Okay. Welcome back everybody for this long session of today, Friday. We are now streaming live on the YouTube channel. It's a pleasure for us to have Ms. Resha Guhtal from the Indian Institute of Science to deliver a set of lectures. And the title of today's lecture is Stop Assisting Best Stability in Ecological Systems Part 1. So as always, you can raise your hand and you can type your questions in the chat. And Ms. Resha and myself will keep an eye on it and then depending on how we especially want to deal with that, you can answer immediately or at the end of the lecture. Okay. So with that, I leave you the floor. Looking forward to hear from you. Thank you. Thank you very much. And it's a great pleasure and honor to be delivering these talks at an ICTP meeting. I was very much hoping to visit ICTP, but unfortunately, as you all know, everything is virtual these days. The last time it was only one time I did really visit ICTP. That was when I was an MSc student and it was in 2002 in the summer. So hopefully, maybe again in the near future, I will do a real visit to ICTP. Okay. So welcome everyone. I'm going to share my slides now. So I would like to keep it interactive. So if you have a question, type it out. If it's something absolutely not clear, just I think you should just unmute and also speak. And that makes it easy for me as well. So I have some plan to cover topics, but I don't have to cover everything, as I just want to make sure everybody's with me. So the broad goal of the three talks that I have planned, these two sort of introduce you all to by stability in ecological systems and the role of stochasticity. How do we apply these models in the context of real data as well? So it's going to be a combination of theory and how do we look at real systems in the context of these theories that are sort of deeply inspired by non-linear dynamical literature, non-linear dynamical systems literature. Okay. So the three part of my talks are sort of planned this way. So in the first part, which is today, I'm going to talk about what we call mean field approach. So I would imagine that you would have heard the word mean field in some talks throughout these last two weeks of fascinating talks. So I'm going to look at a mean field approach in this first hour today. And how do we understand ecological systems, those with by stability using mean field approaches? Okay. So one specific question are the aim for today would be can we predict? So whenever we have this by stability, we have something called tipping points. Okay. Now the question, the main question or the main motivation for today's sort of discussion would be, how can we use these mean field approaches to predict or anticipate tipping points in ecological systems? So the second part, which would be I think next Tuesday, it will be on the same question, but looking at spatial dynamics. Okay. The real liquid ecosystems are spatially spread, spatially extended. What can we do something better than what we do today? Okay. And the part three would be something more general, more interesting, equally interesting, hopefully. So but you know, part three, how much of that I will cover will depend on how much I'm able to accomplish my goals for the part one and part two. Okay. So the main, so let me, this is the outline for today's talk. So the main motivation to study by stability comes from the idea of tipping points and abrupt transitions. So I will provide that motivation why we must be interested in by stability in ecological systems. I will then move on to describe simple mathematical theories of by stability and tipping points. And then I will introduce some mathematical techniques intuitively, you know, there will not be too many calculations and so on. And then how can we use these mathematical theories and intuition that we build to predict or anticipate tipping points in real ecological systems? And finally, the last part of my talk would be discussion on and the demonstration of empirical evidence for these mathematical predictions. Okay. So that's the broad goal. And I understand that, you know, Carla Stavor has already covered some aspects of some examples of, you know, what I'm going to present today. So hope there might be some repetition, some overlap. So please bear me with that if it is there is some overlap. But to make my talk self-contained, I have assumed that you may not have heard her talk or you may not have you may have forgotten some of the details from her talk. Okay. So here is an example of, you know, I'm going to give some examples, empirically documented examples of abrupt transitions in ecological systems. So this is an example of a large scale abrupt change, large scale desertification. So what you are seeing here is the northern Africa. If you look at various proxies for vegetation in this huge landscape, okay, let me see, let me switch on the pointer. Okay. If you look at this large landscape, if you look at proxies for this large landscape over the last 10,000 years, you find that preceding 5,000 years before present day, the vegetation sediments were dramatically different from what you see today. So this current phase, current state of northern Africa, which is the desert, was not always a desert. In fact, the sediments indicate that they were in an entirely different state of vegetation. In fact, it had a pretty decent level of vegetation in the landscape for several thousands of years before it suddenly tipped and became the current state. Here is another example of the opposite type and this is on a much more local scale. So what you're seeing here is not a continental scale, but really a scale of half a kilometer by half a kilometer area in China where over the last 60 to 65 years, the 60 year in this graph represents 2010 also, the grass cover in this area has changed. It used to be a low grass cover area and then it has now settled down to a larger grass cover area. So this was an example of a restoration of grassland ecosystem and it stayed in this low state for several decades before switching and becoming a moderately covered grassland state in the present days. Here is a third example of lake electrification. Data is obviously noisy from many ecological systems, but indicates some very interesting features. For the same amount, same values of drivers, you can have different values of the state variable. The state variable here is the some measure of fraction of the lake surface covered by vegetation. So what you see here is that you know what physicists would call hysteresis. When the lake underwent, when the lake had a very high level of phosphorus concentration, so that's those are the red dots in this right extreme and a very low amount of the vegetation. As this forest values came down, it continued to remain in that state of low vegetation. However, it increased and a reverse transition however happened across a different route entirely, different direction. This is called hysteresis in the using the language of that is also used in physics and magnetic systems. Here is another example, classically cited as that of you know abrupt transition, which is that of stock market crashes. So where the stock markets often are in, even when they are in a very state of boom, there could be sudden crashes in the indices. That is remarkable over a very, very short period of time. So sort of the, if I sort of summarize and look at all these transitions that happened, so they often are abrupt. They are abrupt changes in the state of complex systems and once the change has happened, it's not just abrupt, it actually remains in that new state once the change has happened. It's a persistent change as well. And typically, they seem to have happened for no obvious changes in the driver values. So in every one of the example I gave, no people don't quite know what was the driver that changed dramatically, that could have also caused an abrupt change in the state of ecosystems. So the basic idea is that even for gradual changes in the known drivers, system can respond in an abrupt way. And sometimes these changes are irreversible. For example, if vegetation is lost, if certain species are lost, you can not really recover them back. And even when you can recover, they could still be irreversible on timescales that humans sort of deal with. So in studying this now, these are the empirical phenomena I have just described, we're going to hear a whole bunch of terms that which you may or may not have heard, regime shift. So many of these sudden ecological changes from one ecological state or the other, they're also called regime shifts. They're also genetically called abrupt transitions or tipping point events and more terms like critical transitions. So you will hear some of these terms, I will sort of clarify as and when necessary. Stochastic transitions because they're often driven by large amount of stochasticity in the drivers. And the mathematical concept called bifurcation, which I'm assuming you may be familiar now with many, many of the stocks and hysteresis. So I'm going to use some of these terms and define them more precisely when necessary. So with these examples I showed you, this is the example of this large scale, continental scale, the certification. This is a relatively local scale, the recovery of a grassland and then a stock market crash. So what are the important questions that people in the ecology literature or people more broadly in the complex system literature are interested in? Okay, so one is how do we mathematically model these systems? Okay, so do you build a very detailed process-based model to understand the systems? Yes, that is one approach. Or can we sort of develop fairly simple heuristic mathematical models that only captures essential details? The second question that people have been interested in, can we really have predictions for these kind of transitions? Are there early warning signals before these transitions happen? If there were such warnings, then one can do something to stop these events from happening. For example, in this specific case, imagine hypothetically you are somewhere here and if you had been given data from here to here, likewise, if you were somewhere before this dotted line here, you had somewhere here, let us see if you had been given data, time series of cross cover, could you have anticipated this abrupt transition, likewise in the stock market, which of course has really huge applications? Okay, so that sort of sets motivation for studying these phenomenon and to study them, mathematicians, applied mathematicians, physicists, as well as ecologists, and many of them have applied mathematicians and physicists, turned ecologists. They have been using mathematical theories of bi-stability and tipping points. So I am going to describe those simple models now and see how we can model them and how do we try to do predictions of these models. So here is a very simple model of ecosystem collapse. So here in this model, the ecosystem is represented by a single variable. The ecosystem is really a large interacting system of species. However, one can think of ecosystem or sort of lump all of those into a single quantity called biomass density and one can think of how this biomass density is changing over time or what is the dynamics and the simplest models of these ecosystem dynamics have this concept called carrying capacity and an intrinsic growth rate R and the carrying capacity K. And as long as this intrinsic growth rate is R or is positive, the biomass density V will reach a carrying capacity K over a period of time. This is also called the logistic growth model. Okay, now ecosystems are under constant pressure both internally and externally. One such important one such important process is that of grazing. Grazing could be driven entirely by herbivores within the system. It could also be driven by livestock, that human settlements that are there nearby forests. And usually this is modeled as this sigmoidal rate function. So V square divided by V square plus V zero square, this is represented last due to grazing. This is a non-linear term. So basically this assumes that if the V is low, the grazing rate is small. But if it increases, it increases non-linearly and saturates to a value of C. So what is the, what are the equilibrium values of this, of this simple mathematical model of ecosystem? So equilibrium will be achieved with the logistic growth term will be equal to loss due to grazing. And if you calculate the equilibrium points, what you find for some values of R, K, this is what you will find from the X axis, the driver, which is the grazing rate C here. And Y axis is the steady state or the equilibrium biomass density. So what you find here is, when the grazing rates are low, of course, you will still have, for example, here I have chosen a value of K is equal to 10. So the steady state biomass density will be still close to the original carrying capacity. However, as we increase the grazing rate, that does reduce. But what is really interesting is there is a threshold value of this grazing rate. And once the system reaches this threshold value, the system will collapse into a low, low biomass density state. And now if you do a reversal, it doesn't go back at the same point, but the system will stay in this low biomass density state before undergoing a transition back to a high biomass density states. So to understand these systems, one can think of the simple intuitive picture, which is that of a ball rolling in the landscape. So what is this landscape? Think of X axis on the landscape as, let me see if I can also write. Think of this as the biomass density ecosystem state. And so wherever the ball settles, that becomes the stable state. So in this case, it will settle in this point or in here. So this is a low biomass density state and a high biomass density state. And they both can coexist. And depending on where you start this ball, it can go here or it can go here. So the simple intuitive landscape picture can actually capture how a system can have two stable states. So the one stable state is the low, the deep well here. And the other one is a shallower but a local minima here. So this is an example of what is called a bistability. So you can think of ecosystem stable states as a balling roll in a cup or wherever it settles in this rolling cup or rolling landscape. And then there's also this important concept of basin of attraction. So anywhere, if I drop the ball anywhere to the left of this line, we will have the ball rolling to this side. This is the basin of attraction for this low biomass density state. And this is the basin of attraction for the large biomass density state. And then this leads to the concept of resilience because there is always a possibility that this ball will switch over to other minima. So how resilient is the system is a question that's extremely important from an ecological point of view. So mathematically, how do we define this potential? It turns out that if you have a simple model where x basically is a dynamical variable, you can define x by x a simple integration of this rate function, some x 0 to x. So this function is precisely what I have plotted in the previous graph to obtain the potential landscape. So this potential landscape is not just an intuitive picture. For simple models, one can actually mathematically represent them as well. So I'm going to skip some of those. So no, clearly this landscape picture is quite useful because we can think of minima as corresponding to stable equilibrium, maxima as corresponding to unstable equilibrium. And it captures features like hysteresis, under fact that there is initial condition dependence in these systems. And also that system, many biological systems are irreversible or they take long time to reverse once they are in another state. So all this can be nicely