 Great, so welcome everybody to this day of lectures. Before we start, I want to remind a few rules on how to interact with the speakers to ask questions. So if you are following from YouTube, you can ask questions in the chat. If you are following from Zoom, you can either ask questions and know you are familiar with this but it's always useful to repeat them. You can either ask questions in the chat or use the raise and button of Zoom. So I think we can start with the first lecture. So it's my pleasure to introduce Sandro Azele. Sandro is a professor at the University of Padua and is broadly interested in mathematical modeling of biological systems and has given many contributions to the interface between stochastic processes and ecological dynamics. And today is giving lectures on community patterns and upscale. So thank you very much, Sandro, for being with us and please start when you are ready. Okay, thank you Iacopo. Let me, okay, can you see my screen? Yes, it works. Okay. Okay, I think that we can start whenever you want. Okay, good. Okay, thank you. Thank you very much, Iacopo, for taking care of this and for this nice invitation. Welcome everybody to this second lecture. This is a second lecture out of three. The first one was given by Amos Maritan, the third one will be given by Samir Subbais, tomorrow. The first one was given by Amos on and let's say an independent, modern independent approach to community patterns. Samir and myself are going to give instead a lecture, two lectures on model dependent approach. In particular, I'm going to say something about neutral theory and upscaling biodiversity. And Samir is going to tell us something more about community patterns in consumer resource model. So, this is a kind of the plan of the three lectures. So, why, let me, let me give you a bit of a motivation here. Why are we going to care about modeling community patterns. So, in here, we would like, let's say to understand how, how these communities, how these patterns that we observe in communities emerge from underlying rules. We don't understand, for example, whether they are stable whether why they are complex and in what sense they are complex and if there is a relation between complexity of these communities and stability of their, of their, of their status. We understand in problem but it's important to understand to a certain extent to what degree these patterns that we observe are the result of a contingency to a certain extent or universality. What we observe is, is something is something that is because is universal that I mean goes, it affects all pattern that we observe or is it something that strongly depends on the detail of the underlying individuals on the line components of these patterns. This is, let's say the theoretical side on one hand, but there is also a, let's say a practical sign applied side so we would like to understand how ecosystem react when they are perturbed and whether we can come up with some conservation strategies to improve their, their status. It's important to highlight also the fact that prediction here we would like to understand the mechanisms underlying these patterns on one hand to make predictions but it's also possible to make predictions without understanding. I'm not always safe, but you know we live in the, in the era of the data deluge. And if you have a good algorithm algorithm which can learn from the data, then you can infer some, some properties and you can make some predictions but this is not what I'm going to do today. Although it's an interesting side of this, of this study. So another important thing to understand is that the mechanism, mechanisms that we would like to understand usually operate at different scales with respect to the path to the scale of the patterns which we observe. This is an important point. For example, when we observe pattern, we have to introduce some windows of observations and these windows introduce some, some biases. For example, we could have some final size effects, some spurious correlations or some sampling effects. And this is, we have, it is important that we are aware of this, of this problem. In general, our standpoint is that community patterns in general emerge because there is, they are the result of a collective behavior of interacting units. And from this pattern, we would like to identify key mechanisms, key mechanisms which, from which we can derive this macroscopic, macroscopic factor. All this is a fantastic challenge. But what I've just said, the fact that mechanisms may vary across scales and they reverberate on scales of integration make, make, suggest an important point. We have to be aware that when we observe the world, we are wearing a kind of a thick pair of glasses. Okay. So it may well be that what we observe is not something very interesting. Let me be a bit clearer about this. So we, the first form of the forming lengths is randomness, like everything you can read here. So it may be that what we observe, the patterns that we measure are somehow basically are not the result of mechanism. It may, it might be that they are not the result of mechanism, but simply a result of random associations. This is kind of a problem to a certain extent because of course when we measure something and we want to infer the mechanism, we would like to, we many times assume that there are mechanisms by which we obtain these, these patterns. However, this is not always the case. And there is an interesting paper here by Marcelini et al, who highlighted this problem and showed that basically we have to take care of this problem, because, for example, complex systems, many of them are modular, which means that they are made out of, made out of components, like, like Lego bricks, with Lego bricks, we can do Lego sets. And from this, because, because we have a list of instructions, we can come up with a Lego set and because we have a, let's say, a target