 Okay, welcome everybody to our Friday session of the winter school. My name is Antoni Solani from my STP, and I will be chairing this short session of today. As always, the rules are that you are invited to type your questions in the chat. And at the end of the talk, there will be a discussion session which your questions will be answered. For the viewers who are following us from the YouTube live stream, you can write down your questions in the chat, and I will read them out. For you in the end. Okay. So with that, I will leave the floor to Andrea Ronaldo from the Corporate Technique Federale de Lausanne. We'll deliver the first of his three lectures. Thank you, Andrea. Thank you so much. Thank you so much, Antonio, and welcome everybody. Let me share my screen. I'll be mostly counting on that. And I may be a little bit abrupt in stopping at the 45th minute, but I'm confident that because I have three lectures I can compensate the material should I be somewhat carried away in the presentation because I kind of like the subject quite a bit. So let me, let me show you what's a rationale or networks as ecological corridors, and the subtitle the reads species populations and pathogens and, and in fact, I think at the boy from Bangladesh about which, I might be speculating later on in the third lecture in particular was trying to convince me that the mighty waters of the Magna River couldn't carry the pathogen that in fact it could infect him from cholera and given this happened like a couple hundred meters downstream of the largest the real hospital in the world one wonders whether our limited knowledge. In fact, it's a permanent liability on our ability to put price tag on ecosystem services. So, here I'd be advocating for a particular robustness of our capabilities of predicting what happens in a case of a particular aspect for ecological interactions which is through the channel network and I am told that these modern this is not a seminar supposed to be a lecture. At times I'll try to be somewhat technical and what I can say is that that whatever I'll be doing and I think and I gather that what Marina Gatto will be doing as well in his lectures is taken from the book we published. It came out like this month from maybe the end of November. And if anybody's interested, of course, details and much details, but maybe too much details contained there in and it had no by chance the picture which I took in Bangladesh during Feedwork, in fact, is a very good example. So, even networks as ecological chorus this picture is taking mine, it might be loved hometown it's surrounding the city of Venice these are title networks not really networks, just for those of you more versed in geomorphology would have spotted that immediately. But in any way the coexistence of the built in the natural environment suggests that this is a this is a long standing issue that has been exploited with a limited understanding for a long, long time and in fact, my pitch will be that essentially the intrinsic substrates for ecological interaction, in particular that of river networks of course, of which this is a particular, if you please, geomorphological relevant exception but it's so beautiful the picture that couldn't resist in fact, even showing those wide spots which are nets fishermen's net nets in fact. But anyways, my pitch is for the intrinsic structures for ecological interaction that bear rather fundamental consequences on the number of processes. In particular patterns of biodiversity, but much more in my view, to control the spread of say a number of things, the species of a populations like dynamics of populations into the fluvial connections and throw their into models of infections like COVID-19 as Marina would be doing. So the plan of my first 45 minutes is why in fact I believe that we can move from abstract abstract models of the system yet with some important constraints and why they're important in the quantitative valuation of ecosystem services. In particular, I'd be trying to show you progressively through abstract theoretical work, more refined theoretical work and laboratory and fieldwork that we carried out in my lab in Lausanne that directional dispersal and the spatial ecology, a lot of concert the spatial situation for on an essential bearing on form and function of river network that is through the ecosystems in the sense, but as we'll try to show much more than that, but the issue would be a game of dynamics of ecosystem services not today biological invasions not today populations migrations not today and then transmission of waterborne disease today will set the premise and for these to be verified on practical cases. I think if I may choose a punchline and hydrologist at heart and so I guess we're inching towards a fair distribution of water pillars of hydrology are floods, droughts, and the fair distribution of water and this I think would be an important step in that direction. So let's just start chatting a bit about something which in particular Ignacio Rodriguez and myself have been working for like 35 years at this point. You started at length of the substrate for ecological interactions but through the landscape because one of the issues that we keep having in mind and echoing one of the organizers of these of these winter school Simon Levine, who's it who's the importance of whose work can never be overestimated in ecology and Leon is that although natural ecosystems are characterized by striking diversity of form and function. And then many very quite a few times these often they exhibit they exhibit the deep similarity structural similarities and that the times they emerge across scales of space, time and ecological complexity. So we wonder what the certain universal features that appear in the entity structures like the ones that I'll be pitching in for the fact that are statistically the same regardless of climate, vegetation, exposed lithology, you name it. There is a self organizing process beyond these which is super strong and self organizing the sense that it will generate realizations that statistically identical. Regardless of there's no fine there's no celestial tuning up parameters that allow them to happen it happens, and nonetheless, so it's not a critical phenomenon but it's self organized critical phenomenon. Now these are those of you that being at ICTP for a long time know how important it is being the