Population persistence and species coexistence in the face of environmental uncertainty
Schreiber, Sebastian
The dynamics of species’ densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. I will present recent theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables, and will illustrate their utility with applications to models of eco-evolutionary dynamics.
 So I will start with this quote. Some people have seen it often before. It's this quote from the British poet John Gay who wrote, Less men suspect your tale untrue, keep probability in view. And in this context, less biologists suspect your model untrue, keep probability in view. And an important probabilistic aspect of ecology that we've heard about already, especially in Karen's talk yesterday, is that there's environmental stochasticity. In other words, fluctuations in demographic rates, like survivorship, reproduction, growth, due to fluctuations in environmental factors like temperature, precipitation, resource availability. So the basic way it works is you fluctuate in an environmental factor like temperature. You have potentially a nonlinear response of your fitness to that temperature. And then that creates fluctuations in fitness. Now, the main question I look at today is sort of mathematical methods to address this question, which is, how does environmental stochasticity influence population persistence and the maintenance or disruption of diversity? So there's been a lot of theoretical work. I'm going to try to focus on some mathematical techniques for analyzing models with environmental stochasticity in a rigorous fashion. And the main reference is a recent JMB paper with my longtime collaborator, Michelle Benim, from Switzerland. So I'm going to start really simple, which is a geometric random walk. So you're keeping track of the scalar quantity, the density of the population at time p. Everyone in that population has the same fitness at any given time step. And I'm going to assume just for simplicity for the moment, those fitnesses are IID across time. So then you get the similar simple stochastic linear difference equation where the density in the current time step gets multiplied by the fitness. That gives you the next density. It's trivial to solve for solutions of that linear difference equation, just the iteration. And then if you want to understand what can you say about the growth of the population, you can just apply the law of large numbers by taking the log of the density, dividing by t, then that product, turns into a sum. And then with probability 1 under the right assumptions on the first moment on the log of the fitness function, you get the limit as t goes to infinity is given by the expected log of the fitness function at any point in time, such as time 0. And I'm going to call that throughout this talk the realized per capita growth rate of the population. So r will always mean that throughout the talk. Now, if r is negative, and we run a model where we're randomly sampling the fitnesses and do lots of simulations and plot them in a semi-log plot, we'll get something like this. So you'll see the tendencies to go to negative infinity and log densities, i.e. extinction. On the other hand, if r is positive, you get the opposite trend. So this idea of looking at the expected value of log fitness is not new. In fact, it's very old. It goes back to the work of Daniel Bernoulli in this treatise on economics that was printed in 1738. And in this treatise, you can find loosely translated the following quote, which is dealing not with populations of organisms, but populations of money. So in this case, the fitness terms are the returns on your money. And if you want to know in the long term how fast does your money grow, you take the terms, the fitnesses, multiply them together, and take a root corresponding to the number of terms, which is exactly what you do in the model I described before. If you have a finite number of fitness values that you're taking on with equal likelihood. And so then if you exponentiate that r, you get this geometric mean of fitness. And I'm sure everyone's familiar with. And a key property of the geometric mean is that the fitnesses have some variance to them. Is it strictly less than the arithmetic mean? And several hundred years later, in this classic paper by Lowenton Cohen, they made that observation. And they got a P&S paper out of it, which is that you can have the expected population size to grow infinitely large. But you're going asymptotically toward extinction in any realization due to the difference in geometric and arithmetic means of fitness. But these models, of course, that I've described so far, are way too simplistic and probably get chastised by Kenneth Holding for believing that could go on forever in a finite world. And certainly, I don't want to be called a madman. And given worse, perhaps an economist. Apologies to any economist out there. So of course, people have developed models where you allow for density dependent feedbacks, like negative density dependence. And there's various papers on this. I'm just citing two of them. But one of the key features of these results for these density dependent models is if you look at the expected value of the log fitness at zero density, that tells you whether you have persistence or extinction. And I'll say what that means. Persistence means exactly a little later. And of course, we made models that are more interesting because we wanted to diversify the objects that we're looking at within a population or populations. And one way of diversifying the set of types is to account for differences in age or in stage or spatial location. So then we have these structured population models. And even understanding their dynamics became a really important theme of stochastic demography. And there's this delightful book by Shira Parj Tuljapokar about summarizing what was known as of 1991 about these random matrix models. And a key fact of these random matrix models is that there is also a realized per capita growth rate. And that's due to some work of Ruel as well as others. And one of the things that's sort of amusing about Ruel's paper, which is in a, I've forgotten it's not the annals of mathematics. I think it's the American Journal of Mathematics, which is a very good math journal. He did the analytics of understanding that this realized per capita growth rate actually is analytic with respect to parameter variation. So you could write down a derivative, but he actually calculated the second derivative wrong in that paper. And Tuljap independently calculated it correctly. So this is a very famous small variance approximation of the realized per capita growth rate. And notice again, fluctuations, at least in IID environments, also reduces the growth rate of the population. And again, to avoid being called a madman or an economist, the results about what happens when you add negative density of dependence. Again, the punch line is if R is positive, persistence, R is negative, extinction. OK, but how about looking at diversifying models in terms of any different types that are interacting? So you might be wanting to know, if you have different types, how do you maintain that diversity? And there's sort of two classes of models that's been looked at fairly extensively that's been in the context of population genetics, maintaining a polymorphism, and also in the context of ecology, maintaining species coexistence. So some early theoretical works. I'm going to distinguish here and now theoretical work from mathematically rigorous work. Apologies that people find that offensive, but hopefully knowing that there is a distinction. So there was theoretical work on protective polymorphisms, so how fluctuations in fitness can lead to maintenance of polymorphisms in populations. And some of that work was initially done by Haldane in this paper in 1963 for haploid populations. And then John Gillespie in this delightful paper that I'll pay a tribute to at the end of this talk did stuff for diploid populations. And in fact, he originally discovered, although he didn't realize it, the storage effect in the context of diploid populations. And then, of course, in the context of ecology, species coexistence, there is the quite famous storage effect of a mechanism for maintaining coexistence between competing species through fluctuating environments. And of course, the champion of the storage effect was Peter Cheson. And it's gotten to the point now where we have data sets that we use to calibrate models, that we even do simulations instead of math with to understand how strong the storage effect is, as discussed in Elmer's paper. Now, the mathematical work, surprisingly, was fairly limited on this topic. So the main paper I can think of is a paper dealing with two competing species. These are discrete time models. And it was worked by Cheson and Elmer, both appearing in 1989. And then they were not much done for these understanding persistence of interacting populations. But quite interestingly, an invasion-based math for deterministic models was done for any number of species, any type of interactions. And what I mean by invasion-based, I just mean that you're looking at these realized that your growth rates for one of the species is rare, and using that as a way to determine whether the species coexist or not. So there was all this work done for deterministic models. And most importantly is the paper of Joseph Hofbauer in 1981 and then Butler and Waldman in 1986 that took different approaches to this question. And what I want to do is just show you that we've advanced a little bit in the stochastic theory because we can sort of recover the results of Hofbauer for stochastic models. OK. So what are the stochastic models I'm going to talk about than the ones of this form? So I'm going to keep track of k distinct populations. There might be different genotypes. There might be different species. And then xi is going to be the density of the i-th population. I'm going to have these auxiliary variables, which I'll explain with the odd in a minute. OK. And then again, a sequence of iid random variables. So each species is updating by taking its current density multiplying by its fitness fi, which depends on the state of the community. So x is the vector of densities community. Depends on this auxiliary variables. And it depends on the sequence of iid random variables. And then you have some dynamics for the auxiliary variables that depends on everything. So first thing you might ask, what are these auxiliary variables? I won't do the explicative here. So there can be lots of different things. One thing is we were talking about yesterday is the importance of auto correlation. And I said I'm using iid random variables, but you can encode for auto correlation because the y can be almost any type of Markovian forcing dynamics. So there's some deep theory that shows that almost any type of Markov chain can be realized by a random map with iid random variables. So in particular, you can get a multiplicative auto regressive model using this g function. You can get any finite state Markov chain using this formalization from forcing the dynamics, the ecological dynamics. You can also account for any type of population structure because those auxiliary variables can be keeping track of the fraction of a given type within a population. So it can have spatial structure, a structure, any combination you're of. You can also do trade evolution. So yi might be a trade of the iid population. And then you might have maybe a breeder's equation describing how that trade's evolving. Or you can also do environmental feedbacks. So you might have plant soil feedbacks. So maybe y is the soil dynamics. Or you might have some environmental feature that's characterized by a y that is being engineered by the species and the population. So any of these things can be encoded in these types of models. So it's fairly general, is the point. So for these models, what do I mean by coexistence or persistence? So there's two definitions. One of them is sort of one that I highlighted initially because initially I could only prove theorems for this definition. And the other one was highlighted initially by Peter Chesson in the 1980s. So I'm going to talk about the definitions that I worked with first because that's the one that I think is somewhat more natural. And it deals with the fact that often we're looking at a single realization of a community or a population. So we'll say persistence almost surely means for almost every realization of a model. And you can just look at the right and ignore the left. And so if we have a model with no auxiliary variables and two species, x1, x2 are their densities. And we can say, well, this gray region is within delta of extinction for one of the species. If you run the model for a long time, you can ask what fraction of time does that simulation spend in that gray region close to within delta of extinction set. And stochastic persistence says that fraction of time goes uniformly to 0 with respect to initial conditions. This delta goes to 0. That would be stochastic persistence almost surely. And the complement is the ensemble point of view. So you run many copies of your model, where you have many replicate experiments. And so stochastic persistence and probability, which was introduced by Peter Chesson, is you take many realizations of your model, run them far into the future. So I'm just showing the final time point for those many realizations. And you ask what fraction of those realizations far into the future are within delta of the extinction set. And you want to insist that that fraction goes uniformly to 0 with respect to initial conditions as delta goes to 0. If that happens, you have stochastic persistence and probability. And fortunately, for all results I'll talk about today, when you have one, you have the other. So. Thank you, Kevin. What was that the other reason? Let me adjust that later. I'll have to think about it. I think you can. So Peter Chesson, early on, explained why he thinks this is the right definition of persistence, stochastic persistence. And basically, he realized that many models have good years and bad years. Good years, the population increases. Bad year, the population decreases. And therefore, with probability 1, you're going to have an arbitrary long sequence of bad years, which means you can get to arbitrary low densities. And you don't want to exclude those models from what you're thinking about. OK, so now how do you check the stochastic persistence condition? You do with the sort of invasibility criteria that