 Thank you for being here. Welcome to the first day of this school that is going to take company for the next three weeks. So I see that some of you are connected from the very east. So it's late afternoon. Some of you from the west. So it's very early morning. So thank you for making it. So today we are going to have three slots. So if you have not watched it, please go to the program page and there is going to be a link for the full introductory remarks by the organizers. So today we're going to have three slots. And the first one, Leonardo is going to give a tutorial on nonlinear dynamics and it will be followed by a lecture by Joshua Weitz and a lecture by Carla Statham. So a little bit of an etiquette before we start. So this is a large meeting. So please unmute yourself, unmute the microphone. During the meeting, if you have a question, please raise your hand and the moderator will stop the speaker or when there are enough questions will be stop the speaker to have you ask some questions. Okay. So in particular, the first slot is a tutorial. So it's meant to be very interactive. So please, if something is unclear or you want clarification, please don't hesitate to ask questions. As you have seen in between lectures, we have a 15 minute buffer. So lectures and tutorials are going to be one hour long. And there is a 15 minute break. So in those breaks, I mean, you can of course stretch your legs, get a cup of coffee. But we also will divide all the participants into breaking rooms. And you will have the opportunity to interact very informally with other people that are attending. You are free to stay in the room you are randomly assigned to or change room, interact or not interact. This is just an informal opportunity to know. So, okay. In perfect timing, I think I can introduce the first speaker who is Leonardo Facciani Mori. He is a PhD student from the University of Padua. And today and tomorrow, he will give a tutorial on nonlinear dynamics. So please thank you, Leonardo, for being with us and I'll give you the... Thank you. Thank you for inviting me. So let me share the screen. So you can see my slides, right? Yes. Perfect. So, hi everyone. Thank you, Jacopo, for the introduction. So today and tomorrow, we will be talking about nonlinear dynamics. So let me first show you how I have organized all the material for today's and tomorrow's tutorial. And then we can get started right away. So I will start with a very simple and brief introduction just to remind ourselves what are nonlinear systems? Why are they interesting? And I will also give you a few examples with a particular focus, of course, in ecology. Then I will show you how, for example, by using stream plots, we can actually understand the general behavior of the solutions of a nonlinear system without actually solving the equations analytically. Then we will be talking about stability in nonlinear systems. In particular, I will show you the two main tools that can be used to study the stability of equilibria in nonlinear differential equations. Of course, throughout all these topics, I will show you plenty of examples. We will also do a couple of very simple exercises. So we will get a practical sense of... We will get our hands, let's say, on the topic and so we can better understand what we're doing. Now, as Jakob already told you, these tutorials are... I mean, they shouldn't be just simple lectures, but they are meant to be a moment of discussion. So please, at any time, if what I'm saying is not clear, or if you have questions, please ask me. So if everything is clear for now, we can get started. So what we would like to do in general is, let's say solve or understand something about nonlinear differential equations. Now, by nonlinear differential equations, we basically mean an equation or a system of equations that look like this. So where this function f here is any nonlinear function. So these, for example, will all be simple cases. They can be in one dimension or in two dimensions here, where this function is nonlinear. So these are all simple examples of a nonlinear system. I mean, they are interesting and we want to be able to understand something about them basically because any interesting phenomenon in nature is described by a nonlinear differential equation. Let's see some example. One of the, I think, most simple but also overlooked physical system that is actually described by a nonlinear differential equation is the pendulum. Now, you may be used from introductory physics courses to see this equation with sine of theta approximated by just theta, but that is only an approximation of the true, let's say pendulum equation, which is actually a nonlinear one. Then another important case where of nonlinear system are fluids because fluid dynamics is regulated by Navier-Stokes equations, which generally are nonlinear equations. For example, this is the equation for an incompressible fluid where u here is the velocity field of the fluid. And this term, for example, we can see that is not linear. So again, fluid that an important example of nonlinear system. Now I want to introduce two examples that we will consider over and over again in these tutorials and that are actually relevant for ecology. The first one is the logistic growth equation. You may already have seen it somewhere at some point in your life, but basically this equation simply describes the growth of a population X in a system with, let's say, a limited amount of resources. Now, of course, I'm going to show you this in more detail in a few slides, but basically you see that if we didn't have this term here, this equation would basically be a simple linear differential equation. And so we would have that the population X here is growing exponentially without limit. But this term, as I will show you later, basically makes it impossible for the population to be larger than k. So this system has a maximum population, let's say, that can be sustained. The other system that is relevant for ecology is log-cavolta, log-cavolta equations. Again, this is a very famous system which basically describes the dynamics of the population of a prey, which is X here, and a predator, which is Y here. And these parameters basically measure the interactions between these two populations. You see that in both cases here and here the functions that define our differential equations are nonlinear. So these are indeed nonlinear systems. Okay, so what we would like to do generally with, let's say, an ideal world with nonlinear differential equations would be to solve them analytically. But unfortunately, this is almost never possible, because we don't have like a theorem or a general recipe, let's say, that can give us a direction into how to solve any given nonlinear system. So it is reasonable to ask ourselves if and how we can understand something about nonlinear systems without actually solving the equations. Now what people generally want to do with nonlinear systems is studying their equilibria. Now I'm going to introduce definitions that probably are very familiar to you, but I just want to first refresh their meaning. And then I also want to build a small vocabulary on nonlinear systems that can be useful throughout this course. Now in general, if we have a general nonlinear system like this, a given point X star is said to be an equilibrium of the system if the function that defines our equations here is equal to zero. You see that this means basically that the equilibria of a system are the points where the variable doesn't change because if X is equal to X star, then F is zero, X dot is zero, and so the variable will not change once it is equal to this equilibrium here. Now what people generally want to do with equilibria is understand if they are stable or unstable. So we need some notion of stability and instability to study them. So let me introduce some very informal definitions of stability and instability. I will make these definitions a little bit more formal in a few slides, but I just want you to get an intuitive idea of what they mean. So in general, an equilibrium is said to be stable if any solution of the system that starts with that initial condition that is close enough to the equilibrium will always remain close to X star. Now of course, this notion of closeness, close enough is anything but rigorous, but still I just want you to get the intuitive idea. On the other hand, an equilibrium is said to be unstable if it is not stable. So if we can find solutions of the equations that start close to the equilibrium and eventually go away from it. Now in light of these concepts, we can basically reformulate our initial question as follows. So can we understand something about the stability of the equilibrium of a nonlinear system without actually solving its equation? The answer of course is yes. There are several ways, the several tools that we can use, and one of the simplest one, but also quite effective that we can use in this direction is drawing stream plots, which basically means drawing the trajectories of the system in the state space. Now I think the easiest way to understand how stream plots work is to see how they are done practically. So let's start with a simple example. So assume we are given this differential equation, which is of course nonlinear because this function here is a simple cubic function. So let's draw it. This function basically looks like this, and so we can see very easily. We can also factorize the expression in this way. So we can see very easily that the system has three equilibria in this case, which are the three points where this function is equal to zero. In particular in this case are minus one, zero, and two. Now the basic principle behind drawing stream plots is the following. Now wherever this function f here is positive, x dot will be positive. So the solutions that start from point where f is positive will be characterized by the fact that x is increasing. So for example, if we consider this interval here between minus one and zero, in this interval the function f is positive. So any solution that starts from these points here will be characterized by the fact that x is increasing, and the same will happen for example in this interval here. On the other hand, wherever f is negative, x dot will be negative, and so the solutions will be characterized by the fact that x is decreasing. So if we start from any point in this interval here, we will have that the solutions actually go towards the left and the same here. So in the end what we can draw are these trajectories. So this is the stream plot of our system. And so you can see that by doing this very, very simple drawing, we can already guess which equilibria are stable and which are unstable. In particular, we can guess that this equilibrium here is unstable, because you see that if we take any initial condition that is slightly larger than two, we will have that this solution basically increases without limit and something similar happens here if we take an initial condition that is slightly lower than minus one. On the contrary, we see that solutions here are actually going towards the equilibrium in zero, and so we can guess that this equilibrium is stable. Let's see another. Is everything clear here? Okay, so I think it's a good time if you have a question. Exactly. So please remember that this is a tutorial. So this is really meant to be to have everyone being on the same page on the top. So if something is unclear, don't hesitate to ask questions. Okay. Okay. So let's see basically the same thing applied into a different case. So let's consider this equation x dot equal cosine of x. So in this case, we know that our function here looks like this. So we will have basically an equilibrium in every positive and negative of the multiple of by halves, because these are the zeros of the cosine function. Then if we apply the same principle as before, we will have, for example, that in this interval here, in this interval here, and in this interval here, the function is positive. So the solutions will go in towards the right. On the other hand, in this interval here and in this interval here, the function is negative. And so the solutions will be going towards the left. So in the end, the stream plot that we can draw of this system looks like this. So you see that in this case, we basically have an alternation of stable and a stable equilibria in our state space, which of course is the x axis here because this system is a unidimensional system. We only have x as a variable. So let me now show you basically how we can use these stream plots in cases that are ecologically relevant. So the first thing I want to show you is the stream plot of the logistic equation. Now actually, this is one of the few lucky cases where we can actually solve an equation analytically. So I hope that with this example, it will be clear that, I mean, comparing what we see with the stream plots with what we see with the analytical solution that stream plots can actually help us understand the general behavior of the solution of a nonlinear system. So first thing, let's try to, I mean, let me show you how these equations can be solved analytically. So let me write here in the whiteboard the equation. Okay. So to solve this equation, I'm going to use some physicist's tricks. So if you are a mathematician, I apologize because you will probably be horrified by what I'm going to do. But basically what we can do in this case is we can separate the variables. So we bring everything that depends on x on one side and everything that depends on t on the other one. I'm sorry, this whiteboard is a little bit slow, but I hope you can see everything. So once we have done this, now this fraction here, we can actually decompose it into two terms. So we can write this as dx over x plus dx over k over 1 minus x over k. So you see that if we simply add these two fractions, we get exactly this. So we have this equation, but these are very simple terms to integrate because, for example, this one will be the logarithm of x plus a constant. This one will be minus the logarithm of 1 minus x over k plus a constant. And this here will be simply RT plus a constant. So in the end, we can rewrite everything as logarithm of x over 1 minus x over k equal to RT plus c, which basically means that x