architecture, right. So, in this case, it's clear that the Lego bricks and on one hand, and the list, the Lego sets, the final, the target architecture, there is a strong link between the two. However, in general, when we have functional, when we have ecosystems, usually it's not, it's not clear what kind of functional design or constraints are in place. So these are not clear a priori, right. So, because, for example, we could have some components that are like in Lego, like in the Lego sets, we have some bricks that are small, usually we have very many of them, but some other, which are larger, we have just a few of them. However, when we combine them, the number of these different Lego bricks are different, and this introduces some heterogeneity in the system, and this heterogeneity in the system makes some combinations more likely than others. It's just a kind of an entropy effect, which we, which we must be aware of. So, because, and this is a problem not only, not only problem that is present in ecology, but also, for example, in bacterial genomes, or in linguistics when we analyze book chapters, and the many other, many other patterns. For example, there are situations in which you can have horizontal gene transfer in which the randomness is present, but there are also other components that are not as important as this, or the other way around. So we have to make sure that what we observe is important, is driven by an underlying mechanism. So, how do we do that? Basically, the idea is to build up some, some null model, and now we are going to see one of them, some null models from which we can see, so basically we can derive from whether they work or not. Basically, these null models are informative when they fail. This is the basic idea. When they fail, we see that probably there is a mechanism, there is something, there is something underlying the pattern that is probably interesting. And this tells us whether there are specific architectural features in our, in our patterns. So now, let me start with this kind of introduction, long introduction. Let me start with the first pattern here, which is probably one of the, is very well known probably to many of you here. So it's the species-area relationship here. This gives you just the mean number of species as a function of the area. Okay, and this is quite, is very, is very well studied by ecologists. We know very many things about this, but let me start the idea, and this is something that I think should be clarified, like I said, not only in ecology, but in many, many other areas. So here, let me give you an example. We have a set of islands here, the Galapagos Islands, and here we have the area, and in each and every area, we count the number of species. So here, we are not looking at populations, just number of species. Okay, if you put on one on the x-axis, the log of the area, and on the y-axis, the log of the number of species, you get these points here, these green spots here. And then draw a straight line. This means that you have a power row, right? So, from this, now the question is here. So can we obtain this sort of pattern by placing a random individual space? This is the question. This is what I had in mind before. So do we have this kind of lens, the forming lens? Is this a pattern that comes from something that, from a mechanism on from an underlying mechanism or not? Usually, what ecologists do is basically they don't care about this. Sometimes, not all of them, but sometimes they don't care about this. And they say, okay, let's draw a straight line. This is just proportional to a to the power of z, whereas that is always smaller than one. So always, and usually greater than dot two. Okay, what happens in this case? Let's try to put a random, these individuals in space. So we have species is has an I individuals within some total area, capital A, and then we place a random this over an area. Now, if we want to look at the number of individuals in that fall within the area, small a, we get that the probability that at least one in, at least one of these individual, at least one of individual may fall within small a is given by this expression here. So one minus A over capital A is just the probability that no individual falls here and this one individual and this is for the entire species. Now, if you do the calculation, you end up with the equation one with this formula one, which tells you that Okay, if we assume if we assume to know the distribution of individuals among species. Okay, then the species are a relation that we should get is just is simply given by this expression here. Now, if you want, there is a homework here you can calculate the variance of this. So basically, you can assume that you have your given and the newly random viable with which gets one when the when any of your species I falls within areas moley and the probability is given by this expression here and otherwise is zero. So, this gives you the average value but you can calculate the balance as well it's not difficult and you can find already the, the answer in a question two of this paper here which is very interesting although it is quite old. So, what happens when we go ahead and we try to compare this with the data, this is what we get. So if you go out there and you've collected the data from the Lambert tropical forest of the past tropical forest, you get. I mean the, the, the black dots here are those given from the empirical data but the random placement model is the one is the dashed line. So, as you can see is here and here, and this is kind of a common behavior so basically the idea here as you can see you get a systematic overestimation of the number of species at all sponsors. So it means that somehow here. Let's say this is good news so there is some work to be done. Somehow we see that the patterns that the species area relationship basically has something to tell us something more important