and as soon as great. So welcome everybody to this day of lectures. So before we start I want to remind a few rules on how to interact with the 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 Azzaeli. Sandro is a professor at the University of Padova 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 upskate. So thank you very much Sandro for being with us and please start when you are ready. Okay thank you Iacopo. Can you see my screen? Yes it works. 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, model-independent approach to community patterns. Samir and myself are going to give instead 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 kind of the plan of the three lectures. So 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. So we want to 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 status. This is a long-standing 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. So what we observe 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 or underlying components of these patterns. So 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 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 although this is 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 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 study. So another important thing to understand is that the mechanisms that we would like to understand usually operate a different scale with respect to the part 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 biases for example we could have some finite size effects some spurious correlations or some sampling effects and this is we have it is important that we are aware of this of these problems 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 pattern. 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 duration 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 what let me be a bit clearer about this so the first form of the forming length is randomness like everything you can read here so it may be that what 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 a kind of a problem to a certain extent because of course when we measure something and we want to infer the mechanism 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 Matzolinietal 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 lego bricks with lego bricks we can do lego sets and from this 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 the list the lego sets the the final the target architecture there is a strong link between 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 other 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 is not not only a 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 so 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 the pattern that is probably interesting so and this tells us whether there are specific architectural features in our in our patterns so now let me start with 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 other relationships 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 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 and in a 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 the these green spots here and you can 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 and let's try to put a random these individuals in space so we have species eyes has an eye individuals within some total area capital a and then we place a random this over over an area now if we want to look at the number of individuals in that form 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 it tells you that okay if we assume if we assume to know the distribution of individuals amount of 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 you are given and a Bernoulli random viable with which gets one when the when an individual species eyes falls within area small a 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 Paso 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 sponges so it means that somehow here somehow 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 Paso 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 so okay let me just do a first summary this is a first so this is a first a summary of the first part very simple but I would like to make things clear so we have the species upon those the species area relationship and many other which basically are not just the result of random association we need we need more so 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 a smaller number of species and this is because the Z exponent they're tuned 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 spatial model you can get something that is different from a power law but in some regime is a power law so basically this is just a phenomenological level 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 in 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 hand raised nor any question in the chat but if you have any please now is the time to ask right I think we can move on and okay okay so there is now let me move on to the a second the a second empirical pattern which is the species abundance distribution so in this case we look into the abundances of species so before we have looked just at species species riches now we look into populations and how these populations are distributed across species the usual pattern that the typical s ad 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 longer the log two of the number of individuals this 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 ecologists and here you also 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 a 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 exceedance probability of the other so if you want i can give you the details 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 s ad 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 talk about with something reasonable basically it's too complicated because we should calculate the probability that species one has and one individual species two has and two individuals but if we want to model this we are we have to deal with hundreds of species all of them are interacted in a in a trivial way and this interaction 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-trophic communities when there are many species so the idea here is to introduce an empirical model system so basically we focus just on ecosystems with one trophic level okay so and in here we introduce some simplified assumption here we have species of bees overflies species of breeding birds and so on so in our case we are going to focus just on plants in in a forest when individuals um in within this tropical uh within one uh within let's say this uh ecosystems with one trophic level individuals are roughly can be considered independent or they interact more or less in the same way and they are clamped like I said before so uh in this way we can introduce some simplifying assumptions so in here as um Hubble has uh I mean as as Hubble did at the beginning of the millennium the simplifying 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 uh even though it may seems it may seem that neutrality uh is a very strong assumption basically we are able to introduce some models which can justify and which can tell us more about the patterns and mathematical scale and although this assumption is completely is completely false from an ecological point of view it is very useful when we want to introduce some baselines community parts and some kind of null models null um models from which from um from the from the deviations of of which we can understand something more about the underlying mechanisms so okay so in uh 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 of n1 and to ns to p of n because we have basically one species in the region and here we can basically write down a master equation which is here and this is telling you how it behaves this probability of having a given number of species within individuals we can write this um from one step to the other when you simplify you get the master equation and if uh b0 which is the birth the birth rate when there are zero individuals and d0 are zero then this um master equation may have a steady state which can be calculated in the following way i'm going uh quickly about these technical details but i can leave you all the slides so you can do all the calculation by yourselves so 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 rates 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 b of n is proportional to n and d of n is proportional to n if you do that you plug this into the formula that i showed you before you get what is called the fissure tluxet 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 and the improvement is the following here we can modify just by adding one constant so b n instead of n we have plus r and plus delta and thanks to this new expression we can obtain two different more two different species abundant distributions we can have this which is the negative binomial which goes back to the fissure tluxet series as r goes to zero or we can have this expression here in which when c goes to zero we recover fissure tluxet so why two different expressions okay these have these are good equally good to a certain extent when it comes to model empirical patterns but it turns out that the negative binomial has some interesting mathematical properties which are not shared by this other s demo 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 on 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 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 x the value of x which is very close to one and i i i would like you to bear in mind this value 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 meccato did in his theory of island biography 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 are identity because they are able to partition limited resources on the contrary instead neutral theory confers a lot of importance to stochasticity to ecological drift and basically it washes out the all species identity and because of this it's able to recover some to to identify some underlying universal patterns universal mechanism so as you know all models are wrong but some are useful and i think that neutral theory is somehow useful because it helps identifying these these 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 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 tilman 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 tradeoffs which introduce some equalizing effect on birth and death race and therefore they make look like more neutral okay this is a second part um uh miacopo is there is if there are questions no i don't see any uh in the race um 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 or everything obscure 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 with one uh in with one approach with one model so on one hand when you you see you have this forest when you sample small areas you get these species of unknown distribution with a mode that is basically at when the number of individuals is very small but then when you increase the sampled area these mode goes into higher classes 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 upscaling problem that i'm going to explain later on this is kind of a 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 problem so um is it possible to come up so here i've just shown you species area relationship and species abundance distribution but there are very there are many other patterns uh 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 uh if not all these pattern at least some of them and now i it is uh and now it's what i would like to explain you so in here uh we start with the 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 it's just it has four moves so the first is a death move death moves in which basically one individual here uh or species of one species inside dies at rate r then we have the birth move so we have an individual and this produces one offspring at rate b and this can hop onto one of the nearest neighbor's size with probability one minus gamma finally we have uh that every site is colonized by a constant drain of propagules with the rate b not now it's possible to write down the birth and death rates of this model and and therefore we can also write down the spatial master equation for this model let me give you just highlights of the most important properties it is species independent there are just basic four basic mechanisms birth death diffusion and immigration so basically in here there is only uh stock demographic stockasticity no other effect uh if you if you look at the mean of the the behavior of the mean is is quite trivial so the patterns here emerge simply because of demographic stockasticity 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 b is always must be always smaller than d and uh biodiversity so the states the the stationary state is non-trivial when b not is larger than zero uh because of the term that I showed you here this sum here this tells you that we have uh spatially 