captured from this potential landscape picture. So one can also do the following. Whatever I have been discussing so far is a deterministic picture. So however, what we can do is introduce some stochasticity to these models. So I'm calling them ad hoc stochasticity because what I have done here is I have taken a deterministic mean field model and then I have just added stochasticity, stochastic term to this. Basically here sigma v is a strength of stochasticity and eta v is a random number which is drawn from like our same distribution. And we also assume that the random numbers are uncorrelated over time. So we can incorporate these kind of simple stochasticity. Someone has to be a bit cautious while doing so. For example, you want to ensure that the biomass density is never really negative. So I'm not going to go into those kind of technical details here. So what happens when we introduce the stochasticity is we can now capture much more realistic features of dynamics. For example, here's an example of a vegetation system that is undergoing a collapse from some value of near the carrying capacity to a low value. In all more in this case, really value of zero or close to zero. So this is an example of simulation where a system has undergone a tipping from a moderate or high value of vegetation density to close to zero. On the other hand, you can also have a scenario where the system will sort of fluctuate between two stable states. There is one stable state here, other stable state here and the system can actually fluctuate between the two states depending on the nature of stochasticity. So this is where I want to clarify a few terms here. So in this diagram, this is sometimes called stability diagram. It's also called bifurcation diagram. So in this bifurcation diagram, this point where the green branch is ending or this point where the black branch is ending, those are called bifurcation points or also tipping points and critical points in the context of non-linear dynamical systems. And the transition that happened near this critical point, they are called critical transitions, abrupt transitions and if they have catastrophic consequences, you also call them catastrophic transitions. However, these transitions can also happen when you are not necessarily close to critical points. Imagine you're really, this is a critical point, this 2.6 in the specific case. However, if the system is somewhere near in the middle of this by stable region, so there is this region of by stability from values close to 1.5 to 2.6. If you are in the middle of this region, but I'm going to the stochasticity, the external stochasticity is high, even then a system can fluctuate between these two states and which means that even far from a tipping point, far from critical point, you can actually have abrupt transitions from one state to another state and these are called stochastic transitions and what is interesting is this can actually take you back and forth. It doesn't necessarily take you to one side, it can take you back and forth. So remember these classifications, of course, I will revise these. There are two types of transitions I just mentioned here, one is that of critical transitions or tipping point transitions. The other one is a stochastic transitions. So it's something that happens far from the tipping point or the critical point. So this provides, so whatever I have done so far is some very simple mathematical model which has single variable and it can sort of capture various properties of empirically observed phenomena. So we had in the empirical systems, we had observed abrupt transitions, right? We had observed hysteresis and these two simple empirically observed phenomena, they can be nicely captured with this simple model of vice stability. So that's the point of the my talk so far and then I have also introduced you to this concept of potentials and this was helpful to intuitively understand how do we think of the dynamics. So now can we go for, one purpose of mathematical is not just to reproduce empirically observed features. Of course, we do want to do that, that's the bare minimum but can we do something more? And something more in this context is can you provide, can you predict tipping points? Can you anticipate tipping points? Can you forecast tipping points? That's the question I am now going to address. So for example, in this context of bifurcation diagram, let's imagine there is a real system that exactly follows this bifurcation diagram. However, we want to know, we know that the grazing rates are increasing but I don't know if I am here or if I am here, where am I? Is there some way of knowing where is the current parameter value? Okay, that's the question that we are interested in now. I just want to check if there are any questions at the stage. So I am sort of halfway through my talk today. I have a question. Sure. So in that equation where you go to the equation slides. Sure. Here, what is the functional form of v? You always show a plot v versus c with some bistability and a shift card. So what's the mathematical functional form of v? Yeah. So basically v, I don't have that with me right now but if you set the condition for equilibrium, which would be that, you know, this logistic term becomes equal to grazing term, right? You will have a cubic equation. Oh, okay. Then from that bifurcation analysis, you got not okay. Exactly. You set this and you get a cubic equation to solve. You have, of course, v equal to 0 is 1 equilibrium. Yeah. And then you have a cubic equation to solve. Yeah. Yeah. And those are the roots of the cubic equation. Any other questions? Yeah. And this is the mathematical form you assume but is it also common in real data? That's an excellent question. So these are, you know, sort of, you know, one can think of them as somewhat like toy models. They are inspired by the terms that I have included. They are inspired by ecological processes. But, you know, depending on the ecosystem that I am thinking of, okay, the exact terms can be dramatically different. So the question is how useful are they? Okay. So my answer, so the one way to think about that is to go back to this potential picture. Any, sorry, can we think of another by stable ecological system? Let's say I don't know the equation, right? Yes. But I know that the ecosystem has by stability. As long as I can think of them in this potential well-formed. Okay. No, how do you know that your system has by stability? You are given just a time series with some abundance or land cover data. But I showed you that it does show, right, you know, it shows the abrupt transitions, it shows hysteresis. And both of these are consistent with the existence of by stability. But that's 5000 year long desert time series. Yeah, exactly. Yeah. I showed you several others. I also showed you one grassland data, right, which is much lesser like more than 50 to 60 years. And then we also there are also many, many more data sets, you know, I will show you some references towards the end. So obviously, you know, there is no one model that will capture all of those in one equation, right? So I'm using sort of, you know, what are called stylistic features and try to capture them using the models. So the idea is not to cap, you know, compare the models with data directly. But idea is to compare the predictions and the sort of, you know, with that, you know, with the what happens in data. Okay. So let me go to the next part now. Okay. So, so the question we are trying to go into address now is, you know, if somebody, you know, if you had time series data, could you somehow tell I'm far from, this is the tipping point, right? This is the bifurcation point. I'm a far from the bifurcation point or am I really close to this some way of finding signatures of that from the, let's say time series data of the type that's available in nature. Okay, that's a question we're addressing. So this is, so what you now need to do is look at these potential wells I showed you as a function of the grazing grade in the specific model I showed you. So for example, in this grazing rate, you know, we are quite far from the critical point here, this is much closer to the critical point, right? Now observe this landscape features, which I have computed from the mathematical model. This is the landscape far from the threshold point, is the landscape close to the threshold point or the bifurcation point. Okay. So the landscape here has, you know, is symmetrical, right? It also is relatively deep. In contrast, the landscape in this case, right, landscape where the ball is rolling is actually has two features. One is there is an asymmetric feature compared to this. It also is shallower. It's also shallow around the, you know, minima. So one can ask, do these features of the landscape, do they have effect on the observed dynamics of the data? So what I am now going to do is, whether these two features, the shallower landscape in this case, and also an asymmetric landscape, how do these affect the dynamics of the ecosystem? If the model was right. Okay. So obviously, we're going to assume the model is right. And we're studying within the context of these models. So again, I'm showing you this is, you know, far from the threshold. This is close to the threshold.  What happens? So if you know, now do simulations of this model, what you find is far from the threshold, you know, this, the vegetation biomass, of course, fluctuates the characteristic density around the, around the minima. And so does in this case, closer to the threshold. But, you know, one can visually compare these two time series and observe that there are notable differences. One is that in this case, the amplitude of these fluctuations are large. Right.  In some sense, there are no analogs of that in this, you know, there is an asymmetry in the time series again that you're observing here, which is a consequence of asymmetry in this, you know, potential landscape. Okay. So, and so what this tells you is there is something about the dynamics that is fundamentally different between, you know, these two values. One is that there's another point, which I forgot to mention, sorry, you know, as you are, if you are in a deeper well as compared to a shallower well, the system will take much longer to return to the equilibrium if you are in a shallower well, if there was a perturbation, takes longer to come back. And in fact, that again is quite sort of evident in this diagram. Just look at, you know, there was this perturbation here, right. And it does come back. And then observe the perturbations here, they're, you know, much more closely spaced, the return to the equilibrium value is much more faster. Okay. So what happens is because of the shallower potential, the system responds slowly to perturbations. And one can actually measure this from what is called auto correlation function of the auto correlation, which in the time series, auto correlation coefficient. And secondly, because of the shallow potential again, the system now fluctuates lot more around the equilibrium value. And because of the asymmetric potential, the system also, the time series also shows increased asymmetry. So if we measure the ACF of time series, auto, auto regression coefficient, if you measure the variance in time series, which the skewness in time series, all of them we expect will increase as system is going from, you know, far away to the closer values. So let us plot this, let me demonstrate this principle. This is a very important principle. So here what we have done in this simulation is the following. As the time is increasing, the grazing rate or an equivalent parameter is being increased. System is going towards the, you know, tipping point, the green line is a driver value. Notice the driver value itself is gradually changing. The system is responding in an abrupt way here. It suddenly collapses at around 1000 units of time. So what do we expect if the theory that I have shown you correctly is correct. What we expect is that if I were to plot auto correlation at auto correlation, regressive coefficient, that should also increase. Right? Likewise, if I were to plot the variance, that should also increase. And likewise, if I were to plot the skewness, you know, skewness is a value of symmetry. It could be positive or negative. Basically, the magnitude of skewness basically must increase before the actual collapse happens. So all of these must show these kinds of trends before an actual collapse. So therefore, the idea is if you do observe these, maybe we are approaching abrupt transitions or critical transitions. So in fact, this is an example of simulation from the models. So the auto correlation at lag 1, standard deviation and skewness, they are all showing the expected trends. And these expected trends can therefore be used as, you know, indicators that I am approaching a tipping point. So that's the sort of, you know, how we sort of use this mathematical models to make some predictions about systems that might show, that might show abrupt transitions. Okay, now here is the important thing. So although I showed you one specific model, one specific equation, and then and simulated and analyzed all of this, the mathematical theory behind this is much more general. It only relies on the fact that the threshold point in ecological systems maps on to what is called bifurcations in our models. For example, these are bifurcation points. So basically as we are going towards bifurcation points, these features are sort of universally observed. So therefore, although I have used one specific model, these trends that I am showing you in this, although for this specific model are likely to be true in a large number of cases of abrupt transitions. So that's the theoretical prediction. If a person is approaching, if an ecosystem is approaching tipping points, by measuring these simple dynamical quantities, one may be able to anticipate that you are approaching critical points or tipping points. So that brings me, there is one paper, for those who are interested, in the statistical aspects of it, which I am not going to cover at all in this paper, there is a paper in class 1 in 2012. There are also a toolbox that actually applies these theoretical principles and provides you a statistical estimator of how good can you actually measure these in real data sets. So I'm not going to go into details of this statistical aspects. So now I have basically covered the mathematical theory of tipping points and how we can use them to offer early warning signals. So now the last part of my talk will be, are they really true in real world data? So let's ask the following, how do we ask this question? How do we look for empirical evidence? Imagine if we had a laboratory system. If we can somehow subject a system to tipping point in a laboratory and then push that beyond tipping point, do you actually find those strengths in the autocorrelation values? Do you actually find strengths in increasing variability? Do you find this warning signal before the transition happened in your laboratory system? That's one way to ask this question. Of course, in the, we eventually want to apply these to field systems. So, for example, if you take data, long-term time series data, and if you look at that they have actually undergone the transition, did this system actually exhibit trends that we have predicted from mathematical models? So I'm going to address these two questions from, I'm going to show you what people have found. Okay, so this is a work by John Rake and others. So what they did was they did simple experiments where they, when sort of grew, you know, populations of Daphneam in the laboratories. And what they also did was they subjected these Daphneam populations to increasing stress over time.  