than just randomness. And this is also true for other patterns that we are going to see in a minute. The reason why here I've shown you that the the random placement always overestimates. So the reason for this is that you can see this, you can see the reason from this picture here. These are eight different vascular plant species in the past forest. You see they don't look randomly placed in space. They look aggregated. And this is something that now basically all ecologists have understood in the sense that randomness is one component but it's not the most important one and usually species, vascular plant species are aggregated in space. And if we want to model this we if we want to model spatial distribution of species we have to take care of this of this aggregation in space. Okay, let me just do a first summary. This is the first. This is the first summary of the first part very simple but I would like to make things clear. So we have the species of owners, the species area relationship and many other, which basically are not just the result of random association we did, we need to make sure that there is more work to be done here. On the other hand, the SIR is sub additive in the sense that if we if we double the sample area, then we don't double the the number of species but we get a smaller number and this is smaller and this is because the Z exponent actually before is smaller than one. And usually ecologists use this power law but there is no compelling theoretical reason for that so basically if you if you write down a model, a special model you can get something that is different from a power law but in some regime is a power. Basically, this is just a phenomenological or so don't worry if you if you come up with a model that is not a power law but fits the data is not is not a big and it's not a big problem. And finally, what we have learned is just the species are aggregated to space so this is a summary for the first part Jacob I don't know if there is any question or I can go ahead. So far I don't see any and raised nor any question in the chat, but if you have any please. Now is the time to ask. I think we can move on. Okay. Okay, so there is now let me move on to the second. Okay, so let me move on the empirical pattern which is the species abundance distribution so in this case, we look into the abundances of space so before we have looked just at species species reaches now we look into populations, and how these species are distributed across species. The usual pattern that the typical sad is given by these two plots here so on the left hand side you have on the on the y axis you have the number of species and on the x axis you have the log the log two of the usually you have a peak in the middle even though not always is as big as this, but you have a peak in the middle and and this is the reason why usually we get this kind of log normal behavior which is, which is used by ecologist and here you have this rank abundance so this is telling you that there are just a few species with a lot of individuals but a long tail of fewer. A long tail of many species with just a few few individuals rare species. Here there is another piece of another piece of homework if you want you can calculate the relation between the two. And you can see you can realize that one is proportional to the basically the inverse of the exceedence probability of the other. So, if you want I can give you the days no problem with that so okay these are two ways to present the same piece of information so in here, let me give you a caveat here, because it's important that we understand how we collect this data. You know that the tropical forest live many different kind of species we have creepy crawling creatures and chirping birds and trees and whatever. Are we going to put everything in this sad. All the species are going to be analyzing this species among the distribution the answer is no, no. And the reason for this is that if we, if we wanted to model this, it would be basically impossible to come up with something reasonable. It's too complicated because we should calculate the probability that species one has and one individual species to us and two individuals but if we want want to model this, we are, we have to deal with hundreds of species, all of them are interacted in a trivial way, and this may vary across different spatial scale so no, no way there is currently basically we don't know how to to model quantitatively large scale spatial patterns in multi traffic communities when there are many species so the idea here is to use an empirical model system so basically we focus just on ecosystems with one traffic level. Okay, so and in here we introduce some simplifying assumption here we have species of bees over flies species of breeding birds and so on so so in our case we are going to focus just on plants in in a forest when individuals within this tropical within one within, let's say this ecosystems with one traffic level, individuals are roughly can be considered independent or they interact more or less in the same way, and they are clout like I said before. So, in this way we can introduce some simplifying assumption so in here as Hubble has, I mean, as Hubble did at the beginning of the millennium. The simplifying assumption assumption that he introduced is that individuals basically of any species of all species interact compete in the same in the same way. So they all have the same probability to be birth, dying migrating so on so forth. So the identity in this case is not important. And even though it may seems it may seem that neutrality is a very strong assumption basically we are able to introduce some models which are not justifying which can tell us more about the patterns and much of scale. And although this assumption is completely is completely falls from an ecological point of view, it is very useful when we want to introduce some