 lectures. So the first speaker of today is Vishweshwa Guttal, who is giving the second lecture on Pistability and Socasticity in Ecological Layoffs. So please Vishu, share the screen when you are ready. I don't see the screen. Do you see my, okay, hold on. I think I made a mistake one second. Is it fine now? Can you see my slides? Excellent. Thank you. So welcome back everyone. So yesterday Simon gave a very broad overview that incidentally included ideas of by stability, tipping points, and also early warning signals. So I'm sort of going to continue on those themes today. As I had mentioned in the first talk, so I have this, this is the plan for the three talks I have. So I have sort of talked about basic by stable dynamics, resilience and tipping points, and how can we anticipate these using the ideas of early warning signals. And then today the main focus will be on spatial models, you know, everything I spoke on the last talk assume actually there was no reference to space. All the mathematical techniques I used was basically simple, ordinary differential equations with some noise superimposed on it, then I'd walk away. So but today, let us look at what happens if there is space. Now how do we understand by stable systems? Basically the one question that one can try to, which I try to focus on today would be, can spatial patterns in ecosystems be used to infer resilience of ecosystems? That's the main question I'm going to ask. So the last one, which I'm hoping will be as planned will be to look at, okay, so today I'll just do a brief recap. I'm going to go over this somewhat quickly before I move into some tipping points. So I give this example of large scale continental shift, that is a classic example. This is not the only one. There are many other examples of ecosystems for gradual changes in the driver. Okay. And of course this has been observed in many, many, many systems. Now to my own list of Simon added a whole bunch of other systems as the name is stock. So how do we, so here is the simple ecosystem model I demonstrated the first term in this ecosystem model where we represents the total biomass density in the system actually comprises the classic ecological model to logistic growth to which we add a grazing term which is sort of, you know, which could be intrinsic to the system could also be extrinsic, such as livestock grazing. Okay. So this is a nonlinear term and interestingly the ones you add this and for a whole range of parameter values you get a bistability. There is a region of this grazing rate for which two alternative stable states exist and which state current and there could be these switches, shifts, critical transitions, tipping points, abrupt changes at these two points which are critical points or the bifurcation points of this simple model. So the question, you know, the nice thing about this simple model was that we could think of the dynamics of ecosystems with bistability as a ball rolling in this landscape. So for example, the minima in these landscapes correspond to the stable branches in the bifurcation diagram. So depending on the initial condition, the system could be either in this state or this state. And this helps us understand concepts of basin of attraction and resilience. And this concept of potentials was not just for heuristics. It was also useful for arriving at quantitative metrics that helps us make some forecasts about tipping points. For example, if the system is far away from the critical point as in this, the leftmost point here, these potentials are symmetric. Whereas when you're very close to the tipping points, potentials are shallow with the low curvature and asymmetric. And this led to the phenomenon of critical slowing down and asymmetric fluctuations. Critical slowing down means that systems take no longer to return back to the equilibrium, they exhibit more fluctuations, and they also exhibit asymmetric fluctuations. Put these things together, we could come up with some metrics from the time series, and that could actually forecast. In this example, we found that if the system were to undergo a transition at this point, if we had this time series before the thousand time minutes, if we could, if I could calculate simple quantities like autocorrelation at lag 1, this measures the critical slow down, it's increasing with time as one is approaching a critical point, or the abrupt transition. Standard deviation and measure of fluctuation is increasing. The skewness, the strength of skewness, again, which is a measure of asymmetric fluctuation is also increasing. Now these metrics, if you had no, if you were somewhere around 800 or the 900th time minute, so these combined metrics give us a hint that the system might be going towards tipping point. So that's the summary of where we were. I also spoke to you about how there have been many empirical studies, experimental studies, field studies. Sometimes they work, sometimes they don't work. As I also showed you when there is huge amount of stochasticity, you know, some of these classic signatures can be masked. Okay, so now, so now of course, ecosystems are really spatial, right? Now they also exhibit really fascinating patterns. These are, you know, some of the classic examples of the fairy rings are formed in African deserts. And these are again, you know, patches of laboredentine patterns of vegetation. And, and then, you know, one can go on, there are more types of vegetation. Many of these vegetation patterns are often found in resource constrained systems or systems which actually have a relatively higher amount of ecological stress. So therefore, this sort of prompted