 engine with PD, actually, from Brazil. And I'm gonna talk about this project that was developed here. It started to, it was started to develop here last year with Yacopo, who had a close correspondence with Mercedes-Pasquale, and then we started to study this model. It's mainly about consumer-source models. So it's gonna be more or less like a follow-up of the tutorial that we had last week with some things new on this kind of framework. So consumer-source models. So it was something that started in the 70s, in the late 60s by MacArthur, and it's a very well-known kind of system that tries to explain the ecological dynamics of a community that is competing for the same resources. So the kind of interactions that you have between among species is due to competition for resources. So we start simply with a community. We have our microbes in this community that are also some resources in the community. And you have this dynamics of the abundances of this microbes are changing due to the competitions for the, due to competition for the resources and also the concentration of the resources are changing because of the depletion due to the consumption by the resources. There are many things that we already know about the systems. So this is a very random set of references in the field, some very new references on that. But the fact about this kind of model is that they are always including this important thing that resources are limited and then richness is limited. So okay, I can say in many of these results they are saying that okay I'm going to say that the number of resources is infinite and the number of species is also infinite but there's also this limitation of I cannot never overcome competitive exclusion principle in these kinds of systems. But the interest question is never posed. Like what if the number of resources is not limited? So what would happen in a kind of system? So this is a kind of weird question but it can help us to try to understand different things and these different things can range from something that comes from epidemiology. So we can look for pathogens that are very, that have many, many strains and this is a problem for these pathogens because they are always escaping immunity and we have problems for developing vaccines and efficient treatments for these kind of pathogens. And in this paper by Mercedes and He, we have this, they propose this diversification threshold which is like a basic reproduction number but for diversity. So above this kind of threshold that pathogen starts simply to accumulate many different strains and then it's hard to treat. It's hard to approach this problem. But we can also take a completely different approach for trying to understand why that question of a limited number of resources can be important. So we can go for diversity in marine communities. So we see from fossil records that diversity in marine communities started to increase with an exponential growth in the beginning but then it reached a nice stagnation phase and try to understand why this stagnation phase appears in the fossil record is actually very controversial. So that's a question that you can try. Okay, why these things are happening? So there is this accumulation of resources due to evolution in my system and why it's happening. So to try to attack this problem, we propose this model. So it's an eco-evolutionary model, okay? And it's very based on the works by goods. So it's not based but it reminds a lot of the year model, okay? With some difference. But the things go like this. We're gonna start with a consumer resource model on a chemostat. So we're gonna have our microbes in this chemostat and there is this constant supply of resources and the number of resources that I can have in my system is unlimited. So I simply have a set of microbes. They have their genes. They can consume the resources that are there. But if appears a new microbe that are able to metabolize a new resource, okay, there's a new resource for you. That's what you're saying. Oh, with an unlimited number of resources in the system. And we have our strategy matrix that for some reason I called it consumption matrix. I think the reason is that because it's only one in zero, so it's exactly what I'm consuming or not consuming, okay? So in the next step, we are gonna evolve the psychological dynamics through a set of equations. I'm gonna talk about the equations. And after that, after it reaches an equilibrium, I include a mutant in my system. And then this set of, I run again these equations, these ecological dynamics and the equilibrium of my system chains, the abundances can change. And this mutant, as you can see, it's like a new line as a new column in the strategy matrix. And then a new mutant tries to invade the system again and I'm always just doing the cycle. I include a mutant, I run the system up to equilibrium, and then include a new mutant and here we go. Yes, there is one constraint I'm gonna talk about it, that it appears very naturally, that this question is very natural in the system. And yes, there's gonna be one, okay? Or not, you can include or not, but yes. You can constrain or not, but it appears very naturally, yeah. So in this model, the fact that we have infinite resources means that every new mutant that you insert can technically produce a new resource that other species can use. That's a good question. It can, but it's not necessarily goes because you have different types of mutation that can affect the system. So you can have pure fitness mutations. So my new microbe is gonna simply change the fitness in respect to its ancestor. My new microbe can simply acquire a dean by horizontal-dine transfer that was already there in the pool of deans in this population, or it can be at the novel mutation, which means that it's simply acquiring a dean that was not present before. So this microbe's able to consume something new, okay? I'm sorry, what you're saying is that you can't produce new resources that were not in the basic... The resources was already there. That's the hypothesis, the resources are already there. I just cannot