Okay, and what they do is they basically study them under also a controlled condition where they don't do it, and a condition where they are deteriorating their food supply over time. And what they do is they study for early warning signals in time series, variance, queerness, and correlations in time. And what they do is because they had an empirical system, they were also able to estimate what is the tipping point. And in their experiments, they say, you know, it's a year long experiment, they subject these populations to increasing stress over time over a period of one year. And they estimate that they reach tipping point in their data on day 300. And what they then do is they analyze early warning signals. So what I'm showing you here is coefficient of variation and queerness. What they find, interestingly, this is, you know, the dotted line here represents when the tipping point in their experiment actually happened, the small gray line in the bottom here, that's the control data. And what you're the one subject to stress, they are showing these, you know, much bigger trends. So what they basically show is that they were able to find signals of the upcoming tipping point in their experiments, almost 100 days ahead of actual tipping point. And in fact, this effect was most visible among the four indicators in coefficient of variation and queerness. And of course, since then there have been more experimental verifications of these ideals. For example, in East population, this is an example of East population density, you know, in sort of stable conditions. And this is the dilution factor, again, you know, mimicking stress. And what's remarkable about this diagram is, you know, it looks almost like the mathematical model I showed you, right? It almost shows, it looks like the bifurcation diagram of the mathematical model I showed you. A stable fixed point, unstable fixed point, and another stable fixed point, which corresponds to extinction. And what they do in this, again, they subject these populations to stress. What they find is that the coefficient of variation, standard deviation, autocorrelation, they all increase. However, in this specific experiments, they did not find strong evidence for increasing queerness. So this is then there have been more experiments. And, you know, other microcosm experiments, I won't be able to show you all of them. But the point is that, you know, certain interesting features and the mathematical predictions are actually testable using experiments in this case. Okay. So in the last part of the talk, what I'm going to do is to find empirical evidence for these early warning signals, which are predictions of mathematical models in, you know, in the field, in the data from field. So I spoke about, I mentioned this ecosystem in China, right, where there was a grassland restoration. Okay. So what they found, what they found was that there was this low grass cover for about four decades. And then there is a high grass cover, which is the current state. And, you know, intermediately, there were these strong fluctuations that preceded before the actual transition happened. Okay. So what you find is that if you look at the histogram of this time series, it shows this nice bimodality. And typically, the bimodality is a strong signature of bistability in the underlying system. So this bimodality in the data here shows that maybe the system is indeed quite stable. And what we were also able to show in this paper was that the dynamics of the system sort of dramatically changes around year 40, using something called a change point method. Okay. So what we then did was, okay, now we know that this system has undergone this abrupt change in the grass cover from a low grass cover to high grass cover. Did it exhibit early warning signals that the mathematical theory predicts? So what we did was we sort of drew this line at which the transition happens when we were only looking at data before that, before this year 40. And here we are, we are calculating autocorrelation at lag 1 and we're also calculating spectral density ratio. So now don't bother the spectral density ratio because I haven't described that to you. So what we find is that interestingly, there is no evidence for this signature of critical slowing down in this data set. So one of the main predictions of the mathematical models, we did not find that to be true in this case, in this data set. However, if you look at variance standard deviation, that shows a very clear increasing trend. Likewise, if you look at skewness, we again find a very clear increasing trend. So in other words, there is a very strong evidence for rising variability. But you know, if you remember the mathematical theory predicts that all of these must actually increase, not only this. Okay. But our data shows we rise only one of them to be true. So how do we really reconcile this? How do we explain the fact that there was no critical slowing down before the abrupt transition? However, there was rising variability. Okay. How do we explain this? Okay. So you know, this is where I would like to remind the slide I showed you long back, which is distinction between critical transitions and stochastic transitions. So critical transitions are those which happen at this tipping point or the bifurcation point. So your system is very close to the tipping point and then a small amount of noise pushes it down. On the other hand, you can have stochastic transition where the system is actually far from the tipping point, but a large amount of noise can kick it to the other stage. So now what we did was we looked at all the signatures for critical transition as well as stochastic transition. This is critical transition, the sort of standard theory which I have presented so far. All of the four sort of show these signatures. However, if you have a stochastic transition where a large constant noise pushes system from one stage to the other, you don't find any of the signatures. Excuse me. However, there's a second type of stochastic transition where the strength of noise is actually increasing. If you consider this scenario, you do not find these two signatures. These two signatures measure slowing down of the ecosystem, slow response of ecosystems. However, you do find that the variance and skewness increases. And in fact, this precisely what we found in our data. Our data does not correspond to critical transition not to constant noise stochastic transition, but to the increasing noise stochastic transition. So therefore what we conclude is that the dryland transition that we analyzed is a case of stochastic transition where there was no critical slowing down, but there was a rising variability. One slide that I have skipped because of time is that this rising, this sort of counterintuitive pattern is driven by stochastic rainfall. The rainfall is increasingly becoming variable over time. That seems to explain this phenomenon. I don't know how much time do I have now. I have about seven minutes left. Let me now present one more study we did to look at these early warning signals. So can we apply these tools to anticipate financial market crashes? So our motivation for this came from many papers that look at critical transitions. So for example, if you read this famous and classic paper by Sheffar et al. in nature 2009, complex dynamical systems ranging from ecosystems to financial markets and climate have tipping points at which a sudden shift to contrasting dynamical regime maker. So basically financial systems are sort of often quoted as examples of tipping points, but is it therefore then are the predictions of tipping point models are they valid? That's what we try to understand. Specifically, we asked two questions. Do we find evidence for critical slowing down, which is basically increasing in the autocorrelation as markets approaches a crash? Do these markets exhibit increased variability prior to the crash? What we did was we took a data from a whole range of stock market indices. This is an example of one specific stock market is a famous Dow Jones index. What we know historically is that there are four well studied crashes, 1927, 1987, 1991, and then 2008. So what we did was we took each of these windows and we analyzed all the four metrics. So here also what you find is that the autocorrelation at lag 1, which is a measure of critical slowing down has no clear patterns. In fact, as we are approaching the crash, it actually suddenly reduces. Therefore, there is no critical slowing down in this financial markets. However, if you look at the variance in time series, that shows a very strong and clear trend. So there is a rising variability prior to stock market crash downs. Okay, this is not one market and one index we have stayed. We stayed this for a whole range of markets and whole range of crashes in 1929, 87, 2000, 2008. For all of them, we find that the autocorrelation at lag 1, the critical slowing down doesn't show expected trends, but the variance always shows the expected trends. I don't need to do any sort of statistics for this. You can just see by eye that the fluctuations in the stock market indices are dramatically increasing prior to crashes. And this is true not only for Dow Jones, we also found this for S&P, Aztec and other markets as well. Again, how do we explain this lack of critical slowing down, but with rising variability? Again, I want to remind you about critical and stochastic transitions. In stochastic transitions, we don't find autocorrelation at lag 1 increasing, but we do find variance. So most likely our data corresponds to stochastic transition. The stock market data might correspond to stochastically driven transitions rather than the transition that happened near the bifurcation points. However, I want to highlight that there were also many, many instances of false positives. If one were to rely only on variance, there were also a large number of false positives that occur. Okay, so therefore, one can't just rely on variance as a predictor. So it's only a tool that may provide some signal, but it's not a predictive tool. So therefore, we conclude even in the case of financial markets that they are not critical transitions, they don't happen near a bifurcation point. Their features are better explained by stochastic transitions. Okay, so with that sort of, I want to summarize the first trace talk, which is that there are many, many real world examples of abrupt transitions. And we mathematically model them as tipping points or bifurcation points. And it's important to understand that these models can capture critical transitions which happen near critical points, and also stochastic transitions that are strongly driven by stochastic forces. Okay, and these critical transitions, even stochastic transitions can have early warning signals. And the two classic ones are that of critical slowing down, which is rising auto-correlation and increasing fluctuations measured by variance, queerness and other metrics. And we have, there is no quite a bit of empirical evidence that these metrics may indeed work in real systems. And our specific analysis showed that stochastic transitions are better, you know, many ecosystems and including financial markets are probably thought of as stochastic transitions, because there is a lot of stochasticity in these systems. They are not slowly going towards bifurcation points, but probably they are strongly driven by many stochastic factors. And this also provides an interesting case that, you know, mathematical model models can not only explain and provide insights, they can also provide tools for the assessment of ecosystem resilience. For example, observing these metrics could sort of indicate to us how close one is to, you know, are we approaching low resilience states or are we approaching potential catastrophic transitions. That's a summary of my talk so far. Okay, so I will take more questions now. And in the next talk on Tuesday, I plan to discuss more about the spatial patterns. And before I forget, and you know, before I take on question, I want to thank and acknowledge my collaborators. I start working with Professor Jayaprakash on these problems. And I worked with Nikunj and Srinivas Bhagavan on the financial market problem and chaining on the in the grassland data. So thank you very much. So I'm happy to take questions now. Thank you. So we have a couple of questions from the chat. So how do I open the chat now? And should I close my screen sharing by the way? Yeah. Oh, I can open the chat. We can also cut the chat down. Okay. So one question is the asymmetry and shallowness correlated? Yes. So yes and no. So they're correlated in the sense that they happen simultaneously. Okay. Now, but they're not happening by the same causal factor. So if I were to explain this mathematically, so, okay, I think I have a slide that may even explain this sort of mathematically. Okay. So if I look at if you think of, you know, classic and simple models of catastrophic transitions. So they have this, you know, linear term. Okay, let me see if I can switch on the. Oh, okay. Can you see this? Okay. Okay. In this equation, I have, you know, this bifurcation model with noise. This is linear term r times u. And there is this cubic term, right? So the shallowness is can be entirely explained by the linear term. But the asymmetry requires you to use this cubic term. So you cannot obtain asymmetry without having a cubic term. And so abrupt transitions in complex systems are typically modeled by saddle node bifurcations, right? And saddle node bifurcation does require you to have this cubic term. So, yeah, so they are related, but they are correlated in the sense that they happen simultaneously, but the causal factors are different. And the second question I have is, you know, is it somehow related to curvature? Absolutely yes. So the shallowness is absolutely basically equivalent to curvature. So shallowness means reduced curvature. And that is what is causing reduced return rate to equilibrium. That is also what causes increased fluctuations. So they're all same basically, shallowness and curvature are basically and the third question is my similar question are some of the drivers more often correlated are they vary largely case by case before it's important to consider all. So yeah, I mean, I don't have a good answer for this question because this question, one has to look at a specific system and try to understand what drives or how are the drivers placed. For example, if you were to think of a vegetation system, typically rainfall and fire are very, very important drivers, right? So we need to know the mechanics of the system to sort of, sort of, you know, address this question on a case by case basis. And in all of the model analysis I showed you, they were all assuming that a single driver is changing. Single driver is changing slowly towards a bifurcation point or the stochasticity in the single driver is what is really causing it. Okay, the next question. Stock market indices are often fact-tailed. This leads to divergence of moments. Is it then meaningful to analyze their trends? Yeah, this is a good question.  