cultural activities there. So let me talk to you about the landscape and why in fact this is going to be brief but heartfelt that we have quite a few tools is taken from again Google Maps and what you take a picture, what you can do nowadays you can filter and realize what's vegetation is, and you can actually expose the terrain to the point that you can have lighter or leader what you want to call it maps in which essentially you have the size of the stamp. With centimeters at this point once you have filtered the vegetation, you have the surface of the terrain in an exquisite detail which you see in these cases exactly the same bend of the Amazon. And the accuracy of the vertical direction is now becoming more than enough to generate description of details in geomorphology I really changed the goal game. And these. So this is another picture at the same time once you filter vegetation, the detail of geomorphology that you're having there. Objectively manipulated, and, and automatically retrieved and remotely retrieved. It's really phenomenal over a range of scale which is unheard of. So essentially what you have I'm still attached to the old picture the first one we took out in 1989 over the 90 whatever it was, etc. This is a small catchment at which the restitution was of your few hundreds like square kilometers and this was given the pixel size which you have by general tools, it will be like of the order of this kind of 30 by 30 meters. And the accuracy in the third dimension was of the order like half a meter. So how this is that essentially you have a surface remotely acquired and objectively manipulated that tells you how you can calculate. And gradients means steepest descent directions you have a field Z sub I, which is a scalar, right. And you can calculate the vector which is the gradient which is the steepest descent direction, which by all means in hydrogen geomorphology makes sense to assume that the strongest forces gravity by orders of magnitude so steepest descent direction is also the direction of the flow so aggregation pass can be devised directly the shape of this kind I mean I'm essentially in grossing a particular that so you can delineate a line of this kind and we have a trick which is a useful trick in fact of using a scaling for tracing the black line that shows the mainstream in here, which is a well known Leopoldian rule for the scaling of a with a very good channel this is a trick for showing how the system behave but certainly it has a major implication you can calculate elevations, you can calculate the placients, like, let's say, essentially the average over the nearest neighbors in a place you can easily decide what's the sign of the system that is, whether the surface is concave down or concave up. You have some sort of tricks to the point that we assume early on, and essentially because of a statistically statistical well worn argument, essentially, our theory is that square root of two is equal to one that is because a gradient, the steepest descent the low exceptions can be generated. So essentially you extract a very even network from data over the scales around the order of the meter, even less than that now to the order of thousands of kilometers, still we're talking about the so called runoff generating part of a landscape, where in fact you do have aggregation you don't have a transportational zoning we have no significant injection but still you're talking about six clean orders of magnitude that can be ground truth quite easily. And quite a few at the time we used to go at the physics conference in particular in Trieste, and you see that the data set over which we started learning how nature works. In fact, hardly find a match than in the case of a fluvial basis. So it's something we know, so essentially we define the master variable for these which is total contributing area at the site. Like in this case you have 12 connected pixels upstairs. That is, you essentially have, and it's a typical equation that you had in the aggregation process of injection. If you assume that one is the size of the pixel, whatever square meters you have, you have a connectivity matrix in the system, and I'm getting to the point that now a system of time this is a, well, I skipped an important issue or which I shall return. I'm assuming this is a tree that is in every node you have a unique path leading to that node by different directions, you know, have loops a section and I'd get back to that. What is also historically quite important because that's the assumption that was made in the first manipulation of a system as we see it, and yet it proved a lucky shot, because we didn't search for like optimal configuration of some kind by searching loopy structures, and it turned out to be a fluke. But anyways, what you have is that wji means whether it's something which you have a one if this j is connected to i, I'm sorry, if i is connected to j and zero vice versa, that is, in this case here, you're bringing a guy who carries a weight one a guy who carries a weight one and a guy who carries a weight one plus one that makes four. If you move here, you add one plus four plus three plus three plus one, and you get 12 pixels a second, and this means that the this is a quantity which introduces and the statistical physicist among you has spotted immediately this is a non local interaction, which is applying locally, and this has plenty of consequences of course. Now what is interesting that is, this is a tree, this connectivity matrix has all zero eigenvalues, so you know that immediately. So if by any chance you thought you tried to perturb this configuration generating loops, you would know that we have elementary numerical checks sources area equal to one and the like so one is the basic scale which is delta square the size of this. I'm sorry I trivialized it a little bit, but it shows how we can, in fact, show a remarkable capability we have to remotely acquire that is actually manipulate the scripture is a super accurate of natural over, I would say, up to six orders of magnitude. It was interesting here I removed the scale bar, and one of the main tenants and that's the book where Ignacio and I wrote years back, which was where received in fact is still used in quite a few types of classes or for it's well cited, like so what I'm saying is that if you remove a scale bar, you really don't know whether this is a large, or