 Well, it's not going up. So it's a pleasure to have Daniel Fischer for the second lecture on high-dimensional ecology and evolution. Thank you, Daniel. OK, thank you. So I'm trying a different setup today with my computer, so hopefully it won't cause problems. But let me know if it seems to me any difficulties that I can try to switch back to how I was doing it yesterday with some tundra. So yesterday, I talked about models in which only had one strain. And when a mutant came up, if it was doing better in that environment, then it would replace it. This would change the landscape by a small amount, and that would change the future evolution. And I'm sorry, I didn't actually give this the model a name. This model one could perhaps call the fitness snow scape, a landscape with something which is static, a seascape which people talk about as a dynamical environment. But here, it's the evolution itself that changes the environment. So like when you're going up a mountain and you change the environment that you're walking on as you walk. And what I showed was that the simplest model, and I should emphasize the particular model I wrote down with sort of the cubic interactions, is the simplest model. And that exhibited the red queen phase where the evolution just continued without slowing down. And I said that it was very robust. And what I mean by that is if I change the model somewhat or do a different range of models, then we'll get the same behavior in the limit of infinite dimensions. I raised the possibility that there might be some other behaviors as well, but I didn't talk about those. And then I left to open a question about diversification, which I know almost nothing about, but I think is the most interesting question for those kinds of models. As you mentioned, none of this is written up. I think someone asked me for the name of the model. So I guess this is the closest to being the name. OK, so what I'm going to talk about today and tomorrow is much closer to things people have been, several speakers have talked about so far, which is really talking about a situation where at least I've initially got a fixed environment, but I'm going to have multiple strains. And here I'm going to emphasize calling them strains rather than species because I'm mostly going to be interested in one species with many strains within that species. And I'm going to talk initially about assembled communities where there's no evolution. So again, this is what people have talked about. And introduce a random loss of altera models, which then we need to talk about the statistics. And they have a stable phase, as Stefano Alessino talked about, but then that can be unstable. I'm going to then talk about a special model, which is the perfectly anti-symmetric interactions. Then more generally, the chaos that occurs and that diverges, and then how this can be stabilized by migration. And that's going to be the main thing I'm going to get to. So I'm going to introduce a phenomenology and the basic behaviors with some rough explanations. And then I'm going to go and talk about the theory and using the dynamical mean field theory. And this is a theory. The method is also something which is applicable to what I talked about yesterday, but there I didn't explain how I got anything. And then I'm going to turn to looking at slowly evolving communities that I'll get to tomorrow. And also I'm considering sort of phenotype models, which would determine the interactions. So I should mention, so all of this work that I'm going to talk about today with all the completed parts was done by Michael Pierce and Natish Agrawala, and it's published in PNAS this year, early this year. OK. So the models that I'm going to talk about, the generalized Lotka-Volterra models. And I'm going to call the number of strains K. So I labels those, runs from 1 to K. And N i is the population of strain i. Now I should apologize if this is different notation than Stefano used. He called the population X i and called the number of strains N. Unfortunately, there's not completely standard notation. But I'm going to start by writing the model as he did. Someone has the basic growth rates or growth and death rates here, R i, and then the interactions between the strains. But we're going to be interested in the closely related strains. So if I have closely related strains, then it's natural to replace this by things which tell me how the strains are similar to each other. So I'm going to write that R i then, say, is going to be equal to some R bar, which sort of average over all the strain. That's some common part, plus some varying part, plus some part which is varying, which I'm going to call Si. And I can think of the Si then as being selective differences between them. And the natural one is then to take the Si with an average of 0. So that's going to be the variabilities, the way the strains differ from each other. So that's going to be my A i. Now what about the interactions? Well, the interactions, I'm going to consider two parts. So I'm going to take the A ij. It's, again, going to have some overall average interaction, which will generally be negative if they're competing with each other. That'll be less than 0. And then they'll have a strain specific one. And I'm going to put a factor of n in here, just so I don't have to carry factors around n anywhere else. And this is going to be the interaction with itself. So Q is really going to be the sort of niche interaction, one over the carrying capacity of that strain. And then we're going to put in some random parts. And again, I'm going to pull out a factor of n out in here. So the vijs are the between strain interactions. And so the vij, again, I'm going to take that mean 0. And then I'm going to say something about what the statistics of those are. So the vijs are going to mean 0. Typical magnitude 1 sort of sets the overall scale. Now, since these are going to be considering different strains, these are going to be much larger than the differences. So this and this is going to be small compared to these. And so what does that mean? That means that the R bar and the A bar basically set the total population. You constrain the total population because they're the bits which are the main interactions with the growth and the competition. So they're going to constrain the total population. And that I'm just calling this big n, which is the sum on i of the n i. And then I'm going to consider what they essentially do is make this approximately constant. And then there are variations, of course, between the n i, but with the n being roughly constant. So in that case, then it's natural to define the frequencies. So I'm going to define the frequencies, which are the fractions in the population. So I'm going to find nu i as being equal to n i over the big n. So it's the fraction of the population. So obviously, the sum on i of the nu i equals 1. So that's going to be my model. And then I can rewrite the model in a slightly different form. I'm combining these. I'm now going to pull out the R bar and the A bar. And I'm going to write the model this way is the di nidt. And I'm going to use the notation of a dot for that. So that's going to be nu i times, well, now I've only got the differences coming in here. So I've got the si. And then I've got the plus, the sum on the j of the vij. Oops, let me put them in another order. This has the minus, the specific one, the q times the nu i. And then it's got the plus, the sum on the j, vij, nu i, nu j. But then I'm going to put an extra piece here, where epsilon is a Lagrange multiplier. And the role of that is to make n equals constant, or more specifically enforce this constraint of the sum on the nu i is 1. So I'm replacing the effects of the overall growth interactions. And I'm only going to consider the effects of the differences. So this is the differences in the growth rates. This is the interaction with it itself. And then this is the interaction between the types. So this is the model which I want to understand. And this is going to have average 0. This is going to have average 0. So of course, what we have to talk about is we have to talk about the statistics of the interactions. Daniel, do you really have vij, nu i, nu j there, or is just nu j? Thank you. This is the drawback of writing things in real time, is one keeps the audience on their toes. So thank you for correcting that. Yes. So that's just the standard interaction. And so the first term, this term here, is would be often written as effectively like one over the carrying capacity. But I've rescaled things real scale. So I don't want