So I will not be able to provide a very, very good answer to this question. So my own interest was, you know, more of academic, specific academic question since it is often used as an example of tipping points, does it, and we have really, very, really good high-resolution long-term time series data. Does it show critical slurring down? Does it show the simple feature? Let's ignore the moments part. I agree. You know, moments are complicated. I fully agree. You know, even if you ignore the moments part that I showed you, can we calculate the auto correlation function and the properties of it, right? So I think that can be done without being worried about fact-tailed distribution parts. And we do not find evidence for the mathematical prediction that there is a critical slurring down. And of course, you know, when it comes to thinking a bit more about the fact-tailedness, I think that to think of, you know, mean and variance rather higher moments, you know, to really observe divergence, you need a, you technically need infinitely large data. But we are obviously analyzing data for a finite window. So I think in that limit, it might still be reasonable to calculate that. Robert too, do drastic transitions due to noise happen for specific values of noise? Is it a feature of stochasticism? So I think it is related to, you know, you need a, so basically, you know, if you're the noise term in your model as a Gaussian noise term, right? Technically, the smallest noise can also cause, you know, transitions between two stable states, any small amount of noise. But, you know, you may just have to wait so long that it's meaningless now, right? But if you now say, okay, you know, I want to observe transitions within a given time scale. And I'm going to be interested only in those timescales. Yes, you do need the minimum amount of noise before a transition can happen. And in fact, that's the point that, in fact, my very first paper in my PhD thesis, I try to address that, that specific question fairly, this slide that I'm showing right now. So we basically, we, in fact, we showed that assuming that the noise is bounded, you need to have a minimum amount of noise for you to sort of induce transitions. And you need a minimum amount of noise to also to fluctuate back and forth. And I think it is related to Catholicism. You're right about that too. Okay, there's one more question. Is it possible, at least in principle, to amenable to account for feedback mechanisms? For example, rainfall sustains its own raining rate. Driver is not an external driver. Yeah. So I'm sorry that, you know, in this talk today, I have absolutely not paid attention to mechanisms in some sense, right? I just used a model that showed the features I'm interested in. So what, so what I'm hoping to do in the next one, where I will discuss spatial models is that positive feedback mechanism is really important. And if you have weak positive mechanism, weak positive feedback, you do not have abrupt transitions. Only when you have a strong positive feedback is when you will actually observe, you know, abrupt transitions. And in fact, it is precisely because of the positive feedbacks that, you know, two states, you know, in this slide, right, you know, there are two states, right, you know, the green high vegetation state and the black low vegetation state, you know, so the green line and the black lines, they coexist because both of these forest states are stabilized by their own positive feedback mechanisms. And therefore, you know, it could be, for example, rainfall under rainforest, it could, for example, be the, you know, it could even be much more local scale. For example, the presence of trees, a patch of trees will enhance the water infiltration that will in turn help the tree to go better, better, that will in turn help the local establishment of seeds. And therefore, the entire patch is sustained. On the other extreme, if there is nothing to start with, it's very hard to germinate a new seed because the water doesn't sustain there for long enough for a seedling to arise. So, so the both the alternative stable states are often maintained by these kind of feedback mechanisms. That's a great question. Thank you. Okay, I think we, we had a rich discussion session and thank you again, Bishwesha for a nice lecture. Now we have time for a short break and we'll be back in about six to seven minutes for the next lecture by Andrea Rinovo. Thank you everyone. I'll see you again next week. Ciao. Ciao Antonio. Antonio, non occorre che io faccio altre prove, no? Pritanto, son qui... No, they, you know, the... Hai parola, sono pronto qua? Non bo? Sono... Perfetto. A dopo. A dopo. Ti prenderò. Okay, I think we are ready to start over again and it's a pleasure to have once more Andrea Rinaldo from EPFL for his third final lecture. Please, Andrea. Thank you so much, Antonio, and here let me, let me share my screen and I've had today a few issues with the connection with Wi-Fi, I hope it doesn't happen, but because I, you know, I happen to have a few kids back home so everybody's using the van, but anyways should be okay. So here's lecture three and let me close this up, yep, of the series that we had and let me see where we are. Remember what you may remember where we had discussed like in the first class in the context of neutral meta-community or meta-population model which essentially is something in which if you look now at the right plot only compared to be so-called the pure lattice model of interactions, you have a sequence of interactions in the sense that the dynamics proceeds by canceling at random like a color if you want or an individual or a species if you take a meta-community model, it won't change the main result in a particular place randomly chosen and you keep going with that and you replace it with either an non-existing color slash species slash individual with a certain a probability of a diversification ratio whatever you're going to call it or you replace it with some nearest neighbor or some neighbor which was chosen respect to a kernel of the domain of influence for which you choose the most numerous or in a mean field sense whether you choose it from anywhere else based simply on abundance and that of course entails neutrality in the sense that the preference is simply due to sheer numbers, sheer abundance and that's the essence of a neutrality assumption and the patterns that we've seen in there are quite different depending on a sole modification which is in this case the directional dispersal which is embedded in the network structure, the tree-like structure if you please or the open to any, I'm sorry, nearest direction from a sorry stuff like so let's see how we can exploit this is Ignacio Rodriguez Turbe, emeritus professor at Princeton currently now at Texas A&M after like a couple of decades in Princeton in which he said well why don't we assume that each branch of a network structure here becomes like a meta-population which is called a directly tributary area that is a local community within a meta-community. One of the points that I made in the first class was that the result that the directional dispersal which is embedded in a network structure for interaction that is that defines your nearest neighbors or a way neighbors in a selective manner is rather insensitive of whether you're talking about an individual or a meta-community based approach the result still stands and what later on we have done like field verification of the same result studies theoretical or empirical on migration fronts for instance that feel the structure of a directional dispersal or laboratory in fact experiments we learned in my lab in which was nothing neutral there was a living community the result still stands. So the idea is here is to assume that the large river system this elementary the unit on which you have reactions of physical biological chemical nature in this case biological could be a directly tributary area which is a local community carved within the community like a very each scale and we thought of doing so we referenced to something which is quite important we took on the the whole archive of local species diversity of fish populations in Mississippi Missouri river system and LSR means local species richness however defined essentially can count the observed number of species which you have. So in a DTA in one of the notes of the links in which you can partition I'll be showing you now you can extract that objectively and manipulated remotely acquire and objectively manipulate the structure of the Mississippi in fact acronym means annual average runoff production which is essentially runoff I mean the total volume of runoff that passes through any particular cross section that is the hydrology. Now it doesn't take an expert statistician but simply a not a particularly trained eye to see that there is a correlation between how much runoff you have in a particular patient in different directly tributary area that is the hydrology the hydrology control and the local species diversity but to formalize this what you can do we can extract objectively even humongously large networks to detail which is actually in fact one of the reasons why in fact that over the debate of critical self-organization that is why in fact certain recurrent characters embedded in a power law distribution of some aggregation structure of a catchment is the same regardless of a climate vegetation exposed lithology the scale etc there's something truly remarkable already the networks that generated files like or or datasets like this one in which you can characterize a structure from the scale of one meter or less to the scale of thousands of kilometers and in terms of area even more so that can be done I hinted that that briefly in the first class are not touching on it but I assume that this is something which when you can trust me we've been working for like 20 years in characterizing those network shapes like these ones so let me see how essentially the model proceed it's it's well I have a few notes here about them it's not going to be particularly but the idea is that it's likely experiment you've seen the neutral meta community or meta population experiment that I showed you in the animation I showed before that I showed also in lecture one so these the assumptions in this case is every DTA that is every unit is essentially saturated at its capacity that is no resource available to fish is assumed to be left unexploited now of course you're talking about some sort of an upper limit to the to the fish diversity but that's also quite remarkable how close I will show you this will be reproduced so the model dynamics procedures in the other case which you have seen that is at each time step you randomly select a fish unit selected for all the fish units in the whole system and you assume it to die and the resources that previously sustained that unit are freed and available after grabs for a new fish unit so we have certain probability which we term again the diversification ratio the rate the new unit the fish unit will will be a new species it will probably be one minus new in fact it's going to be one of the existing species that colonizes the spot and the cost of diversification ratio rate in this case could represent external introduction on non-native species we have seen how foreign invasions are so important ecologically for a variety of ecosystems or it could be immigration or re-immigration in fact of a new species from outside region you've seen the case of the example if you may recall from lecture two when we studied the breeding birds of uh of North America as a or the cancerous prairie uh uh species or bassius species in fact that um in fact the the concept persistence time local persistent local extinction has to be seen in the context of the geographical uh area which this is done so um uh with the probability one minus new the new unit can belong in fact to the to the species to be uh to be etc and the idea is that in this case you don't uh touch only nearest neighbors but essentially you characterize a probability pji ij i'm sorry that the empty unit in the i-th uh dta in the i-th node the directly inhibitory area whatever you're not calling is colonized by a species which is elsewhere in the j dta's that appear in the system through a kernel a dispersal kernel that you're having this says which uh specifies and measures the range of species colonization now what is important here is a non-neutral effect um which is uh however not dictated by uh by uh a calibration not dictated by uh uh biological dynamics but essentially dictated by hydrological controls because it's geomorphological so you assume that the habitat capacity determining the resource in the place is the fluvial habitat capacity which is established on the scale of the basis of a scaling geomorphic relations we had hinted at that characterized the fact that um you can decide based on certain metric properties which are remarkably available enough producing area of any river worldwide and are dictated by the aggregation structures essentially the habitat capacity in the place is essentially dictated by how much area you had behind your back how many nodes do you collect through the structure of a network again which is a given now what is interesting that the dispersal kernel has certain features which I shall not discuss but essentially what we have I want to study the particular back-to-back exponential which has a tradition in ecology one or another in fact we tested several of them um they were asked by the nature of yours in fact to uh to run a comparative analysis and we apparently convinced them because we got published but the idea is that for instance if you assume that two units two nodes to DTAs are uh ij what is the distance that separates them now that's interesting for colonizers because if you assume the colonizer are strong fish units right um the path the path that connects you uh i to j could be partly downstream and partly upstream or could be strictly downstream or strictly upstream depending on ij now the question is you may bias the path because if you are a weak juvenile fish for instance a small one you may be way more affected by the velocity that is the drift which is embedded in the stream flow direction that is the oriented nature of the graph then you could be for a strong adult big fish right so either way then you may bias and weigh in fact the downstream direction the upstream direction distances which are in the system which makes it reasonable and this is a tunable parameter in a sense but in a neutral case we kill it and we assume that all species are equally important at the at the at the per capita level so briefly as a result of what you're seeing here is in the frequency distribution of local species richness by letting the model run to stationary state now um it may be uh you may like it you may dislike it etc but for us it was totally remarkable how simply the nature of a connected uh of a connected system and the habitat size which is produced by scaling relations being embedded in geomorphological laws it's the aggregation structure um can allow us to reproduce wonderfully well um how the alpha diversity the distance to outlet i'm sorry this is i made a mistake i was anticipating what i explained here in fact so this is all the frequency distribution this is the how the alpha diversity unfolds from the outlet to the upstream distances with all your differences which you have in a certain place for instance um this is blown out because you may have the the data telling you that some freshwater tolerating uh coastal fish species in fact could or human disturbance in fact or or pollution for that matter um alter what you would have otherwise a distribution that is what the data are showing you that near original new links and the same applies to the same thing at the same time which is also remarkable how the frequency distribution with respect to the distance to be out the frequency of local species with respect to thing which is you count essentially the number of species that equally distance from the outlet you got some sort of a range which is reproduced without any tuning by the model signifying once more one of the main tenets of my classes that there is something inherent in the directional dispersal implied by the network structure which is the substrate for ecological interactions which is given the system granting the system uh reliability and predictability what is interesting also is that if you run the same exercise without changing