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 uh 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 so here you introduce this partition function you write down then the equation for the partition function when where you have uh the partition function in discrete space then because basically this equation four is too complicated and we don't know how to 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 b 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 b over r which is the old x is always close to one and d the spatial diffusion is always comparable to r minus b okay in this starting from this you can introduce uh two different um 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 order one and also b over b naught is order one okay if you do the calculations you do all what uh i mean all the proper uh calculation you end up with this new equation five equation for the partition function and eventually it it turns out that you can introduce a new a new um 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 spatial species abandon distribution this can be derived directly from the partition function and you uh end up with this equation here in which you have this Laplacian here of f of v which introduces the spatial effects so the effects the spatial effects of what you are looking at or what you want to derive are enclosed in this term in this function f of v now to cut a long a long story short basically it turns out that we know this function f of f of v but the the really crucial point the really important point is that when you introduce this when you introduce uh f of v into the equation basically miraculously this uh 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 funnel factor and the funnel factor basically is telling tells you um how your fluctuations deviate from a Poissonian from a Poisson run from Poisson fluctuations so in this you uh you can really simplify things and you can understand that basically space enters uh 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 um 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 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 animal but we did also for other for other forests and we get we got comparable results so okay this is just the mean number of individual species and they 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 percoration function which is given by this model by this expression here this is the exact percoration 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 in the regime that we are considering the model is able to catch is able to to match the the empirical data 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 do you 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 as 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 of under distribution looks like at the global scale and this is what we get from like I said this is a prediction and also that 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 past four so we comparable comparable results so okay third part that we are going to summarize here so in some regimes we can derive analytically spatial spatial explicit models of community patterns and we can link the the patterns in in a consistent way and it seems this model is going to tell us that it's the the model if we want the model to match the data we need to let's say we need basically it's telling us that we are close to a critical point okay although this that doesn't strictly imply that we have a critical behavior from the data but okay this is at least what the the model is telling and if we want this model can also include environmental knowledge so you can get all the patterns that we've seen when we have environmental knowledge 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 uh Alfonso is asking how dependent is the model and its results on the scale of the grid and actually the the results that I've shown you are in the in the continuous in a continuous approximation so in here we used large scales 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 scales with this because we we have taken this continuum limit in space and Ankit 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 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 all the lattice yeah he's asking another clarification about this given the model assumptions do all the species go extinct after long enough times in absence of immigration in our assumptions no because we use 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 okay and Stefano is asking if you could provide more details on how the environmental noise can be inserted in the model um that's a very that's a very nice question this is something that we so basically basic it's difficult to explain without going into 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've 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 the continuous limit the continuous limit in that in the situation but but anyway the patterns can be derived in basically in the same fashion I don't see other question in the chat nor people with the hand raised okay so uh Jacopo how much time left about 15 minutes okay okay so uh may I may move on okay 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 is different and it has some interesting applied size which may some theoretical and applied size so basically let's start let's imagine to have the amazon we have you know the problem is the form we want to um let's say predict how many species we have in the amazon forest okay of course we cannot go onto the amazon and survey everything this is too difficult to time consuming and research consuming so it's we cannot do this so what is the strategy that we could use in order to give an estimate of the total number of species that uh 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 trophic level because they complete the full problem is basically too difficult so in here the idea is to do the following like it's it is suggested by this map so in here we scattered around we scattered some fine scale samples around the the amazon forest and 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 and so on now okay we have we have an incomplete uh set of information we have we have some information we have we don't have a complete information but we have something uh how can we basically link this how can we use uh harness this information in order to upscale and in order to infer the total number of species this is the upscaling problem so basically we collect information from fine scales samples we want to infer we want to make predictions about species species rich richness at core scales 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 by cunin and you have a long list of different approaches but all me more or less all of them have have some problems have some let me go into what we suggested uh 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 the 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 scales so basically what we see at small scales for example here when we have a small number of species the species that the species of on the distribution has one shape but we want to predict something that much larger scale when the shape of the species of on the distribution is completely different so this is quite challenging and it's not difficult it's not it's not usually it's not easy to come up with something that is consistent and it's coherent and controllable to a certain 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 i showed you at the very beginning which has this form here so if we are so let's say for for a 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 it's forming variance under binomial 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 spatially are more or less spatially uncorrelated then we can use a binomial sampling and this binomial sampling when applied to a negative distribution is telling us that even at 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 scaling variant parameter but x changes with scale and you get this x at okay but this x set 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 scales 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 deduced 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 and more 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 forest which is kind of a modifies previous estimations here and here there is a list and now we can go if you want I can explain what we did here but anyway it's possible to find out how many species at least in some specific ecosystem with one trophic 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 a kind of more phenomenological approach but still working and this is and this just going quickly through this model to this approach is the following so we start from a given s ad species abundance distribution in this case we can pick a let's say a two parameters s ad 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 these coupled 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 1000 plots in 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 and with this model we predicted 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 pro 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 so 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 own non-distribution of the specie are relations 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 in depth 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 write in the chat well since no one is asking I have a question myself so oh actually there is one so I need priority so Silvia is asking does the change of the X parameter and the rescaling of the negative binomial distribution reproduce the different shapes of the SAD at different scales yes 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 no no internal modes from basically from internal peak when we have the negative binomial behavior to the fissure log series behavior when the mode is always in the first class great any other question well then I ask 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 yes yeah yes somehow I mean these the negative binomial is at least at least partially justified mechanistically with models right and I mean it's a sort of belief that the neutral models or neutral theory works well at the local community scale because individuals are relatively few and there is demographic because it is more important so I wonder I mean the fact that it seems that it works also for much larger scale but you see here and actually probably I should have been more I 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 dependent than I assume neutrality but in the second part when I explain 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 I could have derived this from a non-neutral model or whatever I we are using the negative binomial simply because it has this forming barrier 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 that's the point that's the point very it's very it's very flexible that's the point thanks any other question Bert uh Bert um are there obvious examples of when the continual approximation fades um when discreteness is important um so this is a question for the spatial model I guess yes okay here uh there are 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 continuum the other continuum it is the continuity in space uh so I mean the approximation the discreteness of population is less important than the other somehow when you use the continuum limiting space yes 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 then 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 um for large scale for macro for macro scale patterns not for more scale great uh so Bert is satisfied so if there are no other questions uh well um thank you usandro uh very much again for giving this lecture um so now we are moving for hello Matteo I never said this how are you doing I'm fine I see you are going to be sharing this unfortunately I'm not very lucky why why no I mean Jacopo uh no Jacopo is is good at this but uh anyhow we'll see how you manage so yes we wait for yeah may I close the breaking room because it takes one it takes one minute okay starting now in one minute thank you yeah I think I only have to to share my screen when everybody is back yes we are waiting just a couple more minutes uh one more minutes to for the other all the participants to join again it's interesting the breakout groups uh I've been at an international meeting where they did a lot of stuff a priority with you know people kind of uh choosing whom they wanted to to perhaps get to meet and make appointments in the end people because you know being online listening and listening takes a lot of so people in general right didn't come to the to the breakout groups so I don't know how how that works but in that meeting it it was a bit like that anyhow no I think uh probably we can start and uh we are going to for the second lecture of uh Mercedes Passapal we are going to learn about this hyper diversity but we are all very interested in so Mercedes can you please uh share the screen yes one moment okay do you see uh my first