baselines and some kind of null models, null models from which from from the from the deviations of which we can understand something more about the underlying mechanisms. So, okay, so in, if we make this assumption then we said we have said that we have said that the identity is not important so we go from P oven one and two NS to P oven because we have basically one species in the region. And here we can basically write down a master equation which is here. This is telling you how it behaves this probability of having a given number of species within the videos we can write this from one step to the other, when you simplify you get the master equation. And if be not, which is the birth, the birth rate when there are zero individuals and be not are zero then this master equation may have a steady state which can be calculated in the following way. I'm going quickly about these technical details but I can leave you all the slides so you can do all the calculation by yourself. P and star can be calculated recursively and you end up with this expression P and start is just the product of these ratios of birth and death rate. And because of an normalization you can calculate explicitly this steady state. Now, with this in our hands we can basically model tropical forest. The first assumption is just to say that be of any is proportional to N and the event is proportional to N. And if you do that you plug this into the formula that I showed you before you get what is called the fish at locks which is widespread and has been studied many times and is in good agreement in several, let's say in several ecosystems, although it can be improved the improvement is the following here, we can modify just by adding one constant so be and instead of any we have plus R and plus delta. And thanks to this new expression we can obtain two different mode to different species. Abundant distributions, we can have this which is the negative binomial which goes back to the fish at lock series as R goes to zero, or we can have this expression here in which when see goes to zero. We recover fish at locks into why two different expressions. Okay, these have these are good equally good to a certain extent when it comes to model and empirical patterns, but it turns out that the negative binomial has some interesting mathematical properties which are not shared by these other as them and we are going to see in in a few slides these mathematical properties so when we compare this model here this model with the density depends to the data we get this very good agreement with the data and this is well known since 2005 more or less. So good agreement, although this is a it doesn't imply that we have neutral mechanism at the fundamental level, we are able to show that there is some degree of universality in these patterns that we observe. Another thing that I would like to highlight is this exit the value of x which is very close to one. And I would like you to bear in mind this value he these values here that are close because it will become important later on when we are going to model spatial degrees of freedom in neutral theory. So, okay, another summary now. So we have neutrality, which is at individual level and it tells us that we have just per capita ecological equivalence among individuals. This is at individual level, not at species level like MacArthur did in his theory of island biogeography geography in the 60s. Then, this is the I mean the niche theory is in contrast to to neutral theory because a niche theory asserts that species coexist because they're identity, because they are able to partition limited resources. On the contrary, instead, neutral theory confers lot of importance to stochasticity to ecological drift. And basically it washes out the old species identity. And because of this it's able to recover some to identify some underlying universal patterns universal makeup. So as you know, all models are wrong, but some are useful. And I think that neutral theory is somehow useful because it tells identifying this this universal drivers. And another warning here neutral patterns here, of course, even though we are able with neutral theory to let's say to match the empirical data, this doesn't mean that there are in place neutral neutral mechanisms. This is important because there are some other mechanisms, for example, some other models which are no neutral like the hierarchical competitive model by Tillman, which can produce log series distribution, even though they are no neutral. But that doesn't mean that is true truer than the neutral theory. Also, there may be some life history trade offs, which introduce some equalizing effect on birth and death race and therefore they make look like more neutral. Okay, this is the second part. Yeah, couple is there is, if there are questions. No, I don't see any in the race. Is there any question anyone that wants to ask something. We can move on no problem with that. Everything is clear. Maybe everything's clear. Or everything obscure. Yes. Okay, so let's move on now. Okay, up to now we we've seen the species abundance distribution and the species area relationship now, we would like to go ahead to move on and try to link them in, in, in with one in one approach with one model. And one hand when you, you see you have this forest, when you sample small areas you get these pieces of on the distribution with a mode that is basically at when the number of individuals is very small but then when you increase the sample area, these species are frozen to higher classes here. And so you move from from small number of species up to large number of species. This is, as you can see this is clearly a kind of a sampling effect which is difficult to, to model and this is one of the reason why, for example in the one that I'm going to explain later on. This is kind of a challenge. So patterns are scale dependent and when we model this we