this is a small catch, but this could be the Amazon that could be small creek in the Dolomites nearby, because nature tends to produce those shapes in a remarkably similar shape regardless again of climate, regardless of vegetation cover, exposure lethology, you name it, or any kind of perturbation system. So do I know that well, actually we can specialize a little bit, the, a little bit more morphological, geomorphological extraction of the geomorphological consideration of how to extract the proper channel part of a landscape. For instance, this is a the network that connects all the concave sides, the one for which nabla square of the eyes larger than zero concave up to eliminate the dots here that other concave sites which has a well defined ecological reason to happen, the concave sides are co colluvium in geology and geomorphology, or you can take a subset of that, which is further refined by some sort of a criteria with a fashion that you're having there you can really the signatures for instance of past climates as seen in a minute so essentially if I enlarge the thing you have the channel pixels, which are the proper parts which are part of the domain of a fluid, the main rhythm it looks as ecological So again, this is what we have seen and the key result is that you take this as a master variable that is, they take you consider that the total contributing area at any pixel, at any site is a random variable whose probability distribution in this case by no chance of a rate of exceedance to avoid a promise of beating, as you may know, or which we may return in case is essentially proportional to this value a to a power, so infinitely popular terms like power laws, which is essentially dictating what Simon Olivier was saying about the deep structural similarities that emerge across scales of space in this case not in time but equally well if you have activity on the network. So essentially oops sorry your direction so this is the what happens, and of course, if you take sub catchments you sub sample within a basin, you had that the maximum area will be limited to the maximum size of the, of the, of the sub basin you have. But what you see that you take nested sub basis bigger and bigger, you see that the argument you had in the system in reality your base, an argument is called the finite size scaling, which is something which is well known to a statistical characteristic which you see that essentially you can have these curves collapsing directly on to one another, showing that, in fact, the deep and remarkable similarity emerging across scales is a well established fact over which there is a significant consequences that the system so it's a well known fractal if you have signatures, etc. A geometrical languages Benoit Mendelbrot, whom we miss a lot and to whom we pay respects all the time for these visionary ideas. It's a common name for that. So it's the language that nature speaks and what you have is that, if you take for instance gross features of a domain for interactions like transverse and horizontal length you have relationships like the distributions are with a remarkably consistent coefficient of lesson point five hiding on trivial or the lengths to the source of any fun are well codified and well known and linked scaling exponents fully characterized the network forms in fact. So these are you have we have there's a zoo of cases that have been investigated in the early 90s and this is a remarkably robust result that we show. Now the finite size argument also allows you to get interesting features for instance, if a finite site argument applies the ratio of consecutive moments of a of a finite size scaling. Distribution produces a particular relationship which shows that regardless of the value of the subsequent moments that you're using. You should have different the same exponent essentially so this is a particular kind of a fractal machine, which allows us to calculate the elongation of the catchment which is something which without math but that Mendelbrot spotted on. And these are the data is for you to show what happens, etc. So essentially, a consistent across scale, it can be done because you can use these to generate, for instance, variations in time of the features that generate the network but what we really know at this point is that essentially especially printing then generates total contributing area which rightly called the master variable for the, for the system is what generates something like this is what happened to Mount St Helens after a few minutes out of the first rainfall in fact. So the printing is being like Kenny or like but he will stay forever the planner in printing of the system that settled the issue or if you take for instance Marshall and forms if you apply to the same things, etc. What you seem to be what you see in fact as something fluid has to be operating in a road in those surfaces. But obviously I should not insist, even for saying that we have tools for and then I'm back to the, not the fluvial but the title networks can tools in which by studying the landscape you can start length of landforms of a very fine scale. And so with these are many other things for me starting for years or for instance if you take like this looks like a photograph but it's a digital terrain map of an accurate money which you can actually start devising or whether you can discuss where these are natural or artificial forms in fact. So that's what we have again this is a digital terrain map or our baby treasure, our data set that you can effect show how nature works from again 10 centimeters to easily in the through the landscape to thousands of kilometers. And that opens the thing to I still allow myself to have like five minutes on networks, and then I move on to the first exploitation of that would be whether all trees are equal. And the ideas that a whether the loopless properties particularly distinctly is something also important. And we see why comparing networks is important for instance I'm claiming that these networks, not this one but the three of them for instance had the same topological properties. And what distinguishes them are metric properties of a different thing but it doesn't take. I mean a scientist to see that the piano network which I'm showing here is different from this one. But they are topologically identical indistinguishable so this deserves some, some extra thinking. This