over 1 minus x over k is the exponential of RT plus c. Now, of course, we have to determine the value of this constant. So we can evaluate this expression for t equal to 0. And by simply rearranging basically the expression that we get in this way, the final analytical solution of the logistic equation looks like this. So this is the analytical solution of this equation, where x0 is, of course, the initial condition on x. Now, you see that here we have an exponential with a negative exponent, because remember that both r and k are positive parameters. So as time passes, so as t becomes larger and larger, we can neglect basically this term here, then x0 and x0 cancel out. And so we can see that the solutions of this equation tend towards the value k. The only case in which this doesn't happen is when the initial condition is 0, because you see that in this case, if x0 is equal to the numerator, is that a question? Yes? Yes, sir. So could you explain what is r and x and k? Yes, of course. Basically, the biological meaning of r is the growth rate of the species. You see that if we didn't have, let me write it down. If we didn't have the quadratic term, so if we had only this equation here, the solution of this equation would simply be constant times, would simply be an exponential. So in this case, we would have a population growing exponentially. The biological meaning on the other hand of this other parameter, and it's going to be clear in a few seconds, but basically this is the maximum population that the system can sustain. Because let me draw basically how the solutions are. So for example, if we start from x0 equal to 0, as I just told you, basically the solution is constantly equal to 0. So this is a trivial case because it basically describes a system with no population. Something similar happens also when x0 is equal to k, because you see that here we would have k minus k. So this is always equal to 0. And again, this and this can sell out. But then if we take any initial condition between these two values, this function basically behaves like this. So we have a population that grows and the growth, the initial growth here is actually an exponential growth. Because you see that when x is very small, we could neglect it. So at the very beginning, the growth is well approximated by an exponential. But eventually this population saturates towards k. On the other hand, if the initial condition is larger than k, what happens is that the population quickly decays towards k. So basically the meaning of this parameter is the maximum population that the system can sustain. Because basically the idea behind the logistic growth equation is that the population here, this could be microorganism, could be animals, could be anything. But it is in an environment with limited resources. So at a certain point, the system will reach a population that is not sustainable anymore. And so x cannot grow larger than k. And even if x is driven to a value larger than k, then it quickly decays back to k. Does that answer your question? I mean, was I clear enough? Yeah, yeah. Please request to all, please don't put the message regarding the witnesses. Okay. Sorry. Okay, it's correct. It's okay. Okay, so the fact that k here is basically the maximum population of the system is also why is this parameter is called the carrying capacity of the system. So now we know how these solutions work. In particular, if we look at what is happening in the state space, so along this axis, you see that these solutions here basically are going towards k. And in the same way, these solutions here are going down towards k. So let's see if we could understand this general behavior already from the stream flow. So again, this is our equation. The function, the linear function that describes the system now is simply a parabola. And so we can draw it. Now, of course, here I am not considering the negative part of the state space simply because this variable here represents a population. So it makes sense only when it is non-negative. So this is the aspect of this function. So we see immediately that we have two equilibria in the end, zero and k. So now let's apply what I've shown you before for stream plots. In this interval here, the function f is positive. And so the solutions, the trajectories of the solutions will go from zero to k. On the other hand, in this other part of the state space, the function here is negative. And so the solutions will move again towards k. So in the end, the stream plot of the system is like this, from which, I mean, we can also guess that this equilibrium here is unstable because the solutions are moving away from it, while this equilibrium here is stable. So you see that if we look at what is happening in the state space, we are actually recovering the same behavior. So of course, by using the stream plots, we can't say, for example, how quickly these trajectories are moving or the exact way in which this is happening. We couldn't guess, for example, that there is an exponential, like I've shown you before, that regulates the speed of this motion. But again, without even trying to find the analytical solutions, we can see how the solutions are moving in general. Now, if everything is clear here, so if there are no further questions, yes, of course. What does it mean of unstable and also stable? By unstable and unstable, in this case, I mean the very informal definition that I have given a few slides ago. So unstable in this case means that the solutions are moving away from the equilibrium. Because you see that in this case, if you pick any initial condition that is slightly larger than zero, the solutions will eventually go away from this equilibrium. While on the other hand, stable equilibrium means that the solutions are remaining close to the equilibrium. So in this particular case, for K, for example, we have that the solutions are actually going towards the equilibrium. But this is not necessary for the equilibrium to be stable. So this is a particular case of stable equilibrium. In general, for an equilibrium to be stable, we just need the solutions remain close to it. Does that answer your question? Yeah, thank you. Okay, no problem. So I hope it is clear with this example that, I mean, what we are seeing with the stream plot makes actually sense if we compare it to the analytical solution? Can I ask? Yes, of course. So ecologically, it means that the K is the point where it achieves equilibrium, right? So there's no resource limitation, like we can say for the growth. No, the resource limitation is, I mean, it is in the fact that this population is not growing exponentially. If we had a system where we had an unlimited amount of resources, basically here we would have only an exponential growth with a fixed growth rate. What happens in this case is that K is the, I mean, is the maximum population that can be sustained by the system, meaning that if, for any reason, the population grows larger than K, there are no enough resources to sustain that population. And so you see that eventually the population will go back to K. Okay. So in this case, in this sense, there is resource limitation. Okay. Okay. So what we can do now is basically apply the same thing. So we can try to draw the stream plots in the case of the log-cable-thera system. So our equations in this case are this and this is our state space. Again, here I am considering only positive values for X and Y because they represent populations, so it doesn't really make sense to look at the other quadrants of the state space here. So if we look at these equations here, we can find very easily two equilibria, which are the origin, sorry, the origin and this non-trivial equilibrium. Now the origin is very easy to see why this is an equilibrium, because if X and Y are both equal to zero, then both X dot and Y dot here are zero. The other non-trivial equilibrium is very easily, let me show you. For example, we have that X dot is equal to alpha X minus beta XY. So if we want this to be equal to zero, we can rewrite this as X times alpha minus beta Y equal to zero. So if X is not zero, we simply have that Y is alpha over beta. If we do this exact thing with the other equation, we get that the equilibrium in the end is this point here. Now let's try to draw the stream plot. The first thing that we can do is see how the system behaves on this axis here. So for example, if we take an initial condition where X zero is equal to zero, so we start from the Y axis here, you see that. Sorry, am I having a problem with the audience? Sorry, yeah, I think I can hear you, yes. I'm sorry, I'm not hitting you well. There is a little bit of background noise. I think that the question was what if the carrying capacity is not constant? What if the carrying capacity here? Okay, you mean here. Well, in this case, we would have basically a different system. K could be a function of X, if you want, or a function of any other variable. But I mean, it would simply be a different system. It would require, I mean, it would have a different stream plot. So we can use the same tools, but it would be simply a different system. Okay, so where were we? Okay, so if we take an initial condition, for example, that is on the Y axis here for the log-table-tera system, you see that our system reduced to this. So the solutions basically will be decaying exponentially towards the origin. Remember that all these parameters here are positive, so minus gamma Y is negative. On the other hand, if we take any solution that starts on the X axis, so if Y0 is equal to 0, we can write our system like this. And so the solutions will be growing exponentially on the X axis. So the first thing that we can draw about the stream plots are these trajectories here. And notice that it would be enough for us to guess that the origin here is an unstable equilibrium, because you see that along this direction we have solutions that actually go away from the equilibrium. So by definition, this equilibrium here would be unstable. So let's try to see what happens in the rest of the state space. What we can do is see exactly when the components of this function here are positive and when they are not. So for example, if we look at when X dot is positive, we have, again, as I've shown you before, that these must be positive. Now we are not on the Y axis, so X is different than 0. And in the end, we get that Y must be lower than alpha or beta. So we can divide the state space into two regions, one where X dot is negative and one where X dot is positive. Similarly, if we look at where Y dot is positive, we get that this is true when X is larger than gamma over delta. So in the end, we can basically divide our state space in four regions. And for each of these ones, we know that X dot and Y dot is other positive or negative. This basically means that we are able to see the general direction, let's say, towards which the solutions are pointing. For example, here if X dot and Y dot are both positive, it means that the solution is growing both in X and in Y. So in general, the solution is pointing in this direction. And this is true also for these other areas. So you see that we haven't even tried to solve the equations, but we can guess already that the solutions of the Lotka-Volterra system oscillate around this equilibrium. Of course, we don't know exactly how this happens, but we know that this is happening. Is that clear? Are there questions? Okay. So I hope it is... Sorry, I was muted. So I hope it is clear now that stream plots are actually useful to understand the general behavior of the solutions of our linear system. But of course, the powerfulness of this approach is limited because we can't always understand something about the stability of the equilibrium. For example, if we consider this case here, we know that the solutions are oscillating, but we can't say anything on the stability of this equilibrium, because we could have that the solutions here are spiraling towards the equilibrium. We could have that the solutions are spiraling away from the equilibrium, or we could have any kind of behavior. So in this case, we can't say anything about the stability of the equilibrium. So how we can study in general the stability of equilibria in nonlinear system. There are two main tools that can be used in this direction, which are liapunov functions and spectral analysis, which is also known as simply linearization. Now, is anything clear up to now? Okay. So before I go on and I talk about how to use the apunov function, I want to introduce the formal version of the definitions that I have given you before about stable and a stable equilibrium. So we consider a generic nonlinear differential equation and assume that we know x star is an equilibrium. Now, using the language of mathematician, now I'm not a mathematician, so I won't be very formal, but I just want you to let you know how the formal definition of stability is. So this equilibrium in the language of mathematician is said to be stable if for every neighborhood A of x star, there is a neighborhood B in A, such that the solution starting from points in B will always remain in A. Now, this is just the formal way to say what I told you before, which means that an equilibrium is stable if every solution starting close to it always remain close to it, where the notion of closeness in the language of mathematicians is given by the use of neighborhoods. So let me show you graphically, because I think it's easier to understand this way. Now, assume we have an equilibrium here. Now, this equilibrium will be stable if for every choice that we can make of a neighborhood A, so a set on the state space that contains this point, we can always find a smaller set B, and if we pick any point inside here and we use it as the initial condition of our nonlinear system, the solution starting from here, it can move around all it wants, but it will never go out of A. So it will always remain close to the equilibrium in this sense. Now, if this is true, I want to introduce a couple of other definitions of stability, because I mean, this way we can build this vocabulary that will be useful throughout the school. Now, if this definition of stability is true, not only let's say in the future, so looking at how the solution behaves for positive time, but if this