have to look, we have to be careful and we have to look into this. So is it possible to come up so here I've just shown you species are relationship and species upon the distribution but there are very, there are many other patterns in that can be measured here is just a list species lifetime distribution is a pattern that involves time percolation in both space so on so forth Taylor's law. Okay, I don't want to look into the details of this, but the idea is to find out whether there is an underlying mathematical theory from which we can derive if not all these patterns list some of them. And now it is now it's what I would like to explain you. So in here, we start with a model which is basically defined on a lattice. Okay, it's a meta community model so in every side, we put several individuals, and there is no threshold on them. So we have and how how does it work is just, it has four moves. So the first is a death move that moves in which basically one individual here. One of species of one species inside dies at the rate are then we have the birth move so we have an individual and this produces one offering at rate B, and this can hop onto one of the nearest neighbor size with probability one minus gamma. Finally, we have that every site is colonized by a constant rain of propagules with the rate be not now it's possible to write down the person death rates of this model and the and therefore we can also write down the special master equation for this model. Let me give you just highlights of the most important properties. It is species independent. There are just best for basic mechanism birth death diffusion and immigration. So basically, in here, there is only stock demographic stochasticity, no other effect. If you if you look at the mean of the behavior of the mean is quite trivial. So the patterns here emerge simply because of demographic stochasticity. There is no environment there are no environmental effects and because the birth and death are linear. There is no carrying capacity. So in here be is always must be always smaller than the end biodiversity. So the states that the stationary state is non trivial. But when be not is larger than zero because of the term that I showed you here is some here this tells you that we have specially aggregated species and finally this model is minimal in the sense that if you get rid of one or more of this parameter basically you end up with something that is trivial. So with this patterns here now I mean this section is a bit more technical so I will, I will go quickly over this part. I just want to show you how it goes but there are papers which where you can find all the details that you want. When you introduce this partition function, you write down then the equation for the partition function when where you have the partition function in discrete space. Then, because basically this equation for is too complicated and we don't know how to use it then we go back to the data and we look at what kind of regime we we are interested in. So what we know is that be over R, which is the X that we've seen in the first part you remember always the X was close to one so here we are going to do the same be over R which is the old X is always close to one and the special diffusion is always comparable to our minus be. Okay, in this starting from this you can introduce two different parameters which are epsilon and eta, which are defined in this way, and you consider a regime in which both they go to zero but in a way that the ratio is all that one, and also be over be notice all that one. Okay, if you do the calculations you do all what I mean all the proper calculation you end up with this new equation five equation for the partition function. And eventually, it turns out that you can introduce a new, a new partition function for the global patterns that for the patterns, basically, for the random variables that you need in order for you to calculate the species area relationship and the special species distribution. This can be derived directly from the partition function, and you end up with this equation here in which you have this Laplacian here of F of B, which introduces the special effects so the effects, the special effects of what you are looking at or what you want to derive are in this term in this function F of V. Now to cut along a long story short, basically it turns out that we know this function F of V, but the important, the really crucial point, the really important point is that when you introduce this, when you introduce F of V into the equation, basically miraculously this integral of Laplacian disappears, and you end up with this function with this equation six, equation six, in which you have just this term, sigma capital sigma, which is proportional to the final factor. And the final factor basically is telling, tells you how your fluctuations deviate from a Poissonian, from a Poisson, from Poisson fluctuations. So in this, you can really simplify things and you can understand that basically space enters the equation only through this function here, and this function here can be solved exactly. Okay, of course in this specific regime. So here we have a stationary distribution which is our probability, our species abundant distribution which depends on space only through this sigma capital sigma function. So, okay, this, okay, if you want, you can also introduce time, but I don't want to look into this. But anyway, it's impossible to look into space. Also, in this regime, which is basically what is interesting from the applied point of view. So in here we have an underlying mathematical model from which we can derive basically all the patterns, so we can link them in a consistent and controlled way. And can we compare with the data? Yes. The answer is yes. And what, okay, here I'm going to show you what we've got for the barocoloral line, but we did also for other forests and we got comparable