many people to think, can these spatial patterns provide some signatures of abrupt regime shifts that might happen in the future? So this is an example of clustering in the muscles, muscles. And this is an example of clustering of vegetation on the sea bed. So there are many, many examples, plenty of examples. So now to understand these kind of state systems, there are three classic approaches. One is called, so the first two of them are actually both reaction diffusion systems. However, the distinction between the first and second comes from the fact that in the first system, there are no regular patterns, there are no patterning in the ecosystem. Systems sort of roughly appears homogeneous. There are second class of systems are regular patterning systems, and you may have heard of Turing patterns. Okay, and on the third class of systems are there are patterns, there is self organization, but there is no periodicity or regularity to these patterns. So there are these three types of spatial ways to think about ecosystem by stability. So I'm going to go one by one and see how we can, you know, do theory of these systems. How can we come up with these early warning indicators, followed by, you know, testing them systems as well. Reaction diffusion system and we think of spatial system, reaction diffusion system with no pattern. So what do I mean by that? Okay, by the way, before I go further, I want to exercise that just one second, my internet seems to be fluctuating a bit, I will see what happens. We show also the connection is a little bit disturbed, so. One second, I'll just, are you able to hear me clearly? Yeah, sometimes it, sometimes not. So perhaps it's better to remove the video. Okay, I will stop my video. Yeah, I'll stop. I have to unshade this for a second. Okay, now my connection, yeah, there's some, yeah, it does say it's a bit slow. Okay, I'll stop my video. I'll share the slide again. Okay. Okay. So, so some of the many of the points I'm going to make today, they are actually nicely summarized in this paper that many of us who work in this area came together and wrote a review article. So, so this is a reference you can refer to to get a summary of the theoretical principles behind spatial patterns and how we can use to use spatial patterns to infer resilience. Okay, so how do we go about this? Okay, let's go back to the same model I showed you. I showed you this model where there was this simple logistic growth term, followed by non-linear grazing term. So I'm going to assume the same term as a local reaction term, meaning in space locally exactly same processes are happening. There's a local carrying capacity in a local grazing rate. Of course, the grazing rate can have some stochasticity in it, which is represented by this eta C term as a Gaussian white noise. And then we add a spatial coupling term via the idea that you know typically see dispersal, which is a mechanism by which you know one of the key mechanisms by which spatial interactions happen is modeled by a simple diffusion term. So what diffusion term does basically intuitively is it spreads things out in space. So this is the model I'm going to use as you do know the mean field part of this. If there was no diffusion, if I did not have the diffusion, this system will show the same bifurcation diagram by stability. And for a, even when there is diffusion under a bunch of condition, this continues to be whole, this continues to hold true. But of course, you know there are some many certain aspects once we add space, I'm assuming those are not in play here. So this is what we showed using a combination of numerical simulations and also analytical calculations. So I'm not going to show analytical calculations. These are actually stochastic non-linear PDEs, right. There are methods borrowed from the statistical physics literature that we can use to compute the same things I'm going to show you today, but I'm going to focus only on the numerical simulations. So what we have that here is over a period of time, we run this simulations and then we sort of stress this ecosystem more and more over time by increasing this grazing parameter C. So by increasing this grazing parameter C slowly over time, we find that a vegetated system becomes a completely bare area. So that happens over a period of 52 years on the time scale that we chose in this model. So let's look at the top graph here. What this shows is the average biomass density. If you were to do a spatial mean of all the vegetation density, you find that it goes, it sort of gradually decays. And then by year 48 or so, it suddenly undergoes, sort of a rapid shift downwards and settle down to a very low biomass density region. So we plot spatial variance and spatial skewness for these spatial images. So what we basically do is for every, imagine every instant of time, we are computing spatial variance, which is basically variance of all the data available in a spatial image. Likewise, spatial skewness is the skewness of all the data available for a given image. So earlier it was computed over time. Now we are computing over space. So if you plot this, what you find interestingly is that even before this nonlinear shift in the mean value happens, there is a dramatic increase, sort of, you know, close to tenfold increase in the value of spatial variance. Likewise, the skewness actually increases from a value of close to zero, all the way to one, and even begins to turn downwards, even before the mean value has shown any significant trends. One can do these calculations again and show that even spatial correlation, the third plot here below also shows similar patterns. So