see it. There's no dean from metabolizing it. That's the idea. The mutations are able to give you the ability of metabolizing a new resource. That's the idea, okay? So going to the equations now. So it's a simple consumer dynamics model. So we have our equation for the consumers. So that A over there is my strategy matrix, okay? It's a linear consumption of the resource concentration. And there's also the log fitness factor there. So that explanation accounts for fitness. And it's a global fitness. Fitness here is not directed to the consumption of a given resource. It could be a modification in the model. And when you look to the dynamics of the resources, we have this constant inflow of resources, eight I for every source I. And you also have the degradation rate or the depletion, the dilution. That's the thing that's washing out the resources from my chemostat, okay? There's also this metabolic trade-off. So the number of resources that I species is consuming is constant, okay? We are not changing the number of consuming resources. So at the moment that I acquire a new resource to consume, it means that I lose one. Yes, it's always constant, okay? So I could, of course, relax this thing and put some, the trade-off as a death rate or anything else that are many ways of doing this, okay? And I'm confused about this rule because you were saying the resources were always there, just passively lying in a chemostat with nobody to utilize. Why ability to consume a new resource means that one of the old ones has to go? I mean, you're kind of not changing the supply to a chemostat because that's the only way a resource can disappear. Yeah, it's like, it's just too costful. The cost for me to consume many different resources is too high. Oh, it's not lost from the chemostat? No, it's lost from my machinery. Yes, yes. Otherwise I think that the evolution will be few strains that consumes everything because it's more efficient. Also, yes, if I don't like this cost. So these Y-sigmas, they're randomly drawn and it's the same for every species or? Okay, so I'm gonna start with the system with a community that has always no fitness difference. So everyone has, in terms of fitness, everyone has the same ability of consuming the resource, but then because of mutations of fitness, so this thing can acquire a just a small difference. It could be higher or lower than its ancestor and it's just acquired from a normal distribution. That it has the center on its ancestor and just a small deviation. So to treat these equations, what you're gonna do is simply to, we're always interested in looking to the equilibrium of these equations so we can do some stuff. And the first thing that you can do is just change timescales. We say that the consumer dynamics is too fast. It's very, very fast. So we can just set the second equation to zero and then we include our result in the previous one, okay? And then we have this final equation. That's the equation we're gonna look, okay? We're gonna treat this equation here. So, and then our question is, and the long time limit, I mean, after I include many, many mutants in my system, what's gonna happen with the number of species, the strains that I have in my system or also what's gonna happen with the number of active resources in my system? I mean, there was resources that are really been consumed by a given species. So is this something that's gonna reach some limit? Is this something that's gonna explode? So that's the question that you're trying to answer now. So we have some analytical results on that equation. Okay, so the first thing is something about uniqueness and coexistence of the solutions of that equation. The important thing here is that it can, if I change, if I use that product there, the delta, which is the death rate of my species of my strains times the exponential of minus the log fitness, then if that guy is different for everyone, then I can have a unique solution and my system is never going to overcome that competitive exclusion principle. But why do I want that? It's just because it's not dependent on initial conditions. So I can really run simulations and all the variability that I had at the end is not due to different initial conditions. It's due to my naturalisticity that I'm using for mutations. So just to talk a bit about the math behind this thing, it's nothing hard. We have a Lyapunov function for the system. So we can show that it has this convex function that it's always decreasing over time, over the dynamics. It's gonna find a minimum. The only thing I need to try to show is that this minimum is non-degenerative for my solutions. That's the only thing. And I can always do that thing if I have my strategies there linearly independent, if I am gonna have the number of species, then it's for sure not true if I have more species than it is source, okay? And the second part of these things, how to prove this non-degeneracy, we're gonna look to the linear equations that can simply come from my system of equations. I simply do use these tricks of finding new things. There's a minus missing there and the exponent of the log fitness. We just analyzed these things. It's something very, just purely linear algebra here, okay? So then as I mentioned, that's the important part. So we always try to treat that thing as something's different. Just for a simulation perspective. And there's also something different that we can prove that's quite trivial, but it's important for us, again, thinking about simulations, that I can only miss resources in the sense that there's a set of species that are consuming your sources in my system and these species can all go extinct, so I'm gonna lose the gene from metabolizing that resource. It can only happen if I have a dilution rate of my chemostat that is non-zero. It's sort of kind of trivial. I'm saying that if I'm never washing out things, my resources from my chemostats, they are always remaining there, so why am I gonna lose it? But that's a result that can make it formal. The math behind this thing is just to dis-consider all the other resources that a strain is consuming