 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 this question from the Indian Institute of Science to deliver a set of lectures. And the title of today's lecture is Stock Assisting with Bestability in Ecological Systems Part 1. So as always, you can raise your hand and you can type your questions in the chat. And each question 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. 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. So 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. I would like to keep it interactive so if you have a question, type it out if it's something absolutely not clear you know, 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 is with me. So the broad goal of the three talks that I have planned these to 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 nonlinear dynamical literature, nonlinear dynamical systems literature. 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 now 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 and ecological systems. So the second part, which would be as in next Tuesday. It will be on, you know, 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 interest equally interesting hopefully. You know, 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, so the main, so let me, so this is the outline for today's talk. The 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. And so 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 demonstration of empirical evidence for these mathematical predictions. Okay, so that's a broad goal. And I'm, I understand that they know Carla stable 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 even 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 transitions in ecological systems. So this is an example of a large scale abrupt change large desertification. So what you're 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. If you look at this large landscape, if you look at proxies for this large landscape over last 10,000 years you find that preceding 5000 years before present day, the vegetation sediments were dramatically different from what you see today. So this, this current phase current state of northern Africa, which is the result was not always a desert. In fact, the sediments indicate that they were in a entirely different different state of vegetation. In fact, it had a pretty, you know, decent level of vegetation in the landscape for several thousands of years before it suddenly tipped and became the current state. So this 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 grass 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 has, it stayed in this low state for several decades before sort of switching and becoming a moderately covered grassland state in the present days. Here is a third example of lake electrification. You know, data is obviously noisy from many, many ecological systems, but indicates some very interesting features for the same amount, same values of drivers, you can have, you know, different values of the state variable. The state variable here is the sort of, you know, some measure of fraction of the lake surface covered by, you know, vegetation. Okay, so what you see here is that, you know, what physicists would call hysteresis when, when the lake underwent, when the lake had a very high level of phosphorus concentration. So that's sort of the red dots in this right extreme and, and a very low amount of the vegetation. As the, as this forest values came down, it continued to remain in that state of low vegetation. However, it increased. And, and a reverse transition, however, happened across a different route entirely different direction. This is called hysteresis in the, using the language of that's also used in physics and magnetic systems. This is a 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're in a very state of boom, there could be certain crashes in the indices that is remarkable over a very, very short period of time. Okay, so sort of the, if I, 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 changes happen, it's not just abrupt, it actually remains in that new state, once a change has happened, it's a persistent change as well. 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 cannot really recover them back. And even when you can recover, they could still be irreversible on timescales that humans sort of, you know, deal with. So in, so in studying this now, these are the empirical phenomenon I have just described, we're going to hear a whole bunch of terms that you may or may not have heard the gene shift. So many of these sudden ecological changes from one ecological state others, they're also called regime shifts. They're also genetically called abrupt transitions are 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 I'm sorry, 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 these talks and hysteresis. Okay, 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 class. So what are the important questions that people in the ecology literature are people more broadly in the complex system literature are interested. Okay, so one is you know how do we mathematically model the 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. So the 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 happens 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 are somewhere before this dotted line here, we had somewhere here let us see if we 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 you know really huge applications. Okay, so, so that sort of sets motivation for studying these phenomenon and to study them. You know mathematicians apply mathematicians physical psychologists, and many of them are you know, applying mathematicians and physicists and ecologists, they have been using mathematical theory theories of by stability and tipping points. So I'm going to describe those simple models are now and 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 he and in this model. You know, the ecosystem is represented by a single variable ecosystem is really a large interacting system of species type how one can think of ecosystem or you know sort of lump all of those into a single quantity called biomass density. And I can think of you know how this biomass density is changing over time. What is the dynamics, and the simplest models of these ecosystem dynamics, have this concept called carrying capacity and an intrinsic growth rate are the carry capacity and and in the you know, as long as this intrinsic growth rate is our are is positive that the biomass density we will reach a carrying capacity k over a period of time is also called the logistic growth model. Okay, now ecosystems are under constant pressure, not on both internally and externally. And one such important, you know, 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. Usually, this is modeled as this sigmoidal rate function. So, v square divided by v square plus v zero square this represents loss to the grazing. This is an all linear term. So, basically, this assumes that if the V is low, the, the grazing rate is small, but if it increases, it increases nonlinearly saturates to a value of sin. 