many people might be familiar with. So what you think is, OK, suppose I have all but a subset of species become very rare. So I have, like, say, J1 through JL that are not rare, and the rest of the species become rare. So that's infinitesimally rare. So they're not there. So we're worried about them not being able to stay in the community. And so the rest of the species, the ones that are there, the J-sub-i indices, they're going to probably approach some sort of stationary behavior. Let's assume that that stationary behavior is even robotic. That's the only stationary behavior we'll have to pay attention to with some law of mu. So that means visually, here's all these species that are not rare. They're approaching some sort of stationary behavior, as well as the feed boxillary variables approaching some stationary behavior. And now you want to know, is one of the infinitesimally rare species able to increase? So what do you do? So you look at this axis with the infinitesimally rare species. And you want to know what is its rate of increase. So it's just this linear question. And it turns out all you want to do is take the log fitness of that species and average across all the fluctuations. So that's the fluctuations in the chi's and the fluctuations in the community state, as well as the feedback or the auxiliary variables. So that z is this whole combination. And if that quantity is positive, that means this infinitesimally rare species would tend to increase exponentially. If it's negative, it would tend to decrease exponentially. But you can also compute this quantity for all the species that are not rare, that are present. And it's not that easy to show. But one can show, and it's fairly intuitive, that their per capita growth rates are always 0. They're present in the community. They're not going off to infinity. They're not going to 0. So on average, it must somehow be 0. OK, so now how do we use these things to state a theorem about persistence? So let me first define the set that we're trying to avoid. That's the extinction set. So that's going to be the set of points in the state space where one of the species is not present. And here's the criterion that's going to be used for stochastic persistence. It's the exact criterion that Joseph Hoffbauer used for deterministic models, very slightly differently, because he didn't do it with respect to measures. But he can do it with respect to measures, even in the deterministic setting. And what it says is, if for each species, you can associate a weight, p1, p2, p3, and so forth. They're positive weights. They could add up to 1, if you want. And you take that fixed weighted combinations of the realized per capita growth rates of the species. So that's saying, somehow I'm taking some community average of the per capita growth rate. That's roughly what that is. If that is for that fixed weight combination, if that is positive for any sort of ergodic behavior of a subset of species, so not all species, then you have the following theorem that if this uptribe of the whole positive condition, then you have stochastic persistence and probability and almost surely. And you even have estimates about how much weight is being put near the extinction set, which I won't show here. On the other hand, if you can flip the condition around, if you can find weights, so that this weighted combination of the per capita growth rates is always negative whenever there's a subset of species supported by the ergodic measures, then you get the opposite result, which is the extinction set is locally stochastically stable, meaning if you start sufficiently close to that set, there's a very high probability that you asymptotically approach that set. And again, these are models that are continuous states, right, continuous densities, so you can only asymptotically approach them. OK, so that's the theorems, or those are the theorems. So let me now briefly show you a few examples, just to give you a flavor of how you can use them, and how they might even relate to things that you're familiar with. So the first example I'm going to talk about is a annual plant model. And I'm not going to write down the model. It's just a model of an annual plant, keeping track of the seeds, OK? And it's a model of two competing species, and it's a model whose can be analyzed using results of a wonderful little piece by Peter Cheson in 1988 and some lecture notes in spring of ergodic. And so it's two species that are competing, and he asks, how do I use that criterion that I mentioned before? So we have species one here. Right now, I'm not having any auxiliary variables. And let's assume each species by itself can persist. That would mean R1 at the Dirac measure at 0 is positive, and R2 is as well. So they both can persist by themselves. You have something like this, OK? Now you want to know, what about each species invade when it's rare? So you have this quantity here. I call this mu1, this mu2. We want to know it's R1 of, so R2 of mu1 greater than 0. We want to know it's R2, R1 mu2 greater than 0. And it turns out this would be the classic mutual invasibility condition. Why is that the same as what I wrote on the board before? Man, that's not obvious. But what do we need? We need to weigh it's P1 since it's R1 of any measure. That's P2 times R2 of any measure is greater than 0. So that's trivially true at the Dirac measure of the origin, because these both are positive. Here, if we look at this measure, by that result I mentioned, the R1 for species 1 is 0. And that's just saying you need R2 to be positive for persistence and similarly for this one. So the mutual invasibility criterion is identical to the criterion I put on the screen earlier. So if we apply that to an annual plant model where it's you have neutral coexistence without any environmental stochasticity, you might not really think, oh, if I add environmental stochasticity, you can get sort of a drift along that line that might lead to extinction. That's not the case. But what you get depends very much on what is fluctuating in time. So you can use the results of Chesson to show for large or small noise. It doesn't matter if it's the size of the noise. It's just there is noise. That if it's fluctuations in survivorship of the seeds, you get both of the R terms are negative. So you get a bistability. So with positive probability, you go lose species 2. With positive probability, you lose species 1. And then you can also show with probability one of those events occurs. On the other hand, if you have fluctuations in germination rates, you can actually show that R1 and R2 are both positive when evaluated at the opposite measures. And then you get stochastic persistence. And you can even numerically estimate the stationary distribution of that model. And in that case, there is a unique stationary distribution. That's one example. But it's already covered by the older theory. It's two species. OK, what about doing more than two types? How about rock, paper, scissors? That's a simple two-dimensional thing, but with three types. So that's like the classic childhood game, except in a replicator context. You have three types of individuals. They replicate true, like rock, paper, scissors. If you win, you get a payoff bigger than one. If you have a draw, you get a payoff of 1. If you lose, you get a payoff less than 1 and those fluctuates randomly. Then what you can show is that you get a heteroplastic cycle on the phase space. So rock gets displaced by paper, which gets displaced by scissors, which gets displaced by rock. And then you can show that if you evaluate these realized per capita Earth rates at the only ergodic stationary measures, which are these Dirac measures at the equilibria, you get the following criteria for stochastic persistence. The product of the expected value of the log winnings plus the product of the expected log losings has to be positive. And that gives you stochastic persistence. Here's a simulation illustrating that. Oops, went too far. There. And if you have the reverse inequality, you get asymptotic extinction. So I started here. With probably one, you're going to approach the heteroplastic cycle. So for the last thing, I've got two minutes, I think for three minutes, based on what I'm starting a little after. So the last one is just what I want to talk about, because I actually just got accepted this morning. But I didn't write that this morning. But it's something that, for whatever reason, I obsessed about for a period of time. And I still find it fascinating. And maybe you'll see there's some merit in that or not. So this is going back to one of the first models that exhibits the storage effect, which preceded the ecological example. And this is John Gillespie's SASCFF model. You can see what a wonderful acronym. No wonder it took a great hold, which it did. And the basic idea of the model is you've deployed individuals. They have kaoleols at a single locus that determines the genotype. And those kaoleols contribute additively to some sort of phenotype or physiological activity of the organism. And then you assume that the fitness of an organism is an increasing function of its phenotype or physiological activity via some C2 function. That's the C and CFF. Actually, it was originally a concave, but it's the C2 in my context. And then just to make it ecological, you can even have the population has density dependent mortality. So you have population growth there as well. So what's the state space for this model? So if we had three alleles, we'd start with the frequencies of those three alleles. So it's just like before in the rock-paper-scissors game. But then we have to account for population abundance. So we have to go out into the third dimension. So that's our state space. It's this solid cylinder. And you can ask various things about, do you get the alleles coexisting? Do you get the population persisting? You can address that with the theory I just described. So in particular, some of the cool results that you get from this is, first of all, get explicit expressions, which we realize per capita growth rates under the right assumptions. I want to tell you what those are now. But some of the results I found really fascinating are you get opposing effects of many factors of this model, features of this model, on maintaining diversity of alleles versus maintaining the persistence of the population. So you get these opposing effects. Specifically, the per capita growth rate of the population gets better the more convex or the less concave the fitness function is. But that has a detrimental effect on maintaining allelic diversity. The number of alleles has also a detrimental effect on another allelic establishing. But it actually, through a bed-hatching mechanism, leads to high population growth. And environmental variability has a negative effect on population growth, but helps maintain the alleles. So you get these interesting opposing effects. And you can even get hidden in these statements of Perando effect, where it might be that every homozygote individual is maladapted and has a negative per capita growth rate. So if you just have one allel, you're going to extinction. But if you have enough alleles around that aren't perfectly correlated in the responses to the environment, you get the population persists. So that sort of losing alleles can make a winning genotype. So with that, I want to finish by saying, hopefully, I've given you some sense of things that you know already. So it's not. In other words, the environmental processes you can inhibit population persistence. I didn't talk about how it can facilitate population persistence, but it can. It can facilitate or disrupt coexistence. And in particular, the mathematical method is that these conditions for persistence and extinction are determined by this weighted combination of realized per capita growth rates. And that applies to discrete time models, like I discussed today. It also applies to stochastic differential equations. In particular, there's a recent paper by Alex Henning in Deng Nguyen that really shows this like was a simpler version of the result and a much more beautiful, more complete result that includes extinction results. And it also follows from a very, very general theory for persistence, but not extinction, by Michel Ben-Eim. And that very general theory, which is available on archive, also covers piecewise deterministic markup processes. So we have random switching of vector fields, but much, much more. I mean, almost any stochastic ecological model that's finite dimensional will fit into this sort of framework. And so it's a beautiful preprint. And I hope someday it shows up somewhere as a monograph. And with that, I want to first thank US National Science Foundation for funding my collaborators. And then more importantly, Andrew, Alex, and Mark for organizing this wonderful workshop, Burrs for Hosting. Even more importantly, you for listening. And then I'd happily answer any questions that you might have. Thanks. It seems like your model is well set up to deal with things that have a trend, sort of temperature is environmental change, sort of your wide variables. And what effect does that have on the kind of analysis you talked about? What do you mean? So like a trend? Like a trend. Yeah. So the trend will not be in the asymptotic analysis, right? So even if it's, you know, even if you're having like some of those models, even if it was an autoregressive model, right, you might be starting not in the stationary distribution, but asymptotically approach the stationary distribution. And only the asymptotics will then determine, right, whether you have persistence or not and the way it's been defined here. Yeah. So it's time goes to infinity. Yeah, so everything is about, has time goes to infinity. Just like we saw the other day, even to deal with how to talk about even transient trends in some models, it was about time going on infinity, right? So you're still doing asymptotic studies. So maybe there's some way of combining these things. Like I've intrigued. Like is there some way of doing what you're doing in this sort of stochastic context, right, to address something where you're still doing sort of asymptotic analysis? Partially. It's not just that. And I realize it's more complicated to sort of understand something about trends. Yeah. But yeah, it doesn't, I mean, it doesn't have an effect, is the answer. As far as the statement that you'd make, it might have an effect on the actual stationary distribution, right? And so understanding that stationary distribution is a much harder question. I don't know good ways in