the habitat capacity per every node embedded as proportional through um predicted geomorphological laws by the uh structure the aggregation structure which is essentially dictated by the total contributing area at any point deciding how big is your river and then what you see and you screw up completely in fact the uh the exercise and you see the hydrology controls embedded in any neutral model which is the simplest possible zero order approximation there's no description of an of the properties on which fish biologists spending lifetimes of scientific work and are mostly explained by the hierarchical size structure of the fluvial network and it's embedded topology which is also reinforcing what we have seen before now things become slightly more complicated if you go into not only starting local species richness by the correlation structure it is a beta diversity that is um what is the probability that the existence of a species in one place is uh matched by the probability of existence of a safe species at a certain distance distance being uh measured in so-called chemical distance that is along the network structure so this can be done you can generate equi probability maps which is the ratio between the number of common species that you have and the species in the central dta to see how the system behaves in fact so this um reinforces the uh the main tenet that I've been hammering on for quite a few times and I am now ready to um move on following uh what Marino Gatto has told you about the evolution of our thinking about spatially explicit epidemiology uh by the original ideas that motivated us to go to jump into covid 19 studies through the same technological tool in as much as some tail some some uh small factories that you had in the milan area we used to make um high fashion dresses converted the alliance of production into massed productions during the covid 19 it was a relatively jump easy jump for us but I'll spell you how in fact this uh thinking this way of thinking and this line of thinking that brought to the book um I talked to you about uh allowed us to move on to jump from fish biodiversity straightforwardly to the study of river networks as ecological corridors for waterborne disease let me see why now um here's a river network and let's assume that those dots are uh human settlements so essentially what we're saying is that um what if we thought notes are human communities where they are in fact with their population on their size etc where disease can spread and um you have a demography of a population of a of a demography of a disease based on the demographic evolution of susceptible individuals in fact individuals possibly recovered and possibly via the as you've seen in the case of shisto quite importantly um through the example that um uh that Marino has pointed out or which I briefly returning by mentioning his mentor because it's important for one of the tenants of my work that is you may in fact couple these with control variables which pertain again water controlled but that pertain the ecology of a disease that is if you have for instance intermediate obligatory um intermediate hosts for the development of a disease so um why this is interesting the example of a shisto I will just briefly show the main results that Marino showed you the help means and you see why this is important right and but the motivator I'm giving it um it's more um it's it's somewhat important that's where we carried out the field work that we carried out with my lab if I did Burkina Faso we had like a 20 year 20 year uh longer experience in um a collaboration in in cooperation with development so um it what is incredible in this area is that in sub-Saharan Africa you have something like in the debilitating disease which is not killing anyone so it's a neglected tropical disease you have like 15 million uh disability adjusted life years uh that do take into account and what you have in the particular regions you have different types of shisto as Marino has shown I'm just going through briefly because and that's what we had uh in the camp that we have in Burkina Faso we deployed and one graduate student in fact carried out his entire PhD thesis on that why this is interesting again it's a complex life cycle and it is interesting for us the existence the need to take into account hydrology controls this is the part which indeed pertains to how form and function of the river network operates because why because if you have like this marriage this fertilization of the successive stages of the of the uh of the disease need to hatch the eggs the best place in which they they are generated within the infected host and they have a worm so they the fat worms that generate the sarcaria or myrassidia effect in live in the and they excrete the three thesis they get into a water environment in which I have to they have to be infecting snails in a given time now that depends of course on shear stresses depends on the flow velocity depends on the habitat size it depends on whether the habitat suitability for the intermediate host is given so something in which as you've seen with marino uh we'll be back into the system so that's what you have and what you have seen and that's the point I'm making um the point I want to make here is that um pricing the planet as a environmental economist pointed out that is um if you don't account for a depreciation of natural capital um that means that um essentially the ecosystem services that are carried out that you cannot quantify precisely are worth zero so pricing the planet means that you have to give the monetary value to the services you may lose as an alternative like in this case making way for a commercial center by destroying a mangrove swamp um that means that in the in gross domestic product indicator GDP like indicators or well-being you will see the advantage in the following year of the of the commercial center of the benefits of it but you don't see the loss which is associated with the ecosystem services you lose from flood protection to uh to uh carmel sequestration to fish nursery areas and the likes so the idea and that's part of the sculptor's main point is that indicators that do not account for the depreciation of natural capital uh put development thinking uh stacked against nature in a sense uh against environmental thinking and the very idea of a misinterpreted kutznet's curve that is that if your GDP goes up uh down goes the inequalities in their place is essentially false as Thomas Piketty in fact has shown quite clearly in his wonderful book uh the capital in the 21st century what happens is that in reality indicators that omit the depreciation of natural capital are totally unsuitable for describing the wealth of nations and here is my point meta studies um that and i'm i'm really sure that marino hasn't spoken about it and then to use my discussion in fact showed that there is a clear relationship between the expansion of irrigation canals and some of the 15 000 small dams so we have seen i took this picture near our field site in Burkina Faso in fact uh the uh the construction of irrigation channels that were possible because of a 15 000 world bank funded uh small dams that uh littered Burkina Faso in fact had the consequences so the water resources developments came of a largely improved GDP over Burkina Faso but in a humongously large expansion of a habitat suitable for the intermediate host of the disease and thus the prevalence of the disease so the idea is that can we put a price tag on the learning impairing disability brought in by a disease of this kind complicated complicated and our permanent our inability to predict for instance the number the increased prevalence um in in unless you have significant and reasonable models of the expansion of the disease then um you know as a matter of fact they won't be able to put the price tag on and this is the same place and this is a picture so they took in the same place how can you keep uh little guys away from the water when you have ephemeral ponds generated by the system mind you that in this case as you're very well known because of the wonderful lectures that truly marino has put forth on the subject that the the those largely penetrate the skin uh in matter of second our student there was here only once put his hands in the water dropping the gloves when he dropped scissors just he totally just picked up in a second the scissors but it was enough to create to to get the system why this is important and related to what you have seen um in the previous slides for the fish diversity it's a picture you're seeing from marino because the set of equations and I'm not insisting on that is something which you have coupled um coupled uh uh odys like in this case is the mean word burden that you have like in the system here the prevalence of infection in the intermediate host in this case is why and these are the the essentially the densities of the sarcaria and mirasidia in node i of a of a dta that is of a single node in which you can characterize the system i'm not pretending that you follow the system but you realize that in here you have a number of extensions which are quite important they pertain for instance human mobility that is if you have a guy that migrates to go to a place to cultivate the feeding that uh uh because of the expansion of the irrigation network human mobility does affect in fact carrying away an exposure and the concentration of sarcaria that generates the burden of disease in an individual that's how the system expands so in a sense you realize that you're moving the study of diseases onto a plane which is completely different and the set of system is a classical system through which used to uh engineering environmental engineering tools and i want uh build on that in particular nor or what you have seen uh with marino about how you characterize the stability of the system and uh possible ideas you have on how to curve again this eigenvector analysis you're seeing with marino is telling you essentially how you can actually generate the patterns of disease and what i really like is the idea is that you cannot make discussion what happens for instance if at random it's an exercise we ran you remove 10 percent of a small dense thereby reducing the distance the mean distance to the nearest water body which is arguably the most important factor of completely geomorphological origins that generates the exposure so and that's the experiments we had in the place and i'm not building any further or uh other diseases that can be treated in this manner what i'll be concentrating here i'm getting back to the first slide in the last 20 minutes of my of my lecture and then i be delighted to answer your questions um you remember the little guy here on the banks of a magna river uh where we did field work on chronic effects cholera i'll be talking about epidemic cholera in a minute we're trying to convince me that it's impossible the mighty waters of the of a magna were the cause of cholera which originated in that region in fact evolutionary um and then from there irradiated worldwide in several ways of pandemics but most significantly 200 meters downstream of the largest derail disease hospital in the world in Bangladesh now um the point i'm making and i'll be carrying out to the end is that the inherent predictability that you grant the system by using directional dispersal embedded in the known a priori and calculated and treated objectively offline remotely acquired and objectively manipulated offline structure of river network grants an unprecedented predictability to disease of this kind and in particular this became absolutely vital when we uh uh started we were just working on that we had published the first spatial explicit model of um epidemic cholera uh in with reference to the outbreak that was in quazulu natal in south africa based on data that were uh uh collected much afterwards in hindsight but what happened is that on october uh uh in the week preceding october 10 2010 all of a sudden in a country where it was cholera free for more than 200 years you've got an outbreak which started propagating downstream the arctic bonita river in the heart of in the heart of the haitian island the island of haiti the part of the left not the dominica republic which is cutting half good example that jarred diamond um in fact in his collapse uh wonderful book was making an example of how in fact the umprobo scene and the bad management of resources uh explains um the uh uh uh not simply environmental factors determine in fact the fate of societies now what's interesting you see the number of cases jumped all of a sudden from day zero to the from 50 to 100 to 200 in places which are small places indeed and right downstream of a un camp of peacekeeping curves what is not only ironic sad and it's it's really uh uh affecting me uh very much is the fact that why we were peacekeeping troops in haiti because a few months earlier haiti the poorest country in the world had been struck by an earthquake that killed 300 000 it destroyed the little infrastructure where it was there uh uh sewer systems were non-existent and roads were destroyed people died um a civil uh infrastructure was demolished it was a it sits on one of those plate tectonics on which earthquakes can be particularly devastating on top of that we planted the disease because it was shown when it was mapped the genome that it was a nepalese strain of cholera when it is endemic brought in by asymptomatic uh peacekeeping forces anyways that was a fantastic exercise in a sense because in a completely naive population as that's a term that you use in these cases um they end um uh but is uh or you can assume safely that because uh no sign of cholera was there for almost 200 years but the entire population was susceptible to the disease and what happens is that then you had thousands of deaths you had the mortality initial mortality we was totally remarkable because there were roadblocks to treat people transported by poor means like uh on the shoulders of a of a younger in fact to be treated to center the roadblock to make it and say that after a year like eight percent of the population of the million people was affected and this picture I took into an hospital in leogand and you see what was essentially the treatment was even I have to say among the sanctity that I've seen in the Medecin Saint-Francaire hospitals organizing the haste in the place or the Cuban brigade that took up the north of the country to assist but essentially put people on the stretcher you cut a hole in the thing and you collect the stools like six times a day and what is totally remarkable that um you survive cholera easily you only have hydration bags to which you should be attached now let me show you the evolution of the daily cases in Haiti for about a year and a half then I think it comes blurred afterwards but that's quite interesting so you have an evolution of the daily cases in behalf of the hispaniola island that is a part of Haiti and I didn't put the data which came later on in fact for the two islands that still belong to the same place etc so this is how the disease in terms of simply reported number of cases with all the inherent errors of the Medecin assistant and that you had in there this is the city of Port-au-Prince here is a long scale of a number of cases if I'm asking you what do you see here well you see the rivers so you even by seeing the most gross indicator number of reported cases what you see in the place is that the avenues of the riverways where the pathogen in fact survives in the environment in the open waters in fact is what generates the system etc so essentially you can't have something in which you essentially can calculate the rate of change of new cases of cholera in every single place that that you have but