 Great. So thank you very much for the organizers, the complementary set of the organizers to ask me to give this talk. So, in the spirit of this workshop, I wanted, I mean, this talk is going to be several speculations, but what is not a speculation is actually due to the people that actually work on this. So Lorenzo and the you work with me on this project. So, I mean, this is the setting of community assembly at least in theory so we explored these in several talks about both in experiments and in theory. And so we have a regional pool of diversity and then through migration. This sort of generates the initial condition for the assembly of community and there are several processes that contribute to the final state of the community. And typically, the most of the focus in modeling is on the processes that happens in the local community that shape diversity given the shape diversity locally given the diversity in the regional pool. So, sort of the one of the key challenge that I think we have to face to understand the limits of diversity in the local police to understand what are the limits of diversity in the regional or in the global pool of diversity. Of course, this process is coupled to the processes of local diversity. And so understanding this coupling is necessary to understand what are the limits of the local scale. The, the other, I mean the other side of this question is, of course, the contingency of the, the regional pool, the pool of diversity is sort of materializes in the fact that trades are not sort of diffusely distributed in the in the regional pool, at least in general, and, and therefore there are contingency in which species, which trades together, right. And again, this is shaped by the history of, of the, of the community. And one aspect that is sort of its interest in mirrors these at the local scale is the fact that, in some sense, on the other, on the other hand, what we see at the local scale is that functional composition is much more robust than taxonomic position. So this is especially in microbial communities that has been a lot of, let's say, observational evidence that sort of goes into this direction that you see a given environment, and the top panel you see that there is a lot of variability in which, in which species taxa, et cetera, are there, while, if you look at, for instance, at trades at the level of metabolic pathways, what you see is that this is way more stable. I mean, I think that here there are much, I mean there are several subtle points, but at least we can sort of start from this observation that it seems that there is a large degree of functional robustness of one end, and the other end of functional redundancy, right, that the same sort of function at the, at the scale of the community can be obtained by several different combinations of, of, of taxa. And the other, I think key challenge that I will not even try to touch is what determines the pool of functional diversity, right, so at the end what limits the local community is which functions are available and there is a much, there is a process on a much longer timescale that determines which functional are present and this is again coupled to the local dynamics in unknown trivial way. Okay, so this was a sort of few speculations and introduction. I don't know if there are questions on this, but I think I can move on with something more substantial. So, the setting I'm going to take here is competition in chemostat so this is interesting because it's sort of related to what happens in places in the gut or to at least a little bit in directed to experiment that can be performed with microbial communities, and the idea is very simple that there is a, there is an inflow of resources from, from the environment that goes into, into this local community, and then there is an outflow of both resources and species through dilution, or that if you want. And so species grow consuming these resources, and in particular one particularly interesting aspect that can be added to these classical models of competition chemostat is cross feeding so when the resources are consumed by by species a part of the energy that is obtained from these resources goes into growth, and another part of the energy goes into metabolizing other compounds that can be used as a resources as a source of energy from other species. So there is also these indirect cooperation through through the environment. Now, there are there is a sort of recent amount of work on these kind of models, which sort of builds on classical models of consumer resource that were starting from a carbon going on where sort of formula. And in this location sort of describes the dynamics of populations of let's say bacteria and the dynamics of concentration of resources. And there are several terms that is an efficiency in which species consume transform the energy from resources to biomass. There is an intrinsic energy content of a resource. There is a sort of maximum intake rate that a species can have on a given resource. Then there is a matrix parameterizing the resource preference, compared to this mass maximum rate. There is a functional response to the concentration of resources. There is a debt of dilution rate, or dilution rate. There is a fraction of how much fraction of the energy taken from a given resource is leaked. There is there is a matrix parameterizing the resource conversion so when you intake a given resource to what can be converted sort of modeling the metabolic network, and there is an intrinsic resource dynamic that I will take constant. So this kind of model, I think that the natural reaction that anyone saying should that is this one right so the clearly, I mean, it's something where there are a lot of details and the reason why I think you should have the same reaction that Munch had perhaps thinking about this model is that there are too many parameters and too many variables, too many choices that can be made, can be made in this kind of model. So we have order number of species parameters and a quadratic number of parameters in the number of resources and so clearly these scales in a terrible way. This is not just a problem of fitting parameters I think it's more from perhaps this is a real statement