slide yes okay let me just see this here okay so okay uh nice to be back with you and uh yes this will be more of a research talk relying on on what I told you last time but first I wanted to acknowledge uh my main collaborators who did a lot of this work my uh previous postdoc Shai Pilosov who now has a position back in Israel and then Shishin He who will soon be part of the faculty at Purdue University so yes today I wanted to touch on assembly or the interaction of ecology and evolution but in a system as I said with a with a lot of pieces and I put here a second arrow coming back from the assemble system uh back to the diversity of the building blocks which is a topic that I like to touch upon towards the end of the lecture and you have seen this plot more times than you probably uh would like I just wanted to remind you that for the purpose of the parasite the uh we will be talking about niche differences differences in traits here that confer an advantage as a function of frequency so an advantage to the rare disadvantage to the common and in pathogens in general we have the advantage I think relative to uh plant systems because we do know which traits belong to these axes in particular that the traits that um essentially relate to recognition by the immune system and and the host gaining information memory about the pathogen belong here so that the distance or the overlap between pathogens in uh this uh in the traits in the molecules that are recognized by the immune system the antigens or epitopes uh that's what matters to this uh difference because of cross immunity and cross protection and access to hosts now of course I recognize that uh some of you of course uh are not so interested in pathogens I like to ask you to make the effort to think that everything I say I think could be extrapolated and of course there are issues of level of organization because I'm going to talk about variation within a species but uh I like to remind people that the very uh interesting and I think a very important hypothesis or on diversity in in rainforests in coral reefs uh rely the Janssen Connell hypothesis is basically a mechanism that relies on frequency dependence um frequency dependent interactions with natural enemies these are specialized interactions with natural enemies that act against your own offspring close to close to you and I'm not going to describe this in detail just to point out that these are uh important mechanisms uh there is a lot of recent literature on evidence for this kind of uh force and um I think it's a fascinating uh topic of course we have the problem and someone yesterday raised the issue of mutualism this is a recent paper for uh these kinds of interactions in the rainforest this Janssen Connell mechanisms but also bringing in mutualists I remember someone asking about that yesterday of course um finding evidence of a particular mechanism doesn't mean that at the microscopic level of the whole community or the whole system they are important in uh the patterns of diversity so we also have a problem in these very diverse systems in asking how should we determine whether those patterns matter and of course the purpose of neutral theory was to help in that respect now I'm going to be talking about a system here um a parasite uh that oops the main parasite of malaria and in particular about this parasite in very high transmission regions these are the endemic regions where there is very high prevalence uh in fact very high asymptomatic prevalence because people gain protection against the parasite but continue getting infected through life so in some places in west Africa with very high transmission you find prevalences in um and I don't mean clinical prevalence I mean prevalence in the population that can be 40 60 70 percent of the population carrying the parasite and uh of course this is a very large reservoir for for transmission so in essence we can think that the challenge of elimination and control in these regions is related to this very high strain diversity so can we try in this case to understand diversity not for the purposes of persistence but for the purpose of elimination and I and for those of you who uh do not come from within biology most people know that malaria is a vector born transmitted and but importantly it has a stage in the blood that's the the asexual state and then a sexual state of the parasite in the mosquito now this is what I like to do today I like to show you that in this system which is I will show you hyperdiverse from the perspective of uh antigenic variation uh negative frequency dependent selection because of competition for a host is really acting this is based on some theory this is computational theory based on stochastic agent based models and network analysis of the structure of diversity that emerges from this assembly I like to touch a little bit on consequences of this structure for resilience to perturbations on consequences for the existence of a threshold in and in diversification that I think is important because below this threshold threshold systems will possibly persist in a more fragile state so just to start we heard about multi-dimensional trade uh trade spaces in terms of niche differences yesterday we also heard Daniel Fischer and we will hear more about uh essentially uh evolution in these very very large spaces here is one and parasites and biology do it in a combinatorial way for the very specific purposes here of our system our parasite has 40 to 60 genes that encode for the same kind of protein this is the major antigen seen by the immune system so for our purposes the phenotype of interest will be a combination of 40 or 60 genes now if you ask these genes that encode for this antigen um how many different genes we find in a local population in these high transmission regions of the order of tens of thousands so you can do the combinatorics of course it's astronomical and so what the immune system sees and gains information on is this very heterogeneous population of of strengths of course this uh molecule I just will say a little bit about it it's it's this molecule PFEMP1 it's it's basically exported to the surface of the red blood cell because it has a function it makes the red blood cell sticky it helps the parasite uh in the stick to the walls of the capillaries etc I will not worry about function I will just say that because it is so exposed it needs to it's very involved in immune evasion and so this is what the parasite does once it infects a red blood cell it it expresses one of these genes then it expresses another then it expresses another and believe it or not there is a certain synchrony of the whole population inside your infection so they are showing their cards to the host in a sequential fashion what this achieves is a longer infection than if we were you're showing everything at once in about the phenotype so our fitness here will be affected by duration of infection now because if a if a host has gained memory the parasite does not express it so you have a shorter shorter infection shorter growth rate of the parasite etc so that's fitness but if you as I said you could think of any any any other way in which the the fitness is affected doesn't matter here it is duration and on top of that what the parasite plays a game of incredible diversification because different phenotypes different repertoires can shuffle their genes during the sexual phase in the mosquito and during the asexual one they can produce true innovation because different genes can produce new genes so all this variation and there is very high of course if there is high transmission there is high recombination and so you can say well there cannot be structured to this mess there is this huge number of genes new genes recombination of the combinations is their structure so this is work in collaboration with my colleague Karen Day who is a molecular epidemiologist the phenomenal one working on malaria and she basically sequels the marker of these genes in every child in these populations from Gabon and what you see in these matrix here in blue is basically all the genes isolated from from each child in each row against the against the same isolates and then you ask with this per wise type sharing how much overlap there is between between these all you have to see is that there is a light color so essentially each infection is almost unique I mean by that each combination of of these antigens that the that the hosts are seen with an infection are almost unique you can tell me well that's not surprising if it were a random system there is so much variation we expect this but if you randomize appropriately keeping the frequencies here on the right on top you see that the the degree of overlap here in red the mean overlap is lower than the one that you expect at random so there is a force pulling these things apart so we will ask whether this is frequency dependent competition and if so how can we tell I mean how should I look at this data to to be able to tell now from my previous lecture and a bit of art you may expect that we should look for clusters that the emergent niches from these kinds of interactions should be clusters of similar parasites this is not exactly what we have so what should we expect to see so existing strength theory as I showed you last time will not encompass this very very large empirical diversity it also does not necessarily in some forms allow for at least in in the versions for malaria for true innovation what I mean by that is a system whose pool is open to new variation and mostly there is no explicit comparison to neutral models by that I mean models that retain the demography of transmission all the birth death processes the extinction but have no specific interaction so we built we extended in fact an individual base formulation we have from the past in which we basically have a pool of our genes from which we get immigration into the system sorry we we start assembling the system we have transmission between between hosts and each gene is composed of two variants this is needed here because we need to be able to have recombination generating new genes but you could do it in a different way so what we have is a system in which each host remembers all the all the I'm calling oh sorry this is advancing by itself each host remembers the the types if the genes it has been seen before I will use the word genes but I mean their product here really and and if they have seen something then the parasite cannot express it so they have shorter infections and essentially that parasite will transmit less so we basically have the mitotic recombination the mayotic recombination we can do that in the computer and we can also build neutral malls we build two malls here on the right we basically have a mall in which transmission occurs but there is no essentially no specific memory there is just a complete neutrality of parasites coming in and infecting the host the one in the middle is is more interesting epidemiologically and it's what models typically do they say well your degree of protection is not a function of whom infecting you but how many times you have been infected so this uh this has a protection but again it's a generalized protection and we parameterize these neutral malls too much too much the duration of infection with age so that they are we are not comparing apples and oranges what I wanted to to tell you is that we basically look at which properties will give away that there is a structure and that there is a structure in which this competition for host is playing a role so for this we decided to look at networks because with the recombination in the system we cannot look at trees and I think these networks are appropriate also in the microbial world we follow the sort of movement of genes but anyhow here we get this network where the nodes are the repertoires and then we have weighted links that tell tell us how much two two repertoires overlap and we have this by the directional links just because we may have some repeated genes that does not matter so much what I wanted to say is that of course if we start with a medium gene pool in this assembly process we will see that we get some clusters in these networks by the way we take the network and we threshold I should say because otherwise everything is connected to everything we only keep the links of the strongest above a certain threshold to sort of have the interactions that are still the strongest so the most similar what's the structure of the most similar parasites you can see in red a typical just here example chosen from from a model where you you see the some degree of clustering and then on the on the blue and yellows the corresponding neutral models now a degree distribution or anything would distinguish these networks but when you get to systems with a very large pool of of variation then the difference between the networks is not so evident this I'm just illustrating here with the degree distribution where of course you cannot distinguish it if you look hard enough you may say there