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 RK, this is what you will find from the x axis the driver which is the grazing great see here. Okay, and y axis is the steady state for the equilibrium biomass density. So what you find here is, you know, when the grazing rates are low, of course, we will still have you know, 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. Once the system reaches this threshold value 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 high biomass density states. Okay, 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 it here. So this is a low biomass density state and 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 the 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. Okay, so this is an example of what is called a by stability. Okay, so you can think of ecosystem stable states as a balling roll in a cup or wherever it settles in this rolling cup or loading landscape. Okay, and then, you know, there's also this important concept of basin of attraction. So, so anywhere if I if I drop the ball anywhere to the left of this line, right, 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 concept of resilience because there is always a possibility that this ball will, you know, 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, x basically is is a dynamical variable, you can define x by x, a simple integration of this rate function, you know, some x 0 to x. So this function is precisely what I have plotted in the previous graph to obtain this, 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. Okay, so I'm going to skip some of those. So no, the 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, you know, you know, and the fact that there is a initial condition dependence in the systems and also that system, many biological systems are sort of, you know, irreversible or you know, they take long time to reverse once they are in another state. So all this can be nicely captured from this potential landscape picture. One can also do the following this was whatever I have been discussing so far is a deterministic picture. So what however what we can do is introduce some stochasticity to these models. So I'm calling them ad hoc stochasticity because I have what I've done here is, I have taken a deterministic mean field model. And, and then I have just added stochasticity stochastic term to this basically here Sigma v is the strength of stochasticity and ita v is the random number which is drawn from that also in distribution. And we also assume that the random numbers are uncorrelated over time. So we can, you know, incorporate these kind of simple stochasticity someone has to be a bit cautious while doing so. And they ensure that the biomass density is never really negative. So I'm not going to go into those kind of technical details here. So, so what happens in 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 a, you know, from some value of near the carrying capacity to a low value and 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, you know, 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 in a fluctuate between two stable states, you know, 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. This is where I want to clarify a few terms here. So in this diagram, this is also this is called sometimes called stability diagram is 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 are also tipping points and critical points in the context of non-linear 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, you know, these transitions can also happen when you are not necessarily close to critical points. Imagine you're really, this is a critical point, right? So this 2.6 in this 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, you know, 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, you know, abrupt transitions from one state or the 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. Okay, so remember these classifications are 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, you know, some very simple mathematical model, which has single variable, and, 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 by 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 far, you know, one purpose of mathematics 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, 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, 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. Is there some way of knowing where is the current parameter value. Okay, that's the question that we're interested in now. I just want to check if there are any questions at the stage. So I'm sort of halfway through my talk today. I have a question. Sure. So, in that equation where you go to the equation slides. Sure. Yeah, yeah, here, here. What is the functional form of V, you always show a plot V versus C with some by stability and a strict card. Yeah, so what's the mathematical functional form of V. Yeah, yeah. Yeah, so basically we, I don't have that with me right now, but you know, 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. Okay, then from that bifurcation analysis you got. Yeah, exactly, exactly. You set this and you get a cubic equation to solve you have of course V equal to zero is one equilibrium. Yeah, and then you have a cubic equation to solve. 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, you know sort of you know one can think of them as somewhat like toy models. They're inspired by the terms that I have included, they're inspired by ecological processes, but you know depending on the ecosystem that I'm thinking of. Okay, the exact terms can be dramatically different. So the question is how useful are they. Okay, so so my answer. So the one way to think about that is to go back to this potential picture. So if I can we think of another by stable ecological system. Let's say I don't know the equation. Right. But I know that I know that the ecosystem has by stability, as long as I can think of them in this potential well form. Okay, so how do you know that your system has by stability you are given just a time series with some abundance or land cover data. I showed you that it does show, right you know it shows the upper transitions, it shows hysteresis, and both of these are consistent with the existence of by stability. But that's 5000 year long desert times. 