basically the message here is that increase in spatial variance and spatial skewness, increase in spatial autocorrelations. And also what I have not shown you here is that even in the spectral properties with sort of, you know, relate to both correlation and the variance, they also increase. All of these quantities increase much more rapidly than the spatial mean value alone would increase. So therefore we can use these as indicators, early warning signals in spatial ecological systems which do not have spatial patterns. Okay, so that provides a theoretical background of, you know, how the same ideas of time series variance skewness autocorrelations can also be sort of, you know, nicely transformed into, you know, can be applied to spatial systems. Now that's a theory, you know, how do you test this in an empirical world? We know what do we need, especially if you have an empirical system that you really want to apply to a large scale ecosystem, how do you do this? So we need three things to be able to do that. One is that I need an ecosystem that has undergone a transition. Okay. And secondly, I need to have spatial data over time at sufficient spatial and temporal resolution for the system. Finally, it turns out that there are very few or I'm not aware of any such data available. At least, you know, five, five years ago when I was looking, there was no such clean data available. So what do we do? Okay, so here is an idea. You know, how do we test this theory, even when you don't have, you know, such an ecosystem that has undergone regime shift or abrupt transition over a period of time. We don't have such a data. What best can we do? You know, I told you the theory, right? You know, we have this state variable as a function of time. We need snapshots over time. And then we measure these metrics, for example. Imagine that instead of snapshots over time, I had snapshots over space. So the x-axis is not time anymore. It's now either space or the driver values. Okay. So this is actually an idea called space for time substitution. This is widely used in the ecology sort of field studies. For example, if you want to understand how shifting temperatures change species compositions, how do species ranges change? One idea that a lot of field ecologists use is, of course, you can't wait for 100 years for the climate change to happen. You can use or look along temperature gradients, altitudinal gradients, where you can find temperature gradients, and then use that space for time substitution to sort of transfer or forecast what might happen in the future if the temperature were to change. So we use this same idea. If we have an ecosystem that goes from one state to other state as a function of space, can I compute these indicators? Do they show expected patterns? So indeed, it turns out that there are such ecosystems. So this is an example of the very famous Serengeti Mara ecosystem in Kenya and Tanzania. And what you find in this ecosystem is the central area is sort of predominantly a grassland. However, the peripheral areas is actually a woodland ecosystem. So we asked the following question. If you now go from the central grassland areas towards the peripheral woodland areas, if you could think of this as space for time substitution, do you find signatures of these early warning signals as measured by spatial variance to sort of correlation and so on, as you go from one end to the other end of the ecosystem? Okay, so we chose this very high resolution data at 30 meters and then we classified each pixel of 30 meters as either wood or grass. And then remind you the scale here is some, you know, we are really talking about a few hundred kilometers by few hundred kilometers. It's really a huge data set. Okay, so what we do is find the relationship between vegetation and the rainfall in this entire landscape. We find that, you know, if we in this landscape, the grass cover sort of shows a non-linear relationship with a mean annual rainfall. You know, it is sort of expected from many savanna forest ecosystems, you know, Carla has spoken about this, Simon has mentioned this. So the data here again supports the same hypothesis. So we find that the grass cover shows this highly non-linear foundation as a function of mean annual rainfall. So what we now do is, let me remind you the theoretical expectation if the mean value of the state variable shows a non-linear transition. Then a spatial variance will increase even before the non-linearity in the mean begins. Secureness will again increase even before the mean will decrease, likewise correlation and the spectral properties. Okay, the black data here, black points here, just to show, you know, that, you know, we are not getting any statistical artifacts. So inside exactly as predicted, we find similar patterns in our data. So even before the non-linearity in the variance have been on the mean has begun, variance has increased quite dramatically. Secureness, spatial autocorrelation and low frequency spectra. And likewise, we find this across many other, many other concepts I showed you there. So sort of, you know, this provides an evidence that, you know, one can use these, you know, mathematical, you know, ideas derived from non-linear dynamical systems and models and apply it to messy real-world ecosystem. Okay, so that sort of, you know, gives one example of how we can use spatial models of ecosystem by stability in reaction diffusion systems, you know, patterns. So let me now move on to regular patterning systems. Regular patterning systems are also called Turing pattern systems. And here are some really beautiful examples. I'm not going to cover