is true also for negative time, we say that this equilibrium here is not only stable, but stable at all times. On the other hand, if this solution here, instead of going around this set A here, at a certain point, it moves toward the equilibrium, so if the limit of the solutions is the equilibrium, we say that the equilibrium is asymptotically stable. So the difference between a simply stable equilibrium and an asymptotically stable equilibrium is that in the asymptotically stable equilibrium, we know that the solutions are moving towards the equilibrium, while if an equilibrium is simply stable, this doesn't necessarily happen. Is this clear? Are there questions? Okay. So the other side of the coin is instability. So again, using the language of mathematician, an equilibrium is said to be unstable if it is not stable. So if we can find at least one neighborhood of x star such that for any choice of a smaller neighborhood, there is always at least one initial condition such that its solution goes out of A, which again is just. Yes? So Shamolina, sorry for misspelling the name, you have raised the answer. Yeah, I have a question. I could not understand the difference between asymptotically stable and normal stable equilibrium. Yeah, of course. The difference basically is in the fact that if an equilibrium is asymptotically stable, it means that we know that the solution is actually going towards the equilibrium. Yes. While if an equilibrium is just stable, so not as stable, but not asymptotically stable, then the solution is not going towards the equilibrium. It can be oscillating. It can be going around the equilibrium, but it's not actually going towards the equilibrium. Then why you are calling it stable? If you don't know its state, I mean, ultimately, when it will go towards that equilibrium, then we can call it stable, right? Then in what sense you are calling it stable? Yeah, I'm calling it stable in the sense that the solution is not going away from the equilibrium. So the idea of a stable equilibrium is an equilibrium where the solutions are always remaining close to it. If on top of remaining close to it, the solutions are actually moving towards the equilibrium, then we call the equilibrium asymptotically stable. Does that answer your question? No, I actually don't know that how could I differentiate this? Say if I have a set of dynamical equations and then I solve that and say after a long period of time, my simulation shows that it is approaching to equilibrium, then it's asymptotically stable. But how could I know from the graphical simulation that it is stable? Because in my system, I don't know where is the boundary of this B and A thing. This is mathematical definition. Yes. In this case, I mean, of course, it depends on the system you are considering, but I would say that you could see if the equilibrium is actually stable by using different initial conditions. So if I don't know, you guess, for example, from your simulations that x star here has a particular value, you start sampling some solutions around this value. So for example, if we have just to make things simpler, if we are in one dimension, let's say you you guess from your simulation that x is more or less equal to one, then you could take, you could sample several points around one. So I don't know, 0.75, 0.80, 0.85, then 1.05, 1.10, and see how these solutions behave. If you see, like starting from those initial conditions, how exactly you take, I don't know, four, five, ten points close to the equilibrium and you see how the solutions starting from these points behave. If you see, for example, that all of them are going towards the equilibrium, then you have a good indication of the fact that the equilibrium is asymptotically stable. So going towards the equilibrium means like how long time or there is no time sense here. In this case, there is no time scale here. I mean, the mathematical requirement is that for t going to infinity, the solutions goes to the equilibrium. So you don't have any kind of measure in this sense. Oh, then we have always played to the asymptotically stable thing. That means we always see the asymptotically stable equilibrium. It depends. I'm going to show you some examples in this tutorial, but for example, if we have an oscillating system, so for example, if we have, okay, so if we have, for example, I mean, I will show you the log-cavolta system or, for example, appendulums. So assume this. Excuse me. Yes. Can you hear me? Yes, I can hear you. So at the last slide, can I go back to it? Yes. Yeah, so you have written x star is stable not only for all t greater than equal to zero, but for all t included in r, right? But time for the sake of its definition, it is supposed to be contained in r plus, right? Yes. I mean, physically, yes, mathematically, not necessarily, meaning that, I mean, if you, let me write it this way, if you have a system like this, so x of t, sorry, x of t equal to f of x of t, yes, you would say that I can solve this for t greater or larger than zero. But then I could define, I don't know, tau equal to minus t and solve this system here, x of tau equal to f of x of tau. This is another nonlinear differential equation which can be solved. But when tau here is larger than zero, t is lower than zero. I don't know if I'm being clear enough. So it is supposed to be, it is supposed to be a transformation on t, but t itself cannot be negative. Yes. If you want, I mean, mathematically, yes. If you want, I can change what I'm saying here by exactly what I've written here. So if you can say that the equilibrium is stable for t larger than zero, and then it is also stable if you make this change of variables, then it is stable at all times. Okay, thank you. Okay, no problem. So about what the question I was answering before. So for example, if we have, this is our system, this is our equilibrium. So for example, if we have a solution that goes in circles like this, this is, for example, what happens for the pendulum equation when we approximate it for very small angles. So you see that in this case, the solution is going around the equilibrium, but it's never going close to it or far from it. So in this case, the equilibrium is not asymptotically stable because the solution is not... Okay, okay, now I understand. The periodic orbit. Yeah, because the solution is not doing something like that. So the equilibrium is not asymptotically stable, but still the solution is not going away from the equilibrium. Okay, is that clear? Yeah, thank you. So just another fact, Leonardo. Yes, yes. So actually, in the linear stability analysis, you are always constrained to near about this stable states, stable or unstable states. Yes. Never, never far from it. And then you have to invoke this large deviation, right? So it has to be all the way... Then you have to invoke, if you far enough, you have to invoke large deviation. Yes, I mean, it depends on the system you are studying, but I'm going to show you something about it, but please continue. So again, all that time, whenever you are applying these analytical results to conclude something, you are again very close, close enough to your steady states. Yes. So this asymptotically stable, asymptotic stability always matters. Yes. Thank you. This is also, this is also the, we can say the price that we have to pay for not being able to solve the equations analytically in general. So of course, if we would be able to solve analytically and in linear differential equations, we would say, we would be able to say everything globally, basically, about an equilibrium. But since we cannot do that, we have to restrict ourselves to the points of the state space which are close to the equilibrium. Does that answer your question? Yeah. Please go ahead. Okay. Thank you. So I was saying the other side of the coin is instability. So graphically, what happens is what I was saying before. So an equilibrium will be unstable if I can find some initial conditions for which the solutions will eventually go away from the equilibrium. In this case, go away is the idea of going away is given by neighborhoods. So if there are no further questions, what time is it? Okay. So we still have 10 minutes left. If there are no other questions, I can start introducing liapunov functions. So, okay. Now, one of the tools that I have anticipated you, we can use to study the stability of equilibria in a linear system are liapunov functions. So let me show you the basic principle that is behind using liapunov functions so that things will be more clear. So let's assume we have a system like this, a nonlinear system, and we know that x star is an equilibrium. Now, here I am drawing this in one dimension simply because it's easier, but what I'm saying is true in any number of dimension. So assume now that we are somehow able, I'm going to explain you how we can be able to do so in a few slides, but assume we are able to define in a neighborhood of this point a function w that has a minimum in x star in the equilibrium. Now, if we had the solutions of the system and we computed the value of this function along the trajectories of the system, for example, we could find out that this function is decreasing along the trajectories. So for example, we would be in a situation like this. So again, we, let me repeat, we define this function here in the proximity of an equilibrium, then we take the trajectories of the system and we compute this function along the trajectories. I'm going to explain you, of course, how we can do that. If, for example, we find out that this function is decreasing along the trajectories, so the time derivative of this function here is negative, then we will be in a situation like this. So you see that necessarily this means that our solutions are moving towards the equilibrium. And this happens because we know that the function here has a minimum in x star. So this would mean, in this case, that the equilibrium of the system here is asymptotically stable, because we know that the solutions are going towards it. On the other hand, if, for example, we find out that this function here is increasing along the trajectory, so if the time derivative of this function is strictly positive, then we are in a situation like this. And so you see that in this case, necessarily the solutions are going away from the equilibrium. So in this case, we could say that this equilibrium is unstable. Is that clear? Okay. So now what I've, yes? Are you approaching from just one side? Yeah. Of course, I'm showing you just from one side, but I mean, if you flip this image vertically, you see that the same is true also from the other side. If you are in this situation, sorry, if you are in this situation, so if the solutions are decreasing, sorry, if the value of the function is decreasing, you see that if I started here, let me draw that probably this will be a little bit clearer. So if this is my situation, this is my equilibrium x star. So let's say that I can define this function here, let me draw it a little bit better. So you see that if I start here, for example, and I find out that the value of the function is decreasing along the trajectories, then I will have exactly the same situation. So the solution will be approaching the equilibrium from the other side. Does that answer your question? Yes. Okay. Great. So, okay. So what I've told you now is absolutely non-Rigorus. I mean, what I've said, of course, can be made mathematically Rigorus, and this is done by what is called the Lyapunov's second theorem. This theorem basically states that if we are exactly in the situation that I have just described, so a general nonlinear system with an equilibrium and this function defined here with a minimum in the equilibrium, then if the time derivative of this function along the trajectories is equal to zero, the theorem says that x star, the equilibrium will be stable at all times. So if this function here is constant along the trajectories, this equilibrium is stable at all times. On the other hand, if the time derivative is non-positive, so it's either negative or equal to zero, the equilibrium is stable. If it is strictly negative, sorry, the equilibrium is asymptotically stable, and if it is strictly positive, the equilibrium is unstable. Finally, a function that I have, I mean, a function like this that I have just introduced is simply called Lyapunov's function for this equilibrium. Now, we still have five minutes, ten minutes, more or less? It's about ten minutes. So, I mean, perhaps we can see if there are more questions from the audience. Absolutely, absolutely. I mean, it depends on whether, because Leonardo is giving a second part of this tutorial tomorrow, so Leonardo on whether you want to, how you want to divide it into parts, but I would say let's see if... Yeah, absolutely. So I can stop here, no problem. So are there questions, clarifications, something that is unclear? So I have one question. Yes. Yes, please. My question is that in this case, the equilibrium is the minimum. Is this property useful for when we have an equilibrium is a maximum? Yes. I mean, you could write basically the same exact theorem when the function has a maximum in the equilibrium instead of a minimum. Of course, you would have to change the names here because of course, if here we have a function with a minimum in the equilibrium and we find, for example, that the time derivative is negative, then we are moving, let me draw this, sorry, probably this will be more clear. So what I've just told you in the slides is basically that if we are in this situation, so I have my equilibrium and I have this function here with a minimum. So what I've shown you in this case is that if the function decreases along the trajectories, then we are moving towards the equilibrium. So the equilibrium is asymptotically stable. But for example, if we decided to define the function differently, so instead of having, sorry, a minimum in the equilibrium, it had a maximum, sorry, okay. So the situation looks a little bit like this. You see that the equilibrium in this case would be asymptotically stable only if the value of the function increases along the trajectories because in this case we would have that the function is increasing like this. And so we are actually moving towards the equilibrium. So