to keep things all over the place. Oh, so I should say one other thing before talking about the statistics. I'm assuming no stochasticity at this point. But I will want to consider extinctions if nu i becomes less than 1, i.e., n i, sorry, less than 1 over n, which is corresponding to n i less than 1, of course, then a kind of less than 1 individual that will go extinct. So I'm going to mostly consider the domestic interaction. Then I'll talk about the effects of the extinctions, which of course, some play an important role, particularly because they'll happen even in the simple model. So this is my basic model. And the thing which I then have to say some things about is what the statistics are. I've said the interactions are going to be random. So the s i's are going to be independent. And I can talk about the variance of those. The variance of s i is just sigma s squared. And these, I would expect that they'll be small, very small, because the strains are all similar. So this is sort of a consequence of the prior evolution, that they're all going to be very similar to each other. If one was better, that strain would diversify and so on. So this is going to be a basic assumption that small. And to often, I'm going to set that equal to 0. But the crucial bits are the v ij and what the statistics of those are. So again, I'm going to take for i not equal to j. So that's the interactions. I'm going to take v ij squared equal to 1. So the variance of the v's is just equal to 1. So this just sets the overall scales of the time. But the crucial bit is then, what are the correlations? So in particular, I want to ask, what is the correlations between v ij and v j i? So this is going to be the only correlation which I'm going to put in. So this is going to be the only correlation. It's going to be between those. So this is how the correlation between the effect of strain i on strain j and the effect of j on strain i. So if this is going to be competitive, then I would expect this to be positive. So I'm going to define a parameter gamma here, which is going to be that correlation. The competitive would correspond to gamma being bigger than 0. So if I correspond to gamma equals 1, it's totally symmetric. That's correspond to a symmetric v ij. And that's often what's considered for models with competition. In general, gamma lies between minus 1 and 1. Sorry, then, it's only to interact with us. Shouldn't the v ij also be small, so the variance of v ij be over the 1 over the number of species? Well, that's a choice of a convention. That's a convention which often people choose for being useful. I want to talk about more things in terms of unre-scaled quantity, so the interaction is setting the basic scale. There's nothing intrinsically that it's going to be small. Now, since the new i's, if there's many species here, the new i's are going to be each of order 1 over k, then this interaction piece is going to be small coming from that with these random of order 1. So this whole piece is going to be of order 1 over square root of k. But I explicitly don't want to scale out the k's. It's convenient for doing theoretical analysis, but I don't want to do it because it sort of confuses what one's assuming when one does that. Well, of course, I can have k, 2, 3, 20. I can add more things. I don't want to be rescaling the v's. Each time I add more things, I'll take more things out. So I'm trying to keep it in terms of the physical or the biological quantities. So gamma equals 1, we correspond to symmetric. What about gamma is less than 0? What would that correspond to? So gamma less than 0. So this has two possible things it can correspond to. One of them, for example, is if I have a direct competition, so if 1 beats 2, what that means is that v1, 2 is bigger than 0. But of course, that means 2 loses to 1. So 2v2, 1 is less than 0. So that's a natural way in which you can get anti-symmetric correlations in the matrix. I just want to add an extra page in here before I have my next bit. OK, so that's one possibility. But there's another possibility in this. I'm going to talk much more about tomorrow. And this is what makes it much more interesting. There's another possibility, which is I have bacteria. So I have bacteria. And so those bacteria will have populations, say bi. And they'll say bk of the bacteria, a number of those. And then I have phage with populations p sub l. And I've got some number of those. So I've got the bacteria and the phage. And these interact with each other. And so if I look at the dynamics of those, then bi dot would be bi. Times some growth rate for the bacteria. But then there will be minus a term which is coming from the interaction with the phages. So this would be some matrix here, h i l times the population of the phage. OK, so that would be that term, where these are going to be positive. Because they have negative effects on the bacteria. And then I've got the phage. The pl is going to be, sorry. I also want to put a term in here, put a term in here, which is a stabilizing term, someone of j of nj. So that's coming from the competition, bj. And then I've got the phage. So the phage are going to die at some rate. The phage are going to die at some rate. But then they gain by eating the bacteria. So they have a someone i here of some other matrix, some other matrix f, now i l i times the population of the bacteria. And again, these are going to be positive the way around I've got. So if I think of that as putting all these together, and I'm going to think of these as being strains of one species of bacteria and one species of phage. So I now have a two species model, but with diversity within the two of them. And if I look at what the interaction matrix is going to be there, so what is the action matrix going to be in that form? Well, it's going to have this form here. If I think of putting the bacteria on top and the phage underneath, it's going to have some minus constant all the way through this part from the bacteria interacting with each other. Then it's going to have minus the h from the bacteria interacting with the phage. Over here, it's going to have the f from the phage interacting with the bacteria. And there are no specific interactions of phage with these other. Now, what do I expect here? I expect that f will be approximately f bar plus something small, so delta f i l. And h will be some average value plus some small variations. So again, in the same spirit, I'm sorry, I've got my indices backwards. So I have these here. And I would expect these will be correlated. So more specifically, f and h transpose are correlated. That's corresponding to the fact that, of course, if one phage does better against the bacteria, that bacteria is worse for the phage. So if I look at this whole matrix here, it has anti-symmetric correlations in the dominant parts here. There's also this part, which is symmetric, but there's a dominant effect here, which is the anti-symmetric part. Again, I'll turn out this model behaves very similar to the simple random multivulteram model that I'm going to mainly talk about. And I'll come back to this one some tomorrow. So you can think of this as being one of the primary motivations for considering models in which gamma can be negative. So one thing we can then start to talk about is what does this model look like? So I'm going to focus on this model here. So we've got here, we've got two main parameters. We've got this parameter gamma here, which we can make in red to find that. I've got that parameter is the key parameter. This epsilon is just the, is the grand multiplier. That adjusts itself. So the only other parameter we've got here is this Q, which is this niche, niche interactions. And that's our basic parameters. So we've got a model now with two parameters. One is the interaction within the, within the same strain between same strain and one is the interaction with others. And I generally have then the S's as well. So another parameter which is associated on the S's, but I'm going to mostly consider the simple possibility where the S is a zero. There's no overall differences. We can add back in the effects of those. So now I'm just going to show a phase diagram and I'm going to say how we, when one gets it in a while. So the S is always equal to zero. So we've got our two, two parameters here. We've got the Q, which is the self, self interactions. That's the strength of the niche interactions. And then we've got gamma, which is the, the symmetry. Well, if the interactions are very strong relative to the, within the strain, very strong relative to the interactions between