you have to account a spatially explicit system we put settlements where they are connectors where they are and the likes not only that if you take the red curve is what happened in the first waves followed like in covid but for completely different reasons a second wave which is clearly related to factors like the tropical rainfall that you have in the place why well the easiest is not simply an overflow of sewers but simply the washout of open air defecation sites which you have to take into account and the fact that the freshly shaped bacteria the bacterium in fact like a single infected individuals expels and through the school through feces like a hundred times more bacteria that in concentration which are also magnitude larger than any any possible survival in the environment so it is the human human host in fact the main reason of a propagation whether symptomatic or asymptomatic whether he moves or not if you have a susceptible person moving on to the wrong place drinking the wrong water and getting back he brings back the disease it happened to me when I was in the market of our tip on it well I'll tell you when I show you a picture later on so the tools of the trade now is knowing ahead of time where settlements are where patterns of rainfall evolve and how the disease can be predicted under these conditions putting cities in human settlements where they are connected by the waterways as we can see them directly so the idea that in a system like that you have two different networks a network which you have like pathogens connecting nodes if they are downstream of a river system or and that's a key place you can have connections among nodes human communities in which with population each population H sub i this is node i in which the disease can diffuse and grow connected by a multiplex network of a different time of a different kind in this case human mobility as you now very well know from marino gato's lecture on covid 19 the spreader the mean mechanism whatever its shape it is and but we had seen even in my set of lectures what happens when we consider from a zebra mussel invasion of a mccp Missouri river system you see that you saw that at times unconnected flare-ups of of those development of those clusters of zebra mussels were generated away from the main backbone of a hydrodynamically generated dispersion why because of the ballast water in which veligar survived where we're taken away and and tucked to different place maybe hundreds of kilometers away from the same place the mechanism of generating the system of this kind so the tools of the trade in this case are the tools of the trade of geosciences of digital information systems or geographic information systems if you want that is we can and we could do it remotely when we predicted the evolution of the of the epidemics of cholera in the place which i'll be showing you in a minute because the digital terrain map from which you extract the river network as i hinted at in my introductory class is something we should do it's a standard exercise that that master students do where we are you can have pixel-based estimate of population density you can have modeling of human mobility which is something which requires some thought and some care in fact generated maybe simplified at times but i'll tell you what is the capability of attracting places like on the main point like all the plants that you have in this system and the set the tools of the trade i mean they different every time but they that's why i showed you before the ones that marino have shown you for schistos so essentially the state variables are susceptibles at node i at time t infected and node i at time t and the bacterial concentration in the reservoir of the i-th community evolved it because of the different factors which is the mortality the survival of the vibrio in the environment with a certain mortality rate of seeing really how long you get survived or transport in a certain proportion coming from connections that are hydraulic and hydrologic connections that depends on the various sizes of the of the water reservoir the local water reservoir which is important because essentially you can assume that one stool a single infected person has a certain probability distribution in terms of absolute number of bacteria shed which is again six orders of magnitude more than you have for the concentration of bacteria in the free living waters and that is the infection thing that generates the P you see P the infection per unit infected person which is here you have the force of infection depends on the local infections plus the of infections connected by a mobility matrix this is a mobility factor which you have in the system so of the i persons that live in a community i sub i infected person whether symptomatic or asymptomatic stay there and pollute the water or you may have persons that because of mobility and its matrix of fluxes generates the infection shed into the place to which you could possibly add like you have it here the rainfall runoff production of vibrios generated by the system again i cannot pretend to explain the details but you see how this is done for instance and you have seen in introduction to the disease ecology that marino put together in this case we assume the timescale of our prediction is one in which recovered persons i mean i put back into the susceptible compartment every time which is one of the role of the order of two to three years and this is the force of infection depends on the number of factors my scope here is not to it's get you curious about the strength but you see that the structure is exactly the same of the shisto is exactly the same of a msccp misery by a diversity model and it's exactly the same ones we have seen before so let me show you how the model works if you assume a piano network very important because if you have like in every node you have a population of the same capacity all of them prone to have the diffusion of the cholera that what you will have in the system right and why this interest is a calculated speed of the traveling wave of cholera under simplifying assumption which depends on the local reproduction number but if you assume as it is meaningful and reasonable the distribution of it in a topologically connected system of this kind you have that node distribution the population distribution is taken drawn from an distribution which is normally i mean almost universally a power log exponent minus two the zipped distribution what you have is that now what you see in the system that you have flare apps in a mechanism which is exactly the same and why because that's the effect of factors that can be remotely measured and objectively manipulated in this case the population size and certain effects of the delays that you have in the system spatially explicit have nothing to do with the disease and everything to do with the geomorphology and the economy of the system so i took to haiti a few times and what these pictures i took they are kind of blurred and i love pictures but the reason being is that my my glasses were shut because people could kill you to steal your camera because there's no police left in the place we have no sewers no streets to speak of and this also shows how large patterns of infection accompanied you consider this safe water bottled water with the guy that handles them uh by the neck in this case quite remarkable or there are places is taking in the again in the bangladesh counseling is taking which the water reservoir which could be a highly abstract uh phenomenal parameter of the system in the case of urban setting possible proportion to the population or in this case you see it or a very physical system like you have in bangladesh as i was telling you and this is the market or car food in the outskirts of of port de prince where i've been sitting in the system and and what i told you and was totally remarkable this lady which you're seeing here blurred against because of a of a safe glass beyond it in a matter of second bought a cabbage at this guy and i couldn't i was speechless when i saw it by showing a nokia 1900 proof of concept i'm sorry knocking i think handle the telephone as a proof of payment so in a place in which you have no sewer system as you see it water flows through the market in this case you have no roads you have no police but you do have a telephone which is a way less uh biased socially biased system as we have seen and this is how in fact human to human transmission this is public transportation in fact takes a place and that's the last thing that i want to show you that models and data in fact uh they are not perfect modeling things etc i mean you have Bayesian estimation parameters but the very fact that you're using specially distributed quantities make sure that in reality the distance within model and data is so small that operational decisions can be taken based on that and i'm skipping this part because it's too late is marino gatto and his idea that in especially explicit models of disease development in fact you can have even an eigenvector can represent the pattern of disease before it happens and quite interestingly you show also that the local reproduction numbers meaning the test for the potential for the outbreak to occur is neither necessary nor sufficient the condition for epidemic disease outbreak if you compare to real cases whenever you have spatial explicit system in which human mobility is a driver not is an embedded driver but it can be calculated and i skip this part i skip also this because i realized i chatted too much about how proliferating kidney in fish can be studied and i jumped from the last two uh fish diversity in that case and and the deadly infections in fish are in fact the proper into the channel network was all that i thought to speak about so my conclusions the whole the general conclusion is that the eco hydrological footprints of river networks as ecological corridors are demonstrated they are pretty strong in fact and from peaks of prevalent in waterborne disease infections to any kind of large scale patterns of species abundance and biodiversity or even the susceptibility to biological evasions that we're seeing it's only the water so in a way it's written is something which could be remotely acquired over over virtually six orders of magnitude and remark will be compelling so in a sense towards a fair distribution of water which is my punchline that is that attaching a price tag to certain things which weren't um material uh materially or observables in economic terms but they are absolutely vital because they don't have a way of predicting what will be the impact for each of the of a an expansion of water resources um uh uh exploitation patterns in the expansion of a disease for each cost so they open um the uh a quantitatively open to a quantitative evaluation on ecosystem services to rethink in a sense um social equality and i thank you uh with that thank you thank you very much on that um we can open the floor for questions you quick question yes please um in the in the model um towards the beginning when you show the alpha diversity increasing as you go downstream um is this is this increase and consequence of accumulation of species going down because they cannot migrate upwards or is it a consequence of just more populations interacting and getting uh going together uh they will reduce the number of canal downstream it's good question the kernel for for the special species are the same everywhere what makes the difference is the fact that habitat size and thereby the carrying capacity of the population of every six species changes uh with respect to be downstream direction because of a natural accumulation so it's an external factor which is dictated by the aggregated structure of a network that gives an inherent predictability even though i mean how could you possibly uh consider all fish species equally uh equally capable of dispersal for instance or insensitive to drift uh at the per capita level and yet neutral pattern um doesn't require neutral process that's what i'm saying so the neutral patterns are more general that's what in fact famously pervis and pacala could hold and if i if i may just a quick follow-up was that pattern completely monotonic i i noticed it was not it was not a straight line it was there was some no that's you have to see the two patterns together so one is the frequency of species distribution uh with respect to the distance to the outlet so essentially you count the number of sites which you have the same distance from the outlet it's a fairly complicated structure so essentially it opens up it closes up and uh and uh and the other one is essentially the uh simply they you simply measure the average that is the local species that are there so you have this different distance from the of course if you are interested in the catfish distribution the neutral model won't work right so you have to go into a serious model of the thing but if you look at large scale patterns for instance for uh conservation reasons that the the neutral pattern gives you robustness reliability and the capability to make decisions actually it's basically a conversation for conservation practice okay if you're not further question i'm totally a bit delighted to go because i have another meeting fairly soon sure sure absolutely if anybody if anybody is interested in any feedback and wants to digest these they can write me anytime thank you thank you very much for your kind of ability and for the beautiful side of lectures and uh uh we will uh return me again uh in a short while for the following lectures okay okay okay excellent we are live again uh we'll come back for the next lecture by Samir Suarez from the University of Padua who will talk about community patterns in consumer resource models please Samir okay so first of all thank you for joining us here it's really for me an opportunity to be here and to try to convey and to share with you some works that we have done in the past two years so let me share the script the slides okay you should be able now to see slide right so this uh what i'm going to present is mainly the work done by Leonardo in his PhD i just got the PhD you met Leonardo because he made some tutorial for you and so these i want to thank you because really a lot of works he he did from experiments to theory so he did really a great great and incredible job and then with Andrea Giometto from Harvard now he moved in another university and and Amos Maritan that you met in the first lectures and of course to all the lab that always in the discussion is very important so i want to start from the from the let's say fundamental questions that is one of the of the important question ecology so wow you can observe so amazing biodiversity and i think you already have been exposed by these questions on the fact that it's actually not so trivial to understand the the incredible diversity that we observe in the natural ecosystem this is one of the most famous cases the plankton so in plankton we have very few resources in the ocean and yet we found thousands of different species and this goes under the name of paradox of plankton but this is in in general this more broad questions about how it's possible to observe the so large number of species coexisting species even in the absence of of many resources and actually in the in the in the last years and again you will have the opportunity to to to hear from Alvaro Sanchez his great work on that you can actually sample directly now from environment DNA and analyze the for example the microbial diversity that you find in a very different types of environment and again you can cultivate this this DNA in in the lab so in a controlled experiment and again just using very few resources even more resources you observe many species 20 30 that