results. Okay, this is just the mean number of individual species, and I think that every, every, more or less every conceivable model can fit this data. Anyway, we can go ahead and we can try to, we try to best fit now the percolation function which is given by this model, by the separation here, this is the exact percolation function that we can calculate from the model. Okay, now we have all the parameters we can make some predictions. And the predictions told us that basically the final factor is always very, very large. The correlation length is large and the correlation time is also large. So in the regime that we are considering, the model is able to catch, is able to match the empirical data. But when is close to a critical point. So when mu is very close to zero. So large fluctuations, so fluctuations are large and they vary on large temporal scales and are correlated on large spatial scales. So now that we have the data, we have the parameters, we can make the prediction. So we have this red curve, which is the true prediction, we can, we can say how the SIR behaves across scales. You can see it goes more or less on top of the data. And also we can predict how it, how the errors or the species upon the distribution looks like at the global scale and this is what we get from, like I said, this is a prediction. And also when we downscale the prediction and smaller, smaller scale, this goes more, is able to explain the data that we observe. And this is also for the pass or forward. So we comparable, comparable results. Okay, third part that we are going to summarize here. So in some regimes, we can derive analytically spatial, spatial explicit models of community partners, and we can link the, the patterns in, in a consistent way. It seems this model is going to tell us that it's the model, if we want the model to match the data, we need to, let's say we need basically telling us that we are close to a critical point. Although this, that doesn't strictly imply that we have a critical behavior in this case from the data. Okay, this is at least what the, the model is telling. And if we want this model can also include environmental noise. So you can get all the patterns that we've seen when we have environmental noise. Okay, I think that this summary. With this, we have the third part, which is, so we, if there are any questions. Yes, there are a couple of questions in the chat. So, also is asking how dependent is the model and its results on the scale of the grid. And actually, the results that I've showed you are in a, in the continuous, in a continuous approximation. So in here, we use the largest scale. So basically, there is, there is no, no dependence on the, on the spatial grid. So basically what we are going to say what we assume here that you cannot go to really fine, fine skates with this because we, we have taken this continuity in space. And I'm kid is asking, is the immigration term indicative of immigration from a source pool. If not, how is it different from the diffusion between grids. No, this, the immigration is different from, because the immigration is something that comes from the outside of the system. So diffusion is telling you that basically individuals hop onto nearest neighbors but are always belonging to the, to the, to the lattice. Instead, immigration is a contribution is contribution of species and individuals come from the outside. So it's kind of a different. It's a, so let's say one is local diffusion the other is global because it affects all the lattice. Yeah, he's asking another clarification about this, given the model assumptions to do all the species go extinct after long enough times in absence of immigration. In our assumptions know because we use the reflecting boundary conditions but if one wants to study this it's possible to understand also this behavior and the solutions that I've shown you before can account for species extinction depending on the area of starting area. But this is something that I didn't, I didn't show, but it's something that can be done. And Stefano is asking if you could provide the more details on how the environmental noise can be inserted in the model. That's a very, that's a very nice question. This is something that we so basically it's difficult to explain without going to the so there is a change of variables, which helps you map one problem to the other. And so basically you can find the corresponding model by using of course an appropriate and appropriate change of variable which maps the problem of the environmental with environmental noise into the one that we have just shown. So basically you can go from one to the other. At least let's say and at least at the course level so when we did the, the, the, the continuous limit, the continuous limit in that in the situation, but anyway the patterns can be derived in basically in the same fashion. I don't see other questions in the chat, nor people with the hand raised. Okay. So, Jacob or how much time left about 15 minutes. Okay. Okay, so may may move on. Okay, let's now look into the the upscaling problem, which is somehow related to what we have looked into up to now, but it's on one hand is different and it has some interesting applied size which may some theoretical and applied size so basically let's let's start let's imagine to have the Amazon we have in our the problem is the phone we want to let's say predict how many species we have in the Amazon forest. Okay, of course we cannot go on to the Amazon and survey everything. This is too difficult to time consuming and research consuming so it's we cannot do this. This is the strategy that we could use in order to give an estimate of the total number of species that live in the Amazon forest. Let's focus on, for example, plant species, because like I said, finding out so let's focus on one traffic level because the complete the full problem is basically too difficult. So