general to compute them for these higher dimensional models, unless you have a lot of symmetry. If you have a lot of symmetry, then you can. Like for Gillespie's model, without the population dynamics, you can actually do a stochastic differential equation approximation and write down explicitly the stationary distribution under certain cases. So there's special cases. But there's always something you have to sacrifice to get the explicit solution. Or at least it seems that way. Yeah, have Sadiq? You showed us for an example that your criteria for persistence is equivalent to the invasibility. And how much generality is that true? Better equivalence? Well, first, the first question to that question, the first question I'd have to that question would be, what do we mean by mutual invasibility when there's more than two species? So that's a question, right? So you could answer it two ways. I thought about it. I'll give the answer. I won't put you on the spot. So you could think of it one way. You say, well, I insist that every missing species has a positive per capita growth rate. That would be a very generous interpretation. And if that's true, then immediately get stochastic persistence. So that's sort of an easy case. The other thing you could say is, well, that's too strong. Because if I look at a predator-prey model, if I look at the origin, clearly not every species has a positive per capita growth at the origin, because a prey has a negative per capita growth rate. So maybe all I need is at least one species has a positive per capita growth rate. That's not true either, because the rock, paper, scissor game gives a counter example. Because every species, every equilibrium there has a species with a positive per capita growth rate, but you don't get stochastic persistence unless something else holds. So I would say, when you say, does it extend? The answer is, I don't know what we want to mean by mutual invisibility. It does, if you say, all missing species should invade, then it's immediate. It's true. And if only one missing species is invaded, then it need not be true. So it's more subtle. And there's still things. It actually, there's a fairly open question to get things more in line with the deterministic theory. So the deterministic theory is much more refined in some ways in the stochastic theory. And one of the ways it is is that Joseph's original theory doesn't cover scenarios that we know how to cover in the deterministic theory. And that's because in the deterministic theory we take advantage of Morse decomposition. And while there exists Morse decomposition from random dynamical systems, it's not clear how well you can use them for this type of analysis. So that's sort of this open piece of work of trying to actually refine this theory so it's actually completely parallel to the deterministic theory, which it's not. So there's a gap that needs to be filled for sure. Yeah. Were you ever ready for that one? Sorry. You had something that's like one of the questions I've been really thinking about. So yes, Chris. So if you can do mutual invasibility, then you can do adaptive dynamics. And then you can do evolution. Yep. Yeah, you can do that. So I mentioned as one example that you could do evolution. I mean, I don't. So I mean, the population genetics model is evolution, right? Changes of frequencies. So I did already evolution example. In the paper, we do another example where we have a trait that's evolving. So we do a version of this model of Landy and Shannon, which is sort of an evolutionary rescue model. It's not really well, just let's say evolution in a fluctuating environment model, but through a quantitative trait with a Landy approximation. So we do an example of that. And there you can compute things explicitly because you end up having an autoregressive process describing the thing. Well, anyways, you just have an autoregressive process that determines essentially the population per capita growth rate of low densities. So you get some interesting predictions there. They're in line with what people seem doing. What I'd say more theoretical. In other words, the same type of computations, but without the mathematics to back up that this gives you persistence or not. So you could argue about whether that's an important distinction or not. You could see arguments in either direction. And I think you had a question. So OK. OK, so how can you extend the theory if you have things like any effect? So that's a great question. So even for deterministic models, you don't have to. Well, it depends how the Ali effect shows up. There are cases where you could have an Ali effect and you still get permanence. But in general, if you start having attractors in the boundary, you don't have persistence. From the perspective of this theory, because the invasion growth rate or the capital growth rate at low density is negative. So that just means this theory isn't appropriate for things like trying to understand systems with Ali effects that really lead to species being lost because they're low in density. And just purely because of that feedback. So there's no question this theory doesn't apply there. That's a criticism that applies to the deterministic and the stochastic theory. One thing I should mention though is if you do, there is a difference in the stochastic models versus the deterministic models. So I have this paper that's coming out in the ecology where we look at a model where there is effectively, we're looking at positive frequency dependence in models of competition. So that leads to Ali effects where you have either competitor being lost. And as soon as you add any sort of non-local environment to cacici, of course, with probably one, you lose one of the species. In some sense, models of the Ali effects, if you have non-local noise, meaning the noise really can push you anywhere in the state space, like the SDEs or like that, for instance, then you're going to always have a result. But it's going to be not an interesting result. You always go to extinction asymptotically, which makes sense. But it might not be what you're looking for. That's not how it shows. OK. Bill, are you ready? Yeah, thank you very much. Very nice talk. What's one question? I mean, in deterministic dynamics, are mutual invisibility normally guaranteed coexistence, but it might not, for instance, if you ever said it on the family? Yeah, that's just like it was, yeah. He was asking the way. Yes, yes. So yeah, but I said, it was quite different. Does it not happen in the east? No, I said it. I just answered that effectively. But I can repeat it. Sorry, I didn't say it clear enough. But if you have a situation, so if the deterministic model hasn't attract on the boundary, for instance, right? Or you used to say, oh, I see what you're saying. It's slightly different. Oh, yeah, this is a different interesting question. Interesting point. Sorry, I just realized what you meant. OK, so you're talking about a situation where you have something like this, right? Is that what you're talking about? Yeah, so you have to have the set of bounds of boundary. Right, so the set of bounds of boundary. Right, so you come back to it. Yeah, so this is a cool little thing about. So the deterministic model, this would mean you don't have permanence, right, just by the definition. However, in stochastic model, you're not, even if you had a little bit of noise, you're not going to pick up this asymptotic behavior. I mean, it depends. I have to be a little careful when I say that. Oh, that's a stochastic system. Yeah, because if I had a stable point here and a stable point here, most likely with small line of noise, just by various results, the stationary distributions are going to concentrate on those stable fixed points. So you won't be evaluating the stochastic growth rates here. And as a consequence, depending on what's happening on these other fixed points, you still might get stochastic persistence. And so you can construct examples like that, where there's a two-bystable prey and a predator. You usually get situations just like you were described with. Or maybe the prey can invade. That would be a weird situation, but you could create it. The predator can invade when there's too much of the prey, when they're at that coexistence equilibrium, but still the whole system is stochastically persistent. So yeah, so that's an interesting point. There is a way that you're getting rid of things that you probably don't think are relevant for the permanence condition. But that's just the way the permanence condition works. That's actually nice. It may be. The possibility of these systems is removed. Yeah, I think so too. I think it's interesting. I think so. OK, Heather. Question. OK. Very inspirational talk. Thank you very much. Just one question. How much of a result would you still fault if you change white noise to color noise? Well, as I mentioned, the y variable can account for color noise. So you can be forcing. You take white noise, you can color it by just applying a function to it. Pretty much color 332, because with color noise, it can put your system over the variation of the walls. Yeah, same with white noise too. But anyways, it does account for color noise, is my point. So you can have the auxiliary variable be like multiple places. Sorry. Both have very unregressive process, and that's what color noise, not any color, but many colors that you might give.
 So I will start with this quote.  Some people have seen it often before.  It's this quote from the British poet John Gay who wrote,  Less men suspect your tale untrue, keep probability in view.  And in this context, less biologists  suspect your model untrue, keep probability in view.  And an important probabilistic aspect of ecology  that we've heard about already, especially  in Karen's talk yesterday, is that there's  environmental stochasticity.  In other words, fluctuations in demographic rates,  like survivorship, reproduction, growth,  due to fluctuations in environmental factors  like temperature, precipitation, resource availability.  So the basic way it works is you fluctuate  in an environmental factor like temperature.  You have potentially a nonlinear response of your fitness  to that temperature.  And then that creates fluctuations in fitness.  Now, the main question I look at today  is sort of mathematical methods to address this question,  which is, how does environmental stochasticity  influence population persistence and the maintenance  or disruption of diversity?  So there's been a lot of theoretical work.  I'm going to try to focus on some mathematical techniques  for analyzing models with environmental stochasticity  in a rigorous fashion.  And the main reference is a recent JMB paper  with my longtime collaborator, Michelle Benim, from Switzerland.  So I'm going to start really simple, which  is a geometric random walk.  So you're keeping track of the scalar quantity, the density  of the population at time p.  Everyone in that population has the same fitness  at any given time step.  And I'm going to assume just for simplicity  for the moment, those fitnesses are IID across time.  So then you get the similar simple stochastic linear  difference equation where the density in the current time  step gets multiplied by the fitness.  That gives you the next density.  It's trivial to solve for solutions  of that linear difference equation, just the iteration.  And then if you want to understand  what can you say about the growth of the population,  you can just apply the law of large numbers  by taking the log of the density,  dividing by t, then that product, turns into a sum.  And then with probability 1 under the right assumptions  on the first moment on the log of the fitness function,  you get the limit as t goes to infinity  is given by the expected log of the fitness function  at any point in time, such as time 0.  And I'm going to call that throughout this talk  the realized per capita growth rate of the population.  So r will always mean that throughout the talk.  Now, if r is negative, and we run a model where  we're randomly sampling the fitnesses  and do lots of simulations and plot them  in a semi-log plot, we'll get something like this.  So you'll see the tendencies to go to negative infinity  and log densities, i.e. extinction.  On the other hand, if r is positive,  you get the opposite trend.  So this idea of looking at the expected value of log fitness  is not new.  In fact, it's very old.  It goes back to the work of Daniel Bernoulli  in this treatise on economics that was printed in 1738.  And in this treatise, you can find loosely translated  the following quote, which is dealing not with populations  of organisms, but populations of money.  So in this case, the fitness terms  are the returns on your money.  And if you want to know in the long term  how fast does your money grow, you  take the terms, the fitnesses, multiply them together,  and take a root corresponding to the number of terms, which  is exactly what you do in the model I described before.  If you have a finite number of fitness values  that you're taking on with equal likelihood.  And so then if you exponentiate that r,  you get this geometric mean of fitness.  And I'm sure everyone's familiar with.  And a key property of the geometric mean  is that the fitnesses have some variance to them.  Is it strictly less than the arithmetic mean?  And several hundred years later, in this classic paper  by Lowenton Cohen, they made that observation.  And they got a P&S paper out of it,  which is that you can have the expected population size  to grow infinitely large.  But you're going asymptotically toward extinction  in any realization due to the difference  in geometric and arithmetic means of fitness.  But these models, of course, that I've described so far,  are way too simplistic and probably get  chastised by Kenneth Holding for believing  that could go on forever in a finite world.  And certainly, I don't want to be called a madman.  And given worse, perhaps an economist.  Apologies to any economist out there.  So of course, people have developed models  where you allow for density dependent feedbacks,  like negative density dependence.  And there's various papers on this.  I'm just citing two of them.  But one of the key features of these results  for these density dependent models  is if you look at the expected value of the log fitness  at zero density, that tells you whether you  have persistence or extinction.  And I'll say what that means.  Persistence means exactly a little later.  And of course, we made models that are more interesting  because we wanted to diversify the objects that we're  looking at within a population or populations.  