is piano network, which is something of which we worked a lot and it was essentially devised by Mandelbrot is an exact fractal, you have a hell of a lot of property which I had a logical of a logical irrelevant they can solve exactly for this construct. Essentially because they map multifractality for instance exactly it's a binomial multiplicative process. In this case, it's a characterizing something which is very important for hydrology is a benchmark but again the topology is the same with the network so it allows you to get exact solution for property that topologically dominated. And we see next class, so there are a few. When I spending some time on optimal channel networks, which is something which we invented the national night. And this is a tool through which you essentially calculate a spanning tree, a tree has to be in fact over a given domain in this case square but it could be anything it could be with boundary condition which are periodic boundary conditions whatever you want to have it and So the idea is the following. There is an exact statement taken from the general landscape evolution equation, which is nothing but the vast balance of the, of the elevation field, Z, in which you have in steady state condition and the small gradient approximation. In the case of parametrization invariance in fact you get that there exists a Hamiltonian of the configuration of the system which is seeing here, which minimizes energy dissipation, which essentially related to the master variable the non local variable at any place, raise to a power gamma. In this case, I shall not spend time because this is that well digested if you, if you want to confine the exact solution in a certain places, etc. But what I'm saying is that what is relevant for us is that there is a zoo or possible cases that we know of, for instance, if gamma is equal to one, these are called random resistor networks that been studied by engineers in particular for a long, long time as is the set of the configuration of the system is essentially the number of L square sides, the total contributing area which is in begging the connectivity structure and the aggregation level. Now, in the case of gamma equal to one, this is a very interesting thing these are called the, well mapping the so called a billion sand piles in self organization that is, you essentially have a network which minimizes the mean length of the outlet. And what is interesting is that all directed networks, in fact, have the same energy dissipation of gamma is equal to one. What's really interesting is that if you assume that this is a problem, a problem which you multiply flow time a gradient, like you have normally the power dissipation the energy dissipation, as you want to call it. And a is proportional to the landscape forming discharge. It's a well known result in ideology, and the slope is proportional to area to a power which is less than one so essentially you have a classical of host optimal channel networks that obey this property. Instead of gamma equal to zero you have spanning networks, and it's a different ball game itself. You can actually keep going in the system what you have in fact in some case, an issue is that from any initial condition can essentially rearrange the system by disconnecting a place, getting a new tree going to other direction and checking whether they, well, you can actually find whether you may accept changes in this case which is the basis from this procedure. For example, you can see some probability are like Gibbs like simulated annealing, and you can actually generate these figures here. Now, an interesting point, and I won't follow up on that is you, you can calculate in the case of trees the true thermodynamic entropy that is you can count the number log of the number of states that have the same energy with scale with a certain condition, and the energy dissipation scales with them with the different expose exponent which is larger than one. So in the thermodynamic limit, if the size of an effort is big enough energy dissipation is also minimizing that thermodynamic entropy. So the figures that we can generate and, in fact, you can show that if you start making good comparisons you can calculate fairly well how these exponents are perfectly matched match scaling exponents are the true tool for comparing different trees. There's also an interesting point that by another lecture that you have almost by now was the lead, the lead author and a former ICTP professor of them, of a ground state, the scaling property of a ground state for this case, and then what is interesting in fact, the scaling exponents for the ground states and other ones were sort of in nature, nor the ones that we get in the case of so called up to a feasible optimality that is, optimality which is readily that is optimality, which is dynamically accessible. Randomness is not a thing. So essentially this is an even growth which is entirely random dominated choice if you make a good comparison of these two networks, look at the different boundary condition that looks similar. Look carefully, they are not like your eyes telling me. And these are the two different conditions, these two scale differently, they are both network and this the chance that the system has to reach a ground state is nil in practice, and this can be seen progressively by relaxing boundary conditions size and the likes. These are the typical networks over which I shall be talking about. And I will jump now to the, I'm showing I'm, I'm skipping details essentially of the fact on why in fact, networks are. I may get back to that. And, and later on in one of the following classes, I think it's more important that I move on, but the idea is that I hope I convey that can give you the material on that is that it's a coherent thinking on how, in fact, we have similarities and also similarities emerging in the substrate for ecological interaction emerging across case of space. So, at the first thing that we had in mind is using on a subsequent this kind of the neutral theory of biodiversity. Why neutral theory by the verse you may know that the neutral theory was originally proposed in complete analogy with the neutral theory of molecular evolution. Which assumes that the gene mutations are selectively neutral that these new genes are demographically equivalent to the old genes. And they do not give any advantage in