the strains, then we've got the standard behavior and the limit of a large number of species. We've got a, so this is now, we're going to want to consider K, the number of species being much larger than one, the number of strains, much larger than one. And up here then, in this regime up here, we've got a single stable community, large community, stable community. And that's actually with the hyperability for the limit of large numbers of strains. There's going to be a unique, unique community there. So this is a result which has been worked out by many people, particularly a lot on recent term years, but it's more or less something which may already, Robert may already knew. But then there's a boundary, which is where that goes unstable. So there's a line I can draw across here. And this is a line here. And this line goes from zero to a value over here of Q of square root of two times K. And so here's where the square root of K that Matteo was asking about comes in. So this is when Q is bigger than this of a single stable community. But when I lower the Q and I cross to here, then the system goes unstable. I lose a single unstable community, okay? And then the big question is what happens down here? Okay, what happens under there? There's not a single large stable community. That's the one thing we know. Now there's a special line, there's a special line which is corresponding to being along here, okay? That's the line where it's the perfectly symmetric model. That's been studied by a lot of physicists also recently. And in that case, what's known, so this is for gamma equals one, for gamma equals one here, there are a very large number of stable communities. There's actually exponentially many different communities. Okay, but that's very special. And it turns out that as soon as you go away from gamma equals one, things change. This behavior over here has a lot of similarities as far as the dynamics to the things I talked about last time of the random landscape is you never actually get to one of these communities. You sort of wander around, things go bouncing down, things come back up again and it never really settles into them. But if you just look in terms of the communities that exist, there are exponentially many stable, uninvatable communities of subsets, okay? So it turns out this community up here is going to have in this case here, it's the number of the size of the community. So the number of which persists in the community is gonna be greater than or equal to K over two, so more than half the types persist, okay? As you go down through here, as I say, it goes unstable and we want to talk about down here. If we ask about putting a little bit of the S's in, so if the sigma of the S squared is not zero, if the sigma S, so the variance of the S's, as long as that's less than or of order, actually to that is one over root K, okay? Then you only get qualitative, quantitative changes to the phase diagram. But it stays qualitatively the same, okay? But I say we're mostly going to say the SI is equal to zero and I can talk more about that. So one thing to note is that you need to have the selective differences being very small to get this behavior. If the selective differences start becoming substantial, then if we go back to our basic model here, that means this term will then tend to dominate and this term will be small compared to that because this is some of a whole bunch of random things. This will be dominate and I won't get the interesting behavior, okay? Now what about the Q? Well here, you notice that in order for the Q to be big here, it has to be bigger than a value which is of all the square root of K, okay? So to get this, you need to get the stable community to get that, you need to have K greater or equal to of order root K, okay? So this corresponds to saying the interactions with your siblings, right? These are all different strength. I think your siblings have to be much, much bigger than the interactions with all your second and third cousins. So this is basically equivalent to assuming that there are niches. You assume that each one interacts with its own strain much more than interacts with others, okay? And there's no a priori reason to assume that, okay? So the particular things that we're going to focus on here, the bit I'm going to want to focus on is actually along here, along this line there where either the Q is very small, so you might as well set it to zero. The natural assumption then is if I have these closely related strains, so with the close relatives, the natural thing is to say that Q is very small, Q equals zero, I've just got the random parts with that as well. If Q is of order one, that doesn't matter, that doesn't matter much as long as it's small compared to a root K, okay? So I'm going to consider that. And then for simplicity, I'm also going to consider the SI is equal to zero, but I say that we'll come back and re-examine. So I think you want to get the questions as they arrive. There is a question in the chat. So you can read it says as anti-symmetry in A still doesn't capture the fact that the signs in the upper and lower triangles or the bacteria phase system are positive and negative. Doesn't it make a difference in behavior? So the interactions by definition, the phages are bad for the bacteria, that's their lifestyle. So these terms here, this has a negative sign and the H's are all positive, okay? Over here, the phages are eating the bacteria and so those terms are also all positive. What the random parts is in the difference. So this is positive, this is positive, but then the random parts are associated with differences between them. Okay, so I'm going to come back and talk more about this one, this tomorrow. Here are the other questions on the model or the sort of basic set up. And I say, I'm going to explain a bit of how one gets this phase diagram, but I mainly want to talk about this part here. There is another question. Is there anybody in the self interaction or is it always exactly? Okay, sorry, I didn't see that one. So the self interaction there, I can have some variability in it. I mean, in particular, if I look at my interaction here, I've got a part where I equals to J, so that would be the variability in it. It turns out if it's variable by similar amount to how the interactions between strains vary, then it doesn't matter much in the limit of large K. It only matters in the limit of large K if it's big, in particular, if it has to be of order root K, that's going to be much bigger. So I can turn the other way around and I can say, let's have fixed Q, that's going to be a property of the biology, a fixed distribution of the S's and ask what happens if I add strains. So I'm assembling a community here, adding more strains, and then what will happen is I add more strains, it'll go unstable. Okay, and where it goes unstable will be associated with how big the Q is. Okay, so that's the basic results of May. He didn't really quite do things right, but the overall result is he certainly got right. So that's this part of the phase diagram up here where there's a single large stable community. And what happens down here for large K really only started to be being investigated in recent years. And I'm going to show some simulations of that, but then really try to develop that theory. So we're going to consider the behavior along here. Okay, so we're going to consider the close relatives, we'll focus on this. And for reasons that sort of motivation of the bacteria phase, we're mostly going to concentrate on this region here. So we're going to concentrate on gamma being less than zero and between zero and minus one, but actually sort of believe that most of the behavior actually persists in this whole region except on that special line. So I'm going to not talk about this stable community, I'll just say something about how one gets it, but I really want to talk about what goes on in here. Okay, so for doing something general, I want to do something very special. So one of the things that one has learned from experience in statistical physics is there's a lot to be gained from having particular models that one can really analyze and understand in detail. And then one can sort of use ways of thinking about what's more general and what isn't to ask which features of those might persist and which ones don't. Okay, so I'm going to first talk about a very special model and this has been looked at by over many years, a special model. And that's the case where I'm going to have the Q equals zero, but I'm now going to have gamma equals minus one. Okay, so this is going to be