yes, we could very well do the exact same thing with a function that has a maximum on the equilibrium provided that we change the definitions here accordingly. Is that clear? Maumat, is the clear the question? The answer? Yeah, it is clear. It is clear. Then there is a question from Juan Jose. Please unmute yourself. Yes, thank you. I wanted to ask you if when we are looking for an attractor of a chaotic system is there around those trajectories that are closed in a kind of space near the equilibrium points? There are we talking about the same notion of equilibrium or it's a bit different? You know, I would say that it is similar but the notion of attractor is actually wider let's say than simple equilibrium because by equilibrium we mean a point, a specific point in the state space while an attractor can be something more, I mean it can be also a different set because for example I don't know if you have ever heard of cycle limits, limit cycles, sorry, but there are some dynamical systems for example where we find out that there is, I mean I'm just drawing an example, there is for example this circumference and any solution that starts for example here instead of tending towards a particular point it will tend towards this trajectory. So the solution will do something like this, sorry I am terrible at drawing, but the idea is that the solution instead of going towards a particular point it goes towards a trajectory. Now the intuitive idea is the same so if we have that the solution instead of tending towards a point it tends towards a trajectory. I mean the idea is the same but mathematically these things are very different and there are, there is a whole different set of tools that can be used to study them. Thank you very much. No problem. The second line for asking a question was Ayan. Yeah I can hear you. Yes, I can hear you. So I was just wondering if we go back from dynamic systems theory to study ecological networks the concept of equilibrium and steady states. So non-equilibrium is something when you have this exchange of energy right? Yes. So then which I guess is always the case in ecological networks. So the proper terminology over there maybe is to use steady states or unstable steady states rather than using these equilibrium states. Yes, when we talk about thermodynamically open systems let's say so system where there is an input of matter or energy or any kind of thermodynamic quantities like ecosystems technically we should say steady states and not really equilibrium because there is no equilibrium. Yes so you're right. Thank you. Okay there was another question by Pablo Lechon. Hello so but when you're talking about not being able to solve the linear system and then in order for you to talk about stability you have to talk about stability in a region near equilibrium. Why is that? Is that because if you don't know how to solve the system would you linearize it around the equilibrium or? Yes exactly. I mean if you don't I mean if we would know the full analytical solution like in the case I've shown you a few slides ago on the logistic equation so if we could find the analytic solution like this like this here we could say anything that we can say about the system. We have like the maximum information possible on the system but when this is not possible and this is not possible almost always basically we have to find ways to determine the maximum information which necessarily cannot be the maximum information because we cannot solve the system so we have to do our best and often doing our best is studying the system close to equilibria. Did that answer your question? Yes thank you. No problem. Great next in line is Debas Mita. Can you hear me? Yes. Yeah so my question is that construction of this Leapunov function like whenever we change the model so there I face problem with construction of Leapunov function. So can you suggest some techniques that some base techniques to assume what kind of Leapunov function we can assume for to test the stability of the participant? Sorry I lost the audio for a moment. Can you repeat the question? Yeah so I'm asking about construction of the Leapunov function for any system. Yeah I mean this is the great problem of this approach. I mean as you can see it's actually very powerful because you can tell us a lot about an equilibrium but the problem of this approach is exactly building a Leapunov function. In general it is not easy at all to find the Leapunov function for any generic system so we have to be either lucky or we have to use some intuition to build the Leapunov function. So there is always this kind of trade-off so we have a very powerful tool but this powerful tool is not easy to use and it's not possible to use it always. Thank you. Sorry you were saying something. Yeah yeah it's fine. So then there was a question by Samson. All right thank you very much for giving me opportunity to ask my question. First of all I want to appreciate the presenter for this wonderful presentation. My question goes straight like this. Where can we have a question of global asymptotic stability? Sorry I don't think I heard correctly because of some audio problems. What was your question? My question is tending towards when are you going to have a question of global asymptotic stability? Well it depends. A case when we can say globally that an equilibrium is asymptotically stable for example our systems with concert quantities for example if we have a particle in a potential so a very simple physical system this particle can move only on the x axis and we know that it is subject to a given potential energy potential v and for example we know that this potential for example is a parabola so this is defined on all the x axis so we know globally how the potential work. We know that I mean from physics we know that the equilibrium the equilibria of this system are the minima of the energy potential so in this case since this is a parabola we only have a minimum and so in this case we can say okay there is one asymptotically stable equilibrium and this is the only one that there is in the system so I mean this is a lucky case in which we can tell anything about the system globally and not only locally. Did that answer my question? Oh thank you. Yes? I think you're right. I'm satisfied. Okay thank you. So there was so there is a question which is quite popular in the chat and I think could be the last question of today's session so the question is is there any general principle guiding gas to the construction of a Lyapunov function? No this is the great problem of this approach. There is no general principle we have to be lucky or smart. I mean there is one big exception I'm going to talk about this tomorrow but if we know for example that we have a system with a conserved quantity so if we for any reason know that there is like I've shown you before potential energy or any kind of quantity that is conserved generally these conserved quantities are a good first guess for a Lyapunov function. This is not always true we have every time we have to check that that they are a good Lyapunov functions but generally when we have conserved quantities these are good these can be good Lyapunov functions if we don't have conserved quantities then we are completely on our own and we have to find them on our own which can be difficult or even outright impossible in some cases. Okay great so I would say that this is the perfect time the end of the today's session today's tutorial so as a reminder Leonardo will give the second part of this tutorial tomorrow at the same time so you can also watch again two or three times as many times as you want the tutorial that Leonardo gave today on YouTube so if you want to something you need something and you want to watch it again please do. So the next slot in the next lecture by Joshua Weitz is starting in about 13 minutes so what we're going to do now is to splitting randomly assigned breaking rooms you are free to stay in the breaking rooms chat we who however you are assigned randomly to you are also you should be able to switch the the breaking rooms if you to switch a room if you want if you see someone you want to say hi I guess Leonardo is staying with us so you can also yes absolutely more informally and but also you are free of course to stretch your legs to get a cup of coffee or take a break from the meeting so with that if I since I'm not host I'm ready okay yes I give there we go yes okay 20 breakout rooms this way yes okay just 150 people yes okay can someone on the ICTP side if you can hear me let me know yes I have answered you in the chat so your sharing is good okay I can see your sharing very good so it's okay okay I just remember you that reminded that we are now live on streaming yeah no I understand okay on streaming streaming people yeah perfectly great hey Josh can you hear me hi yes I can hear you oh great okay so I was uh because we have the participants uh uh paired randomly in group randomly in meeting groups so I was in one of those okay that's fine so how's it going it's going fine you have a big crowd I think we're live on streaming so okay oh yeah just keep keep that in mind hello everyone who's watching us have this chit chat I see okay great yeah if you want to discuss anything before we start we can go to a breakout room but otherwise I think I'm set I'll leave it with your coffee I think you checked everything okay great thanks okay everyone uh we are about to close the rooms um and start uh with the lectures uh again so in a couple of seconds all the breaking rooms will be closed and people will join again the main meeting okay in about one minute I'll start introducing the next uh lecturer so then people have time to join back okay great so uh I think we can start with the um next slot so the next uh lecturer is uh Joshua Bytes who is my pleasure to introduce uh so Joshua is a professor at the School of Biological Sciences and Physics at Georgia Tech in the United States he's an interdisciplinary researcher with a very broad range of interests and perspectives in mainly theoretical ecology and quantitative biology most of his recent research is focused on bacteriophage interactions and dynamics at multiple scales from the fine-scale microscopic details of this interaction to the its ecosystem large-scale consequences so today Joshua Bytes is giving the the first uh lecture of um three uh sorry there was there is can you hear me can I yes I can hear you okay I'm giving one of three lectures great okay because I received comments in the chat so uh great so today's giving the first of three lectures of micro virus micro dynamics so please uh Joshua thank you very much for giving these three lectures please share the screen and let me remind the audience if you have question please use the raise hand uh button um that you can find under participants of zoom and I'll give you the the possibility to talk okay great can you hear me jeffable yes perfect okay wonderful thanks for the invitation welcome to the many hundred plus not 200 plus that are here joining internationally I just have a few preliminary slides before I get started first of all just to let folks know that I'm also the founding director of a quantitative biosense's graduate program at Georgia Tech we are accepting applications now to entry uh for entry in fall 2021 at qbios.gatech.edu and if you want more info again go to the website we have cohorts of approximately eight folks per year looking to welcome a new cohort in the coming year and just a bit about our group again we're located in the United States the southeast united states the state of Georgia and that gives you some indication of the team folks from come from all the world much like in this meeting to work on problems related to virus micro dynamics and theoretical ecology evolutionary biology quantitative biosciences more generally as jeffable mentioned I'll be giving one of three lectures I recognize the background here is quite broad so I'll try to use today's lecture as an opportunity to set a foundation it may cover more material than is humanly possible in this hour but I will try and if it goes too fast I'm uh under the impression that a this is being recorded and it can be reshared and also that the slides will be made available so you can review then at your leisure and again this is collaborative work and I'll try to go over a number of different topics supported by the natural science foundation NIH army research office the simons foundation and others okay so to get started I'll try to start with a boring slide visually at least and although it looks boring to look at I'm going to claim that this is actually a fascinating experiment some of you may be familiar with it you can see here in the rows you have these experiment numbers and these are replicas of the same experiment as you can see there's a lot of variability across a different experiment and I'll just only give the hint that these have something to do with counting bacterial colonies and if any of you have ever done experiments with bacterial growth you would probably imagine that you don't want to report back the results of a of an experiment where sometimes you got no colony sometimes 303 and 483 totally unpredictable but yet this experiment as I'll explain in a moment sits at the very heart of how we think of virus micro dynamics in fact really influences all of modern biology I'll give just one example this particular experiment number five where you see this incredible variability sometimes no colony sometimes hundred plus extreme variability now what this actually was is the number of colonies that were resistant to the action of a particular phage and so this tells us something about the emergence of resistance in a population and it comes from this paper that many of you know by Luria and Delbrook on mutations from susceptibility to resistance against viruses and it comes directly here that one that I highlighted here from experiment number 16 and Luria Delbrook along with Chase won the Nobel Prize in 1969 in large part due to the understanding and advances that they that they were able to push ahead using this experimental setup now to understand a little bit more at the time we have to go back in time and think a little bit about the differences between what people thought