can actually coexist for a long time and this is surprising because in in an ecological fundamental setting we understand that the species that coexist should somehow occupy a niche and that's how species can coexist so if you have a like you can do this experiment actually you take two species one two microbial species one of the two is better let's say in uptaking resources so it's a has a fit it's a higher fit in that environment so what happens is that at the end if you if you wait long enough then the the fittest species the one with the highest growth rate will invade the system and will exclude the other species that's why somehow we need an understanding of how it's possible that so many species can coexist okay let's say a way to study such problem is in a theoretical way it's through the use of consumer resource models in particular one of the fundamental achievement in theoretical ecology of course for sure is a is the model proposed by McArthur in the 17th and so this model as you can see is a model of the system of a couple differential equation one for the species population denoted by here by n and and of course species grows okay and grows by consuming a resource okay so the resource concentration here is given by the c and r is simply the resource uptake rate and typically is considered as a monofunction so this is a classic monofunction so this this term here this alpha sigma i represent the metabolic strategies that has the metabolites that the species needs to actually uptake the resource from the environment and vi is the so-called resource values that is the the amount of energy that the bacteria can extract for some from from such resources so this overall part here is what define the growth rate of the species while we have a death rate delta that here we consider dependent on the species and on the other hand we have the resource concentration that grows here we are thinking of abiotic resources and as i represent a constant resource supply rate of course we have a minus due to the fact that this resource is used by the different species and we can have also the gradation rate here denoted by mu okay typically we will consider this mu equal to zero but this is not i mean we can do that without losing the general generalization of our result so in general you can see that alpha somehow represents so a bipartite kind of networks that tell you which kind of metabolites the species use to consume the resource i okay so if alpha sigma is zero it means that the species sigma cannot use cannot uptake resource of the type i okay okay so through this model is easy to to see so these are the two equations that we retrieved the so-called the competition exclusion principle that i think you have already heard that is to say so consider the stationary states of these two dynamics so you can see that putting zero the first the above equation here we obtained such a condition okay and this condition you can see is these are m equation because we have m species so sigma here goes from one to m so we have m equation and we have a p variable okay because this is a sum over p alpha so alpha now are the variables we have m equation in p variable but therefore if m so if the number of species is greater than the number of resources we have the more equation than variable and so there is no solution for the system except in the generic case otherwise the system it's solvable only if m is smaller or greater than p so if the number of species can be only smaller or equal of the number of the resources and this is the name of competition exclusion principle that i mean is a celebrated result that still we have to fully understand okay as an exercise i proposed to you to show that this couple set of equation if you use a biotic resources instead of a biotic resources that is to say instead of a constant supply rate you have a supply rate that follow a logistic equation okay and you consider a linear resource concentration so rc just depend on c then in this case in the quasi-station approximation so if you put the concentration dynamics to zero and you look at the stationary state for the concentration and once you do that then you put back this result into the population in the question for the population you can see that you will retrieve you will recover the generalized dot-cable terra model okay so the generalized dot-cable terra model can be obtained as a special case in a quasi-station approximation of the mccartel model okay so it's not possible to violate the competition exclusion principle but as pointed out recently there is a very important physical constraint that we are missing that is the amount of energy devoted to resource uptake cannot be unlimited okay and in other words there is a trade-off between the metabolic strategies and this was pointed out by and being in a recent pl so species has a total budget of energy that can spend in a metabolic in producing metabolites basically and so there is a trade-off between the different strategies that can be termed on and in the assumption of this work e was equal for all the species and there is a hard bound so this is actually the sum over all the metabolic strategies should be equal to this energy e and they showed that in this case coexistence of more species down resources is possible in a sum and I will show fine-tune conditions okay what are these conditions well again these are our consumer resource equation and again you can compute the stationary states so a first assumption of that work was that the that rate was species independent that typically they are very small so from this condition you can see that a solution for r star is the following okay so you can find a solution that is a function of the total energy budget then if you put back now r star in the second equation here so again you look at the stationality so you have the supply rate is equal to the sum and you put the solution of r star then you can see that this species all species can coexist independently on the number of resources if such a condition is met okay so this is a condition that once we introduce the energy budget if this is satisfied then we have coexistence of more we can have consist of more species than resources then if you rescale all the quantities so we call x the rescale population as hot the rescale the supply vector okay I pointed out here that with this tilde here I do not the fact that I have absorbed the this y this efficiency but okay as hot is the rescale supply rate and we rescale the metabolic strategy so that all these quantities sum up to one then you can see that this define basically a multi-dimensional simplex and in particular all the the strategies and the supply rates line in this b minus one okay dimensional simplex okay so you can represent the geometrically the solution of the consumer resource model with energy budget and in this I represent here the the resources so these are the axes for the resources and all the metabolic strategies of the species and the supply rate lie on this p minus one dimensional simplex so in general we will consider p equal three or three resources so that the space in which the metabolic strategy and the supply rate leaves is a two-dimensional simplex and so we can visualize it very clearly so let's now see what what what does it mean that this condition for coexistence is satisfied okay so in fact we can have a geometrical interpretation that allow us to understand when the the sum the sum here is is actually satisfied so after the scaling I said this can be represented in in in this simplex and so now consider the points the colored points represent the strategies so red species it's only nutrient two because you can see this is only in in the vertex of of the of the resource number two okay blue feeds equally upon the blue species feeds equally upon species one and species two while the species the the violet and the orange species can feed on all resources okay the star is a supply rate so there is a supply rate that is actually have a component that is different from zero in all the different all the three resources okay so the these conditions imply that the supply rate must lie inside the convex hull composed by the metabolic strategies that is to say if you okay now i'm not was not so good to to be the convex hull but this this region is the convex hull that is to say the region where you have as you limited limited by all the metabolic strategies this star the supply rate in order to satisfy to this condition must lie inside this convex hull so in this case the star the supply rate is outside the convex hull and therefore you will have extinction and the competition exclusion will be satisfied again so in this case only at best three species typically less will coexist so there will be at least one extinction in this case and the competition exclusion principle is recovered if instead the supply rate lie inside the convex hull then this condition is satisfied and in fact that we have the coexistence of all the four species okay so the total energy budget is a fundamental and important and physical ingredient to understand and to allow species to exist at the same time we must say that in this condition in this case if we put a soft bound or if we just perturb a little bit for example the budget based on the energy budget depending on the species we will retrieve again we will retrieve again the competition exclusion principle so in some how this is a condition it is a fine tuning condition so of course in general we understand very important factors that is this total energy budget but still we want to better understand how can species coexist or at least how can species organize to coexist so that they can actually survive even in the presence of your resources okay so now another apparently unrelated aspect is the observation is an experimental observation actually that dates back from Mono in 1949 in his PhD thesis that the growth of of microbial species in presence of more than one resources actually display the so-called the oxy shift what is the oxy shift the oxy shift is the fact that you have you see there are different slope in the growth of the species and that's because basically the species in the presence of for example two resources first use his best his favorable resources and then once he has consumed all that resources then there is a shift in his metabolic strategy so basically turn off some some metabolites and turn on the other one so to start to feed on the second resources and this leads to this bump and this different regime in the growth rate so this is a very strong and in fact then there are of course a lot of evidence since Mono in 1949 that the strategies of the metabolic strategies in bacteria are not fixed in time but they change in time okay so this is a very fundamental ingredient that we are losing we are not considering in consumer resource models typically so they are far not constant but our function of time okay so that's what we have done basically so we wanted to uh in consider in the McCarty's consumer resource model uh metabolic strategies that evolve in time so this means to write an equation for uh for uh the the metabolic strategies okay so how how to to write such such equation well we we used a simple idea maybe the most simple idea that is to use an adaptive framework so that each species changes its metabolic strategies in order to increase its own growth rate okay so this is a um uh in a way uh the possibility that the species can adapt to the to the environment uh and select which kind of of metabolic strategies you want to use so again if this is the growth rate the adaptation so the the the equation for for the metabolic strategies is simply in a simple way the the gradient of of the growth rate so this in this way we optimize the the the growth rate and the one over tau or lambda denotes the velocity of this adaptation okay then we will see how this is related with the parameter debat so now we are so we have a species dependent um adaptation velocity okay so this is now our new equation so we want uh uh to optimize of course the metabolic strategies but it's clear now we need to put a bound okay because again there is no possibility of devote an unbounded amount of energy and if you just increase this uh alpha dot you will have a different increase when we put a soft bound on on a species dependent soft bound on on the on the energy that can be used to produce metabolic metabolic strategies and so now we have to perform such optimization constraints so it's possible this is a a general result if you have to do optimization with some constraint that you can implement so the constraint in in the equation so the idea here okay here you can see there are uh so this is uh uh the phase space of just the two uh two resources so we have a fifth alpha sigma one and alpha sigma two and this line the east star sigma divides the plane in two region one that is the the allowed region we can move inside this this uh half plane because here the uh energy is less than the east star why we are not allowed to to cross such such line we're not moving the in the other plane and to do that what we do is that during the the optimization so while performing the gradient we remove basically the perpendicular uh components of the gradient that is parallel to the gradient of the uh of the energy okay so in this sense we have to to perform this uh evolution by removing all the time these components so that it uh these will allow to not always to move at best around the tangent of the school okay so if you do that this is uh i mean this is just to perform these calculations not so easy and also not allowing the energy to be to negative the metabolic strategy not to be negative you end up with the these two uh condition become this equation here okay so now we have an equation for the metabolic strategies and in this equation is also contained the constraint on the um on the on the metabolic budget on the metabolic trade-off so these are the new equation of the consumer resource model with adaptation so you can see we have an equation for the population the equation for the concentration and our equation for the the metabolic strategies okay so now let's see what we obtain what this model with adaptation can can display with kind of behavior first of all okay so in the red line it's a simple simulation of the model and you know kind of the general setting with general parametrization and you can see that indeed okay so this is the the the model this is the the data that I showed before and actually using a lot of metabolic strategies allow to to reproduce the oxy shift in the growth curve okay so here if we think that there is a strong preference on more resources or we can think that we can turn it off and on to metabolic strategies we indeed observe this kind of behavior this the oxy shift and I stress here that we are completely neglecting the particular molecular mechanism of the of the species metabolism but simply we are for example putting a strong preference on one of the two resources okay with the parameter vi but this is not only qualitative okay so actually Leonardo together with Andrea they performed an experiment so an experiment where they have cerevisia that eats glock glock galactose and then as a as a waste it produces ethanol and and in once a galactose is depleted the cerevisia eats the so feed on the wasted ethanol okay so this is firstly grows on galactose then it grows on ethanol and you can see these these are eight replicas of the of the experiments in the growth so you can actually see very well this this the oxy shift okay now if we try to to describe it want to to this behavior with with the model with the consumer resource model so this is the best fit okay using Monte Carlo chain methods of the model with adaptive on the