the special thing. It's perfectly anti-symmetric. And now I've got the anti-symmetric of V matrix. Perfectly anti-symmetric, okay? So this has a very special property. It has various things which are known about it. It has a unique, it always has a unique stable community and by stable here, I mean an uninvatable community. And this Stefano talked about in some generalizations of this and the size of this community is going to be approximately K over two plus or minus some order square root of K, which all depends on the characterization. So this community always says it's unique stable uninvatable community, okay? And in that community, each of the new eyes will be some fixed point value, new eye star. So this is then the fixed point of the dynamics, okay? Or new eye will be equal to zero for the extinct ones, for ones that go extinct, okay? So if you look at the surviving types, if you look at the surviving types, then it's very special. And you can see that this is very special because this fixed point here, this fixed point is marginally stable because of the anti-symmetry, it means that all the eigenvalues for the stability around that fixed point are imaginary. The imaginary eigenvalues, it's not stable or unstable. It's very special, okay? And that very special behavior is associated with a conservation law, a conserved quantity. I say this is just a mathematical artifact, a conservation law, but which Stefano had talked about. And this quantity here, which plays the role of an energy, okay? It plays the role of an energy. I'm trying to call it that. And it's just gonna be minus some one eye of the new eye stars, okay? Times the log of the new eye, okay? Now you can add another piece to this, which Stefano added. You don't have to include that in this case because the total number is one. So you've got this but the relation here and this is conserved, it's a Lyapunov function, but it's conserved on a situation law after the extinctions. So after all the ones have gone extinct. Before that, it's a Lyapunov function. It'll increase and but when they go extinct, they'll saturate and then become a conservation law, okay? This is a conservation, and you notice this is a sum of a whole bunch of independent terms. The other thing to note is that in the log variables, so if I define Li, which is the logarithm of the new eye, so this is just the log variables, they're the natural ones to think about it. I think about populations growing and shrinking at some rate depending on whether their growth rate, net growth rate is positive or negative. So this is the Li's natural variables and in these variables, the phase space volume is conserved, volume is conserved, okay? Again, it's just a mathematical nicety, mathematical nicety, but it enables one to do certain things, okay? So this means that there is a steady state, steady state of this, which is like equilibrium, like equilibrium stat mech of a bunch of interacting of independent particles. So you can immediately write down the dynamics as a quantity which is like the temperature as a quantity here, which is like the temperature, sorry Daniel, can I ask a question? So the new eye in your equation for the E is a dynamical variable, right? The new eye is a dynamical variable, right? So this is generally a function of time. The new eye is instead of... Is the fixed point. So the dynamics does hover around a fixed point or does it go to a fixed point? Ah, okay, so the dynamics, if I look at the simple cases, okay? So I can look at a very simple case here and I can ask what this looks like. And the simpler situation I can got if three of them survive, okay? Let's go down here. So if I have three surviving, okay? Three of them in the community, in the stable community. Okay, the sum of those is always equal to one so I can draw the flows here in the news, in a phase space where this would be pure one, this would be pure two and this would be pure three and somewhere in there there's a fixed point, okay? So that's a fixed point but it's marginally stable and what that means is it's a family of stable orbit. So there's that one there or there's another one which is here. So these are all limit cycles, okay? These are all limit cycles, the whole family of those. So there's a whole family of the states. And that's where it's very special, okay? In this simple community, they're just periodically like this and this is in fact like the classic Lotka-Volterra model of predator and prey which has a family of cycles here. So this is approximately like the original Lotka-Volterra predator prey model with one of each and its dynamics also looks like this and there's a whole family of cycles but we sort of know that that thing is very, that thing is very special, okay? So what happens more generally, so there's gonna be something which is gonna parametrize this family and there's something which is gonna parametrize this family which is roughly speaking how big a range they go over, okay? So this is gonna, this quantity E, it can be anything. It's gonna depend on the initial conditions, okay? So this quantity E here can be anything. Well, it's constrained as to what values it can take but the V is variable and so just like in thermodynamics this is gonna classify what state one is in. There's a whole family of states, right? So there's a family of states, a family of states here and they're parametrizing something which is like the temperature which is proportional to the average of the energy over all of the overall the species, okay? So the quantity of like the temperature it'll determine how big the fluctuations are in the shows and pictures in a minute, okay? This temperature is going to be conserved. I've got simple statistical mechanics. So if I look at the probability distribution of all of these, the probability distribution then is going to be the probability density of all the new eyes, okay? It's gonna be proportional to new eye to the minus one plus a quantity which is gonna be related actually to the new eye star over theta and that's because of the new eye star appearing here. I exponentiate this, I get new eye to a power and then there's gonna be a factor which is just gonna be associated with a constraint which is the delta function that all the someone J of the new J is equal to one and of course I need a product over all of the I here. So they're all independent of each other they're independent of each other and the distribution which looks like this, okay? Rather amazingly, this is identical to the distribution from purely neutral theory. So this distribution is the same distribution you would get out if you just had a large number of species with a bit of migration into them and they would have a distribution here which would be a distribution which would depend on migration rate in that case but it would have exactly the same distribution they would be independent. So this looks like a neutral distribution, okay? But this is completely if you like coincidental this is just coincidence because the dynamics here is nothing like neutral there's no stochasticity it's driven deterministic dynamic, okay? So let me just, there's a couple of questions here. So one of them is why is E of the sum of the terms with no interaction? So that if I refer you to Stefano's, I was seen as talks where he showed explicitly that this is this quantity is conserved. It does depend on the interactions in the sense that the new I star, right? These new I star are given by the Vs. So the VIJs determine those new I stars. So it does depend on the interactions but the only way depends on interactions though and this particular quantity is on interact. Nevertheless, the species are interacting with each other the strains are interacting with each other. And so there is dynamics and that's what we're going to be interested in, okay? So this is all mathematical. This is just gonna motivate some things. If you don't pick up this, it doesn't matter for what comes later. Yeah, there's another question whether it is a coincidence that the formula for E looks like an entropy. It looks like an entropy. It's not, there's a way of inferfuring it but you think of it like an entropy but it's not very useful, I don't find. It's more like an energy. Yeah, let's see, there's another one there. Yeah, so this is like, these are the neutral cycles in the Kavatera model. They occur when three in the community. But when you get more than three, when you get more than three here, so for K greater than three, well, okay, much greater than one at least. The, so for K much, much greater than one, what happens then is you might