And one way of diversifying the set of types  is to account for differences in age or in stage  or spatial location.  So then we have these structured population models.  And even understanding their dynamics  became a really important theme of stochastic demography.  And there's this delightful book by Shira Parj Tuljapokar  about summarizing what was known as of 1991  about these random matrix models.  And a key fact of these random matrix models  is that there is also a realized per capita growth rate.  And that's due to some work of Ruel as well as others.  And one of the things that's sort of amusing  about Ruel's paper, which is in a,  I've forgotten it's not the annals of mathematics.  I think it's the American Journal of Mathematics, which  is a very good math journal.  He did the analytics of understanding  that this realized per capita growth rate actually  is analytic with respect to parameter variation.  So you could write down a derivative,  but he actually calculated the second derivative wrong  in that paper.  And Tuljap independently calculated it correctly.  So this is a very famous small variance approximation  of the realized per capita growth rate.  And notice again, fluctuations, at least in IID environments,  also reduces the growth rate of the population.  And again, to avoid being called a madman or an economist,  the results about what happens when you add  negative density of dependence.  Again, the punch line is if R is positive, persistence,  R is negative, extinction.  OK, but how about looking at diversifying models  in terms of any different types that are interacting?  So you might be wanting to know, if you have different types,  how do you maintain that diversity?  And there's sort of two classes of models  that's been looked at fairly extensively that's  been in the context of population genetics,  maintaining a polymorphism, and also  in the context of ecology, maintaining species coexistence.  So some early theoretical works.  I'm going to distinguish here and now theoretical work  from mathematically rigorous work.  Apologies that people find that offensive,  but hopefully knowing that there is a distinction.  So there was theoretical work on protective polymorphisms,  so how fluctuations in fitness can  lead to maintenance of polymorphisms in populations.  And some of that work was initially done by Haldane  in this paper in 1963 for haploid populations.  And then John Gillespie in this delightful paper  that I'll pay a tribute to at the end of this talk  did stuff for diploid populations.  And in fact, he originally discovered,  although he didn't realize it, the storage effect  in the context of diploid populations.  And then, of course, in the context of ecology, species  coexistence, there is the quite famous storage effect  of a mechanism for maintaining coexistence between competing  species through fluctuating environments.  And of course, the champion of the storage effect  was Peter Cheson.  And it's gotten to the point now where we have data sets  that we use to calibrate models, that we even  do simulations instead of math with to understand  how strong the storage effect is, as discussed in Elmer's paper.  Now, the mathematical work, surprisingly,  was fairly limited on this topic.  So the main paper I can think of is a paper dealing  with two competing species.  These are discrete time models.  And it was worked by Cheson and Elmer, both appearing in 1989.  And then they were not much done for these understanding  persistence of interacting populations.  But quite interestingly, an invasion-based math  for deterministic models was done for any number of species,  any type of interactions.  And what I mean by invasion-based,  I just mean that you're looking at these realized  that your growth rates for one of the species is rare,  and using that as a way to determine  whether the species coexist or not.  So there was all this work done for deterministic models.  And most importantly is the paper of Joseph Hofbauer in 1981  and then Butler and Waldman in 1986  that took different approaches to this question.  And what I want to do is just show you  that we've advanced a little bit in the stochastic theory  because we can sort of recover the results of Hofbauer  for stochastic models.  OK.  So what are the stochastic models I'm going to talk about  than the ones of this form?  So I'm going to keep track of k distinct populations.  There might be different genotypes.  There might be different species.  And then xi is going to be the density of the i-th population.  I'm going to have these auxiliary variables, which  I'll explain with the odd in a minute.  OK.  And then again, a sequence of iid random variables.  So each species is updating by taking its current density  multiplying by its fitness fi, which  depends on the state of the community.  So x is the vector of densities community.  Depends on this auxiliary variables.  And it depends on the sequence of iid random variables.  And then you have some dynamics for the auxiliary variables  that depends on everything.  So first thing you might ask, what  are these auxiliary variables?  I won't do the explicative here.  So there can be lots of different things.  One thing is we were talking about yesterday  is the importance of auto correlation.  And I said I'm using iid random variables,  but you can encode for auto correlation  because the y can be almost any type of Markovian  forcing dynamics.  So there's some deep theory that shows  that almost any type of Markov chain  can be realized by a random map with iid random variables.  So in particular, you can get a multiplicative auto regressive  model using this g function.  You can get any finite state Markov chain  using this formalization from forcing the dynamics,  the ecological dynamics.  You can also account for any type of population structure  because those auxiliary variables  can be keeping track of the fraction of a given type  within a population.  So it can have spatial structure, a structure,  any combination you're of.  You can also do trade evolution.  So yi might be a trade of the iid population.  And then you might have maybe a breeder's equation describing  how that trade's evolving.  Or you can also do environmental feedbacks.  So you might have plant soil feedbacks.  So maybe y is the soil dynamics.  Or you might have some environmental feature  that's characterized by a y that is being engineered  by the species and the population.  So any of these things can be encoded  in these types of models.  So it's fairly general, is the point.  So for these models, what do I mean  by coexistence or persistence?  So there's two definitions.  One of them is sort of one that I highlighted initially  because initially I could only prove theorems  for this definition.  And the other one was highlighted initially  by Peter Chesson in the 1980s.  So I'm going to talk about the definitions  that I worked with first because that's the one that I think  is somewhat more natural.  And it deals with the fact that often we're  looking at a single realization of a community  or a population.  So we'll say persistence almost surely  means for almost every realization of a model.  And you can just look at the right and ignore the left.  And so if we have a model with no auxiliary variables  and two species, x1, x2 are their densities.  And we can say, well, this gray region  is within delta of extinction for one of the species.  If you run the model for a long time,  you can ask what fraction of time does that simulation  spend in that gray region close to within delta of extinction  set.  And stochastic persistence says that fraction of time  goes uniformly to 0 with respect to initial conditions.  This delta goes to 0.  That would be stochastic persistence almost surely.  And the complement is the ensemble point of view.  So you run many copies of your model,  where you have many replicate experiments.  And so stochastic persistence and probability,  which was introduced by Peter Chesson,  is you take many realizations of your model,  run them far into the future.  So I'm just showing the final time point  for those many realizations.  And you ask what fraction of those realizations  far into the future are within delta of the extinction set.  And you want to insist that that fraction goes uniformly  to 0 with respect to initial conditions as delta goes to 0.  If that happens, you have stochastic persistence  and probability.  And fortunately, for all results I'll talk about today,  when you have one, you have the other.  So.  Thank you, Kevin.  What was that the other reason?  Let me adjust that later.  I'll have to think about it.  I think you can.  So Peter Chesson, early on, explained why he thinks  this is the right definition of persistence,  stochastic persistence.  And basically, he realized that many models  have good years and bad years.  Good years, the population increases.  Bad year, the population decreases.  And therefore, with probability 1,  you're going to have an arbitrary long sequence  of bad years, which means you can get to arbitrary low  densities.  And you don't want to exclude those models from what  you're thinking about.  OK, so now how do you check the stochastic persistence  condition?  You do with the sort of invasibility criteria  that many people might be familiar with.  So what you think is, OK, suppose I have all but a subset  of species become very rare.  So I have, like, say, J1 through JL that are not rare,  and the rest of the species become rare.  So that's infinitesimally rare.  So they're not there.  So we're worried about them not being  able to stay in the community.  And so the rest of the species, the ones that are there,  the J-sub-i indices, they're going  to probably approach some sort of stationary behavior.  Let's assume that that stationary behavior is even  robotic.  That's the only stationary behavior  we'll have to pay attention to with some law of mu.  So that means visually, here's all these species that are not  rare.  They're approaching some sort of stationary behavior,  as well as the feed boxillary variables  approaching some stationary behavior.  And now you want to know, is one of the infinitesimally rare  species able to increase?  So what do you do?  So you look at this axis with the infinitesimally rare species.  And you want to know what is its rate of increase.  So it's just this linear question.  And it turns out all you want to do  is take the log fitness of that species and average  across all the fluctuations.  So that's the fluctuations in the chi's  and the fluctuations in the community state,  as well as the feedback or the auxiliary variables.  So that z is this whole combination.  And if that quantity is positive,  that means this infinitesimally rare species  would tend to increase exponentially.  If it's negative, it would tend to decrease exponentially.  But you can also compute this quantity  for all the species that are not rare, that are present.  And it's not that easy to show.  But one can show, and it's fairly intuitive,  that their per capita growth rates are always 0.  They're present in the community.  They're not going off to infinity.  They're not going to 0.  So on average, it must somehow be 0.  OK, so now how do we use these things to state a theorem  about persistence?  So let me first define the set that we're trying to avoid.  That's the extinction set.  So that's going to be the set of points in the state space  where one of the species is not present.  And here's the criterion that's going  to be used for stochastic persistence.  It's the exact criterion that Joseph Hoffbauer used  for deterministic models, very slightly differently,  because he didn't do it with respect to measures.  But he can do it with respect to measures,  even in the deterministic setting.  And what it says is, if for each species,  you can associate a weight, p1, p2, p3, and so forth.  They're positive weights.  They could add up to 1, if you want.  And you take that fixed weighted combinations  of the realized per capita growth rates of the species.  So that's saying, somehow I'm taking some community  average of the per capita growth rate.  That's roughly what that is.  If that is for that fixed weight combination,  if that is positive for any sort of ergodic behavior  of a subset of species, so not all species,  then you have the following theorem  that if this uptribe of the whole positive condition,  then you have stochastic persistence and probability  and almost surely.  And you even have estimates about how much weight is being  put near the extinction set, which I won't show here.  On the other hand, if you can flip the condition around,  if you can find weights, so that this weighted combination  of the per capita growth rates is always negative whenever  there's a subset of species supported  by the ergodic measures, then you get the opposite result,  which is the extinction set is locally stochastically stable,  meaning if you start sufficiently close to that set,  there's a very high probability that you asymptotically  approach that set.  And again, these are models that are continuous states,  right, continuous densities, so you can only  asymptotically approach them.  OK, so that's the theorems, or those are the theorems.  So let me now briefly show you a few examples,  just to give you a flavor of how you can use them,  and how they might even relate to things  that you're familiar with.  So the first example I'm going to talk about  is a annual plant model.  And I'm not going to write down the model.  It's just a model of an annual plant,  keeping track of the seeds, OK?  And it's a model of two competing species,  and it's a model whose can be analyzed  using results of a wonderful little piece by Peter Cheson  in 1988 and some lecture notes in spring of ergodic.  And so it's two species that are competing,  and he asks, how do I use that criterion that I mentioned  before?  So we have species one here.  Right now, I'm not having any auxiliary variables.  And let's assume each species by itself can persist.  That would mean R1 at the Dirac measure at 0 is positive,  and R2 is as well.  So they both can persist by themselves.  You have something like this, OK?  Now you want to know, what about each species  invade when it's rare?  So you have this quantity here.  I call this mu1, this mu2.  We want to know it's R1 of, so R2 of mu1 greater than 0.  We want to know it's R2, R1 mu2 greater than 0.  And it turns out this would be the classic mutual invasibility  condition.  