a whole lot of this. Again, I showed you some picture earlier. There are many, many parts of the natural world we have refined very nice regular periodic patterns. And these are often modeled by a mechanism called Turing mechanism where there is a very short range positive feedback. And then there is a, you know, an immediate long range negative feedback. So this combination of local positive, short range positive feedback and a longer range negative feedback. Causes these patterns to persist. And this paper in science argued that these patterns may actually provide some indication of a catastrophic shift. So look at this picture here. Look at this sort of, you know, diagram. So what this shows is there is a region of by stability here again, the region of by stability as you move along this act region of by stability. The spatial patterns are changing. So, so, Rick Kirk and co-authors argued that this changes in the, these spatial patterns may offer some hints about the resilience of ecosystem. However, there are other studies that sort of contest this to show that there are so many types of these patterns and the, and the nonlinearity and stochasticity can sort of influence and you know, confirmed of interpretations. Nevertheless, but this is a very interesting idea that spatial pattern can actually offer some signatures of resilience. So all the third type of spatial patterns are not are slightly different. Look at these pictures. So if you look at these pictures, you know, these two have some regularity in the spacing between vegetation, likewise this, however, if you look at this, you know, this cluster is much bigger than this cluster than this cluster and then this cluster. So what you basically see it is see here is that there are a, there's a whole range of vegetation sizes plus sizes that are possible. And this basically gave led to a whole class of new models and new studies that found some very interesting results. In this paper, Scanlan et al including Simon Levin, they show, they found that if you quantify these patch sizes, vegetation patch sizes, and if you look at the probability density functions, they show a power law. The power law basically is a very interesting pattern because power law often indicates that there is no well defined mean and variance, which in other words, what this means that you can probably find really, really large values of clusters, which are impossible in simple coding like models. And of course it's not only in vegetation that people have found such patterns. There are also examples of such power law clusters in muscles, saudras, cigars, and scrubs. Okay, of course, you know, there is a, let me caution, let me add a cautionary tale here by saying that there are many, many studies that also very strongly argue that in a lot of these studies, the statistical evidence for existence of power law is really weak. There's something to keep in mind, a very important point. Okay. And, you know, in, in, you know, early, you know, there was, there has been quite a bit of research on understanding where do these power laws come from because you know, when you have a power law nature of any quantity. And when the power law exponent belongs to certain ranges, especially if the, if that exponent is between one and two, there is the mean value is theoretically infinitely large. Okay, and so is the variance. What does that really mean, you know, how do you explain such sort of seemingly unphysical patterns. So, so in physics, there has been a lot of work, trying to understand our laws and physics in physics. The basic idea in physics is that whenever there is a phase transition, look at the figure see here, whenever there's a phase transition from one state to other state, at the point of phase transition, like what's also called critical point, you see certain universal features. As Simon was pointing out, this is a, you know, irrespective of the complexity of the system, otherwise, at the phase transition, the number of degrees of freedom are often, often very low and so a lot of very different types of systems happen to show very similar properties. And one such property is the existence of power loss. So, there has been a lot of speculation in complex systems whether whenever the power loss that we observe in complex systems such as the foundation system, ecological systems and so on, are the indicative of criticality. And this is a whole field in itself and I'm not going to really touch upon it. I encourage you to sort of read critically on this topic, very interesting topic. So, what I'm going to now focus more on the mechanisms. How is the some vegetation ecosystems show irregular patterns, meaning the cluster sizes are not a well defined average value. On the other hand, there are some other ecosystems where the cluster sizes show very clear periodic pattern, just during patterns. So what's the mechanism that makes them so different. So, how do you serve that to, hello, can you still hear me. Yes. Because my network set the net collection lost. I'm sorry. Yes, on the. Okay. So the question I was trying to address here was, you know, how is that ecosystems of similar types show rather different patterns, spatial patterns. So one way to understand that is to look at the net interaction strength. So here a positive value means a positive feedback, positive feedback means that the presence of, for example, a presence of a tree will facilitate the presence of another tree nearby. So there is a local positive feedback, followed by no net feedback after a distance. So if you have only a local positive feedback what you often find these are these irregular