think that you would get a whole bunch of cycles, different cycles. Remember, everything's interacting with itself. Interaction between cycles tends to be unstable. It tends to drive chaos. And so here one gets a chaotic state. A chaotic state. And there's a whole family of them, I say parameterized by this quantity, quantity theta. Okay. So now I'm just gonna show this what this looks like and then we're gonna try to explain it because this only tells us a steady state. This doesn't tell us anything about the dynamics. It tells us about static snapshots. The static snapshots not very useful because those static snapshots can easily let us think that we're in a completely neutral model. However, these are large populations. These are microbial populations that ends large. The demographic stochasticity effects are small. So the neutral theory just can't be right quantitatively. So this is a coincidence, but it's a real warning. The other thing you note here is that if theta is very large, when there's a lot of chaos, so theta is very large, then this is roughly uniformly spread out on a log scale. It was roughly uniform on a log scale. So then in fact, one sees in the picture. So here's some numerics for this perfectly anti-symmetric model. And here, the characteristic scale of the temperature, the temperature is basically how wide the fluctuations are on a log scale. So this scale is plotting on a linear scale. And you can see here on the linear scale, these come up, they spend most of the time down here and occasionally burst up. So these all have bursts upwards. And I'm gonna say why in a minute, they come up and come back down again. Some other one comes up and so on. And you get this dynamics. To really see what goes on, you need it on a log scale. So this is the frequency on a log scale here. So this was this quantity which I defined as being the LI, which was the log of the new I. So that's looking at that frequency and each of these colors then a different type. And this is a function of time. So this is the log you see the spread here. And this basically scale here. This scale here is this quantity theta, which is like the temperature. If the temperature is small, they will be at a fixed point. So theta equals zero corresponds to a fixed point. We'll correspond to the fixed point with no fluctuations. And large theta will correspond to these well fluctuations. If you make theta even bigger, you get even bigger fluctuations, but they're stable fluctuations, it's stable chaos. Now this picture is only after a lot of others have gone extinct. So if I started with others here, some of the other ones will go extinct and they go extinct quickly and stay extinct. So in addition to this, I've got approximately half of them are extinct, which I'm not showing. And then there's chaos you see by the time you get to a relatively modest number of types. And then you get this chaos behavior and certainly in the limit of a large number. So what is this coming from? What is dynamics is coming from this anti-symmetric nature of the view. And this dynamics one can call ecological, kill the winner. And it's very related to things that Mercedes talked about today, which is that if you're, it's the disadvantage of being popular. Disadvantage of being in big numbers. So let's consider one particular strain here in black. Now at this point here, there's a whole bunch of blue strains, which are the large ones. And if it just happens that the signs of V is such that those blue ones favor the black ones, then they tend to drive the population of the black ones to come up. So these blue ones drive the populations of black ones to come up. So this then comes up. But then what happens when this comes up, then the strains which prey on it, which don't like it, are the red ones here. So then those red ones will come up because this one is now high. So these red ones will now come up. These red ones will now come up. And what will those red ones do? They'll inhibit the effects of the black ones and the black ones will come back down again. Okay, earlier on when the blue one, then the black one first came up, the blue one started to go down. And again, this is because the anti-symmetric sign of the anti-symmetric sign of the interactions. So this is a killer winner. It's something which is there clearly in bacteria phage models. It turns out you automatically get it when you've got these anti-symmetric models. And this behavior, this behavior for this purely anti-symmetric model is going to be a clue to behavior more generally. So the reason I'm gonna talk about this is this kind of behavior with these fluctuations approximately uniform on the log scale. As you can see more here, spread out of some range of the log scale, that's gonna be the ubiquitous behavior. The behavior is complicated. If you look in the details here, you see all these wiggles. Here you just see what looks like bursts. And a crucial part of this is that each type, so each type here, each strain that survives has a burst up to a bloom up to high numbers. So if you went on for long enough time, you would see each of these strains coming up at some time. Some of them come up more often, some of them come up less often. So that's something which is sort of natural in the maybe natural in the bacteria phage content. So this is kill the winner dynamics. Why is it called kill the winner? Well, whichever one is high at that time, whichever one is higher to given a given time, the ones that do well against that like this blue, red, black one here, those are the ones that will come up. And then because the anti-symmetric interactions, then they'll kill that one, that'll come back down again. Okay, so that's where the kill the winner terms come from. This is used in several, both ecological and evolutionary contexts. And so maybe it was, it's not the best term to use, but the basic dynamics here is coming from this anti-symmetric behavior. It pays for phages to attack or to evolve to attack the most common bacteria. They do best. And that drives it back down again. And it pays the bacteria to be resistant to the most common phages. Right, so that's the opposite side of it. And that's what gives rise to this dynamics. Okay, but now we have a problem. The problem is this gamma is, as soon as gamma is bigger than minus one, the behavior is different. The gamma equals one very special. We have no conserved quantity, no conservation anymore. And what happens is we get a behavior like this, that if we look at the log of the new eyes, and we look at them here, okay? So that's the maximum they can go up to. So they'll be fluctuating around. Here's one of them fluctuating around, fluctuating around like that. Bigger and bigger fluctuations. If I do another one, another one will fluctuate around. Also have even bigger and bigger fluctuations. And you get divergent fluctuations that go to extinction. Okay, so you get divergent fluctuations, which drive extinctions. Now this one can very easily see already in with the three types. So if we looked when we just had three types here, you can already see this. So if we've got three types, which are surviving there. And I look again at the dynamics. So this is pure one, that's pure two, this is pure three. Okay, and in this case, we can have a fixed point in the middle, but that fixed point's unstable. That fixed point's unstable. And if I look at dynamics, dynamics gets closer and closer to extinctions. So the dynamics goes around, gets closer and closer to extinctions. So this is unstable dynamics here, heteroclinic dynamics. Here it's unstable chaos. It's divergent chaos. There's no steady state, it just drives extinctions. The dynamics get slower and slower, bigger and bigger fluctuations, but a domestic approximation breaks down. And of course, at some point here, I get extinctions if I go below this. Okay, so if I go through there. So then what will happen is you end up with a few types left. They'll typically have a cycle with a few types in there. So this is behavior that tells us it was very special. It's pretty useless. So why do I care? Okay. So at this point, we have to ask, how does one stabilize the dynamics? And let's add some at a page in here. So how can we now stabilize the dynamics? Okay. So one way to stabilize this, a common thing to do, okay, is by migration. Okay. So the normal thing one would think of is I've got some island that I'm looking at. So that we're looking at and then we've got some big mainland over here. And I have the species coming in from the, migrating in from the mainland. And so species I say comes in at rate Mi