Why is that the same as what I wrote on the board before?  Man, that's not obvious.  But what do we need?  We need to weigh it's P1 since it's  R1 of any measure.  That's P2 times R2 of any measure is greater than 0.  So that's trivially true at the Dirac measure of the origin,  because these both are positive.  Here, if we look at this measure, by that result I mentioned,  the R1 for species 1 is 0.  And that's just saying you need R2 to be positive for persistence  and similarly for this one.  So the mutual invasibility criterion  is identical to the criterion I put on the screen earlier.  So if we apply that to an annual plant model where  it's you have neutral coexistence  without any environmental stochasticity,  you might not really think, oh, if I add environmental stochasticity,  you can get sort of a drift along that line that  might lead to extinction.  That's not the case.  But what you get depends very much  on what is fluctuating in time.  So you can use the results of Chesson  to show for large or small noise.  It doesn't matter if it's the size of the noise.  It's just there is noise.  That if it's fluctuations in survivorship of the seeds,  you get both of the R terms are negative.  So you get a bistability.  So with positive probability, you go lose species 2.  With positive probability, you lose species 1.  And then you can also show with probability  one of those events occurs.  On the other hand, if you have fluctuations  in germination rates, you can actually  show that R1 and R2 are both positive when evaluated  at the opposite measures.  And then you get stochastic persistence.  And you can even numerically estimate  the stationary distribution of that model.  And in that case, there is a unique stationary distribution.  That's one example.  But it's already covered by the older theory.  It's two species.  OK, what about doing more than two types?  How about rock, paper, scissors?  That's a simple two-dimensional thing,  but with three types.  So that's like the classic childhood game,  except in a replicator context.  You have three types of individuals.  They replicate true, like rock, paper, scissors.  If you win, you get a payoff bigger than one.  If you have a draw, you get a payoff of 1.  If you lose, you get a payoff less than 1  and those fluctuates randomly.  Then what you can show is that you  get a heteroplastic cycle on the phase space.  So rock gets displaced by paper, which  gets displaced by scissors, which gets displaced by rock.  And then you can show that if you evaluate these realized  per capita Earth rates at the only ergodic stationary  measures, which are these Dirac measures at the equilibria,  you get the following criteria for stochastic persistence.  The product of the expected value of the log winnings  plus the product of the expected log losings has to be positive.  And that gives you stochastic persistence.  Here's a simulation illustrating that.  Oops, went too far.  There.  And if you have the reverse inequality,  you get asymptotic extinction.  So I started here.  With probably one, you're going to approach the heteroplastic  cycle.  So for the last thing, I've got two minutes,  I think for three minutes, based on what I'm starting a little  after.  So the last one is just what I want to talk about,  because I actually just got accepted this morning.  But I didn't write that this morning.  But it's something that, for whatever reason,  I obsessed about for a period of time.  And I still find it fascinating.  And maybe you'll see there's some merit in that or not.  So this is going back to one of the first models that exhibits  the storage effect, which preceded the ecological example.  And this is John Gillespie's SASCFF model.  You can see what a wonderful acronym.  No wonder it took a great hold, which it did.  And the basic idea of the model is you've deployed individuals.  They have kaoleols at a single locus that  determines the genotype.  And those kaoleols contribute additively  to some sort of phenotype or physiological activity  of the organism.  And then you assume that the fitness of an organism  is an increasing function of its phenotype or physiological  activity via some C2 function.  That's the C and CFF.  Actually, it was originally a concave,  but it's the C2 in my context.  And then just to make it ecological,  you can even have the population has density  dependent mortality.  So you have population growth there as well.  So what's the state space for this model?  So if we had three alleles, we'd start  with the frequencies of those three alleles.  So it's just like before in the rock-paper-scissors game.  But then we have to account for population abundance.  So we have to go out into the third dimension.  So that's our state space.  It's this solid cylinder.  And you can ask various things about, do you get the alleles  coexisting?  Do you get the population persisting?  You can address that with the theory I just described.  So in particular, some of the cool results  that you get from this is, first of all,  get explicit expressions, which we  realize per capita growth rates under the right assumptions.  I want to tell you what those are now.  But some of the results I found really fascinating  are you get opposing effects of many factors of this model,  features of this model, on maintaining diversity of alleles  versus maintaining the persistence of the population.  So you get these opposing effects.  Specifically, the per capita growth rate of the population  gets better the more convex or the less concave  the fitness function is.  But that has a detrimental effect  on maintaining allelic diversity.  The number of alleles has also a detrimental effect  on another allelic establishing.  But it actually, through a bed-hatching mechanism,  leads to high population growth.  And environmental variability has a negative effect  on population growth, but helps maintain the alleles.  So you get these interesting opposing effects.  And you can even get hidden in these statements  of Perando effect, where it might  be that every homozygote individual is maladapted  and has a negative per capita growth rate.  So if you just have one allel, you're going to extinction.  But if you have enough alleles around that  aren't perfectly correlated in the responses  to the environment, you get the population persists.  So that sort of losing alleles can make a winning genotype.  So with that, I want to finish by saying, hopefully,  I've given you some sense of things that you know already.  So it's not.  In other words, the environmental processes  you can inhibit population persistence.  I didn't talk about how it can facilitate population  persistence, but it can.  It can facilitate or disrupt coexistence.  And in particular, the mathematical method  is that these conditions for persistence and extinction  are determined by this weighted combination of realized  per capita growth rates.  And that applies to discrete time models,  like I discussed today.  It also applies to stochastic differential equations.  In particular, there's a recent paper  by Alex Henning in Deng Nguyen that really shows this  like was a simpler version of the result  and a much more beautiful, more complete result that  includes extinction results.  And it also follows from a very, very general theory  for persistence, but not extinction,  by Michel Ben-Eim.  And that very general theory, which is available on archive,  also covers piecewise deterministic markup processes.  So we have random switching of vector fields,  but much, much more.  I mean, almost any stochastic ecological model  that's finite dimensional will fit into this sort of framework.  And so it's a beautiful preprint.  And I hope someday it shows up somewhere as a monograph.  And with that, I want to first thank US National Science  Foundation for funding my collaborators.  And then more importantly, Andrew, Alex, and Mark  for organizing this wonderful workshop, Burrs for Hosting.  Even more importantly, you for listening.  And then I'd happily answer any questions that you might have.  Thanks.  It seems like your model is well set up to deal with things  that have a trend, sort of temperature  is environmental change, sort of your wide variables.  And what effect does that have on the kind of analysis  you talked about?  What do you mean?  So like a trend?  Like a trend.  Yeah.  So the trend will not be in the asymptotic analysis, right?  So even if it's, you know, even if you're having like some  of those models, even if it was an autoregressive model,  right, you might be starting not in the stationary distribution,  but asymptotically approach the stationary distribution.  And only the asymptotics will then determine, right,  whether you have persistence or not  and the way it's been defined here.  Yeah.  So it's time goes to infinity.  Yeah, so everything is about, has time goes to infinity.  Just like we saw the other day, even to deal with how  to talk about even transient trends in some models,  it was about time going on infinity, right?  So you're still doing asymptotic studies.  So maybe there's some way of combining these things.  Like I've intrigued.  Like is there some way of doing what you're doing  in this sort of stochastic context, right,  to address something where you're still doing sort  of asymptotic analysis?  Partially.  It's not just that.  And I realize it's more complicated  to sort of understand something about trends.  Yeah.  But yeah, it doesn't, I mean, it doesn't have an effect,  is the answer.  As far as the statement that you'd make,  it might have an effect on the actual stationary distribution,  right?  And so understanding that stationary distribution  is a much harder question.  I don't know good ways in general  to compute them for these higher dimensional models,  unless you have a lot of symmetry.  If you have a lot of symmetry, then you can.  Like for Gillespie's model, without the population dynamics,  you can actually do a stochastic differential equation  approximation and write down explicitly  the stationary distribution under certain cases.  So there's special cases.  But there's always something you have to sacrifice  to get the explicit solution.  Or at least it seems that way.  Yeah, have Sadiq?  You showed us for an example that your criteria for persistence  is equivalent to the invasibility.  And how much generality is that true?  Better equivalence?  Well, first, the first question to that question,  the first question I'd have to that question  would be, what do we mean by mutual invasibility  when there's more than two species?  So that's a question, right?  So you could answer it two ways.  I thought about it.  I'll give the answer.  I won't put you on the spot.  So you could think of it one way.  You say, well, I insist that every missing species has  a positive per capita growth rate.  That would be a very generous interpretation.  And if that's true, then immediately  get stochastic persistence.  So that's sort of an easy case.  The other thing you could say is, well, that's too strong.  Because if I look at a predator-prey model,  if I look at the origin, clearly not every species  has a positive per capita growth at the origin,  because a prey has a negative per capita growth rate.  So maybe all I need is at least one species  has a positive per capita growth rate.  That's not true either, because the rock, paper,  scissor game gives a counter example.  Because every species, every equilibrium there  has a species with a positive per capita growth rate,  but you don't get stochastic persistence  unless something else holds.  So I would say, when you say, does it extend?  The answer is, I don't know what we  want to mean by mutual invisibility.  It does, if you say, all missing species should invade,  then it's immediate.  It's true.  And if only one missing species is  invaded, then it need not be true.  So it's more subtle.  And there's still things.  It actually, there's a fairly open question  to get things more in line with the deterministic theory.  So the deterministic theory is much more refined in some ways  in the stochastic theory.  And one of the ways it is is that Joseph's original theory  doesn't cover scenarios that we know  how to cover in the deterministic theory.  And that's because in the deterministic theory  we take advantage of Morse decomposition.  And while there exists Morse decomposition  from random dynamical systems, it's  not clear how well you can use them  for this type of analysis.  So that's sort of this open piece of work  of trying to actually refine this theory so it's actually  completely parallel to the deterministic theory, which  it's not.  So there's a gap that needs to be filled for sure.  Yeah.  Were you ever ready for that one?  Sorry.  You had something that's like one of the questions  I've been really thinking about.  So yes, Chris.  So if you can do mutual invasibility,  then you can do adaptive dynamics.  And then you can do evolution.  Yep.  Yeah, you can do that.  So I mentioned as one example that you could do evolution.  I mean, I don't.  So I mean, the population genetics model is evolution, right?  Changes of frequencies.  So I did already evolution example.  In the paper, we do another example  where we have a trait that's evolving.  So we do a version of this model of Landy and Shannon,  which is sort of an evolutionary rescue model.  It's not really well, just let's say evolution  in a fluctuating environment model,  but through a quantitative trait with a Landy approximation.  So we do an example of that.  And there you can compute things explicitly  because you end up having an autoregressive process describing  the thing.  Well, anyways, you just have an autoregressive process  that determines essentially the population per capita growth  rate of low densities.  So you get some interesting predictions there.  They're in line with what people seem doing.  What I'd say more theoretical.  In other words, the same type of computations,  but without the mathematics to back up  that this gives you persistence or not.  So you could argue about whether that's  an important distinction or not.  You could see arguments in either direction.  And I think you had a question.  So OK.  OK, so how can you extend the theory  if you have things like any effect?  So that's a great question.  So even for deterministic models, you don't have to.  Well, it depends how the Ali effect shows up.  There are cases where you could have an Ali effect  and you still get permanence.  But in general, if you start having attractors in the boundary,  you don't have persistence.  