might be the basis for the emergence of viral resistance within bacteria it could be dependent on selection or independent selection really mutations more generally it was not necessarily clear which way these things work was it Darwinian in some sense or Lamarckian in another sense so we can imagine for a moment as a thought experiment the growth of a bacterial population from a single ancestor where you can see these are all susceptible they're denoted as clear they're dividing over time until after a certain number of generations log two of n the final size of the population we have this entirely susceptible population which is then exposed to viruses that should presumably infect and lice all of these bacteria except through a acquisition mechanism if resistance is dependent upon interaction with the virus in other words selection mutations are select are dependent on selection then the subset of these will survive viral infection but in some sense like a Poisson experiment Poisson distribution meaning there's some random chance per bacteria and if we do this again we'll get a small number repeatedly a relatively small number of colonies that will have this particular phenotypic trait alternatively it could be that mutations are independent rather than dependent on selection and therefore early in the proliferation process even before viruses were added a subpopulation had a mutation which were then resistant to viruses only then were revealed when you actually expose them and then a large number of these bacteria again which were already present survived viral infection and that large number really depends on how far back in time how many generations ago this mutation occurred and then proliferated and in light of this one expects actually that in this context you should have significant variability the earlier these mutations occurred the more they may be in the final event and the later they occurred the smaller even none at all and this is precisely revealed here where I look at the number of mutants this is a Poisson fit obviously it's not a good fit this is the number of cultures with this number of mutants and you can see this long tail effect right where you can see that in some cases you get these rare jackpot light events in which there's a number of these cultures that have many mutants whereas this is just certainly not expected in the Poisson case right so this table that looks visually quite boring actually tells a very important Nobel Prize winning story which is that mutations are independent of selection at least in general and we can talk about CRISPR-Cas some other point maybe in the chat so the takeaway from this in the early 1940s that viruses impose a strong selection pressure host mutations that confer resistance are beneficial in some fundamental sense than viruses induced host evolution right we have a change in the frequency of genotypes in a population that is induced by actions of this viral selection pressure but what about the viruses so this famous paper by Lurie and Delbrook is not often seen from the other side right which is viruses are convenient way to impose a selection pressure but what do the viruses do in fact in a later work Lurie explored this question by looking at the inducement of resistance amongst bacteria against phage but also then counter defense in some ways niche expansion by phage against bacteria and here the squares denote the phage these circular oval like things denote the bacteria and what you can see through the solid lines are the intrinsic ability of this particular phage to infect in life's particular strains of bacteria and you'll notice the absence of such lines in other cases where these host range expansion mutants of phage can infect not only the ancestral types but also these newly evolved types so in some sense we have the host range of viruses expanding you can see here twice and then eventually in these experiments a phage resistant host range emerged so yes not only can bacteria induce evolution in phage and vice versa but at least in these early experiments there was an ocean of co-evolution but there was a sense that this interaction might get short circuited in fact really this dogma persisted for decades and this dogma is encapsulated by this idea the co-evolution potential virulent phage is less than that of the bacterial host back in the mid 80s and part of this was also informed by the fact that it just seemed very hard to find abundant bacteria phage in natural systems and maybe that was yet another indication that the bacteria kept escaping their phage parasites but then something happened and what happened was a few years later in the late 80s a group looking at aquatic systems began to take a culture independent approach to try to assay the abundance of viruses in natural systems by taking water samples here from a lake they were also doing this marine systems and then staining anything that seemed to contain genetic material DNA and then if you see these arrows counting the number of these small dots because here's the scale of one micron these things seem to be about 50 nanometers to 100 nanometers in size and there were a lot of them and they went back through and counted how many of these virus like particles there were in these particular systems back and for based on the dilution to the abundance of viruses and found that there were 250 million virus particles per milliliter in natural waters a thousand to 10 million times higher than previous reports right so this is just an incredible difference in part of this because previous reports were using a host and almost certainly the wrong host for the bulk of these viruses as a means to count by looking for plaques in the same way that Lurian Delbrook looked but if we don't know the host and we certainly don't know the virus and we may undercount so there were since since in the late 80s and this is really the jump start of this new approach and new really ecological orientation to thinking hard about virus micro dynamics in environmental systems and it was then realized about a decade later that and only were there a lot of viruses and they could infect and lice particular bacteria but they also had a role in the ecosystem because as viruses infected and lice bacteria they could redirect organic material this this dom dissolved organic material back into the ecosystem so again there was a notion not only where virus is interesting from their perspective potentially as agents of mortality but also in their role in shaping ecosystems and you can see that here again they divert the flow of carbon and nutrients releasing the contents of cells back into this dom pool and of course there were a lot of viruses and part of the challenge is once you have 10 million or 100 million per milliliter these are all not the same type these are not monolithic it's not a monoculture and from the outset and this hasn't changed that much although there's been improvements in understanding viral types the vast majority of these were basically unknown to science so the majority of these weren't characterized much of the diversity in the early 2000s was uncharacterized and obviously that's been improved quite significantly since then so what do we talk about when we talk about viruses I would say in in normal times we might talk about viruses that infect humans whether it's Ebola Zika or influenza and obviously right now I'm giving and I'm aware that I am giving a lecture about virus dynamics when most people are just thinking about this right SARS-CoV-2 and yet I won't be talking about that today or in the next few lectures I'm looking forward to this break and talking about something else some other exciting stuff that we're doing in the group but just to point out that yes over the past year we have been working on COVID-19 quite extensively as I'm sure many of you have in all sorts of ways including what you see in the upper right is some of our collective work to develop a asymptomatic surveillance system at Georgia Tech which is available to anyone student staff faculty on a weekly basis people sometimes test more than once a week saliva-based PCR tests and you see we had an outbreak but detected it very quickly we're able to maintain with this one exception but again we're able to contain that very quickly rates of positive incidents less than 1% based on these population level sampling over the course of the fall term we've also worked on other dynamic models including some work developing a COVID-19 event risk assessment and there's an Italian version that some of you may be aware of it's received multiple millions of views since we launched and happy to discuss that some other time okay but viruses do as you know infect organisms across the diversity of life and also just to point out I believe there is going to be one set of speakers who is going to talk about models of COVID-19 later in this and there'll also be a forum so you will get a chance to learn more about the sort of class of models of infectious disease models by a few speakers later on but viruses do infect organisms across the diversity of life not only charismatic humans mammals birds etc but also microbes including eukaryotes as well as archaea and bacteria and just to give you some indication that these viruses can also be somewhat charismatic as well here are some images at least of the virus particles and the bulk of the work looking at virus micro-dynamics often goes through the same paradigm as Lurie and Delbrook wishes to think about viruses as agents of mortality here are three particular examples where I'm showing what happens when you have a culture alone e-hux algae and here is prochlorococcus merius med-4 ubiquitous cyanobacteria found in the open oceans at high densities globally the surface oceans and then here are examples of what happens when you add viruses and you can see the decay in density and the increase in viruses in this last case actually was timing of looking at the decline of host DNA as well as the intra-cellular and extracellular development of phage DNA so the takeaway here is that viruses act at these microscopic scales across a diverse set of hosts to infect and less reducing the density of the host increasing their own density over time you can learn more about this generally in this very nice popular book and this goes well beyond virus of micros by Carl Zimmer it's called a planet of viruses and you will find that would be a nice evening read it really spans the scale from oceans this is a not to scale pictorial of a sahano phage right here on the surface on the cover it made the cover but also viruses of humans and so on and this will not be an easy bedtime read but if you're really interested and you get inspired and want to learn more i've also written a book called quantitative viral ecology it's meant to be a graduate level intro graduate level textbook but it also can be a guide dynamics of viruses and their microbial hosts it was published in late 2015 okay so what am i going to do in these three lectures with this introductory material in mind today i'm going to try to go through as i said principles of ecoevolutionary dynamics trying to give folks an idea of some of these core ideas of how we can think about virus micro dynamics and begin to think at multiple scales connecting microscopic mechanisms to emergent population dynamics and hint at least a little bit at some of the ecosystem consequences but in the interest of time i won't get into that as much as i've done in some other contexts in the second lecture i will try to extend the scope beyond what i will do today which is largely a predator prey paradigm and then talk more about parasite host paradigms and of course we know phage are parasites but using the language and context and thinking of epidemiology may take us in some new direction so i will do that tomorrow and then on thursday i will connect some of these principles to applications and i will choose just one rather than on an ecosystem context i will choose a biomedical health context and begin to show efforts to use phage as therapeutics and explain in my view why thinking about virus micro dynamics again as a dynamical system may actually aid in efforts to treat multi drug resistant bacterial infections so that's what i'll try to do today principles tomorrow expand these principles into some new directions and on thursday go towards therapy and jocobo you'll let me know if anything goes wrong but otherwise i'm gonna keep plunging ahead that's perfect and again throughout i will try to connect theory modeling uh with both fundamental challenges as well as real world applications as you can see and obviously this will build build towards this real world applications even further as we get to the focus on thursday okay so again principles today more on principles really on the focus the switch between lysis and latency tomorrow and thursday connecting theory to therapy good so there is a question actually from i am so i am please mute yourself yeah am i audible you are i can hear you yeah so professor witz i was just wondering from the luria data perspective so you have a poissonian dynamics coming in so i was just wondering if you have a larger amount of population of these viruses do they actually try to get in the clt that is like a central limit theorem and get it to like a normal distribution or so and i'm just asking this question from a perspective of gene transcription regulation dynamics so when you have this unregulated dynamics of some transcription factors you have this poissonian dynamics coming in but whenever it's something it's getting regulated by a transcription factor you go away from this poissonian dynamics mostly right so is there something similar yeah so thank you for the question as you can see here you brought up a number of different ideas there are certainly notions in which the viral takeover of a cell can be described through the process of stochastic gene regulation just as we can think of the cells dynamics is through that process and if we have some unregulated gene then notions of a poisson like distribution of transcripts and even proteins can emerge and certainly in certain limits that can look Gaussian but that really is is beyond the scope we don't need to invoke that to understand the