terms of decrease mortality and or increase fertility and it's a paradigm that dominated the molecular processes and still does for a long time. And the main advocate in fact for a neutral theory of biodiversity was a colleague from Princeton now at UCLA Hubble wrote the fundamental book of the neutral theory by diversity that is, he was asking at whether the idea, starting from finding a new species that is done on topical forest, in which essentially he proposed to have mutations replaced by the occurrence of new species in the landscape. So, and the idea is that it was a revolutionary one because essentially what you do you assume that all species are considered equivalent to per capita level. In which I shall build more in lecture two. But what I'm showing you and I hopefully I will convey the, the, my main idea is that the, the, if you have a neutral process over particular topology topology of a substrate, this has consequences. The conditions for a species to occupy a site, for instance, and maintain a population are the dispersal ability of a species, the habitat suitability and the susceptibility to any kind of biotic killing of any kind, etc. We want to see okay now, of course we have conditions to grow and maintain a viable population in the place, but what happens that was the key questions knowing that we know a lot now about the recurrent characters of fluvial landscapes in fact which is spanning three of which interactions occur. What would be the consequence of the extension of the neutral theory by diversity to space explicit ecological setting, but in the particular case of the river. So essentially the experiment I'll be showing you now to introduce the following discussions is one in which you have essentially or suppose you have a lattice. Which in a hydrologic term called savannahs because essentially what you have is that every single node is allowed to interact with the nearest neighbors. Whereas in any chunk of the river network, you can have two sites that are neighbors and nearest neighbors but they are completely connectivity wise they take a very long distance from one another. So essentially introduced because of a directional dispersal, which is embedded in the network structure with or without drift because that's a that's a point, and you have a system in which the connectivity matrix is a completely different picture. So we can apply the spatial ecology sensor, like Tillman of the famous work with caravan in 1997 is extended in the model which I'll be discussing now extended to embed the topology of the substrate. So this is what happens in setting the simplest possible knowledge so you have assumed a distribution of colors and assume the color is a species this is what the in social sizes that physical physicals hold the both our model in the beginning. So assume that at random you kill any existing color, any place and then you have two options with probability me, you replace it by one of the colors that is not existing in the particular place. And really one minus me and this is a very small number, you assume that in this case, the color that will be taken out of this will be the one in which all nearest neighbor in the most abundant of the colors in your nearest neighborhood. Now, this is interesting because you assume that there's no stronger species than another right. So you assume that if this is a political opinion, there's no political opinion will be stronger than the others and you'll be simply the, the, what you actually have as the, it's the wisdom of the numbers on a large number of the system. So, indeed, every now and then like every 10 to the five trials, etc. They, they pointed you choose at random and you killed is just parachuted from from outside the domains mimicking either speciation or migration or immigration from different places. So the only difference between the two is essentially the topology of a connection. And when we saw this we we got kind of excited because he said, look, what happens is that the only ingredient you have changed is the matrix is the neighborhood the definition of neighborhood. In one case, the one you see on the left was open, and thereby more open to interactions and the second one was more protected in the sense it was due to the presence of a network itself. And what is interesting that if you just make it quantitative that you haven't here, you essentially have not only you have the different degrees of irregularity the boundaries, which is, but in this case which is the network system here the neutral system here, what is clearly that any statistical measure that you have of the distribution of the colors is essentially completely changed and the rank abundance curve it shows that the biodiversity of a network system is much bigger. And of course, first thing you say okay is this a realistic model, the neutral model of a fluvial ecosystem of a fluvial stream microbial ecosystem distribution or not. Well, it so happens that and patterns may not be neutral patterns do not require a neutral process perhaps, or equity finality issues can pitch in. But the fact is this was an intriguing suggestion early on it happened in 2009 before we embarked in this study, etc. But if for us it was worth checking. Why this is interesting because there's a result that it's absolutely robust respect to a number of generalizations, namely this is 1.1 species. It's actually evolving into a meta population model it's one node because a local community with certain number of species, the same rules of engagement for the interactions. And you can also have this that you can relax the possibility is only the nearest neighbors, giving you by sheer majority, the new colonizer in a place in which you don't parachute the new species. And the result is the same. So a dendritic substrate for interactions changes completely the ball game, regardless of a number of assumptions still in the obvious assumption that you're assuming that all colors are equally valid. And you're assuming the old species are equivalent to the per capita level. But we thought it was interesting. And the first thing that we did, we applied the same rule, individual based, but or meta community based, but with a kernel that is the radius of influence of the area over which you average a progressively to get the majority replacing the place could be immediate neighborhood or mean field that is the one you