From the perspective of this theory,  because the invasion growth rate or the capital growth rate  at low density is negative.  So that just means this theory isn't appropriate for things  like trying to understand systems with Ali effects that really  lead to species being lost because they're low in density.  And just purely because of that feedback.  So there's no question this theory doesn't apply there.  That's a criticism that applies to the deterministic  and the stochastic theory.  One thing I should mention though is if you do,  there is a difference in the stochastic models  versus the deterministic models.  So I have this paper that's coming out in the ecology  where we look at a model where there is effectively,  we're looking at positive frequency dependence  in models of competition.  So that leads to Ali effects where you have either  competitor being lost.  And as soon as you add any sort of non-local environment  to cacici, of course, with probably one,  you lose one of the species.  In some sense, models of the Ali effects,  if you have non-local noise, meaning  the noise really can push you anywhere in the state space,  like the SDEs or like that, for instance,  then you're going to always have a result.  But it's going to be not an interesting result.  You always go to extinction asymptotically,  which makes sense.  But it might not be what you're looking for.  That's not how it shows.  OK.  Bill, are you ready?  Yeah, thank you very much.  Very nice talk.  What's one question?  I mean, in deterministic dynamics,  are mutual invisibility normally guaranteed  coexistence, but it might not, for instance,  if you ever said it on the family?  Yeah, that's just like it was, yeah.  He was asking the way.  Yes, yes.  So yeah, but I said, it was quite different.  Does it not happen in the east?  No, I said it.  I just answered that effectively.  But I can repeat it.  Sorry, I didn't say it clear enough.  But if you have a situation, so if the deterministic model  hasn't attract on the boundary, for instance, right?  Or you used to say, oh, I see what you're saying.  It's slightly different.  Oh, yeah, this is a different interesting question.  Interesting point.  Sorry, I just realized what you meant.  OK, so you're talking about a situation  where you have something like this, right?  Is that what you're talking about?  Yeah, so you have to have the set of bounds of boundary.  Right, so the set of bounds of boundary.  Right, so you come back to it.  Yeah, so this is a cool little thing about.  So the deterministic model, this would  mean you don't have permanence, right, just by the definition.  However, in stochastic model, you're not, even if you had  a little bit of noise, you're not going to pick up  this asymptotic behavior.  I mean, it depends.  I have to be a little careful when I say that.  Oh, that's a stochastic system.  Yeah, because if I had a stable point here  and a stable point here, most likely with small line of noise,  just by various results, the stationary distributions  are going to concentrate on those stable fixed points.  So you won't be evaluating the stochastic growth rates here.  And as a consequence, depending on what's  happening on these other fixed points,  you still might get stochastic persistence.  And so you can construct examples like that,  where there's a two-bystable prey and a predator.  You usually get situations just like you were described with.  Or maybe the prey can invade.  That would be a weird situation, but you could create it.  The predator can invade when there's too much of the prey,  when they're at that coexistence equilibrium,  but still the whole system is stochastically persistent.  So yeah, so that's an interesting point.  There is a way that you're getting rid of things  that you probably don't think are  relevant for the permanence condition.  But that's just the way the permanence condition works.  That's actually nice.  It may be.  The possibility of these systems is removed.  Yeah, I think so too.  I think it's interesting.  I think so.  OK, Heather.  Question.  OK.  Very inspirational talk.  Thank you very much.  Just one question.  How much of a result would you still  fault if you change white noise to color noise?  Well, as I mentioned, the y variable  can account for color noise.  So you can be forcing.  You take white noise, you can color it  by just applying a function to it.  Pretty much color 332, because with color noise,  it can put your system over the variation of the walls.  Yeah, same with white noise too.  But anyways, it does account for color noise, is my point.  So you can have the auxiliary variable be like multiple  places.  Sorry.  Both have very unregressive process,  and that's what color noise, not any color,  but many colors that you might give.
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So I will start with this quote. Some people have seen it often before. It's this quote from the British poet John Gay who wrote, Less men suspect your tale untrue, keep probability in view. And in this context, less biologists suspect your model untrue, keep probability in view. And an important probabilistic aspect of ecology that we've heard about already, especially in Karen's talk yesterday, is that there's environmental stochasticity. In other words, fluctuations in demographic rates, like survivorship, reproduction, growth, due to fluctuations in environmental factors like temperature, precipitation, resource availability. So the basic way it works is you fluctuate in an environmental factor like temperature. You have potentially a nonlinear response of your fitness to that temperature. And then that creates fluctuations in fitness. Now, the main question I look at today is sort of mathematical methods to address this question, which is, how does environmental stochasticity influence population persistence and the maintenance or disruption of diversity? So there's been a lot of theoretical work. I'm going to try to focus on some mathematical techniques for analyzing models with environmental stochasticity in a rigorous fashion. And the main reference is a recent JMB paper with my longtime collaborator, Michelle Benim, from Switzerland. So I'm going to start really simple, which is a geometric random walk. So you're keeping track of the scalar quantity, the density of the population at time p. Everyone in that population has the same fitness at any given time step. And I'm going to assume just for simplicity for the moment, those fitnesses are IID across time. So then you get the similar simple stochastic linear difference equation where the density in the current time step gets multiplied by the fitness. That gives you the next density. It's trivial to solve for solutions of that linear difference equation, just the iteration. And then if you want to understand what can you say about the growth of the population, you can just apply the law of large numbers by taking the log of the density, dividing by t, then that product, turns into a sum. And then with probability 1 under the right assumptions on the first moment on the log of the fitness function, you get the limit as t goes to infinity is given by the expected log of the fitness function at any point in time, such as time 0. And I'm going to call that throughout this talk the realized per capita growth rate of the population. So r will always mean that throughout the talk. Now, if r is negative, and we run a model where we're randomly sampling the fitnesses and do lots of simulations and plot them in a semi-log plot, we'll get something like this. So you'll see the tendencies to go to negative infinity and log densities, i.e. extinction. On the other hand, if r is positive, you get the opposite trend. So this idea of looking at the expected value of log fitness is not new. In fact, it's very old. It goes back to the work of Daniel Bernoulli in this treatise on economics that was printed in 1738. And in this treatise, you can find loosely translated the following quote, which is dealing not with populations of organisms, but populations of money. So in this case, the fitness terms are the returns on your money. And if you want to know in the long term how fast does your money grow, you take the terms, the fitnesses, multiply them together, and take a root corresponding to the number of terms, which is exactly what you do in the model I described before. If you have a finite number of fitness values that you're taking on with equal likelihood. And so then if you exponentiate that r, you get this geometric mean of fitness. And I'm sure everyone's familiar with. And a key property of the geometric mean is that the fitnesses have some variance to them. Is it strictly less than the arithmetic mean? And several hundred years later, in this classic paper by Lowenton Cohen, they made that observation. And they got a P&S paper out of it, which is that you can have the expected population size to grow infinitely large. But you're going asymptotically toward extinction in any realization due to the difference in geometric and arithmetic means of fitness. But these models, of course, that I've described so far, are way too simplistic and probably get chastised by Kenneth Holding for believing that could go on forever in a finite world. And certainly, I don't want to be called a madman. And given worse, perhaps an economist. Apologies to any economist out there. So of course, people have developed models where you allow for density dependent feedbacks, like negative density dependence. And there's various papers on this. I'm just citing two of them. But one of the key features of these results for these density dependent models is if you look at the expected value of the log fitness at zero density, that tells you whether you have persistence or extinction. And I'll say what that means. Persistence means exactly a little later. And of course, we made models that are more interesting because we wanted to diversify the objects that we're looking at within a population or populations. And one way of diversifying the set of types is to account for differences in age or in stage or spatial location. So then we have these structured population models. And even understanding their dynamics became a really important theme of stochastic demography. And there's this delightful book by Shira Parj Tuljapokar about summarizing what was known as of 1991 about these random matrix models. And a key fact of these random matrix models is that there is also a realized per capita growth rate. And that's due to some work of Ruel as well as others. And one of the things that's sort of amusing about Ruel's paper, which is in a, I've forgotten it's not the annals of mathematics. I think it's the American Journal of Mathematics, which is a very good math journal. He did the analytics of understanding that this realized per capita growth rate actually is analytic with respect to parameter variation. So you could write down a derivative, but he actually calculated the second derivative wrong in that paper. And Tuljap independently calculated it correctly. So this is a very famous small variance approximation of the realized per capita growth rate. And notice again, fluctuations, at least in IID environments, also reduces the growth rate of the population. And again, to avoid being called a madman or an economist, the results about what happens when you add negative density of dependence. Again, the punch line is if R is positive, persistence, R is negative, extinction. OK, but how about looking at diversifying models in terms of any different types that are interacting? So you might be wanting to know, if you have different types, how do you maintain that diversity? And there's sort of two classes of models that's been looked at fairly extensively that's been in the context of population genetics, maintaining a polymorphism, and also in the context of ecology, maintaining species coexistence. So some early theoretical works. I'm going to distinguish here and now theoretical work from mathematically rigorous work. Apologies that people find that offensive, but hopefully knowing that there is a distinction. So there was theoretical work on protective polymorphisms, so how fluctuations in fitness can lead to maintenance of polymorphisms in populations. And some of that work was initially done by Haldane in this paper in 1963 for haploid populations. And then John Gillespie in this delightful paper that I'll pay a tribute to at the end of this talk did stuff for diploid populations. And in fact, he originally discovered, although he didn't realize it, the storage effect in the context of diploid populations. And then, of course, in the context of ecology, species coexistence, there is the quite famous storage effect of a mechanism for maintaining coexistence between competing species through fluctuating environments. And of course, the champion of the storage effect was Peter Cheson. And it's gotten to the point now where we have data sets that we use to calibrate models, that we even do simulations instead of math with to understand how strong the storage effect is, as discussed in Elmer's paper. Now, the mathematical work, surprisingly, was fairly limited on this topic. So the main paper I can think of is a paper dealing with two competing species. These are discrete time models. And it was worked by Cheson and Elmer, both appearing in 1989. And then they were not much done for these understanding persistence of interacting populations. But quite interestingly, an invasion-based math for deterministic models was done for any number of species, any type of interactions. And what I mean by invasion-based, I just mean that you're looking at these realized that your growth rates for one of the species is rare, and using that as a way to determine whether the species coexist or not. So there was all this work done for deterministic models. And most importantly is the paper of Joseph Hofbauer in 1981 and then Butler and Waldman in 1986 that took different approaches to this question. And what I want to do is just show you that we've advanced a little bit in the stochastic theory because we can sort of recover the results of Hofbauer for stochastic models. OK. So what are the stochastic models I'm going to talk about than the ones of this form? So I'm going to keep track of k distinct populations. There might be different genotypes. There might be different species. And then xi is going to be the density of the i-th population. I'm going to have these auxiliary variables, which I'll explain with the odd in a minute. OK. And then again, a sequence of iid random variables. So each species is updating by taking its current