thanks Jacopo and thanks every one of you. I think you come up with extremely good questions. And I think a proof of that is that these are the questions that we are encounter when trying to publish papers. So in that regard, I think that you are thinking very well. So I've really enjoyed and had a great time. So I'd like just to add that feel free to email if some of these ideas start developing or you have further questions. So just drop an email and it would be my pleasure to keep discussing some of these ideas. And best luck to every one of you. Hopefully you have a nice school and a great career. Thank you. Thanks, Jordy, also for your availability. So what we're going to do now before the next lecture, which will start in about 20 minutes, is that we're going to split again in breakout rooms. So feel free to chat