fact that what we're looking here is that outcome and if resistance here has a largely to do with surface resistance so the viruses aren't even getting in that we don't need to ascribe all the processes that could be interesting to explore in some other context but we could just think of it as the outcome of those cells those colonies didn't even enable or permit the the viruses to get inside we could have a separate discussion on uh intercell resistance whether resistance modification mechanism or CRISPR-Cas immunity and that would turn out different and we might need to think about gene regulation there certainly have done in other contexts so hopefully that helps this is strictly the colonies are resistant they're we don't necessarily need to invoke the details the intercell or details to understand that then that colony its offspring inherit that same feature of being resistant and that's why we get a poisson distribution at the level of resistant columns okay thank you good why don't i keep moving ahead jacobo is that good yeah great okay so what i'll try to do today uh is explain a few things and i will essentially give you an introduction to how viral infection can change my curvil population dynamics because but you've already seen that we can't stop at population dynamics we have to think about evolutionary change so i will then go in that direction and then we'll see uh how far i go i have about 35 minutes i'm going to try to use it all to really give you this broad introduction and expand some of this in the direction of virus host dynamics in complex communities okay so how does viral infection change microbial population dynamics well to do that and again we're since i have the i'm lucky to be here at the early start of this thing and you've just had one introductory lecture i'm sure many of you heard of block of voltera and a long time ago vito voltera and it's always fun to say that name when we're we're here in a Trieste Italian conference was convinced by a son-in-law in bertha donkona to examine fluctuations the adriatic fisheries why is it that we see these fluctuations this is all exogenously driven or can there be the result of endogenous interaction and the first case that voltera consider was a two associated species one would multiply indefinitely because it would just keep growing but the other one died if it was left alone but the second one the predator feeds on the first the prey and the two species can coexist together in modern language we would write down the model that voltera and then alper lock independently proposed as a coupled system of nonlinear differential equations where the dots notes the derivative here we have the indefinite proliferation here we have predation conversion of prey biomass into predator biomass and the death of this predator left alone the outcome of this are these predator prey oscillations where the predator peaks tend to be shifted to the right of prey peaks so we have a prey peak then predators rise in abundance driving down prey as prey are driven down the predators decline prey go back up again and we see the cycle continues we could then superimpose this on a prey predator phase plane and note that we get these cycles and they're counterclockwise and the reason they're counterclockwise is again we have a prey peak which leads to the rise of predators the rise of predators leads to the decline of prey and the decline of prey leads to the decline of predators allowing prey to restore and so on you will notice that this seems to be a closed orbit that is true which means that if you had a different initial condition you'd have a different closed orbit which in physics would be terrific news we'd have a conservative system but that is not good news for biology because it means that the initial condition is remembered forever these are not true limit cycles so just keep that in mind as a caution of course later models did have true limit cycles and that was often because there was some handling time or other features of the interactions again which is that prey peaks before the predators they're lagged and then the oscillations again appear counterclockwise in the prey predator phase plan as you can see here on the right these ideas of Lacan Volterra really sit at the heart of how the field of virus micro dynamics has emerged beginning in work by bruce levin in the late 1970s along with frank steward mathematician and lin chow bruce is now here at emory and lin as at university of california san diego and in their view they viewed this system an experimental system of having a phage a bacteria and something for the bacteria to eat on as a resource a prey the vector and a predator the virus and the idea really goes back even to alan cambell in the 60s that these phage these viruses that exclusively infect and lice bacteria we can think of them as a simple predator and the reason or rationale is that they act to convert prey biomass into predator biomass and also they never lead to the death of the prey through that process. So this Lacan Volterra model in the more simplified form again is the basis for these virus host population dynamic models and here again you can see this simple resource prey predator model and this is a model in which we're envisioning a chemostat in which there's a rate omega in which resources are flowed into the system and a rate omega which everything is flowed out including resources prey and predators the bacteria and the viruses the prey take up resources converting it into new prey the viruses infect and lice bacteria leading to beta this birth size of new viruses and what you can see is that if you start a in silico chemostat with a certain number of prey and resources and out of virus you get oscillations you get a decay a decline excuse me in prey density because now it's being top down rather than bottom up controlled and you also see these oscillations although it's hard to see here in this log space we can project this onto the phase plan ignoring the resource level and what we see are these counterclockwise dynamics right and the counterclockwise dynamics are for the same reason that we have these locovolterra like dynamics that we have a peak and viruses driving down the host density leading to a decay in the viral density we're allowing to the restoration of prey density and so on in this particular set of equations this should actually relax back to any equilibrium to a fixed point so just to remind everyone so that we're on the same page this is a bit of the life history of a bacterial virus i'll go through this probably once in each lecture just to make sure everyone remembers it in case you showed up for one i'll probably do it again but also do it a little bit faster each time because i'll assume you will have remembered it here we have this 50 to 100 nanometer size passive bacteriophage it's diffusing a natural environment comes into contact with the bacterial host injects its genetic material into the bacterial host takes over the cell machinery as was alluded to by ion and then through a time process leads to both the encapsulation of the genetic material into the capsule of the host the self assembly of these excuse me of the virus of the self assembly of these viruses and through the release of both hole and lysine genes there can be a hole made in an inner membrane a hole made in the cell wall and out go the viruses and the life cycle continues okay so we're this is just a process by which a parasite takes over a host a microbial host and you can see there's a time that it will take between when this happens and when this happens right between adsorption encounter infection injection etc and lysis and so obviously the earlier model didn't have this there was an assumption of an immediate trans conversion of prey bacterial density biomass into that of the virus so obviously we need to make some corrections there and this is all done in a particular context right this chemist that context in which we're envisioning that media is being flowed in there's some dynamics here but everything is going out through waste which we can then measure and observe so these types of models can also be extended to include models with an infected class and the point I'm showing here for those of you familiar with reading these is this susceptible host infected host and viruses the only difference is that rather than immediate lysis now we have an infected class which decays at array 8s in other words 1 over 8 is the latent period and we get new viruses the point is that in this model we get a true limit cycle again counterclockwise dynamics but leading to a true limit cycle you might not like that because you might say well a rate eta of decay means that we have an exponentially distributed distribution of latent periods and the peak of such a distribution is at zero and that's too soon we could also make an explicit delayed set of differential equations in which the infected bacteria are produced at array 5 and v but then this subscript tau means the number of bacteria and viruses tau before are those that are released now these become more complicated to deal with for various reasons as you can see you instead of having a finite number of initial conditions here you need an infinite number of initial conditions because you have to go back and all the times between zero negative tau there's also this decay because some of the bacteria that were infected before not all of them survive because some of they're being washed out through the chemist that which is why we have this even the minus omega tau factor nonetheless in this particular model the quantitative details may differ a little bit but the qualitative features remain we again get these counterclockwise dynamics so we have this robustness of an idea from lockable terror that could be applied to virus micro dynamics which is the same notion of a simple predator prey system should lead to endogenous oscillations of this counterclockwise type and just to put a little mathematical checkpoint here as I assume that you're going through in these non-linear dynamic tutorials that it's not inevitable just because the virus can infect the host that it can invade but rather it we need to think of this as in some ways a destabilization of a otherwise stable fixed point which we just have bacteria and so you need to look at most simply the linearized system and check to see if the eigenvalue was positive and that would imply invasion I go over this in the book and on Tuesday's lecture I actually elaborate on intuitive criteria that can explain this as well okay so just to remind folks that it's not inevitable not every virus just because it infected bacteria the ecological conditions have to be sufficient which depend on life history traits as well as the abundance of the host so just can I finish this thought or you had a question jackable there is a question from the audience if you want to go on you can wait otherwise let me just finish this one side this a few more slides and then I'll take the question just because I don't want to lose the train of thought here which is these same types of dynamics can be observed in the laboratory these are lockable pteralike cycles between pht4 and E coli b you can see the population density time so this is about a 10-day experiment where we have large-scale endogenous oscillations and you can see that the ratio of virus those can be quite large on the order of hundreds if not thousands there's also a time shift here so that when the bacteria peaks it's usually followed by the virus peak and again this is in a chemistat system where it's otherwise homogeneous shaken and not being driven by some exogenous change in resources so just to point out here that this really is a demonstration that there can be a feedback between virus and hosts that lead to endogenous oscillations that's why I just wanted to get this one idea out then I'll take the question these original models of virus host dynamics presuppose a simple one virus one host relationship if viruses act like predators we should expect cyclical dynamics whether we use the simple lockable pteralike models or ones that take into account the microscopic details of infection and lysis and reminding post that invasion is not inevitable it depends on traits and this has been observed in experimental systems so this would be a good time to take the question yeah thanks I just had a very quick question a few slides ago when you were talking about the delays and you had this n sub tau can you just you went quickly over that can you just say again what is tau and are you integrating over tau in that equation right so tau is a fixed latent period so you should think of this as n of t minus tau so it's this is interpreting this would say that the change in the number of infected cells now is equal to new infections minus infections that happened tau ago that are now lysing good and when you say an incident number of initial conditions you just mean you have to keep track of tau's tau's worth of time data to be correct correct and so just to point out for people are not so used to it that it's just a different way of thinking about this particular system and oftentimes what people do if they don't want to do with delay differential equations is to set up you know a finite number of stages which replicate these kind of shapes of distributions and then you can go back to this ordinary differential equation approach good and is the is the set of solutions to that differential equation really an incident dimensional parameter space or I mean really different for I guess it's different for every n tau function so so just keep in mind remember n tau is not a function that says n of t minus tau so you're just getting a one set of niv solutions but to figure what happens now you need to go back tau ago so just because of that delay revolutionaries would have n as a function of tau for that period of time and if n of t between zero and minus tau minus tau ago in order to integrate for thanks yep yep great we're good jackable yes yes I'm plunging ahead here I go part two but I'm I'm on pace we're gonna finish uh so I've set things up here intentionally so we have ecological dynamics but I've already told you luring and delbra told us that we should expect evolution