are differences in this network can we tell them apart so we built a network classifier based on an ensemble of network properties for example we can give them we can give we can consider three-way motifs in the networks and other properties and I will not go into details if you look here at the bottom in sorry in the right just to mention that we use a number of network features not just on motifs but distance reciprocity transitivity a bunch of as I said a bunch of features and here on the left you can see for high diversity the scale going from light blue to purple show you shows you how many times we properly classify a simulation if we give it blindly to the system can we basically classify it to the right model for different parameters and you should just see that we do and that how much the network features contribute to the classification is here on the bottom right you see that the motifs are important another but also some other distance related properties so we said okay we have this classifier what happens if we apply it to real data and this is work of course in collaboration with current day we have a long longitudinal study now in its seven years so that's that's nice here in northern Ghana in the bongo district so we built this classifier and then to be able to classify the empirical system we used a method called this called discriminant analysis on principal components that's what we use for the classification and if you I gave the the reference here at the at the bottom it's this method is is a very nice method to find groups in genetic data it was specifically developed for this and it combines these the multi the two multi variant methods discriminant analysis and principal components I will not get into details I would be glad to to discuss it but you can see here in the colors the two principal axes that are from many many simulations the points are different simulations in yellow and blue the two neutral models in red all the the simulations in a certain feasible parameter regime of for our system where you see the difference and the black dot is the empirical data which classifies with very very high probability to this immune selection regime so this is telling us the networks in the the network we observe of this similarity between repertoires seems closer to what we expect under those conditions now as I mentioned there were no no clear clusters it's not a simple cluster structure and the way to see it is that we are looking at limiting similarity but in this limit of a very high dimensional space and it looks it doesn't look cleanly like clusters so we did look at the structure that was the structure in a given season in a given time we can look at the structure over time where each network is built for let's say a different season and we can connect them we can connect them also to similarity the same similarity the same threshold as we did before and what I'm going to do to show you very quickly in this paper is if we look as a function of time and we look at modularity of these networks with a network approach that basically relies on a random walk basically we identify clusters and we we can well modules so groups of repertoires that are more similar to each other than two members of other groups and then we track them in time so these are modules in in time with a multi-layer network and I wanted to show you that although we don't have clear clusters if we look at the modules there is a large number of modules on top here of the left when we have immune selection we see these modules persist for a while at some point they go extinct and others are arising in this assembly process to the right in B on top you see these much less persistent in a cluster well modules in the neutral model so there is a certain in this very sparse I would say in this system where everything is so different from each other there are there is something you can call strengths but it's perhaps not what we have been used to to think about so anyhow let me just pause here and just say that of course we don't have the simple clusters but we have some limiting similarity structure that is affected by this cross immunity and we seem to have it in nature so I said this structure is both non-random and non-neutral we have some sort of clusters that would constitute it's the best you can find that you could call strengths and it's very dynamic so okay you can say does this matter at all and this is largely now I'm going to move to unpublish work what we I like to show you one example that this matters to a response to perturbation so the kind of perturbation we have here is one that reduces transmission so if you look here to the right you see transmission is seasonal in our model and in the real system and we parametrize this from the the real system including during an intervention that can last a certain number of years from two to five years we use this intervention that reduces it's known as IRS for indoor residual spraying we are essentially decreasing the number of vectors the transmission rate for a while and then we release it again I like to show you that in the model if we compare the response of the system one of the neutral models the one the generalized immunity model where there is no specific interaction just a generalized interaction then what we find is some increased persistence under this negative frequency dependent selection so increased persistence during the intervention so if I reduce transmission which is equivalent here to reducing the growth rate of the parasite right for two five or ten years in the plots and I vary for example the initial tool size of the system you can see that the extinction probability in blue for the generalized immunity tends to be higher than for when you have this kind of frequency dependent selection so there is this longer persistent this higher persistence of the parasite through the intervention and it's not due to abundance because the prevalence as you see at the bottom is very similar between the two systems so this this you can ask well why is this and I did oh sorry perhaps I did let's see well this is sorry I should have added here the reference this is a paper now in reviewing frontiers in ecology so you sorry I will add that but what I wanted to show you is that if we look at this quantity I showed you before which is a measure of overlap the distribution of overlap if you look under selection here at the bottom the distribution of overlap before intervention is in yellow during intervention in purple and then after intervention and you see that there is there are in the population parasites with very very little overlap because of this competition and that is quite robust during the intervention of course it changes afterwards because of the transience but also because you have lost a lot of diversity during intervention but this reduced overlap gives you longer duration and therefore higher fitness which allows you to make you to make it through the intervention and this is not the case for generalized immunity you cannot maintain this fraction of the population that is extremely different from each other so in some way this structure would this limiting similarity would enable persistence now I like to show you and this is the part that I'm most interested in showing you at the moment we discussed this at the end of Daniel Fisher's lecture yesterday what about the diversification the accumulation of innovation in this system and this is the connection between a force this this frequency dependent selection is acting at the level of the the individuals that are our phenotypes of interest but it's also acting at the level of the genes right that are part of the pool of variation so I'm going to kind of try to show you that the things are connected and I'm going to speculate even farther and say that a system that is hyperdiverse in nature will always be built from many many pieces a lot of genetic variation and phenotypic variation at the lower level what Daniel yesterday called the nano phenotype because the same force you cannot have one without the other and here is the idea there will be a threshold I will show you that there is a threshold this is in a paper now in review by Shishin and I and you see that this idea of a threshold in the diversification of the genes right so below the threshold a new gene comes in but it goes extinct so this invading genes come in but cannot stick around a lot above the threshold by the time a new gene goes out others have come in so you can accumulate novelty in the system we wrote this by analogy to perhaps not a complete analogy to with r0 we wrote it as a something we call rd and this is the the sort of a reproductive number or an innovation number for new genes they have you have two components g new and the new rate of of gene accumulation and t new the time the average time that one of these new genes stays around in this paper we derived an expression for g new I don't uh I don't have time here to show the details but it relates uh on the basis of population genetics on the size of the population size of the parasite new the the rate of change imitation rate in the in the broad sense here recombination rate and then this p invasion which is the probability that something new invades and that depends a lot on how many susceptibles you have so that will be influenced by again this kind of selection and then t new the expression for t new we got an approximation from some very non-linear PDE in the in the supplement but we don't have a close form solution for t new so computationally I like to show you that there is a threshold that that threshold so when this number is smaller than one you cannot accumulate diversity so here it is in the log the log of rd so the threshold is at zero and then what I'm plotting from my simulations in the y-axis is the percentage of new genes during a window of time so choose a window of time and then count how many new genes have uh accumulated by the end of this simulation and I should say that the points here are different simulations with very different both uh parameters and assumptions and we see that below this number there are no new genes accumulating below zero and above they do and what is interesting is that on the right I have plotted something similar in the y-axis but for a quantity that measures the intensity of transmission this is the number of infected bytes per unit of time in in malaria and and this is interesting because this tells us that as we reduce transmission we are going to push the system below this threshold this threshold is well above our zero of one so this is a system where transmission occurs but you have lost the ability to accumulate essentially the building blocks of diversity and this is illustrated here in the simulations each color is a new gene coming into the system I'm plotting the frequency and this is over time in a simulation over the transients of this system the the the gray color we are accumulating all the genes that are very very uh at very low frequency the colors are those that have higher frequency and you can see below threshold things are coming in but they are not sticking to the right this system is very happily accumulating diversity so you can ask does this matter for population dynamics for what I'm going to call epidemiology well we only have some preliminary results on this and what you see on top is just over time a system now with an intervention we're at time this time zero here we are going to reduce transmission in a way that for example doesn't cross this threshold in the left or crosses this threshold so we see before in the the genes that were there the frequency of the genes that were there before intervention and those that now are there building up after intervention and again when we cross that threshold in here on top in the right you see that you cannot accumulate new genes we did uh here in different colors you see in the plot with the bars how much we have reduced the this quantity in these four different simulations only in the green one we have crossed this threshold and I will focus now on the center plot here at the bottom to look at what has happened to prevalence after that intervention and you see that in the in all the simulations that have not crossed we get a reduced prevalence and then the system rebounds very characteristic of all the efforts to intervene in these very high transmission regions you release intervention the system comes back interestingly in this uh when we in the the green seeds the green simulation that does not happen so in practice there is nothing nothing in the system at the population level that is easy to measure that will tell you you have crossed this threshold but it appears to make a difference and a difference we need to investigate further to how the system responds to these kinds of interventions and in particular how how fast the system rebounds or whether it rebounds so let me note here of course that for a while here these systems because we I should have said uh this is not a simulation