but from the mental is a statement about what are the properties of the species pool so the, the, how should we choose parameters to describe these these the, what we call parameters is actually a property of the species pool. Okay. Any question about this. No question in the chat. Okay, great. So one approach to study this, this kind of models and this kind of system, which is, I will say, is the new standard approach is the one that was described by guy. And there are several works also on this model, trying to study the case when the number of species is large. And the parameters are chosen and so they, they come from some distribution and the sort of strength of these of these approach that was highlighted by guy is that the details don't matter so it doesn't matter really. So in terms of distribution, what matters are the broad statistical properties of these parameters and the focusing is on what are these typical properties right and one of the key parameter is how big is the species pool compared to the diversity and the sort of the interesting limit in these approaches when the size of the pool is order the size of of the number of resources so the limit of diversity in the local. So I think that there is another approach, which is somewhat specular to this, and this is much less explored so I think that the only one who explored this is a mission to come up and increase my knowledge, and this is sort of the approach to the limit of very large number of species in the pool, where somehow the diversity of the pool is combinatorial in the, in the potential diversity that we have in the local community. And for instance, indeed, in the setting the idea is, you have these resource preference and you can take all the possible combination of research. So let's say that we put ourself in the simple case where these resource preferences are just zero or one. Then there are the species pool is as two to the R number of species, you could take the case where these value can take zero one and one and you will get three to the R. And then these other parameters, just to start can be sort of set to respect some metabolic trade off. And this metabolic trade off sort of taking to account the trade off between being a specially than being a generalist. So, if you are a specialist, you sort of pay a low cost so you have an higher efficiency or a low that doesn't really matter what you put it. And you are more efficient and you have a competitive advantage when you grow on the resource you are a specialist, while the generalist is of course efficient, but can grow on a much larger set of substance. Great. So this model, there are two limits of these of these models. So the limit where these cost a key, which is sort of cost per trade per part of the gene, however you want to call it, the limit when this cost is zero. So it means that basically a jack of all trades is a master of all so the generalist is the one that takes all that is no diversity not coexistence, only the one that forms all the function and survives, while in the regime, where this key goes to infinity. So essentially this becomes a model of sort of strategy partitioning where the amount of effort you put into sort of the resource preferences is total resources is cut. So that is reduced to the model studied by Nisha and the work by we in green and co workers in the case of without cross feeding and without separation anyway. So it's sort of a simpler model. So the, so this is the kind of model you want to study so this limit of the pool of diversity being extremely large, where everything is possible. And the reason why we can solve this is that, even if that model is quite complicated, we can write. So this is a model of function, and there is a global stable equilibrium, and this can be obtained sort of generalizing a classic work from the 70s. And, in particular, when you write is the opponent function what you realize is that it's convenient to express the variables in terms of the total biomass, and the trade occupancy which is basically the fraction of individuals that are able to rely on substrate I, and sort of all the parameters about the environment can be captured into a parameter which is a resource quality which is basically how much energy from a given resource is available in an individual lifetime in that sense of competition. So, the point is that we can exactly solve these, the system when diversity is large enough, compared to the number of resources. And basically, the solution is the following so these functional occurrence so how many, what is the fraction of individuals able to grow on a given resource, sort of as the solution where there is a core of resource quality where everyone can grow on. And there is a non core set of resources, whose, which current, whose occurrence correlates is linear in the number of resource quality, and we can get an expression for the total So what about diversity well diversity is actually, you have to count the number of resources that are known core and diversity is exponential in the number of known core resources. And of course these immediately recognize that this is not possible under competitive exclusion. And in fact this is just a property of the metabolic trade off and is due to the fact that we have a margin is really. So we have these marginally stable equilibrium we do to this number of non core resources. Now what, what happens if you have cross feeding well, if you have cross feeding. So it seems that the result breaks apart. And so this is the same input of resources with different cross feeding matrices so with different structure of the metabolic network so when one resources compared to what else, but eventually you can recover the same the same result by sort of rotating as a resource quality based on the resource competition on the, on the conversion. So, also in the case of cross feeding, you get exactly the same result and the effect of cross feeding is just to rotate the resource quality in this rotation where you conserve the, you want the one normal so you can serve the total amount of resources supply. Now, of course, given this result what you might suspect given that this is a marginally stable equilibrium, what you can suspect is