if not co-evolutionary change so I'll try to give some sense of this question how does co-evolutionary change and evolutionary change affect these population dynamics so let me go back now to this experiment which I showed you in which we have these endogenous oscillations lockable terror like oscillations between viruses and e coli and what I want to now show is something different and I'll switch back between them which is that for the first 200 hours or so it all looked like theory was fine and we'd be done but past that time something else happened this is the same chemostat but now we see essentially a flat line of hosts and oscillations of viruses and given the size of this crowd and the context I'll just point out that the fact that viruses are increasing implies this is a chemostat they would otherwise decay they're replicating on something but what's interesting here is that the host density remains apparently flat which doesn't seem to make sense if you saw this time series in this time series you wouldn't necessarily relate them and this is why this is often called cryptic dynamics what happened in fact is that they went and observed that this host population was not homogeneous anymore but in fact was comprised of two different populations one that seemed more susceptible and resistant to the virus that isolated it marked those and then we're able to repeat the experiment in some sense by being able to track the two types independently and what they found is that what happened was that there was the emergence of resistance so a subpopulation was resistant rose to a high abundance but there was still a susceptible subpopulation the viruses were replicating on this subpopulation and if you'll notice these dynamics look like a Volterra like between the virus and the susceptible host in terms of being phase shifted but also these susceptible host oscillations note the log scale would just be noise in the background of the resistant host density implying that we do have the emergence of resistance and there's a essentially an evolution of the system in which the virus of induced evolution but not lean to their own extinction but rather than to a new kind of dynamic so again just to point out that these resistant hosts invade obviously they have that benefit but they don't exclude the susceptible host because they come with somewhat of a cost and if there is a tradeoff between growth and resistance or defense then these two different types can coexist which leads me to a more general point which is that when you have predator prey like dynamics in a virus microbe system you can get these canonical predator prey cycles lock of Volterra cycles when you have evolution you can get cryptic cycles or anti-phase dynamics and that's a more general problem in the field of rapid eco-evolutionary dynamics and has been seen notably in other predator prey systems including that of rotifers and we also have a multi-type possibility as well so this is no evolution evolution I'm about to explain why something even funkier can occur where it seems like the prey peak is following the predator peak in other words prey eating predators not the case but something about the system has fundamentally changed when there can be co-evolutionary dynamics and just to make this more explicit here I've taken a system now of two viruses and two hosts so we have co-evolution in the sense that there's the change of the frequency of genotypes in a population over time you can see the shading here these dash lines denote the virus strains and the darker lines denote the host strains and when you add them together the total virus and total hosts seem to have a predator prey like dynamic but shifted where the virus peak precedes the host peak okay so I hope that's clear you can see also this in a phase plane where when you get one of these clockwise cycles what this implies is that rather going around this way it goes around this way which is not what we expect because it means that the peak in predators is then followed by an increase in prey and we can understand this by recognizing that at the same time the total population is changing we also have changes in the fraction and the frequencies of genotypes here hosts and here viruses and I've denoted them by having these high vulnerability types and these low vulnerability on others defense specialists and then we also have viruses that may have low offensive types and high offensive types in terms of being able to get in more efficiently and otherwise not but they differ in their decay rates between the types and what you can see here as at the peak of of the system from the perspective of predators we have a lot of high vulnerability types which leads to an opportunity for these low offense types to invade but as these low offense types invade on the viruses then the system can shift to low vulnerability and these low vulnerability hosts can do exceptionally well in the system of low offense page now when there are many prey around that happen to be low vulnerability but because there are very few phage then there can be a benefit for the high vulnerability types to invade and as they do that also comes with a rapid shift to high offense types restoring then the virus population right so we have a joint dynamics of population dynamics and evolutionary dynamics and it's really driven by the concurrent change in genotype frequencies does this happen well in work by way at all working with Bruce Levin they had observed what they viewed as complex dynamics of vibrio cholera and its phage and cholera and phage have its own interesting story which you can see here our dynamics that look as I said complicated but we notice something interesting here when we focused on a few sections and I've highlighted them and raised them in the line with here so that you can see them and I will then project these on the next slide into the phase plane and what we found were there are multiple examples of clockwise cycles and there was actually further evidence that the system didn't just have a single host or virus but in fact they found this t and b phage making turbid or big plaques and multiple kinds of resistant hosts and what you can see here is that we have these clockwise like dynamics where a system not only goes around the wrong orientation but gets back nearly to where it started of course one can raise a question is this sufficient evidence for a clockwise cycle and so what we did was in some sense take that time series and then create random time series that had the same point-to-point correlations reconstruct synthetic time series and ask how often would we see these kinds of short clockwise like cycles and then see if the fact that we found these should be a surprise and the answer is yes it was surprising despite the shortness of these cycles the idealized clockwise cycle has a winding angle of 2 pi and goes back exactly where it starts here's the distribution of all these synthetic time series what we observe in red and just to point out that this is essentially as good as the link's hair evidence as well which is the canonical example for counterclockwise cycles or prayer pray dynamics so we have as much evidence as we have essentially for these phage vectors we did for the link's hair and so we have examples now of clockwise dynamics in this co-evolutionary system so just to sum up the second part that more generally whether we're talking about phage microbe dynamics or prayer pray dynamics with evolution in general sense without evolution we can only have these counterclockwise cycles in a qualitative sense when either evolved but not both we can have this or we can have anti-phase cycles of the kind that you can see or even these cryptic cycles where one is changing and the other is not and when co-evolution is included then either of these two things can happen but also it's possible to have clockwise cycles okay so the take home here is simply that a rapid change in genotypes can impact ecology right so when we think about virus microbe dynamics we can't think as evolution has just turned off and again this is a natural consequence of going beyond Luria and Delbrook and asking what happened next like Luria did in the 1940s but not just on a plate that's static but actually thinking of it as an intrinsic part of the dynamical system but in this last part in the last 15 minutes I'm going to ask what other dynamics emerge in even more complex communities and I should pause here again in case there are questions yes there are two questions so Miguel please mute yourself hi hello Joshua I have a question for the clockwise dynamics are we assuming in the system that the the aggressiveness or the vulnerability have a fitness disadvantage or advantage beyond the viral micro dynamics or or why else will the the low aggressiveness genotype for example catch up again right so we are assuming that there are trade-offs in traits so what I and I just said before it's not inevitable just because you have co-evolution doesn't mean you have to have clockwise cycles right there could be a replacement and we could just knock out one of the other kinds maybe the emergence of a doubly resistant bacteria and even the fact that there's some trade-off in growth it still knocks out the first kind but there are particular conditions which we go into in this paper in which these cycles can generically be seen but you're right that it involves trait differences which I'll elaborate on next thank you and there was another question by maybe we can just because we I've heard from ion once and I appreciate that but just to make sure that given there are hundreds of people here um maybe I'll just keep going to part three and ion you can send me a or something in the chat just to make sure we hear from multiple folks is that okay jackable yes okay for me so I am why don't you just send me a put it in the chat and I'll answer after or just to make sure that I finish up part three sound good sure sure sure thank you so much very good okay so let me go to part three and ask this uh the last question which what is the relationship between the infection networks and host viral dynamics and complex communities as you can see I've structured this talk focusing on single virus host dynamics then two and one and two and two and this would be a very long talk but I only have a 10 or so minutes left so we're going to try to make a leap of a kind and this leap has been possible with really almost a decade of work by Cesar Flores who began some of these projects continued by Luis Jover and continued even further by Ashley Kuhnen in the group now and all these are physics students doing this kind of quantitative bioscience stuff okay and again I'll go back to this notion that there are many viruses in these systems and pointing out that when you have this many viruses uh they're going to be quite diverse we went back to some of these original data sets to really hone in on this question as part of a collaborative effort to analyze a virus microbe dynamics in natural systems finding that if we look at the number of prokaryotes largely bacteria and viruses per milliliter you can see relationships you may have heard of this notion that there's 10 to 1 viruses per bacteria in natural systems that's not necessarily the case so that's a decent median there's natural variability between 1 to 100 to 1 there's more of both in the surface than there are depth but the point I'm trying to say here is that there's a significant amount of variability and there's also just a significant number of viruses there tends to be many more viruses than there are bacteria in a typical ocean community which means there's a lot of different types and this has often been very hard to estimate because of the problem a rarity so there's some early work 15 years old now which estimated the diversity in terms of species richness between 10 000 and 1 million viral genotypes that's a lot also says that estimating is hard and that estimating problem remains hard there's something fundamentally hard about understanding the abundance of rare things because you don't see how many rare things there are because they're rare so just a side note but sufficient to say that these really are complex abundant men diverse communities so how do we actually figure out who infects whom we can't just keep going with these simple models we actually have to look into uh natural systems and so that's what we began to do to get a sense of what some of these relationships might look like and how that might explain coexistence so what folks in the field have usually done is like we're in delbro collect bacteria and phage from the environment or from experiments or evolution experiments and then see who infected whom as you can see here we have a plate of bacteria and you can see as different kinds of bacteria exposed to different viruses sometimes these viruses make these clear holes these are plaques which denote clearing and then here they added the phage but there was no clearing to note that this particular bacteria was resistant to this particular phage people repeated this many times over characterizing these as susceptible or resistant and here you have one example of a paper in our digital version just to point out that these are the same thing where the blue background denotes the fact that the phage and the column can infect the hose in the rows and the white scares denote the fact that they can when we looked at many such patterns across many different systems we weren't sure what to expect and if you look at this you can't really see much of anything at first we were disappointed we realized of course there are many ways to display these particular results of phage bacteria infection experiments because you have row and column orders and this becomes a large exponentially large number of combinations so we used heuristics used to analyze network systems more generally to look for latent structure and in fact that's what we found examples in which it seemed like nestedness was quite typical and what i mean by nested is that we have these remember that viruses are on the columns and host in the rows you can let me see if i have an example yes good here's the original data here and here's the restructuring this is exactly the same data i just