comes into there. And this is the island I'm focused on. So we lose the, we lose the possible extinctions here because I always get the extinctions. So this is a big pool with all of the types. It's strange in it. Okay. So what is that? A corresponding I'm just adding a little bit of migration on the top. However, this is cheating. I consider this completely cheating. It's cheating because the problem of diversity on the island is just replaced by understanding why there's so much diversity on the mainland. Okay. Of course, if there's geographical structure and so on that can happen, but we want to understand sort of which things happen in principle and in simple, simple models. Okay. So this is really cheating. So we don't want to do that. However, we can still think about migration that I'm now going to have many islands. So I'm now going to have I islands. I islands. Okay. Where, which are labeled by alpha is equal one up to I. So those number of islands. Okay. And I'm now going to have migration between all of the islands and all the other islands. So I'm going to have migration going in both directions here. Migration going in all directions from each island to every other island. So on. Okay. So all to all migration. Okay. But things are going to be simple is the interactions are only going to be on the islands. So the interactions are only on the island, on the island. Okay. And all these islands are identical. So I'm not allowing myself to have different environments. Okay. Now it turned out already two islands is interesting. But we're going to consider the case where this is going to be very, very large mostly. They migrate from all to all of them. And so what does the dynamics look like? Well, if I look at the dynamics for a population on one island. So I've got now new I alpha. So that's the ice type on the alpha island. So that's just going to be new I alpha. Okay. So it's going to have the terms I just got. It's got before I'm going to ignore the S and the Q here and just write down the interaction term. So this just has the sum on the J. Okay. And it's got the VIJ, which is the same on all of the islands. But then it's only the ones on the same island that it's interacting with. Okay. So it's got the Lagrange multiplier, which is just going to be for that island, which depends on time. That keeps the total population on that island fixed. Okay. But then it's got another part. It's got a part, which is M migration rate here. And I'm going to normalize it this way. So it's someone all the islands of the, someone all the other islands of the same species on the other islands. Okay. And then it's going to have a migration out. New I alpha. Sorry. And this bit doesn't come, this bit comes of course outside here. This is not a growth rate. This is migration in and migration out. Okay. So that's the, that's the migration effects. They come all to all. And this quantity, this quantity here. Okay. This quantity here, this is the average over all the other islands. And I'm going to call that quantity new I bar. Okay. So that's the average over all the other islands. Of course that can itself depend on time. So that's the island average. That's saying I get input from all of the other islands and I get migration out. Okay. So now we have to ask what happens. Okay. We have to ask what happens here. So we now model some number of islands. You can simulate the different islands. They're all the same on each and we can ask what goes on. So here's this, sorry, this was meant to be down here. Added the page in the wrong place. So here is now I'm looking at a situation with 10 islands. So we've got 10 islands here. We see an example here where we've got global extinction. This is showing the one type. So one strain across many islands. So this is dynamics of one strain I'm looking at here. And the different colors are now the different islands. And you notice it's bouncing up and down on the different islands. It starts going down. If I look at the island average, so this is the quantity which is the island average given coming from the total migration right into each island. Sorry, I'm cold and use N here. I didn't have the normalization. And if that fluctuates around, if that fluctuates down, they don't get much migration anymore. And then if these died down, they just go extinct. So here's the extinction threshold when the frequency on an island reaches of that strain which is one over N. So this is going to global extinctions. So I haven't made the assumption of the mainland. In this case, in this situation that I've got here, this particular one we've shown is if I look at the island average for this type, this island average goes to zero at long times. Okay, so this is just going extinction. So I've got a global extinction. So that's boring. But you can also have the more interesting phenomena here's now looking at another strain in the same population, the same community of the strains. So we're looking at another strain here. And this strain, you notice it comes up and down. You can get local extinctions. Here's something which has dropped all the way down to extinction down there. In fact, you can actually get the total population to go down small enough that it would go extinct. But it actually doesn't because it happened to be some strains that are doing well at that time, they come up, they repopulate the other islands and everything goes along and stays. So this looks as if the dynamics is stabilized. A crucial part here is that the chaos on different islands is desynchronized across the islands. Okay, now it's generally to, if you take two chaotic systems and you put a weak coupling between them, so it can particularly gonna be interested in the cases where the migration is very small, we're gonna be interested in the small migration, sorry, let's stop the pages. So we're interested in M being very small, less than basic growth rates times. So when M is very small, we get this desynchronization. It doesn't have to be that small to get that. And we can get this behavior. Not only that, but you can actually have a new type. So here's a new type. It's initially coming in on one island, it comes in there, rises up. Rises up enough that it starts seeding other islands. It actually comes back down again, it goes extinct on its island, but meanwhile it's seeded some other islands, it rises there, some of those go extinct, but eventually you get to something which is a steady state that looks like this. Okay, so what we've got here is one find from the simulations at least, but we have a possibility for stable chaos here. We can get stable chaos by desynchronizing all the islands and we can provide a pool. So our theoretical challenge is to try to understand this behavior. Okay, so that's gonna be our challenge. That's gonna be what we want to try to do. Now I should be honest here, being an old fashioned theorist, I tend to think that the one of the roles of theory is to confirm simulations rather than the other way around. If I give you a calculation in detail, you can check whether it's right. If I give you a simulation, it's much harder to check whether it's right. Okay, and in fact, in this case, the theory actually came before the simulations and then there was a very nice back and forth between the theory and the simulations are some of the things that I'll talk about tomorrow. Okay, so we really want to try to develop theory for understanding this. We want to understand the simplest situation which is the chaos on the one island in the perfectly anti-symmetric case. So we wanted to understand this and then we want to build on that to understand the dynamics of the models with the on the many islands and I should be able to move well, yes, so we want to be able to understand all this. So this is the basic phenomenology. If I look at any given species, it can spend a lot of time down here, down at the sort of floor that's set by the migration, the migration in, as long as it's coming in here, this is the migration I'm coming in sort of black rate, which will set the sort of floor here. And so this floor coming from the migration, they won't tend to drop below that. So this thing, this dashed line, we can sort of call the migration floor, that's the lowest it's going to go between on an island. But where this floor is, where this floor is is set by this new I bar, which itself can be a function of time. Okay, so the questions on the basic model, the basic phenomenology. So a question here, how much does the floor fluctuate based on how many islands are? Ah, okay. So this is a very good question. One of the things we're