density multiplying by its fitness fi, which depends on the state of the community. So x is the vector of densities community. Depends on this auxiliary variables. And it depends on the sequence of iid random variables. And then you have some dynamics for the auxiliary variables that depends on everything. So first thing you might ask, what are these auxiliary variables? I won't do the explicative here. So there can be lots of different things. One thing is we were talking about yesterday is the importance of auto correlation. And I said I'm using iid random variables, but you can encode for auto correlation because the y can be almost any type of Markovian forcing dynamics. So there's some deep theory that shows that almost any type of Markov chain can be realized by a random map with iid random variables. So in particular, you can get a multiplicative auto regressive model using this g function. You can get any finite state Markov chain using this formalization from forcing the dynamics, the ecological dynamics. You can also account for any type of population structure because those auxiliary variables can be keeping track of the fraction of a given type within a population. So it can have spatial structure, a structure, any combination you're of. You can also do trade evolution. So yi might be a trade of the iid population. And then you might have maybe a breeder's equation describing how that trade's evolving. Or you can also do environmental feedbacks. So you might have plant soil feedbacks. So maybe y is the soil dynamics. Or you might have some environmental feature that's characterized by a y that is being engineered by the species and the population. So any of these things can be encoded in these types of models. So it's fairly general, is the point. So for these models, what do I mean by coexistence or persistence? So there's two definitions. One of them is sort of one that I highlighted initially because initially I could only prove theorems for this definition. And the other one was highlighted initially by Peter Chesson in the 1980s. So I'm going to talk about the definitions that I worked with first because that's the one that I think is somewhat more natural. And it deals with the fact that often we're looking at a single realization of a community or a population. So we'll say persistence almost surely means for almost every realization of a model. And you can just look at the right and ignore the left. And so if we have a model with no auxiliary variables and two species, x1, x2 are their densities. And we can say, well, this gray region is within delta of extinction for one of the species. If you run the model for a long time, you can ask what fraction of time does that simulation spend in that gray region close to within delta of extinction set. And stochastic persistence says that fraction of time goes uniformly to 0 with respect to initial conditions. This delta goes to 0. That would be stochastic persistence almost surely. And the complement is the ensemble point of view. So you run many copies of your model, where you have many replicate experiments. And so stochastic persistence and probability, which was introduced by Peter Chesson, is you take many realizations of your model, run them far into the future. So I'm just showing the final time point for those many realizations. And you ask what fraction of those realizations far into the future are within delta of the extinction set. And you want to insist that that fraction goes uniformly to 0 with respect to initial conditions as delta goes to 0. If that happens, you have stochastic persistence and probability. And fortunately, for all results I'll talk about today, when you have one, you have the other. So. Thank you, Kevin. What was that the other reason? Let me adjust that later. I'll have to think about it. I think you can. So Peter Chesson, early on, explained why he thinks this is the right definition of persistence, stochastic persistence. And basically, he realized that many models have good years and bad years. Good years, the population increases. Bad year, the population decreases. And therefore, with probability 1, you're going to have an arbitrary long sequence of bad years, which means you can get to arbitrary low densities. And you don't want to exclude those models from what you're thinking about. OK, so now how do you check the stochastic persistence condition? You do with the sort of invasibility criteria that many people might be familiar with. So what you think is, OK, suppose I have all but a subset of species become very rare. So I have, like, say, J1 through JL that are not rare, and the rest of the species become rare. So that's infinitesimally rare. So they're not there. So we're worried about them not being able to stay in the community. And so the rest of the species, the ones that are there, the J-sub-i indices, they're going to probably approach some sort of stationary behavior. Let's assume that that stationary behavior is even robotic. That's the only stationary behavior we'll have to pay attention to with some law of mu. So that means visually, here's all these species that are not rare. They're approaching some sort of stationary behavior, as well as the feed boxillary variables approaching some stationary behavior. And now you want to know, is one of the infinitesimally rare species able to increase? So what do you do? So you look at this axis with the infinitesimally rare species. And you want to know what is its rate of increase. So it's just this linear question. And it turns out all you want to do is take the log fitness of that species and average across all the fluctuations. So that's the fluctuations in the chi's and the fluctuations in the community state, as well as the feedback or the auxiliary variables. So that z is this whole combination. And if that quantity is positive, that means this infinitesimally rare species would tend to increase exponentially. If it's negative, it would tend to decrease exponentially. But you can also compute this quantity for all the species that are not rare, that are present. And it's not that easy to show. But one can show, and it's fairly intuitive, that their per capita growth rates are always 0. They're present in the community. They're not going off to infinity. They're not going to 0. So on average, it must somehow be 0. OK, so now how do we use these things to state a theorem about persistence? So let me first define the set that we're trying to avoid. That's the extinction set. So that's going to be the set of points in the state space where one of the species is not present. And here's the criterion that's going to be used for stochastic persistence. It's the exact criterion that Joseph Hoffbauer used for deterministic models, very slightly differently, because he didn't do it with respect to measures. But he can do it with respect to measures, even in the deterministic setting. And what it says is, if for each species, you can associate a weight, p1, p2, p3, and so forth. They're positive weights. They could add up to 1, if you want. And you take that fixed weighted combinations of the realized per capita growth rates of the species. So that's saying, somehow I'm taking some community average of the per capita growth rate. That's roughly what that is. If that is for that fixed weight combination, if that is positive for any sort of ergodic behavior of a subset of species, so not all species, then you have the following theorem that if this uptribe of the whole positive condition, then you have stochastic persistence and probability and almost surely. And you even have estimates about how much weight is being put near the extinction set, which I won't show here. On the other hand, if you can flip the condition around, if you can find weights, so that this weighted combination of the per capita growth rates is always negative whenever there's a subset of species supported by the ergodic measures, then you get the opposite result, which is the extinction set is locally stochastically stable, meaning if you start sufficiently close to that set, there's a very high probability that you asymptotically approach that set. And again, these are models that are continuous states, right, continuous densities, so you can only asymptotically approach them. OK, so that's the theorems, or those are the theorems. So let me now briefly show you a few examples, just to give you a flavor of how you can use them, and how they might even relate to things that you're familiar with. So the first example I'm going to talk about is a annual plant model. And I'm not going to write down the model. It's just a model of an annual plant, keeping track of the seeds, OK? And it's a model of two competing species, and it's a model whose can be analyzed using results of a wonderful little piece by Peter Cheson in 1988 and some lecture notes in spring of ergodic. And so it's two species that are competing, and he asks, how do I use that criterion that I mentioned before? So we have species one here. Right now, I'm not having any auxiliary variables. And let's assume each species by itself can persist. That would mean R1 at the Dirac measure at 0 is positive, and R2 is as well. So they both can persist by themselves. You have something like this, OK? Now you want to know, what about each species invade when it's rare? So you have this quantity here. I call this mu1, this mu2. We want to know it's R1 of, so R2 of mu1 greater than 0. We want to know it's R2, R1 mu2 greater than 0. And it turns out this would be the classic mutual invasibility condition. Why is that the same as what I wrote on the board before? Man, that's not obvious. But what do we need? We need to weigh it's P1 since it's R1 of any measure. That's P2 times R2 of any measure is greater than 0. So that's trivially true at the Dirac measure of the origin, because these both are positive. Here, if we look at this measure, by that result I mentioned, the R1 for species 1 is 0. And that's just saying you need R2 to be positive for persistence and similarly for this one. So the mutual invasibility criterion is identical to the criterion I put on the screen earlier. So if we apply that to an annual plant model where it's you have neutral coexistence without any environmental stochasticity, you might not really think, oh, if I add environmental stochasticity, you can get sort of a drift along that line that might lead to extinction. That's not the case. But what you get depends very much on what is fluctuating in time. So you can use the results of Chesson to show for large or small noise. It doesn't matter if it's the size of the noise. It's just there is noise. That if it's fluctuations in survivorship of the seeds, you get both of the R terms are negative. So you get a bistability. So with positive probability, you go lose species 2. With positive probability, you lose species 1. And then you can also show with probability one of those events occurs. On the other hand, if you have fluctuations in germination rates, you can actually show that R1 and R2 are both positive when evaluated at the opposite measures. And then you get stochastic persistence. And you can even numerically estimate the stationary distribution of that model. And in that case, there is a unique stationary distribution. That's one example. But it's already covered by the older theory. It's two species. OK, what about doing more than two types? How about rock, paper, scissors? That's a simple two-dimensional thing, but with three types. So that's like the classic childhood game, except in a replicator context. You have three types of individuals. They replicate true, like rock, paper, scissors. If you win, you get a payoff bigger than one. If you have a draw, you get a payoff of 1. If you lose, you get a payoff less than 1 and those fluctuates randomly. Then what you can show is that you get a heteroplastic cycle on the phase space. So rock gets displaced by paper, which gets displaced by scissors, which gets displaced by rock. And then you can show that if you evaluate these realized per capita Earth rates at the only ergodic stationary measures, which are these Dirac measures at the equilibria, you get the following criteria for stochastic persistence. The product of the expected value of the log winnings plus the product of the expected log losings has to be positive. And that gives you stochastic persistence. Here's a simulation illustrating that. Oops, went too far. There. And if you have the reverse inequality, you get asymptotic extinction. So I started here. With probably one, you're going to approach the heteroplastic cycle. So for the last thing, I've got two minutes, I think for three minutes, based on what I'm starting a little after. So the last one is just what I want to talk about, because I actually just got accepted this morning. But I didn't write that this morning. But it's something that, for whatever reason, I obsessed about for a period of time. And I still find it fascinating. And maybe you'll see there's some merit in that or not. So this is going back to one of the first models that exhibits the storage effect, which preceded the ecological example. And this is John Gillespie's SASCFF model. You can see what a wonderful acronym. No wonder it took a great hold, which it did. And the basic idea of the model is you've deployed individuals. They have kaoleols at a single locus that determines the genotype. And those kaoleols contribute additively to some sort of phenotype or physiological activity of the organism. And then you assume that the fitness of an organism is an increasing function of its phenotype or physiological activity via some C2 function. That's the C and CFF. Actually, it was originally a concave, but it's the C2 in my context. And then just to make it ecological, you can even have the population has density dependent mortality. So you have population growth there as well. So what's the state space for this model? So if we had three alleles, we'd start with the frequencies of those three alleles. So it's just like before in the rock-paper-scissors game. But then we have to account for population abundance. So we have to go out into the third dimension. So that's our state space. It's this solid cylinder. And you can ask various things about, do you get the alleles coexisting? Do you get the population persisting? You can address that with the theory I just described. So in particular, some of the cool results that you get from this is, first of all, get explicit expressions, which we realize per capita growth rates under the right assumptions. I want to tell you what those are now. But some of the results I