in here, the idea is to do the following like it is suggested by this map so in here we scattered around we scattered some fine scale samples around the the Amazon forest. So when we do that we collect all possible information within one sample and all the out so we collect all the species identities, all species post population so on so on. Now okay we have, we have an incomplete set of information we have we have some information we have we don't have complete information but we have something. How can we basically link this how can we use harness this information in order to upscale and in order to infer the total number of species this is the upscaling program so basically we collect information from fine scale samples. So we want to infer we want to make predictions about species species rich richness at core scale at very large scales. What can we do, I mean, here there is a long history of different models and for a recent monograph you can look into this final paper and you have a long list of different approaches, but all met more or less all of them have have some problems have some let me go into what we suggested. Starting from what I just told you now so here. Okay, there are several problems that we have to face so the species are relation is not additive. These species are usually specially correlated. Instead we use some uncorrelated distributions up to now so there are several problems here and most and one of the big problems is that there are. I mean species populations varies across spatial scale so basically what we see at small scales, for example here when we have a small number of species, the species that the species upon the distribution has one shape. But we want to predict something that much larger scale when the shape of the species upon the distribution is completely different. So this is quite challenging and it's not different, it's not, it's not usually it's not easy to come up with something that is consistent and is coherent and controllable to a certain extent. However, what we have now I'm going back to what we what I told you before. We can start from the negative binomial distribution which I showed you at the very beginning, which has this form here so if we assume so let's say for for it, we know that this can match the data quite well at the global scale. Okay, so when we go down scale so when we look instead, instead of looking at the entire area, if we look at smaller scales, how, how does this distribution behave. So there is an important property there is an important mathematical property, which is the following which basically it is what we are going to use and to explore. So basically, if we start from a given negative binomial distribution, when all species are assembled, then if we go down to smaller scales, then the distribution at smaller scales is still a negative binomial, which means that is forming an under binomial same. So, so basically if we can, to a certain extent, start from a rough independence assumption so basically we assume that for the time being, let's say specially are more or less specially uncorrelated, then we can use a binomial sampling. And this binomial sampling, when applied to a negative distribution is telling us that even the smaller scales, the negative binomial remains still a negative binomial. This is a very, very nice property which can be used and can be with this we can somehow avoid the problem that I was talking about before. So what happens so in this case, if you so here, this is what you have to do when you apply the binomial sampling to your negative binomial distribution. When you do the calculation you end up with this expression here so it is still a negative binomial but now, as you can see, there are two interesting features are one of the two parameters does not change across scales. So it is an invariant, a scale invariant parameter, but X changes with scale and you get this X at, okay, but this except now can be inverted so now you can get X, which is the parameter at the global scale as a function of the parameters calculated as smaller scale. So this is telling us the following interesting suggestion. So here we can, when we sample with, when we sample our area with fine scales samples, we can measure this X hat and P and R. And then by using this formula which, which can be reduced straight away from the approach that I just shown, you can upscale and basically say what's going on when. So basically you measure SP which is the number of species at scale P, and by measuring X hat, and then X as a function that is outlined here, you can infer the total number of species. The number of details are here but basically this is the, this is the idea, and this, we can do this simply because of this nice property of the negative binomial by using this, we can infer, for example, the total number of species in the Amazon in the forest, which is kind of a modifies previous estimations here, and here there is a list and now we can go, if you want to explain what we did here but anyway, it's possible to find out how many species, at least in some specific ecosystem with one traffic level, what kind of distributions and what kind of predictions we have. Of course, we have also the species abundance distribution. So when instead we have, we do have some correlations, then we have to use kind of more phenomenological approach, but still working. And this just going quickly through this model to this approach is the following. So we start from a given SAD species abundance distribution in this case we can pick a, let's say, a two parameters. Then, from this we can pick another, we can pick another spatial correlation here with two parameters as well. And from this we can calculate the spatial variance here. At this point, we can make a link. So we substitute to alpha and beta, alpha and beta here, two functions, alpha of r and beta of r, and we set this couple of