where we let the system rebound I'm sorry we have just reduced transmission to a lower level and this is why you see that the prevalence basically reaches a different sort of steady state after intervention we have sort of decreased diversity and therefore and we have decreased transmission so we get to lower levels so let me sort of try to extrapolate some thoughts here that I think apply to the system but may apply to others these hyper diverse systems may well occur at the opposite end of neutrality where coexistence is assembled under interactions that are specific and frequency dependent so interactions that will lead to this negative frequency dependent selection which is a form of balancing selection and therefore should enable diversity it is in these evolutionary systems along the lines of the stabilizing competition for example of Chesson I think more importantly large phenotypic diversity in this system is built from a large pool of diversity at the lower level of organization and this is not by chance this is not but by chance because I just showed you that the ability of the system to accumulate diversity with this kind of critical threshold is going to also be influenced by these same interactions and selection so this would set the stage for the existence of an unappreciated threshold that concerns the accumulation of genetic variation on which high biodiversity is built so that when we lose diversity of species for example we may be losing much more because we may be losing the diversity of the traits that allows the system to have that coexistence in the first place and the question that is worrisome is how do we know that we have crossed this threshold and transition to increase fragility of the system I think that's important there is a question in the chat by Aditya so how do you parametrize mutations are mutants related to the parents at all or are they totally new types yes thank you very much that's a very good question I know they are related to to the parents because remember and of course this the two that each type has two two elements so it's very hierarchical allow us to do recombination and so so there will be some similarity and in fact we have we use an empirical sort of result from recombination on that that for example if the I'm trying to remember the details but not all the recombinants are viable right like something very far from the parents is not going to be very valuable so so we are kind of it's not a completely new type and then you can decide you can sort of measure similarity at that level of the epitopes we have some evidence that in this part of the gene there are two epitopes so two parts of the the molecule that the system recognized there may be more but we recombine them and then we we have some similarity to the parents as a function of that we also have mutations but they are not so important in this system this the details of that mall are described in the paper by he and collaborators in nature communications 2018 so if if you like to see more details of this that's the place to look did you say thanks you thank you thank you for the question I forgot I am trying to give here the big picture but and yeah I obviously I think some of these results will not be dependent on these on these specific assumptions but but I think we have to consider them in particular when trying to to parametrize the system and so on so this this question of the increased fragility of course here at the moment we have only preliminary results on on this I have shown you some preliminary results we know in malaria that when we intervene we move the system towards lower diversity of the parasite when when we reduce transmission and we know that the low transmission systems that are so for example the low transmission systems in continents where for example transmission is lower like South America or locate geographic locations of low transmission are systems that respond more easily to to intervention than these very high transmission systems so there's an interesting question of could we determine how could we determine that we have crossed this kind of threshold so that then we could say the system is at a stage where we should hit it with further intervention because it is in a susceptible place of course for conservation we would like the opposite let me now mention that I like to end with some references to other hyper diverse systems with this kind of selection or interaction other equivolutionary systems like this for some of you who may be interested this appeared very very recently in this work we look at the diversification strain diversification of a microbe and a virus through the CRISPR induced immune memory of the mic of the microbes so we know microbes can acquire memory of who infects them of the viruses and this is also a very piecewise system a very combinatorial system where the microbe basically takes parts of the genomes into its genome and that constitutes memory so we work here with bipartite because for microbes and viruses we may we can look both at the host and at the viruses we can look at bipartite networks and if you are interested in this connection between the equivolutionary dynamics and how you see the network structure that arises we looked at this here it's a much more dynamic system with different dynamical regimes that alternate until the viruses go extinct and that may be because this is much more of a predator system a predator prey system than the than the disease systems I was showing you before just to end this another this is another paper it's not it's not of mine it's but it's it's a phenomenal paper by the group of in Harvard, Mark Lipset and colleagues on another pathogen that is incredibly diverse streptococcus pneumonia they have been looking a lot at how these systems respond to vaccination you vaccinate with specific strains how does the system respond why is this paper fascinating and I think it is because it gets to this very important question of these niche differences or these frequency dependence how high dimensional are these these spaces they look at the genes of these microbes that are non-core genes so they are believed non-essential they are not present in all the in all the genomes they looked at I forget if it's three thousand or four thousand at that point it doesn't matter and they wrote a system of equations based on the replication equations which on the basis of frequencies will predict which strains would come out from this from this vaccination in the in the well in the empirical system in the real world and they managed to do so so this means this means that some that very that some very effective frequency dependence is being played by genes that we don't even think are so important it must be through the through myriad interactions maybe with viruses with who knows I can't be I can't begin to to kind of think how this is happening very very interesting paper and I recommend that you look at it so I like to end here and just uh record I don't know if I went too fast or too slow but maybe there is time for questions this is uh I like to acknowledge as I mentioned current day and her group for all the very uh interesting conceptual and empirical work and thank you for listening thank you very much so it's open for questions so maybe I can start I think you show this data from Congo and where there is there was this very high diversity and which you could quantify and could one use data of this type to quantify this at least maybe not the treasure but relative I mean how populations tend relative to each other yes yes that is that is the case by the way I I didn't put the slide here I thought I had put the slide here if you compare population of course if you compare uh systems uh parasite populations in different geographical regions right uh it has been done I showed you I wanted to show you where we got this uh information on the number of genes in different places so what Karen does is she sequences the genes uh looks at uh counts as different genes those that differ at a certain level of of uh sequence divergence and then she counts them and she does the typical cumulative curves that we do for species but we do we can do them here for genes so I wanted to show you those curves but I didn't put them here so they are typical cumulative curves and you can do some extrapolation on on on how uh at what level they should saturate if you go to South America or New Guinea those curves saturate at much lower levels than for West Africa very very clear difference so yeah biogeographically we know that now of course we are saying that um that this transition is not just a continuous transition that there is this uh this uh threshold that should apply right now the so can you apply that in time to assist them in a transient state where you intervene and now the the the problem is you could monitor of course you can monitor as we are trying to do the changes in the gene diversity but in terms of the accumulation of new genes that are coming in right you need to do that over a certain amount of time so it's not a very practical way to to find out whether you have crossed this threshold there must be other there must be other ways um yeah to determine this but certainly the point we are trying to make is that we should be monitoring this diversity which is not typically done because this is such a complex system that uh it's typically not even considered in epidemiological moles of malaria right i mean we at best deal with generalized immunity yeah yeah sorry for a long answer question from the chat uh the question is uh from Pavel Yomi when uh we are below the threshold limit uh is it possible for the generic system to go to extinction yes of course it is and the the thing that is uh interesting maybe she meant the ones with uh when you say generic perhaps you meant uh maybe i'm going to to answer that in two ways for the system for the genetic system okay thank you so so it is possible but this is what is interesting right the and i think this touches on some open questions at the moment is that that threshold so that when you mention extinction you are talking about demographic extinction right and to some degree at least from the from some perspectives r0 or r would be more important there right so that's a demographic number and and of course you can view you can look at extinction in a stochastic way and so on that's purely a demographic phenomena right uh here uh this this uh system as i said you can cross this threshold at levels where you are transmitting still very well right but you are doing so with much less diversity so it is somehow of a hidden threshold from the population dynamics but we are saying that there are that it should be important that we have begun to to look at that it's very early work when i showed you that the system that has crossed it essentially stays at the lower level right and has a different response to the perturbation so presumably we know i'm going to speculate here the biogeographic regions that that steady state have very different levels of diversity have over very long evolutionary times and large regions in some sort of metapopulation right have assembled much less diversity so they live in some of those may exist below this threshold right uh and we believe the ones that that uh in west africa that are still transmitting at very high level uh are on the right i i like to mention that some of the pieces of these genes that we find in humans of the var genes can be found in parasites of primates so that the phylogeny of these var genes can also have very very deep uh branches because this kind of balancing selection will promote persistence not just of the parasites of the parasites that coexist but of the once you have become part of the established genes you can exist for a long time and so so that's why also you can accumulate diversity and i think that uh yeah these are these are all open questions sorry i i took long enough are there other questions thanks so there are no other other questions so you should you can either raise your hand and uh open up your microphone or write in the chat it looks like uh there is no other questions so so i have a curiosity Mercedes if i may so you spoke about this uh like a synchronization phenomenon that occurs inside the body where this parasite expresses different epitopes at different times so how does this uh how is this synchronization uh achieved yeah that's a fascinating question by the way there is a body of theory but i cannot uh i don't think there is a full answer i mean there isn't a an understanding of that right there is a literature on that there are different uh hypotheses and the the question of course one critical question is on which the different moles differ is whether the immune system is involved right so is it is it something just from the parasite or is it that falls to the immune system right and so of course it's not completely perfect but there are waves of parasitemia within the host that are dominated by some of these types right so we have uh