that this is not structurally stable. And so what if the trade off is not perfect well you can introduce some variability, again either in the data evolution term or in the efficiency, which eventually affect is a sort of source of fitness difference. What I find is that, even for larger fitness difference is a fitness difference of 10%. You see that the same functional composition is obtained, and the same biomass is, is realized. And not surprisingly, what becomes what is traction is stable is diversity by introducing this fitness effects, you effectively select a subset of species for existing. And instead of having these exponential diversity, the diversity is linear in the number of resources. Any question at this point. Any question in the chat. Okay. So, now, this solution is exact for finite number of resources and for small fitness differences. And so we can calculate the total biomass the number of species and that these functional occupancies. And this is valid for any choice of the, of these parameters of these resources preferences, as long as they are bound. Of course, this is computation in much harder to explore. When you take more, more complicated. I don't really look for these resource metrics but all the results are the, are the same. And in this context we have R plus one free parameters so we have sort of encapsulated all this complicated parameters that appeared in the original model into one parameter per resource. These quality is effective quality plus one free parameter which is the cost, the metabolic cost. Well, just if I may ask a question on the clarification for the question for a star. So that's the one stand for a general list of species and the summer or the specialist. So the sum is basically you, you, this delta function, you look at which functions are what I said core so the functions that are at one. Okay. So you have the function that are at one. And so with these delta function you basically you are counting the function that are not at one so it's the functions that are not core. Okay. Is that clear. Yes. So, so this is a star is the number of surviving species. Yes. Okay. This is with the, let's say small fitness differences. And of course the fitness differences also affect these f star and the total biomass but let's say in the limit where these fitness differences go to zero, you can neglect their effect. Okay. And the point is that the case where there are no fitness differences are in the language of the structure and stable so you, you basically as soon as you perturb you have a lot of extinction because you have this margin stable equilibrium and you have a lot of again but Sorry, could you remind us what the interpretation is of if I start. If I start is the fraction of individuals that are able to grow. Okay, so what if these are is larger than one much. Well, so we have this one parameter per resource and one free parameter which is this cost. So what if these number of resources is much larger than one well we can assume that these resource quality which is effectively the only parameter that there's an emphasis around variable and distribution is, I mean, it's a couple distributed to some distribution and basically one can rewrite this equation, given a distribution of the of the resource quality, and one of things some sort of relations between, for instance, the diversity and the diversity of the activity so the, the discontent distribution function function enters in the position of these threshold here where resources are known core and then determines both the diversity and the number of species and therefore you get such diversity relationship which is a sort of equation of state of the of the local community which depends on the distribution of resources so in particular on the variance of resources quality and on the on these metabolic costs. And so what you do is that you can do some sort of a thermodynamic transformation where you ask what happens to the state of my community as I change the property of of of my system. So the same spirit of having isothermic or isobaric transformation you can take a dictionary with in Greek and you can define the isotomic transformation where you have a constant cost but you change the resource quality heterogeneity and you find for instance in this case positive productivity diversity relationship so you basically fix this parameter chi which is the cost of consuming one resource and you change how heterogeneous are the resources or you have these other transformations where you fix the resource quality and you change the cost and you find in this case negative productivity diversity. So this is just one example of a much richer phenomenology and this model and approach can be sort of extended to includes include a cost which is resource-dependent. We are working to extend it to beyond the chemostat to sort of describe serial dilution experiments because they're more relevant for experiments with alternative form of trade-off and I think that sort of this approach of infinite species pool can be used beyond the case of competition to other interaction types like host parasites. Good, so any question at this point? No question. Great. So one important point and comment about this is that in absence of fitness differences the dynamics select sort of goes to a manifold which is represented by the functional occupancies or these trade occupancies so how many individual what is the fraction of individuals in a community with a given trade and so depending on the initial condition you go to a given state on this manifold and so the position where you go of course on this manifold depends on the initial condition but the manifold is sort of what it's by definition general and one interesting point which is also was made by mission newspaper is that somehow the dynamics depends I mean full dynamics depends on species identity so whether a given trade of course together with a given with another trade so if you are able to grow on the resource side are you also able to grow on a resource j let's say let's call it a sort of linkage right and the dynamics depend on this variable so if you write the dynamics for the fraction of individuals that are able to grow on a given resource well the