because they are always adding something new to you and you're just gonna say that that resource is gonna be consumed by a single strain and you prove that the rate of change is positive. That's it. So if you want to go to negative change and negative rate of change, you need to pass for zero. So you're gonna stop the evolution there, okay? You can lose resource from the system. That's the only way you can lose. I'm not saying that you will lose, but you can. Then in the simulations, we see that we lost, okay? So just to summarize these two things, so from the uniqueness and coexistence theorem, we're gonna always try to work with death rates different for everyone. We're just gonna draw the death rates from our normal distribution. It looks kind of, many people who are working in the system say, okay, but we're always doing these things, drawing things from random distributions and what's the new here, the novelty here? The thing is that, okay, I'm trying to find the most simple system to simulate and now I'm just saying that no, you cannot look for the most simple one. You need to make some stuff random there, okay? The second thing again is choosing a value for the dilution rate, so it cannot be zero again. So the simplest case would be just choose it zero, but no, we cannot choose it zero. We need to put some value if you want to check for different responses of my system, okay? There's another result that we can do here. It's just a sufficient condition for invasion of a new resource. So if your new resource, quality, the inflow of this new resource is greater than a given threshold for that species, it's gonna invade it. That's it, so, and this has, to prove this thing, it's also not hard, we also neglect things that are not, the other resources are only focused on these things for the evolution. We can show some stuff again, that it can be positive if that exact condition is satisfied, but the important thing of that equation is that it leads to this important result on the evolution of the system that says that if I have fitness evolution, I can always increase the number of species in my system. That's the important result here. So just to see that in the previous equation, we just see that the log fitness there, so if I increase the log fitness, the resource quality can decrease as much as you want. So you just need to have a high fitness to be able to the next random resource invader system. We just need to give it time. So let's go to the computational results to see, and here comes your question. So the fact is that if you just look to the equilibrium of that equation, we can see that the carrying capacity of my system can increase with the number of resources that have an inflow. So if this thing is just an open thing, I just, as soon as I find a new resource to consume, the carrying capacity of my system increase, so I can expect that I am always increasing the richness of my system now, because everyone is able to increase their abundance once the carrying capacity is higher. So, question? So we're gonna divide our simulations into case. So the first case, we're not gonna fix the carrying capacity, so the sum of the inflows of the active resources is not fixed. As soon as I discover new resource simply, I let it invade the system, I let it to be there, so the richness can increase, but I can also fix that guy, so I can renormalize as soon as a new resource appears, I just renormalize, all the inflow rate is in a way that's always fixed. I have my critics to that, but we're gonna reach that. So this is the result for in the case that we don't have a fixed energy. So what we see is that when I don't have energy conservation, so that's sigma y here is just how much the fitness mutations are changing when you have a fitness mutation, how much they're changing fitness. So what you see is that even in the neutral case where I don't have fitness mutations, the richness is able to increase. So in this case, there's a de novo mutant, so it's carrying a new gene in the system and it's able to be there, but even so I'm losing resources there. So it's not that the invasion of a new mutant is doing nothing in the system, it's doing something. It's shaking the equilibrium of my system, I'm losing things, but at the end, the richness is increasing, okay? But now, if I use the trick of saying that the energy of the system, like the sum of all the inflow rates is a constant, then a few different things can happen. So to start with in the neutral case, now there's a limiting, the number of resources that can evade my system and the second thing is that, okay, now I have fitness evolution. As expected, fitness evolution is always overcome, effects of degradation is always overcome, the effects of dilution and then I can see the resources accumulating in the system as also the number of strains are accumulating in the system but now in a linear way. So it's much lower than before. We can see actually this value on the number of resources. So we just use that lemma that you had for invasion, using that you can find the maximum number of resources that my system would acquire and it really matched the simulations when you had the neutral case. For the other cases, then you need to do this balance of all the mutation rates to try to see at which rate I'm actually including the resources and then you could see what would be the maximum number of resources for that time, okay? K, K, K, K, K is the number of resources that a single species can consume. It's the metabolitate of parameter, okay? So it's like I'm saying in the equation below, I'm just saying that I'm using the sufficient condition for invasion for all the resources that are strains consuming as if there was no competition for this resource. So that's why it appears a K multiply and because I'm neglecting the competition, there's the result is approximately to be. So, but I don't like this thing of limiting energy like this limiting summing all the inflow rates and saying that it's a constant because it's something weird, the inflows were already there. I have my experiment, there's already