i must say of course you know no it looks like this uh i mean this synchronization inside the host and the diversity they are a key factor right yes yes of course the synchronization is very important because if you show if you showed your your 50 i'm going to say 50 if you showed your 50 faces at once right then the immune system could react uh you know could protect here and could shorten the infection much much faster right um and in fact i should say but of course this sequential for the theory of course here because the fitness is related to duration of infection but i showed you models in which uh the other day in which uh the the the memory affected the risk of infection given contact so the transmissibility and not the duration it doesn't matter because the fitness of the parasite is the product of the tool so i find i strongly believe that the results i just showed you would extrapolate to two other systems with different assumptions as long as you had this competition that depends on overlap and distance okay thank you very much so so i think we can stop here and maybe go into breakout rooms and uh then we reconvene uh at uh 15 past six european time for uh the new fish second lecture thank you all thank you Mercedes thank you let me stop sharing am i still sharing no no i'm not Mateo you can see my screen and and hear me i can see your screen now i'm going to call people from breakout rooms okay no breakout rooms i don't know i got invited to one which i didn't join but okay i was in one two and then i read but i was by myself so something something is going on yes so they they've been closed uh yes people are slowly coming back i just wait uh one more minute actually can you see me also when i talk because i don't seem to see myself is it yeah we see you i have the same problem that if i don't leave everybody like you know in the in the view that has everybody vertically or horizontally i don't see myself when i'm speaking so i had to leave everybody which covers part of the oh i see okay i don't know why otherwise you see just Mateo but we see you okay usually i've been able to but i don't quite know um quite know why but well yesterday i wasn't you couldn't see me talking anyway so today is um i'm trying a different setup which hopefully will work so it's okay i think probably we can start because i think the number of participants is not going up so it's a pleasure to have Daniel Fischer for the second lecture on high dimensional ecology and evolution thank you Daniel okay thank you um so i'm trying a different setup today with my um computer so hopefully it won't um cause um cause problems but let me know if it seems too many any difficulties and i can try to switch back to how i was doing it yesterday with some time delay so um yesterday we talked about models in which only had one strain um and when a mutant came up if it was if it was doing better in that environment then it would replace it this would change the landscape by a small amount and that would change the future evolution and i'm sorry i didn't actually give this the model a name this model one could perhaps call the fitness snowscape a landscape with something which is static a seascape which people talk about is a dynamical environment but here it's the evolution itself that changes the environment so like when you're going up a mountain and you change the environment that you're walking on as you um as you walk and what i showed was that the simplest model and i should emphasize the particular model i wrote down with sort of the cubic interactions is the simplest model and that exhibited the red queen phase where the evolution just continued without um without slowing down and this i mean i said that it was very robust and what i mean by that is if i change the model somewhat um or do the different range of models then we'll get the same behavior in the limit of infinite um of infinite dimensions i raised the possibility that there might be some other behaviors as well but haven't um um didn't talk about those and then i left it open a question about diversification which i know almost um nothing about but i think is the most interesting question for those um for those kinds of models as you mentioned none of this is written written up i think someone asked me for the name of the model so i guess this is the closest to being the um the name okay so what i'm going to talk about um uh today and tomorrow it's much closer to things people have been um several speakers have talked about so far um which is really talking about a situation where at least i've initially got a fixed environment but i'm going to have multiple strains and here i'm going to emphasize calling them strains rather than species because i'm mostly going to be interested in one species with many strains within that um uh within that species and i'm going to talk initially about assembled communities where there's no evolution so again this is what several people have talked about um and introduced a random um lots of altera models which then we need to talk about the statistics and they have a stable phase as Stefano Alessina talked about but then that can be unstable i'm going to then talk about a special model which is the perfectly anti-symmetric um interactions um then more generally that show that the the chaos that occurs that diverges um and then how you this can be stabilized by migration and that's sort of going to be the main thing i'm going to get to so i'm going to introduce the sort of phenomenology in the basic behaviors with some rough explanations and then i'm going to go and talk about the um uh the theory and using the dynamical mean field theory and this is a theory the the method is also something which is applicable to what i talked about um yesterday but there i didn't explain how i got anything and then i'm going to turn to looking at slowly evolving um uh communities that i'll get to tomorrow and also i'm considering sort of phenotype models which would determine the um the interactions so i should mention so all of this work that i'm going to talk about today with all the completed parts was done um by michael pierce and natish agaruala and is published in in pna s um this year early this year okay so the models that um uh talk about the generalized um uh lot of altera models and i'm going to call the number of strains k so i labels those runs from one to k and n i is the population of strain i now i should apologize is this a different notation than um savano used he called the population x i and called the number of strains n unfortunately there's not completely standard um um notation okay but i'm going to start but i'm writing the model as um as he did someone has the basic growth rates or growth of death and his death rates here r i and then the interactions between the um between the strains okay but we're going to be interested in the closely related strains okay so if i have closely related strains then it's natural to replace this by um things which tell me what how the strains are similar to each other so i'm going to write that r i then say is going to be equal to some r bar which sort of average over overall strain that's some common part plus some varying part um plus some part which is varying which i'm going to call s i okay and i can think of the s i then as being like selective differences between them and the natural one is then to take the s i with an average of um of zero so that's going to be the variabilities the way the strains differ from um uh from each other so that's going to be my a hours now what about the um um uh the interactions okay well the interactions i'm going to consider two um uh two parts so i'm going to take the a ij is again going to have some overall average um uh average interaction which will generally be negative if they're competing with um um with each other that'll be less than zero and then they'll have a strain specific one and i'm going to put a factor of n in here just so i don't have to carry factors around n anywhere else and this is going to be the interaction with it with itself so q is really going to be the sort of niche um uh niche interaction um or one over the carrying capacity of that um um of that strain okay and then we're going to put in some random parts and again i'm going to pull out a factor of n out in uh in here so the vijs are the between strain interactions and so the vij again i'm going to take that mean zero and then i'm going to say something about what the statistics of those of those are okay so the vijs are going to mean zero typical magnitude one sort of sets the overall scale now since these are going to be considering different strains this is going to be these are going to be much larger than the differences so this and this is going to be small compared to these and so what does that mean that means that the r bar and and the a bar basically set the total population you constrain the total population because they are the bits which are the um uh the main um interactions with the growth and the competition so they're going to constrain the total population and that i'm just calling this big n which is the sum on i of the ni okay and then i'm going to consider what they essentially do is make this approximately constant and um and then there are variations of course between um with between the ni but with the end of being roughly um roughly constant okay so in that case then it's natural to define the frequencies so i'm going to define the frequencies which are the fractions in the fractions in the population so i'm going to find new i as being equal to ni over the big n so it's the fraction of the population so obviously the sum on i of the new i equals one okay so that's going to be my um uh my model and then i can rewrite the um the model in the in the slightly different form i'm combining these i'm now going to pull out the r bar and the um and the a bar and i'm going to write the the model this way is the di and id t and i'm going to i'm off going to use the notation of a dot for that so that's going to be new i okay times well now i've only got the differences coming in here so i've got the si and then i've got the plus the sum on the j of the vi j oops let me put them in another order um this has the minus the specific one the q times the new i and then it's got the plus the sum on the j vi j new i new j okay but then i'm going to put an extra piece here okay where epsilon is a Lagrange multiplier and the role of that is to make n equals constant or more specifically enforce this constraint that the sum on the new i is one okay so i'm replacing the effects of the overall growth interactions and i'm only going to consider the effects of the differences so this is the differences in the growth and okay rates this is the interaction with it itself and then this is the interaction between the between the types so this is the model which i want to um i want to understand and this is going to have average zero this is going to average zero so of course what we have to talk about is we have to talk about the statistics of the interactions yes then so do you really have a vi j new i no j there or it's just no j thank you thank you i um this is the drawback of writing things in real um is real time is one keeps the audience on their um on their toes so thank you for for correct me yes um so that's just a standard um the standard interaction and so the first term this term here is would be you're often written as like one over the it's effectively like one over the comparing carrying capacity but i've you know rescaled things um uh real scaling is up so i don't want to keep things um um all over the place oh so i should say one other thing before talking about the statistics i'm assuming no stochasticity at this point okay but i will want to consider extinctions if new i becomes less than one i.e. n i sorry um less than one over n which is corresponding n i less than one of course then i kind of less than one individual that'll go extinct so i'm going to mostly consider the domestic the deterministic interactions then i'll talk about the effects of the extinctions which of course some play an important um um role um particularly because they'll happen and even even in the simple um um the the simple model so this is my basic model and the thing which i then have to um uh say some things about is what the statistics are i've said the interactions are going to be interactions and the si's are going to be random okay so the si's are going to be independent okay and i can talk about the variance of those um the variance of si is just sigma s squared okay and these i would expect that they'll be small very small because the strains are all similar okay so this is sort of a consequence of the um the prior evolution that they're all going to be very similar to each other if one was a better that strain would diversify and so on so this is going to be a basic assumption that small and to often i'm going to set that equal to zero but the crucial bits are the vij and what the statistics of those are okay so