essentially go and check what happens in the system. You can do that on a 2D lattice, you can do it on a network, you can do it on a 3D lattice on a 3D network. And much to our surprise, what have we found. Okay, one good measure to see what happens in this case would be essentially related to how long would a certain color last from onset when it's created, or in a particular place, or it has local extension. And this is called the persistence time of the color or the species into a landscape, only in this case affected by technicalities like the structure of the network. And much to our surprise, the distribution, whatever you want to call it, certainly there is a huge, there's a huge range of scales that covered by power law in this case show there is a distinct and unmistakable effect of the topology of the substrate for topological interaction. So, in our case, we took on to a first empirical check because it's interesting. I think we did too, in fact, so we started the this the we had like 41 daily values 41 year of population of breeding birds in North America into a place, and you see you see the, what is this, the persistence time in this case meaning suppose that you are in this area here. I should add in this area here you count in those observational sites account, how many times here you spotted for first time a particular bird, and here for a number of years you haven't. So these are independent persistent science. And you can study the value that you have a different level of aggregation, for instance, or you can do the Kansas pre her boss, her boss use plants, etc. So you can characterize it. It's long story short, what I'm saying is that you can isolate the effect of the finiteness of the sample, which you have an atom or probability for the species that's been continuously observed, for instance. And you can characterize this interesting feature. So long story short what you have is that you confirm what we have seen topology means a lot and topology of the substrate for interactions has an effect. So the point that you start for instance aggregating a different spaces you can transform this scaling effect, exponent of a lifetime into a distribution of biodiversity so essentially by taking course grained versions of the same space, you can find the probability distribution of the survival how it behaves on the course grading, and thereby generating a true species area relationship. And the next step which I want to show you and on this will be relatively fast it's only the last five minutes of a class we said okay say look our course and you talk to an ecology say hey but this is bullshit I mean species are not equal and per capita level how you can do that, etc. And yet, we thought early on there was something to, I mean, manifest to be completely false a topological effect of a substrate on the nature of a recursive nature of a former function which is embedded in the structure of the universe regardless of whatever it is truly a universal feature. So we decided to decide to test on in my lab. These with living communities, in which was nothing neutral. There is absolutely nothing neutral. So essentially what we do is just, we choose, for instance you see how rudimentary the city is we should have had the money to buy a robot and the in the absence of robot to have so this is what we have in the, in the lattice in which nearest neighbors are the four nearest neighbors. I mean, the in which nearest neighbors are the topological ones, the adjacent things and the structure in which these two guys are not nearest neighbors. And we did that the brute force this is Francesco Carrara now it is said, and this is a Ricco Bertuzzo now a professor in the University of Venice. Now, long story short, this is what we got replicas words as many as needed. And this is the experiment that you have in the random network system that you have, or in the thing that you have and the color essentially is a measure of species richness. So, in these, the theoretical idea was telling us look guys. There's something which pertains to the fact that you have a certain topology of the connections. So regardless of everything else, it's, but the prediction came directly from a neutral model. If you have like a lattice of interactions. This doesn't happen experimentally. And this, again, this is a living communities. I'm talking about, in this case the, the, the, and then give you the details that took a note someplace here about what is the experimental community. I'm talking about 21 protozoan species, in which we had, I'm sorry nine protozoan species, one rotifer which is a multicellular one, and a set of freshwater bacteria as food resource in the place. And the dispersal was done manually, but it was done essentially according to the things and I can give you details if you care for any interesting tool was also published in another set of painfully conducted experiments. In which essentially get okay now suppose that we take into account the fact that habitat capacity in a river is hierarchical. Okay, so essentially you assume that the medium, in any single well depends on the number of upstream sides. So you have a hierarchical structure, or you get system in which the same total amount of medium is randomly distributed, or in which you take a uniform distribution of the system. The hierarchical system is a completely different structure. So, and that's the take for the next classes are saying that whether neutral models or models of anything as modeling or a party and vegetation modeling or species diversity, etc. We're talking about the landscape quite a bit. We have an environmental matrix which is made by nodes and connections. The substrate matters, and I'll show you how these, in fact, has a say on the oriented graph is something which we give you don't assign a network interaction this is given by the by the geometry of a system. And, and in good timing in fact I'm showing you the conclusion a general conclusion but I'm essentially what I have shown is that eco hydrological footprints of rivers as ecological corridors were suspected from early abstract theoretical and confirmed by empirical analysis on a particular set of persistence times measure for breeding birds or plants in the cancer spray or experimental work done on living communities in this place. So I'm ready for your questions. Of course, even following classes what are we