want to emphasize same data we've just shifted the rows and columns revealing that this is in fact more of a nested structure where we have these specialist viruses which tend to infect a few hosts it's not perfectly nested but it tends to be so and they infect the host that everyone else can infect so the specialists seem to be infecting the easiest to infect of the host and likewise the generalists seem to infect the hardest host to infect right so the virus generalists are infecting both the easiest as well as the hardest and you see that then the infection ranges are nested one within the other and the same with the host so even though this wasn't apparent in the original studies it was clear that nestedness seemed to be quite typical which raises questions and the question it raises is how do we have coexistence potentially between these specialists which seem to infect the host that everyone else can affect and the generalists right and likewise why do we have hosts that seem to be infected by everyone hanging out with hosts that seem to be infected by almost none or very few and just again to point out this was unexpected the nestedness in the original displays were quite low compared to the ones that we found with this revised sorting and again this means that we have these phage bacteria networks that are typically nested and it's common in both ecology and evolutionary studies so how can they coexist so let me give a particular example here of one which why doesn't this most resistant host I'll compete the rest right it's only susceptible to one virus but so is everyone else and it's not susceptible to the others and likewise why isn't this most specialist virus outcompeted by the others same reasons only in fact one thing so can everyone else what we've done is take these dynamical systems that I've already explained but then using this letter i and j denote the fact that we have many types so the hosts are competing they're being infected in lice by viruses which are then releasing new virus back virus particles back into the environment and I'm only going to point out the fact we have this matrix M which we now are going to use something based on what the measurements imply rather than early kill the winter models which say there's one virus for every host and clearly that could lead to co-existence we're going to replace that with these nested models raising the question of how do we have coexistence when we have overlapping niche or overlapping host ranges and what we can see is that if we take a very simple community when we have two viruses and bacteria together in one case only one coexist you can see host one survives excuse yes host one survives and here we have virus two persisting and virus one going away this other host may be at low density and here we have both of them co-existing together and they both have a nested network but now the strains differ in their life history traits so one case one gets excluded but in the other all of them coexist and they have the same structure but it's the life history traits again that make the difference if we solve for the steady states in the model i'm just going to point out you see these kind of differences of traits r2 minus r1 the differences in the infection efficiency so it seems like these steady states depend on differences in terms of growth rates and differences in terms of viral life history traits and the key points without going into all the details we've elaborated this for quite some time is that if there is a growth rate that decreases with defense so in other words most vulnerable grows quickly least vulnerable grows slowly and likewise that the efficiency of the specialist is high in other words it draws down host to low densities whereas the efficiency of these generalists is not as high it doesn't draw down the host to low densities you can get arbitrary levels of coexistence so viruses and hosts can coexist with arbitrary complexity now this does raise the question is there evidence for such tradeoffs and just to point out that typically when one looks for tradeoffs you see a tendency for there to be a growth rate resistance tradeoff and a fitness and host range tradeoff for viruses but i'll also point out here's strain and growth rate all of these seem to be resistant to this particular virus that there are examples in which there doesn't seem to be any cost and likewise here fitness on original host no cost even though now it has this niche expansion there are other interesting stories here about contextual benefits of fitness able to infect some but not other hosts but again points out that these relationships are statistical in nature just like those of the of the nest in this in these communities and so on a separate work we looked at the relationship between biodiversity in other words richness of this community as a function of nestiness and only to find that nestiness does seem to promote diversity in the system even if the nestiness isn't perfect and this depends on the life history traits and we've done this for various kinds of life history trade assumptions and tradeoffs but again the tendency remains so just to finish up here i'll just make one last thought and then i'll conclude which is i've made a point here from single to multiple to these complex communities showing that there is mechanisms of coexistence whether through evolutionary dynamics or through tradeoffs but as nestiness the only feature and i don't have time to fully elaborate but i'll just give you a caution that most of the studies looking at host range and infection dynamics focus narrowly on a particular phage and bacteria within a species or genera but clearly if i have a phage of E. coli it's unlikely to also able to infect prochlorococcus and synococcus i can get it to you won't happen right so we clearly have differences at other scales and just to point out that a long time ago and we continue to work on related problems we found that in these large scale studies of ocean systems here was the original data which we then restructured and identified modularity and found large scale systems did have modularity where the bulk of interactions happened within these modules and so this just again suggests that it's not only nestiness that there is separation some phage are affecting some types what we found is these modules were correlated with geographic diversity this is from an open ocean Atlantic sampling system and even within modules there tended to be nestiness so in reality what we expect is that true network structures multi-scale and that remains a challenge for the field as a whole okay so i think i'm just on time and wrapping up just to point out that tradeoffs facilitate coexistence in this last part that the abundance depends not just on who they can infect but on like history traits that these tradeoffs need not be perfect for coexistence obviously we saw that in terms of the nestiness it was not perfect nestiness nor was the tradeoff necessarily perfect but just as a caution large-scale networks may have different structures and i anticipate i'd run out of time but we continue to work on actually the joint analysis of genotypes and phenotypes and if you want to see some of that work this is with justin myer it just came out on bio archive going beyond just these phenotype assays and linking them to genotypes as well and with that i want to thank you for your time i went over a lot of material but my hope is that for many of you who are new to this area this gives you a broad overview of some of the challenges as we move from microscopic dynamics to build population models evolutionary models and begin in the direction of trying to understand virus micro dynamics and ecosystems thank you and i think it's probably time nearly for a break yes so thanks a lot josh that's what was a fantastic broad introduction so there are actually two questions so if we can i think we can take five minutes for question okay so there is one question from ronardo yes thank you thank you for the lecture professor i wanted to ask and i'm sorry if i'm asking something trivial but i wanted to ask if there are explanations or intuitions as to why the relationship between phages and hosts is nested right so that's that's a terrific question and one that has bugged me for quite a long time because it seems to imply a kind of arms race dynamic is it a question of evolutionary constraint or is it something about the ecology and i i won't open up my slide deck to be tempted to kind of give you yet another lecture on the questions but if you look in our paper by gupta at all we begin to ask this finding that in some cases it seems like it's really the ecology that's driving it not necessarily a fundamental constraint from the evolution let's say the biophysical side it implies that there's an ability for a phage to then have this host range mutant without necessarily this earlier cost right and likewise for there to be the possibility of becoming resistant to prior phage and yet still then being infected by a new phage type so in a co-evolutionary sense it implies an arms race like dynamics which would give rise to nest in this but yet in this paper gupta at all we show that even though that's what the phenotype looks like the genotype implies that something else might have happened but there was some trade-off or constraint ecologically that didn't permit a fluctuating dynamic in which case you might expect modularity rather than nested it's an open question a terrific question so i'll just leave it there that it's one that we continue to try to explore really gets the heart of what we're trying to figure out okay thank you very much there is another question from martina yeah um thanks josh from it was really interesting and my question is more a curiosity i would say because from what you told us it seems that there are if there are very few generalist viruses and a lot of specialist viruses and i see a sort of parallelism with the distribution of generalists especially in bacteria because again in natural systems there are very few generalist bacteria and a lot of specialist bacteria and i want to know if you have a comment on that or you can yeah if you can comment yes so i guess i'll comment from the theoretical side that i didn't explain necessarily the relationship between the host range and the abundance but you can actually go do that and it turns out it's the model is flexible so i know that may be unsatisfying but it could be in the model predictions that the generalists are more about or even less above than the specialist and it depends in some sense on their difference with respect to the most similar host or virus i'd also caution that the generalism here is still narrow from the virus side because we're talking about generalism within the context of probably a species or perhaps general not somehow beyond that so just to caution but you're right that there it really depends on the life history traits and the trade-offs but at least in the model it doesn't say that absolutely one or the other will be the more abundant kind but again the cautionary you're probably thinking about metabolic generalism and so on and here it's just still narrow so we don't yet have a sense i don't think enough quantitative data to give a definitive answer on on the virus side it's something we're still working on thanks a lot there is one last question by mitz and then we'll take a break yes hi Joshua thank you for the talk i was um i was hoping if i could get your opinion on particularly this connection between virus host dynamics and scaling up to ecosystem level consequences the example that i had in mind was the unfortunate fact that we are currently destroying biodiversity at unparalleled rates and how might the virom respond to this obviously coronavirus is one example of this but i just wanted to hear kind of your professional opinion on this how we could understand equal evolutionary consequences of this yeah so that is probably not a question i can answer even in one minute to the extent to which we're thinking about that question just because it's such an important broad question we have begun work which i'd hope they would almost be ready at this point but still maybe a few weeks away but hopefully we'll be out soon on the bio archive to embed these virus micro dynamics in the context of ecosystem models in which viruses are not the only agent of mortality but also includes flagellates and other protists that might be grazing upon target hosts and so we really need to think about viruses as part of these complex communities and once we start changing things and particularly one of the things we could change are things like salinity right as well as co2 levels or even temperature that's going to change virus micro dynamics in ways that frankly concerns me but i don't know the extent to which that will start to impact long-term let's call them microbial loop mediated processes but i think we should be concerned that in addition to pampering or somehow upsetting biodiversity we're also messing with the ecosystem function i'll point out that we do have one little speculative paper on micromonus viruses and hosts and affects putatively of temperature and long-term scenarios where it implies that at least current viruses may have a hard time keeping up or infecting with certain bacteria and to the extent to which they play a role in shifting the fate of dissolved organic material that may also mean that it's really beginning to transform and change carbon and nutrient cycles at global scales but at this point that's all i can say that we're at the beginning of that process but that's one of the reasons we're focusing so much on these ocean systems because of that shared concern thank you great thanks a lot joshua so joshua will give a lecture again tomorrow and what we're going to do now is to take a seven minute break so we are going to be again randomly assigned to breaking rooms so if you want to stretch your legs get a cup of coffee i mean please get away for a few minutes from the screen but also if you want to take the opportunity to chat and discuss informally with other participants please take this this occasion as an opportunity to do that so yes so we'll be assigned randomly assigned to breaking rooms in a few seconds