again i'm going to take for um um uh for i not equal to j so that's the interactions i'm going to take vij squared um equal to one so the variance of the v's is just equal to one so this just sets the overall um uh the overall scales of the time okay but the crucial bit is then what are what are the correlations okay so in particular i want to ask what is the correlations between vij and vji okay so this is going to be the only correlation which i'm going to put in so this is going to be the only correlation it's going to be between those so this is how the correlation between the effect of strain i on strain j and the effect of j on the on strain i okay so if this is going to be competitive then i would expect this to be positive okay so i'm going to define a parameter gamma here which is going to be that correlation the competitive would correspond to gamma being um uh bigger than zero if i'm correspond to gamma equals one is totally symmetric that's correspond to a symmetric vij and that's often what's um what's considered for models with competition okay in general gamma lies between minus one and uh and one sorry then then it's only to interact yes shouldn't vij also be small so the variance of vij be over the one over the number of species or well that's that's a choice of a convention that's a convention which often people choose for being used for i want to talk about more things in terms of unre-scaled um unre-scaled quantity so the interaction is setting the basic um scale there's nothing intrinsically that it's going to be small okay now since the new eyes if there's many species here the new eyes are going to be each of order one over k then this interaction piece is going to be um small coming from that with these random of order one so this whole piece is going to be of order one over square root of k but i explicitly don't want to scale out the um the k's it's convenient for doing theoretical analysis but i don't want to do it because it sort of confuses what one's assuming when one does that okay of course i can have k two three 20 i can add more things i don't want to be rescaling the v's each time i add more things or take more things um or take more things out so these are i'm trying to keep it in terms of the the physical or the biological the biological quantities okay so gamma equals one would correspond to um symmetric what about gamma is uh less than um um less than zero what would that correspond to okay so gamma less than zero so this has um uh two possible things it can um correspond to one of them for example is if one if i have a direct competition so if one beats two okay what that means is that v one two is bigger than zero but of course that means two loses to one so two v two one is less than zero okay so that's a natural way in which you can get anti-symmetric correlations in the um in the matrix okay i just want to add an extra page in here um before i have my next bit um okay so that that's one um one possibility but there's another possibility in this i'm going to talk much more about tomorrow and this is what makes it much more um much more interesting there's another possibility which is i have bacteria um so i have bacteria and so those bacteria will have populations say bi um and they'll say b and k of the bacteria a number of those and then i have phage with populations p um uh sub l and i've got some number of um of those so i've got the bacteria and the phage and these interact with um uh with each other and so if i look at the dynamics of those then bi dot would be bi okay time some growth rate for the um uh bacteria but then there'll be minus a term which is coming from the interaction with the phages so this would be some matrix here um h i l times the population of the phage okay so that would be of that term where these are going to be positive because they're the negative effects on the bacteria and then i've got the phage the b the pl is going to be sorry i also want to put a term in here um put a term in here um which is a stabilizing term most um someone um j of um nj so that's coming from the um uh the competition okay and then i've got the the phage so the phage the phage are going to die at some um at some rate the phage are going to die at some rate but then they gain by eating the um uh bacteria so they have a someone i here of some other matrix um um some other matrix f um uh now i um l i times the population of the bacteria okay and again these are going to be positive the way around i've got so if i think of that as putting all these together and i'm going to think of these as being strains of one species of bacteria and one species of phage so i now have a two um uh two species model but with diversity within the two of them and if i look at what the interaction matrix is going to be there so what is the action matrix going to be in that form okay well it's going to have this form here if i think of putting the bacteria on top and the phage underneath it's going to have some minus constant all the way through um through this part from the bacteria interacting with each other then it's going to have minus the h from the bacteria interacting with the phage over here it's going to have the f from the phage interacting with the bacteria and there are no specific interactions of the phage with these other okay now what do i expect here i expect that f will be approximately f bar plus something small so delta f um i um l and h will be some um average um average value plus some small variations so again in the same um um spirit um sorry i've got my my indices backwards um the um um so i have these here and i would expect these will be correlated so more specifically f and h transpose are correlated okay that's corresponding to the fact that of course if one phage does better against the bacteria that bacteria is worse for the phage okay so if i look at this whole matrix here this whole matrix here it has anti symmetric correlations in the dominant parts here there's also this part which is which is symmetric but there's a dominant effect here which is the anti symmetric ones okay now turn out this model behaves very similar to the simple um random lot cabalterra model that i'm going to mainly talk about and i'll come back to this one um some tomorrow so you can think of this as being one of the primary motivations for considering models in which um gamma can be um gamma can be negative okay so one thing we can then um start to um uh talk about is what does this model look like so i'm going to focus on this model here so we've got um here we've got two um uh main um parameters we've got this parameter gamma here um um which we can make him read to find that i've got that parameter is the key parameter this um epsilon is just the um is the grand multiplier that adjusts itself so the only other parameter we've got here is this q which is this niche um niche interactions and that's our basic parameters okay so we've got a model now with two um parameters one is the interaction within the um within the same strain between same strain and one is the interaction with others okay and i generally have then the um the s's as well so another parameter which is associated on the s's but i'm going to mostly consider the simple possibility where the s i's is zero there's no overall differences we can add back in the effects of those so now i'm just going to show a phase diagram and i'm going to say how we when one gets it in a while so the s is always equal to zero so we've got our two um uh two parameters here we've got the q which is the self um the self interactions that's the strength of the niche interactions and then we've got gamma which is the um the symmetry okay well if the interactions are very strong relative to the um um within the strain very strong relative to the interactions between the strains then we've got the standard um behavior and the limit of a large number of species we've got a um so this is now we're going to want to consider k the number of species being much larger than one the number of strains much larger than one and up here then in this regime up here we've got a single stable community large community stable community and that's actually um um with uh hyperability for the limit of large numbers of strains there's going to be a unique um uh unique community there okay so this is a result which has been worked out by um uh by by many people particularly a lot on on recent um years but it's more or less something which may already um uh Robert may already already knew okay but then there's a boundary which is where that goes um unstable so there's a line I can draw across um across here um and this is a line here and this line goes from zero to a value over here of q of square root of two times k and so here's where the square root of k that Matteo was asking about comes uh um uh comes in so this is when q is bigger than this I have a single stable community but when I lower the um when I lower the q and I cross to here then I then the system goes unstable I lose a single unstable community okay and then the big question is what happens down here okay what happens under there there's not a single large stable community that's the one thing we know okay now there's a special line there's a special line which is corresponding to being along here okay that's the line where it's the perfectly anti perfectly symmetric model that's been studied by um a lot of physicists also um recently and in that case what's um uh known so this is for gamma equals one um for gamma equals one here there are a very large number of stable communities um there's actually exponentially many um uh um different communities okay but that's very special and it turns out that as soon as you go away from gamma equals one things change this behavior over here has a lot of similarities as far as the dynamics to the things I talked about last time of the random landscape is you never actually get to one of these communities you sort of wander around things go bouncing down things come back up again and it never really settles into them but if you just look in terms of the the communities that exist there are exponentially many stable uninvadeable uninvadeable communities of subsets okay so it turns out this um this community up here um is going to have in this space here it's the number of the size of the community so the number of which persists in the community is going to be bigger greater than or equal to k over two so more than half the types persist okay as you go down through here let's say it goes unstable and we want to talk about down if we ask about putting a little bit of the s's in um so if the sigma of the s squared is not zero um if the sigma s um so the variance of the s is as long as that's um less than or of order um actually tonight is you order one over root k okay then you only get qualitative quantitative changes um to the phase diagram but it's it stays qualitatively same okay but I say we're mostly going to say the s i's equal to zero and I can talk more about that so one thing to note is that you need to have the selective differences being very small to get this behavior if the selective differences start becoming substantial then if we go back to our our um our basic model here that means this term will then tend to dominate and this term will be small compared to that because this is some of a whole bunch of random things this will this will be dominate and I won't get the interesting um um the interesting behavior okay now what about the q well here you notice that in order for the q to be in big here it has to be bigger than a value which is of all the square root of k okay so to get this you need to get the stable community to get that you need to have k um greater or equal to of order root k okay so this corresponds to saying the interactions with your siblings right these are all different strength interacting with siblings have to be much much bigger than the interactions with all your second and third cousins so this is basically equivalent to assuming that there are niches you assume that each one interacts with its own strain um much more than interacts with others okay and there's no a priori reason to um assume that okay so the particular things that we're going to um focus on here the bit I'm going to want to focus on is actually along here along this line there where either the q is very small so you might as well set it to zero the natural assumption then is if I have these closely related strains so with the close relatives the natural thing is to say that q is very small um q equals zero I've just got the random parts with that as well if q's of order one it doesn't matter um uh doesn't matter much as long as it's small compared to um root k so I'm going to consider that and then for simplicity I'm also going to consider the s i's equal to zero um but I say that will come back and um uh re-examine so I think uh you want to get the questions as they arrive there is a question in the chapter so you can read it it says as anti-symmetry in a still doesn't capture the fact