left on the left and with fixed metabolic strategies of course here we have to constrained so we can measure some parameter of the model independently from the and we know from the biology that we have constrained on the parameters let's say and given such constraints we we perform this this best fit of the model and you can see that as expected the consumer resource model without metabolic adaptation follow a simple and let's say with the one-slope growth curve while with the adaptive strategies you really can see that we can quantitatively describe the the experimental data okay now let me make a link an interesting link this is a suggestion rather than a proof with the metabolic theory of ecology okay so the metabolic theory of ecology is a fascinating topic i think that almost told you something about that and you can i mean one of the most celebrated equation describing this metabolic theory of ecology is the so-called the clever law that described the relation between the mass of species and their metabolic rate b okay and indeed over several order of magnitudes you have that such relation is a power law with exponent three three over four now is discussed about this exponent of course this is just average is a relationship so these are average mass and average metabolic rate but this is quite a strong evidence in many in many different fields of the existence of such a metabolic rate that is the fundamental trait that govern many patterns in biology so if we assume that clever laws holds then in our equation our in our in our physics of the by physics of the model we have different rate one given for the death rate another one given by the adaptation velocity another given by the rate of metabolic production well all these rates will depend on the clever law and so finally will depend on the on the biomass on the mass of the of the of the species indeed it's easy i mean it can be shown that if we assume that the the metabolic theory of ecology holds then we have that both the total energy budget and the death rate scale as the biomass to the minus one over fourth okay now in this condition what we have as a consequence is that the ratio between the energy budget and the death rate is species independent okay and also we have that the death rate uh i mean the the the the adaptation velocity can be written as a function of the death rate okay so this leads everything to one only one characteristic type of scale okay so this is not mandatory okay we can relax this hypothesis and if you want i can in the question you can ask me what's happening if we relax this hypothesis but now we are assuming this hypothesis so we are considering in the following that the ratio between the total energy budget and the death rate is a species independent okay so again now we can write for our model a condition for coexistence and in fact we have that in this case all species survive again so these are just different little mathematical details but the the final point is that again the supply rate the scale supply rate must be inside the convex hull of the metabolic strategies okay so different uh derivation but the same result so now the point is that now the alpha depends on time okay so here i forgot to explicitly make a time dependence here of the of the of the so let me do it just to stress this point so basically here this alpha okay here the alpha are courageous okay let me see if i can put it depends on time now okay and okay so now if so now this is the initial condition for example okay so after scaling we set initial condition alpha t equal to zero and the supply rate is outside the convex hull okay so we have four species three resources same condition as before and the supply rate is outside the convex hull so the question is what about species coexistence okay so i remember to you that in the case of fixed static metabolic strategies then in this case if the convex hull if the supply rate is outside the convex hull then you can actually you can see we have extinction of many species and only two in this case survive and chap is recovered the commercial principle is recovered let's see what happened now if we allow this maximization of growth rate constrained by total energy budget in our equation okay so this is what happened it happened that there is a dynamics of the metabolic strategies along this simplex and finally you have that all metabolic strategies self-organized in a way that the supply rate now is in a stationary condition is inside the convex hull and in fact as you can see all the species survive as you can see the the wild species is the closest one is the most abundant one but it's not trivial because for example you can see that this is not i mean the orange one is not the it's not the closest so i mean it's not trivial to find based on the position of the metabolic strategies the the abundance of the species these are open problem as far as i know but we can see that the strategy self-organized and coexistent is allowed okay so we can now look at the other other experiment for example you can think about perturbing the environment so that supply rate now is the star and for a for a given time and then you turn it off and you turn it on a new supply rate like having different kind of source of resources from the environment and you change it okay and in the case of fixed metabolic strategies of course this leads to a stress of the population dynamics that you see they start oscillates larger and larger until most of them will reach extinction and only a few of them actually if you wait long enough maybe none of them will survive okay so let's see what happened the same condition for adaptive strategies okay so what you see is that adaptive strategies increase the community resilience and so stabilize the population dynamics of our microbial community that is able somehow to follow to adapt to this external environment okay so we have done a lot of different tests so for example what happened in the presence of resources that are heavily degraded or what happened in the presence of very inefficient resources and all the time you see that they just implementing this optimization principle with the constraints the community self-organized so to have the best response to the kind of perturbation you implement to the community so this was very cool okay so finally one can say okay but so here we have a new paradox everybody always survive okay well the answer is this is not true in the sense that depends on the velocity of the adaptation and non-velocity of the perturbation that is to say that so here I we are we plotted the rank abundance curve so this is the log of the stationary abundance of a community of 20 species and three resources for different uh adaptation velocity and also for no adaptation at all so you know for no adaptation at all of course you recover check okay so you only have three species survive but you see you do not down a little the adaptation then more species survive but not all the species survive only six if you never increase the adaptation around 13 species survive and then if you increase again the adaptation velocity then all the species survive so the adaptation velocity is a fundamental control parameter in controlling the the observed biodiversity of the system okay so let's let's first make up first part of my conclusion of my of this presentation we have introduced adaptive metabolic strategies that maximize the species growth rate and we have we observed that this allows to to describe the oxy shift growth pattern and moreover that this adaptation drive self-organization of species toward coexistent pattern and also that metabolic adaptation increased stability of the ecology community against environmental perturbation okay finally we have seen that adaptation velocity is the parameter controlling the actual number of species that will coexist at stationality okay so this is the first the first uh conclusion that I hope uh you have uh if there are questions about this first part maybe we can take it now yeah there's a question in the chat can you read it yes how would one design experiment to test the uh adaptive strategies prediction uh okay so first okay uh our first test was that we have done was uh to to to perform the experiment about uh try to describe the oxy shift okay so in this case uh I mean this maybe I mean it's not a prediction in the sense that we know that we observed the oxy shift but if you wish you can actually see one experiment to test the strategy about uh adaptive strategy prediction would be to be able to cultivate microbial community with species maybe engineered species characterized by different adaptation velocity or you you actually false some some inefficiency in some species and actually what you what you would expect is that being less efficient in adaptation and somehow measuring this velocity adaptation you can actually test that which species survive which not we have done a kind of similar experiment and I will present in the second part but of course I mean you you may think of many experiments that's what we are doing right now so not an experiment at least I don't want to speak uh out of uh of rigor but uh yes we you can actually do uh I think many experiments to test this this prediction so second question was uh would you say that this model solve the planton paradox well okay same definitive words about the science is always uh I mean uh to to demanding so I would not say that solve the paradox of the planton I would say that suggest uh strongly suggest if you wish that I mean quantitatively suggest that adaptation is a very important mechanism for species at least the microbial species to exist yes in this regard I would say yes we what well I think we have understood that dynamic I mean having adaptation is a fundamental ingredient for having a high biodiversity even in the presence of your resources okay if it's not the only explanation it's not probably the only contribution to to to to the solution of the paradox but for sure is one one one part one important part of that okay so if not I go to the second part I hope to to be able to to be on timer there's a I don't see a question I didn't see can you read that I don't see the chat it disappeared have you checked whether these results are robust to noise uh to noise in in in in where so yes we we have checked that these results are robust to noise depends where you what you put in the noise so I will at the end maybe you remind me this question I will show you that if you perturb the the condition so if you put the noise in the ratio between the energy budget and the deteriorated so if you allow the ratio of the energy budget and the deteriorate uh to uh not to be species dependent so in this case it's not fixed to a constant but may vary like also I mean like you you you consider a variance then in this case this is in the long time to competition exclusion so as soon as that condition is is not observed if you look if you wait long enough you will obtain uh competition exclusion but this will occur in uh uh time scale of the order of 10 to the 8, 10 to the 9 depends on the variance okay so this is a important point the effect of the noise is like when you when you go from the deterministic to to you add a noise to I mean you have a solving state of course in the deterministic case maybe you are stable uh in the in the stochastic case if you have that you you pass the barrier but maybe in exponential time okay so in this sense it's a robust to noise of course we have you have a quantitatively different result are there any other questions what happens if some resources are not substitutable uh what does it mean I would love to answer but I have to ask uh you can and uh ask yeah my question I was thinking like okay we heard the the other day from James O'Dwyer of non resource non-stitutable resources it means that like one species for example plankton it needs nitrogen so the adaptation of its strategy I mean it can adopt strategies but still these resources is needed it cannot stop using nitrogen so could you like did you think what will happen to the model if this happens well I think that this is in the model would uh would means that I mean this is not to incorporate in the model so I I I don't know in the sense that from one side you can think that there is some metabolic strategies from alpha sigma i so if i is nitrogen this must always be greater than zero so you put constrained on the entries of this metabolic matrix but how to put this non-stitutorial resources in the growth rate this is not I mean this is not I didn't think about okay so this is not trivial so you can constrain the model to use if you wish some resources but I don't in this model in this moment there is no I mean preferential use of one of resources so there is a no this possibility to constrain the use of one resources can I also question here yes or oh sure thank you so my question is about the the assumption of the growth rate maximization I mean think about a community with um you know consists of the two species that have the obligate interdependent like you know amino acid offshore um so how does in that case that your model needs to be you know changed or you know added a term to account for um the cases you know the the mutualism and and not not the species want to optimize its own but the whole community okay yes this is not this is a good point I mean this is really something that we are doing now we are not considering any kind of cooperation and so any kind of I don't know for example cross feeding type of effect for sure this will have an important effect also on on on the possibility to coexist so that's why I said before that this is not the only mechanism okay this is not that I mean with that we can explain everything it's a this is right now it's it's just adaptation in a complete in a poor competitive communities but for sure in the same framework actually you could and we are doing we want adding also cross feeding and other cooperative mechanism okay so I'm I'm just reading the there is a consider now two questions about physiological adaptation is physiological genetic changes or so let me just go to the second part because I think that this is a enlightening on this second part of the question in fact my second aspect is bridging try to bridge we try to bridge the cellular and the ecological scale because in fact there are evidence that the abundance of microbial species is strongly correlated with the metabolic function so that you can actually predict community composition by assembling microbial species in metabolic blocks that are specialized in particular metabolic function and we have already seen that the metabolic adaptation is very important in in in determining the the the evolution of the population so what we want to understand is the function is of course the function performed by a species depends on the protein it is producing and so the balance between the function that is the metabolic trade off depends on how the proteome of species is allocated so now we would like to try to understand what's the origin of the metabolic budget and of the adaptation of the fact that these strategies the evolving time okay so how does a location of the proteome affect the dynamics of microbial community so we take a step in a in a in a smaller scale and we look at so we consider a species and we consider the the complexity of its proteome that is the all the proteins that can be produced by that species and it's known that the total proteome can be divided into three let's say three important functions or regions so they that are denoted by q p and r so all the proteome dedicated to housekeeping function such as transcription factors are the fee are the fee q so fee q is the fraction of the proteome allocated for housekeeping function and typically