clearly going to want to understand is this fluctuations of the floor with a large, but finite number of islands tends not that big. And of course, if this goes down far enough, then we can get extinctions as we did here. Here, the total population across all the islands, which was the sum of all these, that went down low enough to that extinction. So one of the things we would like to ask is about, when do we have global extinctions? And the when to those global extinctions, how do they depend on the number of islands and other things? So that's one of the things we need to understand. And that's one of the things I say, which the understanding of came later and really with the sort of back and forth of the theory and the simulations. Okay. So now I'm going to talk about the theory and let's see how much time left, maybe 10 minutes before. Well, no, I should probably stop now and take questions. Let me just write down one thing, which is just sort of to lead as to where we're gonna go. So this is now a general method, which is dynamical mean field theory. It was developed in a lot in the context of spin glasses. And what we're gonna do is we're gonna focus on one eye, on one island, okay? And the dynamics of this is gonna be coming from something that looks like noise from the others, okay? But this noise from the others is gonna be determined by the dynamics of all the others. So this is coming from all the others. So I need to understand this, because of course this comes from all of the others, all the other types on that island and the new eyes on all of the other islands. So this is gonna get affected by migration. This is gonna affect it via the interactions. So it's coming from all these and so determining what this noise is is a big challenge and one has to do that self-consistently. So this noise then has to be determined self-consistently. So this strategy is exactly like mean field theory for a magnet, it's exactly in the same spirit as that. There one assumes that one spin what makes the approximation that one spin here has an effect of a field coming from all the others. Magnetic field coming from all the others. That magnetic field depends on the magnetizations of all the others. The magnetization will always give you this field that gives this magnetization. You don't have to average this to give it to this spin. We have to average that to get the magnetization. So it's exactly the same spirit as doing mean field theory for a magnet and it's gonna be valid in the limit that K is very large. So K is going to infinity and when we do the things with the islands we're also gonna initially wanna take I to infinity and then of course we have to ask the crucial thing is what happens if those aren't infinite. So this will be the spirit and I'll start next time with explaining really in detail how one does this and trying to explain the results that we've got and then I'll follow that with things on more open questions and talking about evolution. Okay, thank you very much. So other questions? Hey Daniel, I have a small technical question. So this might be silly but does the fixed point corresponds to the dynamical mean at all? Is that what's going on? Okay, so when I'm in the anti-symmetric model okay, so the anti-symmetric model where there's a fixed point, okay? So here in this anti-symmetric model this has the fixed point associated with it and that as Stefano worked out theta equals zero is the fixed point. Here when I've got positive theta in general the average of the new i and this is averaged over time, the time average of that will be equal to the fixed point value, okay? That's true here when I have migration it's not gonna be true and when I've got migration it's not going to be true because I can't straightforward the average things. So where that came from was dividing this by this and averaging the l i's, the logs and if you average the logs this averages out and you get the fixed point condition but you notice here, if I average the logs then I have to pull out a one over new i down here and it becomes non-linear becomes extra non-linear. So this is not true with the migration, okay? So that statement is only true with no without migration. As soon as I've got my migration then there doesn't correspond to that and in fact if I look at the dynamics here when I'm looking at all these things there's no fixed point there's no stable fixed point in this case, okay? There is a possible behavior of all the islands being in sync. If all the islands are in sync then it's exactly like one island but then it'll just fluctuate wildly and drive the extinctions like this, okay? But the chaos will tend to go a little bit and the differences will tend to make the chaos go non-asynchronous between the islands and this crucial bit of the chaos desynchronizing which is what causes this enables it to persist, okay? So it's no longer true here that there is a fixed point even a stable fixed point and each island is doing sort of its own thing but they're coupled to each other via the migration. So we have another question from Miguel and then Mercedes, Miguel Rodriguez. Yeah, thank you, Daniel, that was really cool. I have a question that doesn't really affect much of the math of the development of this model but mostly of the biological assumptions of the phase. Oh, okay. Okay. So in this matrix that you use you say that by definition, the interaction between phase and bacteria is basically an interaction of parasitism. Right. And we know that if we consider the prophage phase of the viruses, actually many viruses have a mutualistic interaction with the bacteria and more recent work by Jett Furman, for example shows that in the wild, this is the norm, not the exception. How would that affect this stability of the model? Okay, so as in everything with biology there are all kinds of complications and I'm not claiming it all here to understand specific biological systems, okay? However, I would make the following statement. If all phages were prophages that had symbiotic interactions and they were not parasitic, they didn't attack, okay? I personally think there would be a hell of a lot less bacterial diversity or phage diversity. Those are being differently, then you can get stability and so on and that stability can be there but it will pay for some of the phages to adopt a different lifestyle and then those are the ones that will drive this kind of dynamics or can and will drive, I think, the longer-term evolution, okay? So I mean, my, and this is I say this is just an instinct at this stage but it's sort of based on developing understanding is it really the absolutely crucial thing for diversity and for longer-term evolution is really the sort of antagonistic interactions, okay? One certainly can get a lot of diversity and complications coming from interactions via resources and so on. I think one has to cook things up much more to do that and what I'm trying to convince you here is that something can happen with very little assumptions, right? So I made very specifically, I'm making assumptions here that I don't have the niche interactions, okay? I don't have niche interactions. I'm not assuming anything special to have interaction with self. Now the analogous thing for the niche, for the phage would be to have specialist phages, each phage is attached to one bacteria, okay? And then those will interact with each other, then they'll tend to have cycles or can have cycles or can be stable. Again, there can be pressures for the phage to start doing something different but there you're assuming in some sense the answer, you're assuming that everything has a niche and if I have a slight variant of the bacteria or a slight variant of the phage, that will no longer be exactly in that niche and then you can get back into these kinds of situations. So I think that's a really important question. I hope that'll be something which will come up in the round table discussion next week but I should be clear. I'm really trying to ask about what possible, which things not surprising, if we can get things in really simple models, then we can say, geez, maybe they're not so surprising that we see them in nature but that doesn't mean that we can apply it to particular biological systems and there are always very large numbers of extra complications. The same is true in physics. I should say, I'm a condensed matter physicist, not an atomic or particle physicist. So I've always dealt with complications and one has made tremendous progress by saying, okay, we're gonna try to ignore a lot of the complications, look at simple models and then