that the signs in the upper and lower triangles or the bacteria phase system are positive and negative doesn't make a difference in behavior so the the um the interactions by definition the phages are bad for the um uh bacteria um that's their lifestyle so these terms here this is not a negative sign and the h's are all positive okay over here the phages are eating the um um the bacteria and so this is the those terms are also all positive what what the random parts is in the difference so this is positive this is positive but then the random parts are associated with differences between okay so I'm going to come back and talk more about this one um um this tomorrow okay the other questions on the model or the sort of basic setup and I say I'm going to explain a bit of how one gets this um phase diagram but I mainly want to talk about this um focus this part here there is another question uh is there anybody about the self interaction or is it always exactly okay sorry I didn't see that one so the self interaction there um I can have have some variability in it I'm in particular if I look at my interaction here I've got a part where i equals to j so that would be the variability in it it turns out if it's variable variable by similar amount to how the interactions between strains vary then it doesn't matter much in the limit of large k it only matters in the limit of large k if it's big in particular if it has to be of order root k that's been much bigger so I can turn the other way around and I can say let's have fixed q that's going to be a property of you know the biology of fixed distribution of the s's and ask what happens if I add strains so I'm assembling a community here I'm adding more strains and then what'll happen is I add more strains it'll go unstable okay and where it goes unstable would be associated with how big the um how big the q is okay so that's the basic results of uh of may he didn't really quite do things right but the the overall result is is um um he he certainly got got right so that's this part of the phase diagram up here where there's a single large stable community and what happens down here for large k really only started to be investigated um um in recent uh recent years and I'm going to show some um simulations of that but then really try to develop the theory okay so we're going to consider the behavior along um um along here okay so we're going to consider the um uh the close relatives um we'll focus on this and for reasons that sort of motivation of the um bacteria phase we're mostly going to concentrate on this region um um on this region here so we're going to concentrate on gamma being less than um uh less than zero and between um zero and minus one but actually sort of believe that most of the behavior actually persists in this whole um whole region except on that special um special line okay so I'm going to not not talk about this table community I'll just say something about how one gets it but I'm really want to talk about what goes on um uh what goes on in here okay so for doing something general I want to do something very special so one of the things that one has learned from you know experience in statistical physics is a lot to be gained from having particular models that one can really under analyze and understand in detail and then one can sort of use ways of thinking about what's more general and what isn't to ask which features of those might persist and which ones don't okay so I'm going to first talk about a very special model and this has been looked at by um over many years um a special model and that's the case where I'm going to have the q equals zero but I'm now going to have gamma equals minus one okay so this is going to be the special thing it's perfectly anti-symmetric and now I've got the anti-symmetric um uh uh v matrix okay perfectly anti-symmetric okay so this has a very special um uh property it has a um various things which are known about it it has a unique um it always has a unique um stable community and by stable here I mean an uninvatable um community and this um Stefano talked about in some generalizations of of this and the size the size of this community is going to be um approximately k over two um plus or minus some um order square root of k which all depend on the particularization so this community always says it's unique stable uninvatable community okay and in that community the each of the new eyes will be some fixed point um uh value new i star so this is then the fixed point of the dynamics okay or new i will be equal to zero for the extinct points for ones that go extinct okay so if you look at the surviving types if you look at the surviving types then it's very special and you can see that this is very special because this fixed point here this fixed point is marginally stable because of the anti-symmetry it means that all the eigenvalues for the stability around that fixed point are imaginary so the imaginary eigenvalues it's not stable or unstable it's very special okay and that very special behavior is associated with a conservation law a conserved quantity and say this is just a mathematical artifact um a conservation law but which Stefano um had um talked about and this um um this quantity here which plays the role of a um of an energy okay it plays the role of an energy i'm sorry i'm gonna i'm gonna call it that and it's just going to be minus someone i of the new eye stars okay times the log of the um um new i okay now you can add another piece to this which um um Stefano added you don't have to include that for in this case because the total number is is is one so you've got this but the elation here and this is conserved it's a Lyapunov function but it's conserved um a conservation law after the extinctions so after all the ones have gone extinct before that it's a Lyapunov function it'll um increase and but when they go extinct they'll saturate and then become a conservation law okay this is a conservation you notice this is a sum of a whole bunch of independent terms the other thing to note is that in the log variables so if i define li which is the logarithm of the of the new i so this is just the log variables they're the natural ones to think about if i think about populations growing and growing and shrinking at some the rate depending on whether their growth rate net growth rate is positive or negative so this is the the li's the natural variables and in these variables the phase space volume is conserved um its volume is conserved okay again it's just a it's just a mathematical um um uh nice city um mathematical nice city but it enables one to do certain things okay so this means that there is a steady state steady state of this which is like equilibrium um like equilibrium stat mech of a bunch of interacting um of independent particles so you can immediately write down the dynamics there's a quantity which is like the temperature there's a quantity here which is like the temperature okay sorry Daniel can i ask a question so yes this so the new i in your equation for the e is a dynamical variable right the new i's are dynamical variables right so this is generally a function of time is uh is a fixed point so so but so the dynamics does hover around a fixed point or does it go to a fixed point ah okay so the dynamics if i look at the simple cases okay so i can look at the very simple um um a very simple case here and i can ask what this um um uh what this looks like and the the simpler situation i can guard if three of them survive okay um let's go down here so if i have three surviving okay three of them in the community in the stable community okay the sum of those is always equal to one so i can draw the flows here of in the news in a phase space where this would be pure one this would be pure two and this would be pure three and somewhere in there there's a fixed point okay so that's a fixed point but it's marginally stable and what that means is it's a family of um a stable orbit so there's that one there or there's another one which is here so these are all limit cycles okay these are all limit cycles the whole family of those so there's a whole family of the um um of the state and that's where it's very uh um uh very special okay and in this simple community they're just periodic like this and this is in fact like the classic lotcavolterra model of predator and prey which has a family of um cycles here so this is like approximately like the original lotcavolterra predator prey model with one of one of each and its dynamics also looks like um looks like this and there's a whole family of cycles but we sort of know that that thing is very um um that thing is very special okay so what happens more generally so there's going to be something which is going to parametrize this um um this family um and there's something which is going to parametrize this family which is roughly speaking how big a range they go over okay so this is going to this quantity e it can be anything it's going to depend on the initial conditions okay so this quantity e here can be um anything um well it has it's constrained as to what values it can take but the the v is um variable and so just like in thermodynamics this is going to classify the um what state one is it there's a whole family of states right so there's a family of states a family of states here and they're parametrizing something which is like the temperature which is um which is proportional to the average of the of the energy over all of the um overall the species okay so the quantities like the temperature it'll determine how big these fluctuations uh the fluctuations are going to show some pictures in a minute okay this temperature is going to be conserved I've got simple statistical mechanics so if I look at the the probability distribution of all of the um all of these the probability distribution then is going to be um um the probability density of all the um of all the new i's okay it's going to be proportional to new i to the minus one plus a quantity which is going to be related actually to the new i star over um theta and that's because of the new i star appearing here I exponentiate this I get new i to a power and then there's going to be a factor which is just going to be associated with a constraint which is the delta function that all the someone j of the new j is equal to one and of course I need a product over all of the i here so they're all independent of each other they're independent of each other and this distribution which looks like this okay rather amazingly this is identical to the distribution from purely neutral theory so this distribution is the same distribution you would get out if you just had a large number of species with a bit of migration in um into them and they would have a distribution here um which would be a distribution which would depend on migration rate in that case but it would have exactly the same distribution they would be independent so this looks like a neutral distribution okay but this is completely if you like coincidental this is just coincidence because the dynamics here is nothing like neutral there's no stochasticity it's driven deterministic um dynamic okay so let me just um there's a couple of questions um here um so one of them is is why is e of the sum of the terms with no interaction so that if you I refer you to Stefano's um I was seen as um um uh talks where he showed explicitly that this is this this corner is conserved it does depend on the interactions in the sense that the new i star right these new i star um are given by the v's so the v i j's determine those new i stars so it does depend on the interactions but the only way depends on interactions though and this particular quantity is not interactive nevertheless the species are interacting with each other the strains are interacting with each other and so there is dynamics and that's what we're going to be um um interested okay so this is all mathematical this is just going to motivate some things if you don't pick up this it doesn't matter for what um um uh for what comes later yeah there's another question uh whether it is a coincidence that the formula for e looks like an entropy it looks like an entropy it's not there's a way of inferfuring it that it's you think of it like an entropy but it's not very useful I don't find it's more like an energy um yeah um let's see was the another other one there um yeah so this this is like the these are the neutral um cycles in the um kavatera model they occur when three in the community okay but when you get more than three when you get more than three here so for k greater than three well okay much much greater than one at least um the um so for k much much greater than one what happens then is you might think that you would get a whole bunch of cycles different cycles remember everything is interacting with itself interaction between cycles tends to be unstable it tends to drive chaos and so here one gets a chaotic state um a chaotic state and there's a whole family of them I say parameterized by this quantity um uh quantity theta okay s