doing I'm talking about species about biological invasions I'll be showing you why biological invasions slowed by bifurcating structure of the system, and then I'm talking about disease, which is an interesting point. disease is tackling this manner of spatial space in my back. Okay, let me check. Chats. Okay. Hi everyone I'm having serious problem with electricity supplies. No, no, I'm sorry that's a question about the disease supply cannot help you I'm sorry. So we are ready to take questions from the audience if you want to speak just raise your hand in the list of participants and I can give you the floor. Okay. We have a question from Miguel Rodriguez. Hello. That that was a that was a fantastic lecture thank you for this last part where you do this experiments with micro organisms moving from well to well. Will that will that pattern be robust to different degrees of base diversity will that be true for much larger values of richness for example in those super crescent super crescent now. Well, we have experimented with the varying number of protests in fact because you needed to have a side of the end and I have to say the protest that being a good model organism effects of these kind of studies for a long, long time. And if you look at what there's a number of holy oak Florian altermath which was actually working with us on these I said that they work a lot on so called the dendritic substrates for interaction. But what they had the dendritic were just a pipe and a few bifurcations. That's not the river network. That's why I keep saying and I think I convinced Florian who followed us et cetera to the point that in fact that he put on your find an old, he put an open source code for having his own networks in the place because you have to have the number of sites. The number of sites that you have are constrained by the total area is not that you can assign them independently the connectivity, which is suited to a through the ecosystem is a specific one highly constrained one, a highly recurrent one. There's no one you just had like a pipe and a few branching patterns in there. And so the question is that we have experimented with different number of processes, but we haven't explored it a hell of a lot. All I'm saying that as you will be seeing soon after these kind of confirmation we got more and more enthusiastic. I got an ERC grant, and we started having lots of people working on different aspects of that. I'll be talking the next two classes. And I'll be showing how we did species we did a population migration which is fascinating subject I'll be talking to you because we start from migrations human migrations in the 19th century. So long time. And, and then I'll be talking about disease, believe it or not. We have the same tools. We started the simulating spatial explicit models of color. Of course it has to be right pathogen and whatever has the pathogen living in or proliferative kidney disease in fish. So anyways, we haven't experimented a lot a lot more in the lab mind you, at the time I didn't have the money for the robot. So Francesco and Erica stayed out long nights in by painting from one side to the other. And that's what it's typically know it's it's a typical a professor say what are these details. It's not detailed but works a lot. We have a question from Susie. Do you want to speak for yourself or do you wish us to read your question on the chat. Susie I saw I see it. I see it here want to repeat it. Susie yourself, otherwise I read it. You can go ahead if you can understand the question. Okay, the question is, are there evidence for biodiversity reduction whenever rivers are diverted or dams or construction costs are of course fantastic, fantastic questions. Yes, there is in fact a group of fish biologists in Spain. And it's paperbarkin I think is a guy in Santander that published it published quite extensively on the study of how biodiversity is impacted by it's actually common sense. It's common sense or number of counts for instance I could tell you that there are a number of studies, especially the biofilm guys are very prominent of that. If you change if you modulate the natural sequence of stream flows you change the ecosystem period. It's a fact. Can we use it for for example impact studies before prior to constructing dams to say that you know, oh yeah, oh yeah but that's I haven't done it but the paperbarkin and his group has done it quite a bit actually. I mean, it just but even a common sense. I mean, it's, it's known by the work that Dan does besides the interruption and the change in connectivity which is directly reproducible in the case of this, because essentially you change at any downstream point the total contributing area. Right. But the sequence of stream flows is not proportional to area anymore. So essentially you change the structure of the engine for interactions. And I mean, it's known people to study the biofilms know that which is the basis the engine for the ecological interactions in the in the microbial ecosystems or through the microbial ecosystems and so there is evidence for that yes the answer is yes I haven't done it. No, I haven't done it. You have a question from Victor. Okay, within this model the river capes could we add the effects of factual changes in river curvature, due to coastal erosion or even okay, or reverse driven by deforestation or Riverside vegetation okay the two different issues. First, the, I'm referring to separate issues. The first one the answer is no why, because the recurrent properties we see in rivers do not apply generally they apply the so-called runoff producing areas. So the river structure is essentially they're not producing area with the landscape form and discharge is proportional to the total contributing area at the point the local quantity. Then, for instance, if you enter the desert like the Colorado River does after Glen Canyon Dam, for instance, they had a place in which I have no injection of water. So essentially the transportation thing is dominated by geomorphological process and morphodynamics which is not related to the aggregation process at all. And then you have a delta of a history, which is the distributary system, which is again governed by different rules. So the thing here is that they're not producing areas we mind you, it's not the whole of the river basin, but it is something which can easily extend to scales of the