an inflow of resources there and then out of nowhere, a mutant appears and it changes everything. So it's like I'm the lab guy, just watch, okay, there's a mutant, let me change everything now. So it's not nice, this thing. It's really, but I can try to include this energy conservation a different way. So the other way that you can do that's due to cross feeding. So the energy is only appearing with one, a given set of resources and I am consuming this resources, metabolizing it and what's appearing from that metabolism is actually my new resource. So I can do an infinite chain, an infinite tree of this metabolites and these are my resource and the system conserves energy. It's gonna conserve all the inflow rates, okay? Actually, I am always limiting because the last guy I'm not using that energy, okay? So to do that, the consumer dynamics just changed a bit because we just need to say, okay, it's not all the biomass that I'm consuming from a resource that's actually becoming my biomass. I need to consume a fraction L over that and then for the consumer dynamics, I'm just saying, okay, I'm consuming that thing but now there are some inflow of that resource just coming from the metabolization of another resource. That's the last term in my equations and this is a general equation that are many things we can, okay. So this is the long equation. We can just do many things here but we can simplify this thing. So we're gonna use a linear chain of resources. So I just have one resource being converted in another resource. So I have my set of bugs there. They are consuming a resource. They are secreting the other resource and so on. When you do that, we just can, whoops, we just change our set of equations in a very simple way and now we have the final equation. It's this very ugly thing but the important thing about this ugly stuff is that that term there looks like an inflow rate exactly as the previous equation that we have. So if we kind of just simplify this thing because this equation does not have all the properties that we have, all the analytical properties that we have calculated. So we just change things. We just say that our resources in this linear chain are exponentially decaying their quality, okay, through this chain of consumers. And energy in this case is still limited. I just sum over all the infinite resources. I have a limited energy here. And in this case, we can show also a sufficient condition for evasion and that's obesity equation. And then we can go to the simulations. And the simulations is something weird because now we are just running simulations and we're seeing that there's a limit in the number of active resources even if I have fitness evolution. That was not what we had before. And the thing here is because when you're doing simulations, when you're doing computational stuff, we need to decide what's the threshold for zero. So what's an species that went extinct in my system? So if we just use this constant threshold for extinction, then you find this result. It's exactly having that sufficient condition and including a constant threshold there. When you do that, we really find a maximum number of resources that can be acquired in my system. That can actually not be lost in my system. And that's there. That's exactly what you see. You change this threshold and you actually change what's the number of resources that are there. But then if I allow my extinction threshold to evolve and I'm using all the equations to actually see what should be this evolution of the extinction threshold, and then we can simply see the evolution of this. Yeah, but in the real world that at some point there's one cell and you're okay. But so isn't the, I thought you were gonna say if you're not demanding a linear chain, but you allow sort of let's say more random graphs, then L is gonna be much less, right? There's gonna be less steps that you're down from the most abundant guys. So wouldn't that sort of give you any more? Yes, that's something that I want to do. I still want to do that to just include a very random graph or a very random tree just to see what can happen there, to see what's the result. But I think that things would be slower just that or maybe faster because if I am acquiring resources that are on the top of the tree, now they have a, they are having much more energy than in this case that they're always harvesting a very low energy. Yes, you can change that. Yes, yes, that's true. That's true. Yeah, no, that's something to explore. That's something to explore for sure. Go, I'm really reaching the end. So another thing that's quite interesting here is when you look to that sufficient condition for invasion and that thing, it depends on the species that is trying to invade my system. So it depends on its fitness. But if I just change that equation to the maximum fitness in my population, it really fits all the curves. So it's really like the highest fitness strain is the one that's driving all the invasions in my system. Other things that you can do, we can try to study in the cross-fitting system is to try to see how the system is fragile to random events in the environment. So I'm not gonna explain what you did here, but the fact is that I'm gonna have many species that are consuming only one resource. So if it is a species, a random species simply disappear, the following resources in this cross-fitting chain are not gonna be produced and then something can happen at the end. And when you see the simulation, that's really something that's happening. So I'm now randomly extinct in a species. It's an environmental fluctuation. I simply start to see the number of activity sources and species, they are simply more or less reaching something constant. Is this still a work in progress? So there are many simulations just to confirm these results but it's something that you already have in mind. Like it's really gonna acquire some equilibrium in the system. So just to sum up, just to finish this thing, to finish this story. So we start with the question, what if I have an unlimited number of resources in my system and to approach