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. You 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 today is not to cap, you know, compare the models with data directly, but ideas to compare the predictions, and sort of you know, with that, I know with what happens in data. Okay, so let me go to the next part now. Okay. So, so the question we're trying to go into address now is, if somebody, you know, if, if, 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 the 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. So observe this landscape features which have computed from the mathematical model is the landscape far from the threshold point is the landscape close to the threshold point of 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 like studying within the context of this model. So what I'm showing you this is, you know, far from the threshold. This is close to the threshold. And you know, basically we need to think of this as you know this ball rolling in this landscape. What happens, so if you know now do simulations of this model, what you find is far from the threshold. So one of this vegetation biomass, of course fluctuates the characteristicity 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. And secondly, there are these, you know, sort of, you know, you know, spikes that you're observing towards the lower values. 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 says another point which I forgot to mention sorry. Now as your if you're in a deeper well as compared to a shallower well, the system will take much longer to return to the equilibrium if you're in a shallower well if there was a perturbation takes longer to come back and in fact that 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 we measure this from what is called auto correlation function or 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. So it's an asymmetric potential. The, the system also time series also shows increased asymmetry. So if you measure the ACF of time series auto core 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 me just 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 our equivalent parameter is being increased system is going towards the, you know, tipping point, the green line is a driver value. The value itself is gradually changing and 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, 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. Likewise, if I were to plot the variance, that should also increase. And likewise, if I want 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 kind of trends before an actual collapse. So therefore the idea is if you do observe these, maybe we are approaching abrupt transition for 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're 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 analyze 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 the, in the, in our models for example these are bifurcation points right. So, so basically as, 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'm showing you in this, although for the specific model are likely to be true in a large number of cases of abrupt transitions. So that's the theoretical prediction, right you know, if, if a person is approaching if an ecosystem is approaching tipping points by measuring these simple dynamical quantities. You know, one may be able to anticipate that you are approaching critical points or tipping points. Okay, so that brings me guess there is one paper enough for those who are interested I know in the statistical aspects of it which I am not going to cover at all in this paper. In plus one in 2012. There are also a toolbox that actually applies these theoretical principles and provides your statistical estimator of how good can you actually measure these in real data sets. Okay, so I'm not going to go into details of this statistical aspects. Okay, so now I have basically covered the mathematical theory of tipping points and how we can use them to offer early warning signals. Okay, 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 you ask this question, how do we look for empirical evidence imagine if we had a laboratory system. If you can somehow subject a system to tipping point in a laboratory, and then push that beyond tipping point. Do you actually find those trends in autocorrelation values, do you actually find trends 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 two field systems. So, so, for example, if you take data long term time series data. And if you look at that they have actually undergone up 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 Drake and others. So what they did was they did simple experiments where they were sort of grew, you know populations of Daphne in the laboratories and what they also did was they subjected these Daphne populations to increasing stress over time. Basically increasing stress can be sort of managed, you know, out of a mimic by reducing their, you know, food that's given to them. Okay, and what they do is they basically study them under also a controlled condition where they don't do it, and the condition where they're 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 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 signal. So what I'm showing you here is coefficient of variation and skewness. What they find, interestingly, is this is 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 this, 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 skewness. And of course, since then there have been more experimental variations of these ideals. For example, in East population, this is an example of East population density, you know, in sort of, you know, 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 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 skewness. 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 less stable 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, it's 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 the typically the bimodality is a strong signature of by stability 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 the system sort of dramatically changes around year 40. Using something called a change point method. Okay, so what it then did was okay now we know that the 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. But our data shows we raise 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. The 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 you know standard theory which I have presented so far, all of the four sort of you know show these signatures. However, if you have a stochastic transition where a large constant noise pushes system from one state 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 dry land 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, you know, stochastic rainfall, the rainfall is increasingly becoming variable over time. That means to explain this phenomenon. I don't know how much time do you have now, how about seven minutes. Let me now present one more such 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, you know, 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. We try to understand specifically we asked two questions. Do we find evidence for critical slowing down, which is basically increasing in auto correlation as markets approach as a crash. Do these markets exhibit increased variability prior to the crash. 