dynamics its dynamics depends on to which trades what sorry how frequently the trades you are focusing on i is in combination with trade j right and this is sort of mirrors in some sense niche overlap between species but on the other end when you look at the the the attractor of ecological dynamics the final attractor is completely insensitive to these overlaps and depends only on the trade occupancy okay so the attractor is completely insensitive of course the position where you go depends also on that because of the initial condition but the final attractor depends also on the function and so this is I think goes beyond the idea of ecology without species and is even ecology without individuals where somehow what matters is just the distribution of traits in the in the and when you have small fitness differences what happens is that you have a fast direction which brings you to these manifold and then you have a slow direction driven by fitness differences that sort of pushes you to one of the boundaries of these manifold where you have a subset of species coexisting and so if you think about different realization of the noise what you obtain is that you have different communities and you have different communities where you imagine that a different community has some sort of small variability in these fitness differences that corresponds to completely different taxonomic composition while corresponding to exactly the same or very similar functional composition and similarly if you think that these fitness differences are time-dependent so you you have some noise some sarcasticity then you have a fast direction that brings you to the manifold and then you have fluctuations on these manifold driven by these fitness that depends on time and again what you observe is that the population abundance fluctuates widely on these over time and these fluctuations are driven by fitness differences while the functional abundances are approximately constant over time because they corresponds to the fast direction good any point on these with us when before I say some sort of reckless ideas okay so now what is a large pool so the the point is that we started from I started saying from having much many more species than the number of resources make sure a combinatorial number and if you take a realistic number of resources you get of course a number which is completely unrealistic but I mean these being much larger I think should not be taken as a as a sort of strong requirement is more a question about how much the the pool the species pool is representative of what is selected in the local community and representative means whether the attractor can be satisfied by the species that are present in the in the regional pool and also I mean in cases where you have rapid evolution and there is a lot of recombination for this and through horizontal gene transfer you can imagine that sort of this state where you you find effective ecologically you find that these these these attractors can be can be obtained through evolution and I think that the interest on this is is the following so Guy was describing the case of random matrices and sort of the case of a species pool of the order of the number of resources with the random interaction as a new model and they think that the more successful new models are the new models when there are two new models that are alternative right and they think that the random interactions versus the infinite species pool represents some sort of two sides of or two very alternative properties of the species pool so in the case of random interaction the pool composition is somewhat the least informative of ecological dynamics so is really an informative of the ecological dynamics while the infinite species pool there is some information or it's a sort of there is a coupling with the local species dynamics and when you have the random interactions basically your attractor is as close as possible according to some metric to this functional manifold while if you have an infinite species pool you have you are on the functional maintenance so in the case of random interactions the property that emerged in that context are the properties of how far and how close can you be to the to the manifold where you cannot satisfy given the species pool the manifold exactly and in some sense and I'm not really I don't really understand these two sentences that I wrote but I think that there is something there is that in the case of random interaction the functional composition of the community is informative of the species pool because it's sort of carrying these these information about the limits of diversity of the species pool while in this case of the infinite species pool the functional composition the community is itself an informative of what is sort of a necessity as an informative of what was there in the species now of course the key challenge is to is that we should go beyond to having the species pool as a initial condition and try to understand how the the species pool is is itself determined and how it is the outcome of the evolutionary processes at multiple scales and the key aspect to characterize in respect to what I said is the functional similarity with the local community where these similarities the ability to to to respect the constraint imposed by by dynamics so a sort of conclusion so the not the take home because they're all at home so they keep at home messages that somehow apparently complicated or convoluted ecological models with many parameters and and complicated structures can simplify in the limit of these infinite species pool and contingency is exactly what makes the models complicated uh that even in this limit which is uh in some sense very simple there is a rich microscopic phenomenology and the functional composition emerges as the natural variable to describe this community much more and and I think that the fact that emerges the national variable is way more uh deep that the fact that you can predict uh functional rebounds it really means that what matters for ecological dynamics is that uh functional composition so uh with that I'd like to thank all the people that were directly or indirectly involved in this and thank you for your attention