shaped patterns. On the other hand, if you had a local positive feedback immediately followed by a short range negative feedback. Okay, this difference leads to highly regular patterns which are also called during patterns. So therefore, as Simon was pointing out yesterday, when we try to understand spatial patterns mechanisms become really crucially important. So what are the biological processes, and often these are integrated with many geophysical processes such as water that flows in the landscape. So therefore eco hydrology plays a very important role in shaping these feedback as well. So, so moving on from these you know irregular. So let's go back to the original questions that I set out, which is, can we look at these spatial patterns and then make some inferences about stability. So the paper by Sonia Caffee and co authors, they made this following claim. They said that, if you have, if you observe a power loss scale free clustering, such an ecosystem is a stable ecosystem. On the other hand, if you observed a, if you observed as, I was trying to use mine. On the other hand, if you observe that the, if you observe that the cluster sizes are not proper law but if they're exponential, then they are more likely to be less resilient. So basically the argument was that you can use the cluster size properties to infer something about the resilience of ecosystem. So to make that point again, what you're seeing here is a level of low, low, low stress system. This is to high stress ecosystem. And what you're seeing here is this power law is bending away bending away meaning, you know, this is not a power law anymore. So, so these, these parts are likely to represent more stable ecosystem, whenever you see a power law or a heavy tear distribution. If you find a exponential distribution, this could be an case of a highly resilient ecosystem. So, you know, so sort of summarize, you know, this idea of how one can use spatial patterns as signatures of tipping points. So I want to emphasize there are three types, you know, first is that of no spatial pattern, right? When there are no spatial patterns, I discussed, I also showed you empirical evidence. And then when you have these periodic patterns, okay, and then you have scale free patterns. So when you have this no patterning system, you can use the what we call generic early warning signals such as you know, variance, skewness and correlations. Those provide early warning signal. When you have periodic pattern, the very presence of the nature of the periodicity and this, you know, usually this spotted patterns, they are often thought to be maybe an indicator that we are very close to critical transitions. On the other hand, when you have systems with irregular patterns, as you move from scale free pattern to, you know, you know, you know, not so fact that a thin tailed distributions. Okay, then, then you infer that those systems are less resilient. That's a summary of various theoretical models. So so far in my, in my understanding, there is a reasonably good evidence that these sorts of indicators work well in real ecosystems. There's quite a bit of debate about these two. And there are, there have not been too many empirical studies that look at these two. So now let me, let me sort of provide a twist or spin here. Basically, to sort of show that these results are a lot more tricky and complicated. So for example, in this paper that was led by my PhD students Sumitra Shankaran, what she showed was that this sort of claim that cluster size distributions are the patch size distributions to provide early warning signal actually have nothing to do with this phenomenon of critical slowing down, which I have already sort of summarized. And in fact, the claim that loss of power law clustering is an indicator is not true. So basically in my this loss of power law clustering is not a generic indicator of ecosystem resilience. So in fact, there is a quite a bit of, you know, depending on the model that one uses, and the mechanism that one incorporates, we can actually show sort of, you know, test the limitations of some of these metrics. So how do we go about doing this, let me, let me sort of try to describe how we build these simple models. So these are very different type of models, these are not reaction diffusion systems. So we call them send, you know, they're popularly known as cellular automata models. So in these cellular automata models, you have just one second. So you have, so you sort of assume that the space is divided into a whole large grid like in the central, central figure here. Okay, and different cells either can be occupied by a, for example, a tree or it could be a muscle bed. So tree is only a representative organism here. And it could also be empty. So it could basically be one, you know, we represent occupied cells as one, and others as zero. So you sort of implement certain stochastic update rules. For example, let's say if we choose this, this pair of a tree and an unoccupied area next to each other. So two possibilities happen. So one is that of death. So there is a probability of death here. And then there's a probability of birth here. For example, you can see that in this case, if this arrow is chosen in the simulation, this original tree has been is now dead. On the other hand, if the birth was chosen, you know, there is now an addition of three here. So there could be a birth event or a death event. Likewise, there could be additional complex interactions. So these are called facilitation or positive feedback interaction. For example, if we choose to update for a pair of trees which are next to each other, two