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 is what we know historically is that there are four well studied crashes. In 1987, 2000, 1991, and then 2008. So what we did was we took each of these windows, and we analyzed all the four metrics. Okay, so here also to find is that the auto correlation at lag one which is a measure of critical slowing down has no clear patterns. In fact, as we're approaching as we're approaching the crash, it actually suddenly reduces. Therefore, there is no critical slowing down in this financial markets. However, if you look at the variants 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 studied. We studied 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 auto correlation at lag one the critical slowing down doesn't show expected trends, but the variance always shows the expected trends. You know, I don't need to do any sort of statistics for this. You know, 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 and other markets as well. Again, how do we explain this lack of critical slowing down, but with raising variability again I want to remind you about critical and stochastic transitions in stochastic transitions we don't find auto correlation at lag one increasing but we do find variance. So most likely our data corresponds to, you know, 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's also a large number of false positives that occur. Okay, so therefore, you know, 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. Therefore, we conclude even in the case of financial markets, they're not critical transitions, they don't happen near a bifurcation point, their features are better explained by stochastic transitions. So, so with that sort of, you know, I want to summarize the first race talk, which is that there are many, many in the real world examples of abrupt transitions, and we mathematically model them as tipping points or bifurcation points. And, you know, it's important to understand that there are 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 the critical slowing down, which is rising auto correlation, and increasing fluctuations measured by variance q-ness 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 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 resilient states, 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, the next talk on Tuesday I plan to discuss more about the spatial patterns. Okay. 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 work with Nikunj and Srinivas Vagaventra on the financial market problem and chaining on the in the crossland data. Okay. 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. Okay. So is one question is the asymmetry and shallowness correlated. Yes. So, yes and no. So, they're correlated in the sense that they happen simultaneously. Now, but they're, 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 look at if you think of in a classic and simple models of catastrophic transitions. So they have this, you know, linear term. Okay, let me see if I can switch on. Okay, can you see this? Okay. Okay, in this equation have, you know, this bifurcation model with noise. This is linear time 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, you know, the asymmetry requires you to use this cubic term. You cannot obtain asymmetry without having a cubic term. And so abrupt transitions in complex systems are typically modeled by saddle node bifurcations, right. A saddle node bifurcation does require you to have this cubic term. So, so yeah, so they're 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, so the shallowness is 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 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. What are the drivers place? For example, if you were to think of vegetation system typically rainfall and fire are very, very important drivers, right. So, 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. A single driver is changing slowly towards a bifurcation point or the stochasticity in the single driver is what is really causing it. Okay, next question. These are often fat tailed this leads to divergence of moments. Is it then meaningful to analyze the trends are top trends. Yeah, this is a good question and I'm not an expert on financial stock markets. So, so, so I will not be able to provide a very, very good answer to this your question. And 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, really, really good high resolution long term time series data. Does it show critical slow down. Does it show the simple feature let's ignore the moments part. You know, moments are complicated actually agree. You know, even if you know the moments part that I showed you, can we calculate the autocorrelation function and the properties of it right. So, I think that can be done without being worried about fat tail distribution parts, and we do not find evidence for the mathematical prediction that there is a critical slow down. Of course, you know, when it comes to so thinking a bit more about the fat tailedness, I think that to think of, you know, mean and variance rather higher moments. You know, you to really observe divergence you need a, you technically in financial data, but we are obviously analyzing data for a finite window. So I think in that limit, it might still be reasonable to calculate the proper to do drastic transitions due to noise happen for specific values of noise. Is it a feature of stochasticism. I think it is related to, you know, you know, you need a. If you're if you're the noise term in your model is 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 no I want to observe transitions within a given time scale. And I'm going to be interested only in those times kids. 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, try to address that specific question fairly. And that's the slide that I'm showing right now. So we basically wish. 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. You're right about that. Yeah, there's one more question is it possible, at least in principle to amenable to account for feedback mechanisms, for example, reinforced sustained so in great. Driver is not an external driver. Yeah. So, so I'm sorry that you know in this talk today, I have absolutely not pay attention to mechanisms in some sense right, I just used a model that showed the features I'm interested. 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, we positive feedback, you do not have the transitions, only when you have a strong positive feedback is, is when you will actually observe, you know, those transitions, and, 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 patch of trees will enhance the water infiltration that will in turn help the tree to go better that will in turn help a 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. Okay, I think we, we had a rich discussion session and thank you again for a nice lecture. Now we have time for a short break, and we'll be back in about six to seven minutes for next lecture by Thank you.