other 1000s of kilometers inside. If you're talking about deforestation or Riverside vegetation of course yes, and I work on that I'll be talking about briefly about models of vegetation, over which we worked a lot. Yes to the second one. Is that okay, Victor? Do you hear me? We cannot say, please assume that Victor is satisfied. Yes. We have any further questions from the audience. Can I ask one more, one more question? In your persistence time distributions patterns, you showed that these networks, these river networks sit somewhere between 1D and 2D fully connected grids. Can all these be predicted by the dimensions in that fractal structure? Again, a very good comment. What happens is that it's the topology of a substrate that matters in determining the persistence time. So if you sit in a point, you close your eyes, close your ears, and you measure for how long the color stays there. Okay. You find that if you do that, of course, every side is a different persistence time from the onset of a species, we're not measuring abundance here. You don't measure the number of colors. So color right in this place. Yes, for how long it stays and you measure it. So every particular from onset to local extension is one value of a random variable persistence time, local persistence time. You do that for every single thing. But what I'm saying is that the game of simulating that in a couple of hours, you can do that. Because absolutely trivial, right? If you give by hand, for instance, the connectivity with which is slightly more complicated, but okay, but the essence is that one. What happens is that what you see that depending on the topology of the system that you have, you don't know anything about what's going on in the model, etc. You simply count. And you see that those distributions once you start to make it big enough to have enough statistics, etc. behavior completely different. And that is the counterpart in which this is telling you, look, there may be something deep going on here, because what we are saying is that depending on how I'm connected. It's how I'm exposed to interactions. And how in fact, the distribution of my friends, it completes completely different changes radically. And what is super interesting is that this topological effect has been a single doubt exactly. But what is interesting if you look at data, you know, it's a tricky thing, because if you look at the data, for instance, look at breeding birds in an area like of Georgia, God knows what, you know, all the most society started like 50 years ago in collecting those that the network of the birdwatchers is a serious business in the North, in the North, United States and Mediterranean one. What happens is that what you had is that okay, let's suppose the cardinal. You can see this area now that he hasn't been seen on on like this year he's been spotted. Next year he hasn't. So, essentially, you measure what happens locally, and then what you do, you can make it a bigger area. So it changes because it may be local extension can be here and not be near said so at the continental scale. Okay, if you assume there's no migration or carnals, etc. is PC a show that you're measuring. You can call it the speciation rate or a true speciation rate. So what I'm saying is that you can empirically, there is also an issue because if you have a finite sample in terms of length. What happens is that you may ask yourself, okay, look, if you see throughout always this, you always spot the same bird in the same place, you have an atom probability of one. Okay, and that is very important because you see how this atom is reducing with the size of a sample standing a hell of a lot of the underlying distribution. And so being careful, but you see I'm showing you in 45 minutes like four or five years of work. And, but again, if you if you if you care for that I'm asked your library to buy the book because we spent a hell of a lot of time putting it together. We have time for one couple of questions more than we'll close. Okay, one to one is asking, could you please remind me what is the difference in topological structure between the savannahs of the ocean. Okay savannahs means that in every side, your nearest neighbors are the four dearest neighbors in the coordinate direction. North, South, East and West. That's all you check for. Okay, so once you kill in a place a system, you replace the color you have in there by the most abundant color in the nearest neighbors, and if all three, if all four values are different you choose a random. Okay. In the ocean, you have a directional dispersal so you have essentially drainage direction that I tell you who's connected with whom. So you may easily have that because of this directed structure nearest neighbors topographically may not be nearest neighbors in terms of interaction, and that's a very nice direction, which you get directly from the, from the landscape elevation. Zebsa rubber, always I think that there is a limit number of species because it's in the structure, then all about special structure that's a good question to Zebsa what I'm saying is that they I'm talking about the abstract model now in which one site is one species. But you can generalize this into a meta community model which I'll be explaining next class, in which for every node, you have a meta community and local community, it is a community of communities. So you can actually calculate carrying capacity. If you're worried if you're worried about that you can calculate logistic. You can put this into it into a special ecology framework, like the special ecology framework that Tillman, the mighty Tillman and Kareva put together. Only thing, what we provide it's those recurrent features of channel network is no better than anybody would work for 20 years on those. This is a three a question is, of course, there's a carrying capacity on each node, you bypass it, but you see the pattern explained by the, by the individual based say kernel one that is only nearest neighbors thing etc. It's telling you a story, which is conferred by making the kernel same mean field, like the entire structure of the catchment or number of species fixed or number of pieces variable or whatever. The result that is the pattern is affected, resist and is general. Thank you. Okay, so maybe it's time for us to call an end to this first lecture and thank you, Andrea. Thank you so much for the lecture.