things can happen. One of them can die like this happened here. So of the two here, one of them has died. On the other hand, the two together will facilitate birth of another tree nearby. Okay, and they happen with another set of probabilities. Okay, so now we can convert all these sort of intuitive rules of how birth and death happens in probabilistic terms, and depending on the neighborhood. Okay, so and so and in these type of models, typically, there are many, many, many parameters. For example, in many of the papers I was showing you earlier, often there are five to 10 parameter values, and those large number of parameter values make it very hard to understand what parameter is causing what effect. In contrast, in this model what we have done is we have chosen a very simple model inspired by statistical physics that has only two parameter values. One is P. P represents basically local birth rate as I demonstrated here, or even local death rate. P controls the baseline, the production and death rate. Q controls the strength of facilitation, how a plant will influence the birth and death of plants nearby. There are two parameters and therefore, because there are just two parameters in this simple model, one can sort of do an extensive set of simulations and find out what happens. So what we then have done with these type of models is do simulates, simulates them for a very large amount of time. So for example, I'm just showing you two representative simulations here. On the left side you're seeing a simulation for, I don't remember the exact parameter values, no they're probably low P and the low Q. Okay, on the other hand, I, if I'm right here on the right side we have kept the value of P which is the baseline birth rate, same as the previous one. However, we have a much higher value of Q, which is high positive feedback. So, so now we can very easily control two parameter values, and then see the effect. So you can already see very clearly in this simple simulation that all else being equal, increasing positive feedback increases clustering in these ecosystems. Okay, so therefore one can study how the clustering properties change as a function of positive feedback values. So here is the classic phase diagram. So what we do here in this case is consider two cases, Q is equal to zero represents a case when there is no, no, no, no, no, no facilitation. So in this case you find that as you vary this driver value P which is the baseline birth rate, the system undergoes a phase transition from a bare steady state which means everyone is dead on the entire spatial landscape. And there's a continuous phase transition to a vegetated state are also called active phase in the, in the physics literature. And the crucial point here is this phase transition is a continuous phase transition. On the other hand, look at the value with large value of positive feedback. You find that the response of the ecosystem to reducing value of the driver values highly nonlinear. There is a very small range of P the steady state density drops fairly nonlinearly sharply. And in fact, there is actually a gap, there's actually a jump for extremely tiny values of driver values. So this shows an example of how positive feedback in space affects the abrupt regime shifts. So unlike the sort of you know, phenomenological model I showed you in the previous talk. This is an example of a spatially explicit model where one can incorporate some realistic features and mimic features of by stability. Again, this would be the region of seen this model the region of by stability would be roughly as much as we saw very clearly shown here. So let's change this model what we do is the following you know we do simulations. And then you know from the simulations like this. We calculate at steady state, what are all the different clusters, what are the customer sizes, and what are the statistics of cutter sizes. So I'm going to show you one result from that right now. So what we have done here is you know just assume that we have chosen a value of P, which is in the middle of this phase diagram continuous phase transition, which is low positive feedback. We have chosen a system far from the critical point zero here. And then you find that at this point there is a power loss in the class sizes. Okay, so this much simpler model than the previous ones can also reproduce existence of power law in the ecosystems. And he said when the, when the positive feedback is high. You can find this power loss, even at the point of critical threshold here so this is I know, so this is the parameter value for this graph here corresponds to the exact parameter value with the system will abruptly collapse, and you find that you find a power law there. This basically means is some of the previous claims, such as cluster size distribution can be used as a how far or how resilient the ecosystem is really not robust for example, I'm finding the same power law distribution of course the exponents are different. They are originally experiments are different. And yet, this in this case, the system is really far from the threshold, right. However, in this case, this time is quite close to the show. So one can find the same cluster size distribution is irrespective of how far or how close you are to ecosystem. And what really matters is the positive feedback. The positive feedback is important mechanisms are important. So depending on the values of the server with my slides spent away. So depending on the value of positive feedback, we can find my, my, my tablet is behaving a bit awkwardly. What is this, there must be some option to raise. Yeah, I guess it's not