this question, we use this consumer resource dynamics with an unlimited number of possible resources in my system. We find that the evolution of fitness, the fitness evolution is always able to overcome degradation effects and that quality effects. And then we separated our results into parts. So we're gonna have the systems that have limited energy, those that don't have limited energy, the systems with no energy conservation, they're always accumulating resources even in the neutral case. But if I have energy conservation then we can see that the accumulation of resources is lower and I can also have different ways of conserving energy like cross-fitting systems and cross-fitting systems, they really show us the problems of the simulation thresholds, how we do the simulations, how we perform some medical stuff and they are very susceptible to random events so that are happening in the environment. And there's a very nice to do least things here. We can, you can give ideas also, we can include dilution stuff here that are many things everyone can, I don't know, could put a box here, just putting our ideas. This is, there are many things that we can do research and investigate in this kind of system. But the take-home message is that in this kind of model with its limitations, fitness evolution always allows diversity to increase. If you want to limit to diversity in this kind of things, you really need to put some exogenous effect in the system, exogenous actions in my system. So I just want to thank all the organizers, thank you my funding institution and thank STP for receiving here and thank you all. Right, very, very nice work. I'm just trying to put it in context of what I know. So in this last model with cross-fitting, right, where you have a cascade, it looks very, very similar to what we did with Akshit Goyal in 2018. The only difference is that we fed the ecosystem with only one resource and let it cascade down. And we also saw this increase in diversity as a function of time. So what is different in your model where you have, as far as I can tell, the main difference is that you feed it not with one resource, but with a certain number of resources in the beginning. And then everything else is generated through this cross-fitting cascades, like the new resources appear at the first, second, third-trophic levels. Yeah, I don't know. I think that the only thing that's changing here is what are the mutants doing? So if I have a gene that's lost in my system, so all the forward part of the cascade is gonna disappear. And then what's the next mutant that is gonna appear in my system? And here I am sure that the next mutant, the novel one, it's gonna be the one that fits in that position just to reconstruct all this cascade. I see. Maybe this is the only change. Got it, got it. These BKIs, right? They determine how much of each resource, K, each strain, or no way. Yes, BK sigma. Okay, I'm not, are producing. So can those evolve? And this question, no, no, in the system, no. In this model, no. We are not putting it in there. But the idea is that some guy, while he's eating, he's excreting something. For sure. For sure. And in that case, probably, maybe this is also subject to evolution, and maybe this can then destabilize the thing because maybe you want to evolve these things to zero. No? For sure, for sure. Yes, the fact is that I can change these things each one of them I can change. So, and they are always related to fitness mutations. It's just my efficiency in converting resources in one to another. It's my efficiency in harvesting resources that's always related to that. In this case, the fitness mutations are global. So if I simply, I can change, I can simply put the fitness mutations directed to a given resource. So I'm gonna change only the efficiency of the streams. That specific resource. This is something that can be done. We discussed that, but we thought that it wouldn't lead to something new, actually. If we look only to the, to this conversion rate, that can be something new, for sure. Only to that, instead of just putting global fitness on efficiency, different kinds of efficiency in harvesting the resource. But that's, again, the general result. So if I still allow this fitness to increase, I'm still gonna allow a species stream rate. I need to have a mechanism to stop that. So, for sure. If I simply get one of these conversion rates and set that to one or to zero, then I'm also changing how everything is happening there. And then for sure. That's, then that's what we're not looking. That's what it is. If you allow somebody to evolve that eats everything and doesn't discreet anything else, then they can always push out everybody else, right? But that's exactly the question that we're trying to not find. The answer that we're trying to not find, because this is trivial. We're looking for something not trivial. And then at the end of the equations, we really show the triviality of this result. That's what we see. That's just that. One more quick question, and then we go to have a coffee break. So, as a small comment, when you have the same e to the y sigma for everybody, you can just rescale. And do you have a consumerist model with a slightly different death rate delta sigma e to the minus y sigma? So, since you have the same metabolic trade-off plus five, if the death rates are equal, you have more consumers and resources in some cases. Yes, but at the end, what you're trying to see is only the equilibrium system. It's only the equilibrium point. So, when I'm changing timescales of the ecological dynamics, it's not changing actually the results that I'm seeing. Because it's like the timescale between two points. No, but just to say that, when you say if we choose the same death rate for everybody, it's not interesting to us. Yes. Right. There are no more questions now. If somebody wants to ask something, he can ask or she can ask on the coffee break. All right. We meet here at 11.05, all right? For sure, please. Y is usually yield. Every consumer resource, what I saw, Y means yield. Don't use it for fitness and other things.