equations, integral equations, integral functional equations actually. So in here you impose that these equations, self consistent equations satisfy these two equations. And these are equations for alpha of r and beta of r. It turns out that in some situations, alpha of r and beta of r can be fine, can be found exactly. And when you have this you can make some predictions. How does it work? I mean, it works fine. And here you have an application to the breeding birds. Here you have some more or less 1000 plots in France. You apply this machinery so you try to best fit the data with this function, but you can use another function. It doesn't matter. You can do it again. You can do it anyway. And then you can predict. So here you have more than three order of magnitude here and we, we in France, I have been told that there are more or less 300 species expected even though I didn't find official predictions. So this model we predict in 330 species with by taking into account this kind of correlations. So, okay, this is the upscaling model. This is the upscaling approach and I'm going to finish now. So we have this modeling community. This is has interesting theoretical aspects which are related to the hyper dominance of species, which is still an open question if we if we want, we can go into the details of this. We can also, if there are special correlations, if there are not, we can use the negative binomial distribution, which is robust under scale, under scale variation. And this is because we have these four new variants. And in any case, if we don't have this advantage and we do have special correlations, and these correlations are important, it's still possible to make some predictions with a more phenomenological approach, and we can predict the species of on the distribution of the species. This approach can tell us basically not only how many species, we are, we are missing, but also what is the distribution of populations among species that are yet to be observed. Okay, I think that I'm done. And I just wanted to thank you everyone. Here is our lab, the leaf lab and here are the collaborators. Of course, what I presented is the result of deep interaction with all people. So I'm debt with all these people without whom I couldn't get anything of what I presented. Thank you very much to everyone. And thank you, Jacopo. Thank you very much, Sandra for the very nice lecture and overview of these results. So there is time for a few questions so please if you have any raise hand or right in the chat. Someone is asking, I have a question myself. So, oh, actually there is one so I'll leave priority so Sylvia is asking, does the change of the X parameter under rescaling of the negative binomial distribution reproduce the different shapes of the different scales. Yes, yes, yes, yes. In fact, it goes from one mode from one let's say internal mode to the distribution where there are no internal modes. Basically from internal peak when we have the negative binomial behavior to the feature log series behavior when the mode is always in the first class. I have a question. Well, then I asked the question I wanted to ask. So, one thing that it's surprising for me is that it seems that the same distribution hold across several orders of magnitude in scale and somehow it's Are you are you referring to the upscaling problem. Yeah, yes. Somehow, I mean these the negative binomial is at least, at least partially justified mechanistically with models. And it's sort of believe that the neutral models or neutral theory works well at the local community scale because individuals are relatively few and there is a demographic because it is more important. So I wonder the fact that it seems that it works also for much larger scale. You see here. And actually, probably I should have been more should have been clearer when I address the probe the upscaling problem. I'm not making any assumption about species. So in the first part, it was more the dependent and I assume neutrality. But in the second part when I explained the upscaling problem. In that case, I just use the negative binomial. So I'm not saying that I'm not saying that I'm using the negative binomial because I'm assuming neutral, neutral theory. This is just an I'm just using a distribution because of its nice mathematical properties. You derive this from a neutral model or whatever. I will we are using the negative binomial simply because it has this form invariance. So there are no assumptions in using that. So that's probably answers your question or I don't know. The reason why it works is because of the flexibility of the negative. Yes, yes, that's the point. That's the point is very is very is very flexible. That's the point. That's the question. Bert. Bert. Are there obvious examples of when the continual approximation fades. When discreteness is important. And so this is a question for the spatial model. Yes. Okay, here. And there are two, two things that there are two continuous limits. One continuous limit is referring to the basically to the individuals. So I, this is one continuity. The other continuous limit is the continuity in space. So, I mean the approximation the discreteness of population is less important than the other somehow. When you use the continuum limiting space. There is a limit basically when you go down, when you go down to scales as small as, let's say, compared to meters or tens of meters. Basically, this, this doesn't doesn't work very well. So basically, when you go down to scales that are much, much less, much, much smaller than the correlation lens. There are, there are differences. And basically you cannot use, you cannot use the approximation that we used. But anyway, this is a model which was devised for, for large scale for macro for macro scale patterns, not for more scale. So, but he's satisfied. So if there are no other questions. Well, thank you, Sandro, very much again for