the annotation of zoom, otherwise you have to go to annotate clear. It's called. Okay, I'll come back. So, you know, power law in the cluster sizes, therefore, is not a good measure. However, you know, there are other types of power loss that do emerge at the, you know, at the, at the point of phase transition. Unfortunately, the resolution of this figure is a bit unsatisfactory. But what you're really seeing here is a measure of correlations, spatial correlations. Basically, what this means is that, you know, how perturbations in space are correlated across the entire landscape, what we find is that typically, at the critical thresholds, correlations have, there are very, very long range correlations in ecosystems. And, and this manifests often as a power law in the correlation function, or the power spectrum. Therefore, those could be used as indicators of proximity to critical points. So in fact, we now have a manuscript that is currently being written up where we find some empirical evidence, some supporting evidence I wouldn't say very conclusive, but something consistent with this theory. Okay, so basically sort of, you know, if I have to sort of summarize this patient patterns are fascinating mechanisms matter. Therefore, however, the interpretation of the resilience from the patterns alone are certain one cannot naively use the patterns to make interpretations so the. So, for example, if they're one by by either by looking at the nature of patchiness or the question size distributions, we have shown in our papers that some of the previous simulation studies were probably finding a set of parameters where it seems to have worked but it's not really a general result. So, so this, so with that sort of, you know, demonstration of various simulational results, let me sort of briefly allude to what kind of analytical techniques are can be used in these studies. So, unfortunately, I'm not going to go into much detail. So, for example, we can take the space the cellular automata model I have described, one can write down a mean field approximation, and the mean field approximation makes a very nice set of predictions of the existence of critical, you know, critical points, whether the phase transition is continuous or discontinuous, it often underestimates the values of positive feedback, one needs to find a discontinuous phase one can also do something called a stochastic demographic approximation. Again, I'm not going to go into these details, I'm just going to mention to you, and you know, I would, if any of you are interested more in this, feel free to get in touch with me, I will be happy to share some of our own labs as well as more general manuscripts that sort of describes these methods. So basically there are a whole variety of methods one can use so classically the idea of a mean field approximation is that we have a well connected system, and I'm looking at a deterministic limit where in the number of sites are infinitely large number of individuals are infinitely large. However, we know that real ecosystems have finite numbers. How can we know the effect of those finite numbers so to do that something called a finite cell expansion, it's a very powerful technique. And there are methods of methods called focal plant equation Langevin equation, using which one can find the steady state distributions. Okay, so often in this approximation, you don't have a deterministic equation of this type, what you have is a stochastic differential equation of this type. And the stochastic differential equation would then need to be solved using the methods of focal plant equation on Langevin equations. And in some cases, the stochastic differential equations can also capture the effect of the system sizes that you are considering. So I'm not going to go into any further details of this. So I'm coming to sort of end of my talk so a lot of the work I presented from my lab today was done by these two PhD students they have finished their PhDs now. Really, very, very fabulous work done by them. And then also, many, many collaborators in particular Sonya Kefi, and her students Alex and Miguel. And of course, you know, there are many, many more collaborators and funding that has made this work possible. In particular, Amit Ashwin and Stephanie and it option they were all involved in some of the work I presented today. And so finally, before, before I take any question as I had mentioned, so I'm going to talk about intrinsic noise and by stability in collective so that's the sort of you know, prelude to tomorrow's talk how we can actually assess of noise and by stability and and how noise has some very counterintuitive effects. When we study collective sort of plants and trees, but instead of, you know, animals. So I'm now happy to take question I think we have quite a bit of time so I'm happy to go to details of anything that has sort of skipped over. A few questions in the chat, which I read for you. I just remind that if anyone wants to ask a question please use the quite a bit of I should have seen this midway through. Sorry. He doesn't pop up on my window when somebody chats. Okay, so go ahead with the questions please. Yes, yes. So I was just saying that if anyone wants to ask a question about can use the raise and tool. So, I'm going to start reading the question so the one question from to one is, so there is no universality for specially correlated systems found in ecology in general. So is there no universality for specially correlated stuff question. I can only give a lousy answer. See clearly, you know, there are many common principles and common patterns right you know you look at zebra