we can sort of add the complications in one by one. And I think that really would be the goal here. In particular, I can ask, okay, maybe the way the evolution goes is that I drive things which go in the direction of they get more niche-like. That's a possibility. The other one is this assumption here, which is that I've said, I'm assuming it's very small, I could say what's gonna happen is it's gonna be generalists that evolve. What does a generalist correspond to? A generalist corresponds to it's S getting larger, right? It's doing well against all of the others. So the S is evolving would correspond to generalists. And I'm gonna say something briefly tomorrow about that possibility, what happens at an evolution, okay? So the crucial thing here is we really shouldn't be talking about assembled models. We have to ask, can this evolve? And of course, if it evolve elsewhere in the world, it can come together, assembled, and that can be relevant, but then it will continue to evolve, bacteria and phages evolve fast, especially when there are new conditions like being with new 23rd cousins instead of being with their close relative. So the crucial thing here really is to ask whether you can get to these kinds of things from evolution with again, reasonably simple assumptions about the evolution process. So that's what I'm gonna sort of end with tomorrow. So we have Mercedes, please Mercedes. Yes, quickly. I just, yeah, you just said that the important question about can you assemble it with evolution? So it was a bit my comment, but in reality, I think a lot of, it will be very enlightening to connect these kinds of analysis to evolution on trade space because where you connect the VIJs. Yes. To the VIJ, it's not just evolving particular parameters, is that the result of that evolution gives structure to the VIJ. My last thing, my last topic here tomorrow is to talk about phenotype models. So that's exactly going to be evolution in trade space and I have some preliminary things to say about that. So what this would be, this would be where the VIJs of course are determined by the traits of strain I and strain J. If I, I's a phage and J's a bacteria, those are really the traits associated with the direction. So I'm gonna say a little bit about that at the end, so that's absolutely crucial. I think it is crucial and I think it is the challenge because I think like it will also enable the empirical problem, which is if we look at some distributions when you say this could look like neutrality, right? In reality, we have to ask what are these macroscopic properties that differentiate from neutrality and that tell us something insightful about the processes. So I'm gonna say something as a very brief and very preliminary, very conjectural still about the simple phenotype models and the context of the bacteriophage system and how that connects or might connect to what I'm talking about. So I'm gonna say the things which I've just shown here today, those are all solid, there's one paper on those with a quite long paper and the methods I'll talk about tomorrow. After that, everything is very conjectural, very much work in process and particularly to continue work with Michael Pierce. And really, these are exactly the questions that I want to come to. So thank you for advertising my talk tomorrow, which I guess is the last of six talks tomorrow or something or is that today? So thank you for being here, those of you that have survived the earlier ones. So we have one question in the chat. Yes. Do you have an idea of how experimentally test cows? Yes. Okay, so one of the crucial things from here is that snapshots of abundance distributions can be very misleading. And I showed that just with this simple, idealized anti-symmetric model, you've got snapshots that look very close to a neutral model. You put in numbers and it just doesn't make sense for microbes. And that's also true with this state, which I've been talking about where I've got many islands and so on. Again, if you take snapshots on one island, you will see things that look neutral. So the crucial thing is to look at the dynamics. The crucial things look at them, just to follow the dynamics of the strains of time. Now, this has been done particularly in planktonic systems, Forest Roar and others have done that and looked at that and you tend to see kind of dynamics of things coming up and down. Okay, that's a more complicated system but the planktonic systems are just the kind of ones I want to think of in this context. They're much simpler than the things like human guts. And things mixed together, they compete, the spatial structure, things can move around. But really the secrets are in the dynamics. Now, in the long run, they're in evolutionary dynamics as well as the ecological dynamics. And so you want to understand the sort of relations between the strains and so on. And you'd really like to be able to track the genes that were responsible for the traits that dominated the interactions. So coming back to Mercedes points. So you would really like to be identified those and track those genes so that even if those genes were not always in the same organism they were moving around, you could track the dynamics of those genes that were associated say with the phage receptor and the tail of the phage that binds to that. The simple example. So really one would like to be able to identify those and track those. And then you can use all genetic tricks to be able to track those. So that's really thinking about the sort of future and how it might hope to make contact with reality but contact on sort of the conceptual ideas. And this is sort of a scenario for getting diversity and stabilizing and evolving diversity. It's not something which is, think of a predictive theory in any detail. A detailed thing. But it really is a scenario and it suggests what things to look at. So let's take one last question from Almond. Daniel, so I've seen a lot of work on into using second quantization with, I mean, using, I say like, what's only been done in spatial settings in geography. I was wondering, especially in this context of DMFT, if there's any real advantage or if it's just something purely aesthetic because you say that, for example, well, from what I understand of DMFT, I mean, you need small fluctuations around the average. But then again, I think would having a second quantized form. Yeah, yeah. So you can write things as a field theory. You can start with the field theory for the dynamics and that's a useful way to sort of drive things. But I want to make a comment about this because people coming from physics, as I do, one tends to like to put things into a form that one can then beat on and use the standard tools. Some of those here in this particular context, one can do and these ideas that came from spin glasses and so on. And some of that you can say do with field theory, but in turn that doesn't help much. You can just sort of check that you're doing things in a consistent way. There's another much simpler problem which I've worked on a lot which is trying to understand dynamically, continually generated diversity coming from evolution in large populations. And this is a large bacterial population to lab or viruses within a person, within a host and they just evolve, they're racing the whole time against other than this continual evolution. For that, you can quickly write down asexual evolution looks like a field theory. I have never seen anybody get anything useful from the fact that it looks like a field theory. The things that I find I bring from statistical mechanics are some of the conceptual things and some of the ways of sort of thinking about asymptotics of how to approach problems and how to ask about questions of robustness to convince oneself or try at least that what one is doing is not very, very special. So a lot of the things you can't do by sort of cranking out the methods that one has from statistical physics or from field theory, which makes it a lot of fun, makes it hard, you can't just assign a problem saying, okay, here, go do this and have a means of doing it. This is still very much and the things I'm gonna talk about here, to some extent it's still very much an art and it's not sort of methods that you can directly take to be able to do it. You can write down things like this dynamical mean field theory but you just can't get anywhere with it without a sort of lot of extra conceptual ideas and mathematical to some extent trickery where one's only used it once but one hopes that becomes a method. Okay, so thank you very much. I think we can stop here and...