found really fascinating are you get opposing effects of many factors of this model, features of this model, on maintaining diversity of alleles versus maintaining the persistence of the population. So you get these opposing effects. Specifically, the per capita growth rate of the population gets better the more convex or the less concave the fitness function is. But that has a detrimental effect on maintaining allelic diversity. The number of alleles has also a detrimental effect on another allelic establishing. But it actually, through a bed-hatching mechanism, leads to high population growth. And environmental variability has a negative effect on population growth, but helps maintain the alleles. So you get these interesting opposing effects. And you can even get hidden in these statements of Perando effect, where it might be that every homozygote individual is maladapted and has a negative per capita growth rate. So if you just have one allel, you're going to extinction. But if you have enough alleles around that aren't perfectly correlated in the responses to the environment, you get the population persists. So that sort of losing alleles can make a winning genotype. So with that, I want to finish by saying, hopefully, I've given you some sense of things that you know already. So it's not. In other words, the environmental processes you can inhibit population persistence. I didn't talk about how it can facilitate population persistence, but it can. It can facilitate or disrupt coexistence. And in particular, the mathematical method is that these conditions for persistence and extinction are determined by this weighted combination of realized per capita growth rates. And that applies to discrete time models, like I discussed today. It also applies to stochastic differential equations. In particular, there's a recent paper by Alex Henning in Deng Nguyen that really shows this like was a simpler version of the result and a much more beautiful, more complete result that includes extinction results. And it also follows from a very, very general theory for persistence, but not extinction, by Michel Ben-Eim. And that very general theory, which is available on archive, also covers piecewise deterministic markup processes. So we have random switching of vector fields, but much, much more. I mean, almost any stochastic ecological model that's finite dimensional will fit into this sort of framework. And so it's a beautiful preprint. And I hope someday it shows up somewhere as a monograph. And with that, I want to first thank US National Science Foundation for funding my collaborators. And then more importantly, Andrew, Alex, and Mark for organizing this wonderful workshop, Burrs for Hosting. Even more importantly, you for listening. And then I'd happily answer any questions that you might have. Thanks. It seems like your model is well set up to deal with things that have a trend, sort of temperature is environmental change, sort of your wide variables. And what effect does that have on the kind of analysis you talked about? What do you mean? So like a trend? Like a trend. Yeah. So the trend will not be in the asymptotic analysis, right? So even if it's, you know, even if you're having like some of those models, even if it was an autoregressive model, right, you might be starting not in the stationary distribution, but asymptotically approach the stationary distribution. And only the asymptotics will then determine, right, whether you have persistence or not and the way it's been defined here. Yeah. So it's time goes to infinity. Yeah, so everything is about, has time goes to infinity. Just like we saw the other day, even to deal with how to talk about even transient trends in some models, it was about time going on infinity, right? So you're still doing asymptotic studies. So maybe there's some way of combining these things. Like I've intrigued. Like is there some way of doing what you're doing in this sort of stochastic context, right, to address something where you're still doing sort of asymptotic analysis? Partially. It's not just that. And I realize it's more complicated to sort of understand something about trends. Yeah. But yeah, it doesn't, I mean, it doesn't have an effect, is the answer. As far as the statement that you'd make, it might have an effect on the actual stationary distribution, right? And so understanding that stationary distribution is a much harder question. I don't know good ways in general to compute them for these higher dimensional models, unless you have a lot of symmetry. If you have a lot of symmetry, then you can. Like for Gillespie's model, without the population dynamics, you can actually do a stochastic differential equation approximation and write down explicitly the stationary distribution under certain cases. So there's special cases. But there's always something you have to sacrifice to get the explicit solution. Or at least it seems that way. Yeah, have Sadiq? You showed us for an example that your criteria for persistence is equivalent to the invasibility. And how much generality is that true? Better equivalence? Well, first, the first question to that question, the first question I'd have to that question would be, what do we mean by mutual invasibility when there's more than two species? So that's a question, right? So you could answer it two ways. I thought about it. I'll give the answer. I won't put you on the spot. So you could think of it one way. You say, well, I insist that every missing species has a positive per capita growth rate. That would be a very generous interpretation. And if that's true, then immediately get stochastic persistence. So that's sort of an easy case. The other thing you could say is, well, that's too strong. Because if I look at a predator-prey model, if I look at the origin, clearly not every species has a positive per capita growth at the origin, because a prey has a negative per capita growth rate. So maybe all I need is at least one species has a positive per capita growth rate. That's not true either, because the rock, paper, scissor game gives a counter example. Because every species, every equilibrium there has a species with a positive per capita growth rate, but you don't get stochastic persistence unless something else holds. So I would say, when you say, does it extend? The answer is, I don't know what we want to mean by mutual invisibility. It does, if you say, all missing species should invade, then it's immediate. It's true. And if only one missing species is invaded, then it need not be true. So it's more subtle. And there's still things. It actually, there's a fairly open question to get things more in line with the deterministic theory. So the deterministic theory is much more refined in some ways in the stochastic theory. And one of the ways it is is that Joseph's original theory doesn't cover scenarios that we know how to cover in the deterministic theory. And that's because in the deterministic theory we take advantage of Morse decomposition. And while there exists Morse decomposition from random dynamical systems, it's not clear how well you can use them for this type of analysis. So that's sort of this open piece of work of trying to actually refine this theory so it's actually completely parallel to the deterministic theory, which it's not. So there's a gap that needs to be filled for sure. Yeah. Were you ever ready for that one? Sorry. You had something that's like one of the questions I've been really thinking about. So yes, Chris. So if you can do mutual invasibility, then you can do adaptive dynamics. And then you can do evolution. Yep. Yeah, you can do that. So I mentioned as one example that you could do evolution. I mean, I don't. So I mean, the population genetics model is evolution, right? Changes of frequencies. So I did already evolution example. In the paper, we do another example where we have a trait that's evolving. So we do a version of this model of Landy and Shannon, which is sort of an evolutionary rescue model. It's not really well, just let's say evolution in a fluctuating environment model, but through a quantitative trait with a Landy approximation. So we do an example of that. And there you can compute things explicitly because you end up having an autoregressive process describing the thing. Well, anyways, you just have an autoregressive process that determines essentially the population per capita growth rate of low densities. So you get some interesting predictions there. They're in line with what people seem doing. What I'd say more theoretical. In other words, the same type of computations, but without the mathematics to back up that this gives you persistence or not. So you could argue about whether that's an important distinction or not. You could see arguments in either direction. And I think you had a question. So OK. OK, so how can you extend the theory if you have things like any effect? So that's a great question. So even for deterministic models, you don't have to. Well, it depends how the Ali effect shows up. There are cases where you could have an Ali effect and you still get permanence. But in general, if you start having attractors in the boundary, you don't have persistence. From the perspective of this theory, because the invasion growth rate or the capital growth rate at low density is negative. So that just means this theory isn't appropriate for things like trying to understand systems with Ali effects that really lead to species being lost because they're low in density. And just purely because of that feedback. So there's no question this theory doesn't apply there. That's a criticism that applies to the deterministic and the stochastic theory. One thing I should mention though is if you do, there is a difference in the stochastic models versus the deterministic models. So I have this paper that's coming out in the ecology where we look at a model where there is effectively, we're looking at positive frequency dependence in models of competition. So that leads to Ali effects where you have either competitor being lost. And as soon as you add any sort of non-local environment to cacici, of course, with probably one, you lose one of the species. In some sense, models of the Ali effects, if you have non-local noise, meaning the noise really can push you anywhere in the state space, like the SDEs or like that, for instance, then you're going to always have a result. But it's going to be not an interesting result. You always go to extinction asymptotically, which makes sense. But it might not be what you're looking for. That's not how it shows. OK. Bill, are you ready? Yeah, thank you very much. Very nice talk. What's one question? I mean, in deterministic dynamics, are mutual invisibility normally guaranteed coexistence, but it might not, for instance, if you ever said it on the family? Yeah, that's just like it was, yeah. He was asking the way. Yes, yes. So yeah, but I said, it was quite different. Does it not happen in the east? No, I said it. I just answered that effectively. But I can repeat it. Sorry, I didn't say it clear enough. But if you have a situation, so if the deterministic model hasn't attract on the boundary, for instance, right? Or you used to say, oh, I see what you're saying. It's slightly different. Oh, yeah, this is a different interesting question. Interesting point. Sorry, I just realized what you meant. OK, so you're talking about a situation where you have something like this, right? Is that what you're talking about? Yeah, so you have to have the set of bounds of boundary. Right, so the set of bounds of boundary. Right, so you come back to it. Yeah, so this is a cool little thing about. So the deterministic model, this would mean you don't have permanence, right, just by the definition. However, in stochastic model, you're not, even if you had a little bit of noise, you're not going to pick up this asymptotic behavior. I mean, it depends. I have to be a little careful when I say that. Oh, that's a stochastic system. Yeah, because if I had a stable point here and a stable point here, most likely with small line of noise, just by various results, the stationary distributions are going to concentrate on those stable fixed points. So you won't be evaluating the stochastic growth rates here. And as a consequence, depending on what's happening on these other fixed points, you still might get stochastic persistence. And so you can construct examples like that, where there's a two-bystable prey and a predator. You usually get situations just like you were described with. Or maybe the prey can invade. That would be a weird situation, but you could create it. The predator can invade when there's too much of the prey, when they're at that coexistence equilibrium, but still the whole system is stochastically persistent. So yeah, so that's an interesting point. There is a way that you're getting rid of things that you probably don't think are relevant for the permanence condition. But that's just the way the permanence condition works. That's actually nice. It may be. The possibility of these systems is removed. Yeah, I think so too. I think it's interesting. I think so. OK, Heather. Question. OK. Very inspirational talk. Thank you very much. Just one question. How much of a result would you still fault if you change white noise to color noise? Well, as I mentioned, the y variable can account for color noise. So you can be forcing. You take white noise, you can color it by just applying a function to it. Pretty much color 332, because with color noise, it can put your system over the variation of the walls. Yeah, same with white noise too. But anyways, it does account for color noise, is my point. So you can have the auxiliary variable be like multiple places. Sorry. Both have very unregressive process, and that's what color noise, not any color, but many colors that you might give.
So I will start with this quote. Some people have reproduction, growth, due to fluctuations in environmental factors like temperature, any realization due to the difference in species. These are discrete time models. And it for the auxiliary variables that depends on forcing the dynamics, the ecological dynamics. within a population. So it can have spatial only prove theorems for this definition. And the species, x1, x2 are their densities. And we can You have three types of individuals. They criteria for stochastic persistence. The product frequencies of those three alleles. So it's just you get the population persists. So that sort of that these conditions for persistence and differential equations. In particular, there's a determine, right, whether you have persistence or