code patterns are these some of the patterns I showed you in the beginning right, for example, sorry, I can, for example, sorry on this. Look at look at these patterns right you know some of these patterns, or even the code patterns in zebra, you know, we now know that the code patterns in many animals, and on this kind of vegetation can both be explained by mechanism such as during instability during patterns. So there is lot of there is their generality, and there is universality in the spatial pattern formation. But, but there are some. Also, there are also some details that are important you know, you know certain additional parameters like strong positive feedback and the scale and the strength of positive feedback can confirm these results therefore I think one has to use a combination of these mathematical general mathematical theories, together with the mechanistic approach I think one has to use a combination to sort of understand the limitations of universality here. I don't know if I answered your question right. Well there is no sign from. Yes. Thanks for the answer. So there is a question from Miguel Rodriguez, please. Yeah, I, I am thinking a little bit of the lack of data sets that you mentioned to test the this this principles and I was thinking if maybe land covers super pose like land cover maps super pose with carbon sequestration maps could be used or that's just two cores. So I think there are a resolution, because for both of them there are contemporary distribution and also historical distributions around. So, so, so I think for the, so I spoke about three types of special patterns like one is the first one being where you course crane so much that you don't know the special patterns anymore. Right. So the second one is there are very fine scale special patterns you know those images I showed you those were patches over tens of meters of approximately those scales. So the same with third one as the regular patterns they're also need to be observed on those case is the first one can be done using some of the data sets we mentioned. So those you know Landsat kind of data sets, right, which we are, which we are currently working on as well we are using a whole bunch of satellite based vegetation metrics to not only look at the patterns of various spatial metrics, but also can we construct model from data, can we derive, you know model from data as we are also looking at those questions but if you want to look at these you know fine scales patient tuning like patterns, I could be totally wrong but you know, I think that the data sets that clearly show evidence for the theory are lacking right now. Thank you. There are more questions on the chat box actually I can see that. Yeah, there were a few questions about the new model. Yeah. What is what is the model one without by stability is that's right for the model. So I don't know at what instant you asked me this question. So I think you are maybe referring to the model in a had in one of those slides. For example, what one can do is imagine that the density, the, let's say there is a huge spatial landscape, and you find that 40% of the area is covered by trees. Right. So 60% is background right. So, now how do I know it's driven by positive feedback. Okay, so what one can do is, we can create a null model which maintains the same cover in the landscape 40% but it's randomized the distribution of the entire trees. So the clusters that will be formed in the randomized data set will be entirely not because of the interactions in ecology. They're entirely because of the density itself. So that's one way to arrive at null models for some of these kind of questions. So it will not have by stability. So it's just entirely, I made up the data right so the second there's a question on what causes the sharp negative feedback. I think it's the steering system said so you find that there's a positive feedback for short range, which becomes negative feedback for longer range. So what happens in these systems is that, so for example if these are clusters of trees. So that is, they draw water from neighboring regions. So the neighborhood of trees will have higher water retention. What that also means is that the slightly farther away from this cluster will be devoid of water. So the water is sort of conserved, if you were to think of water as conserved on very short timescales. So the rainfall falls homogenous in the landscape. So there is a cluster of trees, they not only retain locally well they also draw water from the neighbors local neighborhood. And because of which are slightly larger distances, there will be a lesser amount of water than the, than the average. So that causes the negative feedback that this slightly longer distances. Oh, somebody has answered the questions. Yes, this is the beauty of the job. Yeah. Another question by Zoray, who is asking which, I mean, what does feedback bifurcation mean ecology, ecological modeling. I don't think I understood the question. What was the positive feedback can cause new types of bifurcations. So when we have a week by when we have weak positive feedback, you have typically have continuous phase transitions or continuous transition from one state to other states. On the other hand, when you have strong positive feedback, the transition can become abrupt. So that was what I was saying. I was not sure if I use the word feedback bifurcation though. Great. So we have, if there are no more questions, I don't see any hand raised. Great. So there will be the last lecture by visual on Thursday. So today's from now. Yeah, I think you will have opportunity to ask a further question. Now.