 So welcome, everybody, for this new day of with many exciting lectures. So before we start, just a reminder of the few rules on how to ask questions. So if you're following from YouTube, you can ask questions in the chat and I'll read them for you. If you are connected to Zoom, you can either post them in the chat or use the raise and the feature of Zoom. Great, so it's my pleasure to introduce again Andrea Rinaldo, who is giving the second of three lectures. So thank you very much, Andrea, for being with us. Please, you can share the screen when you are ready. Thank you so much, Iacopo. And let me go back to the screen. I assume that unless I hear you lamenting that you don't see, I would assume that my full screen would be visible to you all. So I come back to the second of three classes on river networks as ecological corridors. I will subtitle species, population, and pathogens. You may remember, if you attended the previous one, that essentially the light motif of my talks is that we claim that dendritic substrates, like river network, instead, with certain recurrent properties that we have analyzed, however briefly. But those dendritic substrates for ecological interactions bear important consequences for a number of processes that we'd be analyzing. That is essentially patterns of biodiversity, which I haven't done, I'll do today in part. The spread of species along the network is also biological invasions, in a sense. And the spread of waterborne disease, which you will see in my last class, with reference to epidemic cholera and dendritic disease, like proliferative kidney disease in fish. And the idea is that the system is so constrained by the nature of the substrate in the recurrent properties, no local properties embedded in it. But in reality, it would endow the system a certain degree of predictability beyond what was originally thought. And so what we have seen during the first class was that once you put those constraints in place, and you use even the simplest possible model, in that case, was a neutral model of biodiversity. Whether you're using a meta-relation model or a meta-community model, whether you're using nearest neighbor interactions, or you're taking a kernel which goes from a certain discrete length or a mean field approach, in which you average over the whole ensemble for replacing, for getting replacements, or randomly selected species removed from a single site, the system tends to have certain indication. That is, the very nature of the substrate of interactions with the dendritic network has important consequences, which we have later verified by a number of different tools. So the main argument that I put together, and it's a one that said that what these lectures tried to do, were to draw together several line of arguments to suggest that the integrated eco-hydrology framework that blends laboratory field and theoretical evidence has contributed substantially to our understanding of the form and function of the river network. What shall I do today? I shall begin when talking about biological invasions, possibly biological invasions on fractal supports, like a tree, which is the ones that we have seen in the case of river networks. And what you're seeing here is a map of the invasion of the zebra mussel along the Mississippi Missouri River system in the United States. And you see different coders by different colors what happened to a species was completely alien and introduced by accident into the system. We're talking about hydrochery. And to make a first example, we're talking about human range expansions and the population migrations in the past, which is an interesting idea by the group of Spanish physicists which I'll be putting forth today. So first, let me start with the standard things many of you have seen in a number of contexts. And it is the fissure comorbore of a tumbling wave approach to a system. So if you have a model in which you have dispersion role, in this case, it's like a density of a species, or you name it, of any kind, moving along a one-dimensional support, remind you that in the case of river networks, makes absolutely good sense to think of a one-dimensional system in intrinsic coordinates because the width and the depth of a river cross-section are much smaller than the longitudinal length you have to travel. So the assumption that making open channel flows and flows in pipe, et cetera, is a one-dimensional thing. It's absolutely standard assumption verified by hundreds of years of practice. So essentially, what we say is, suppose that the system would obey a simple diffusion with constant coefficient. So the term d square rho over dx square coupled to a significant of the timescale of diffusion reproduction system. But it is, in this case, a logistic equation in which essentially the reproduction of a density follows initial phase, which is the exponential growth phase, the Malthusian so-called dynamics, later on curd by reaching an asymptote. So the growth term is nil, either where the density is 0, or where rho reaches a carrying capacity k. Why this is interesting? It's been studied for a long time and notably from the most gifted applied mathematician of the last century, Kolmogorov, in fact, and Ray Fischer, the inventor of so many results that are crucial now to modern genetics and molecular medicine, et cetera. They started to get under different angles of the same result. What happens if you look, if you suppose that, for instance, you start to generate at a single point in space over a length L, and a single, you inject a single shot of a certain number of organisms of the species whose density we plan to analyze. The system here is known to generate a soliton that is an undeformed wave after a short while that propagates without the constant celerity with the basis equation, which is solely dictated by the value of the growth rate inverse of time and the diffusion coefficient length squared divided by time to generate a characteristic velocity, which is twice the square root of a times d. How do you prove this result? Well, it's interesting point which you essentially take the diffusion equation under this limiting term. And you do that initially by assuming that this term is negligible. And what you have is that you can calculate the expansion radius of the most of a mass or a assigned fraction of a mass, which you have in your system and how this expands in time. Thereby, what you're studying is the behavior that has an important feature for what concerns our capability to describe the network effect. What happens in the vicinity of a low density when this starts propagating actually. So the idea is that if you calculate the expansion radius that in a regime in which this log of a quantity whatever you want to have it pales compared to the term in which you have the square dependence of a square root of a radius. So the velocity of a speed is essentially twice d times r, which is the growth rate in the Malthusian model in which you eliminate the carrying capacity effect. And by no surprise the carrying capacity even if you extend this, it can be shown that the front speed is essentially in the case of the Malthusian dynamic still what happens in the vicinity of this reaction term for which the density approach is zero. And that's not by chance because what we are saying is that so essentially the same result of the Malthusian population which if you take a proper computation of the expansion radius for a Malthusian population and through the curve here and the actual solution of the system which you have, you see it quite soon the system behaves exactly in the same manner. So what is interesting is that in a two-dimensional system for instance or in an isotropic migration system one, two, three n-dimensions, whatever you have it the system will have essentially that the speed of a traveling wave or the argon is generated that will reach a plateau and stay there and propagate on the form it's proportional to be these two coefficients ignoring the effect of the carrying capacity which is immaterial but only depending on the fusion coefficient and the rate of reproduction, reproduction rate one at a time. This thereby in say Eulerian support but is in one two-dimensions, et cetera. And this will have some importance in what follows. Now what is interesting thereby is that what matters and that is important for the particular substrate we are studying which is a tree that is a unique path for any node to any other node it's a directed graph in which there is a sense of direction which is embedded in the structure of a system of those diffusion. And so essentially you can map the evolution of a density of organisms in space and time as a rescale variable of an under form property which is the typical structure of the characteristic curve which I'm in there. So the fact that what matters is what happens in the vicinity of a zero density is important whenever you have to look at systems like this one this is Piano Network which I introduced during the last question there. Why am I using Piano? I'm using Piano because it's the deterministic fractal I rediscovered Giuseppe Piano, the Italian mathematician who wrote a famous paper over a curve that fills the entire plane which was considered like a blunder by mathematician of this area and we're still studying it a hundred years afterwards. When one Mandelbrot, a genius of Mandelbrot rediscovered it to support this idea that mountains are no cones coastlines are not broken lines nor does traveling straight lines. The fact that after 2000 years after Euclid we started looking at the geometry of nature with different eyes was embedded in the curve which is strictly deterministically self-similar because it's created by a process where it's from the originator you essentially do the system which have a broken down half of a segment here adding it into a system of this kind and that allows you to calculate exactly a number of properties. Why this is an interesting feature? Because if you look at topological properties of course if you look at it and say this is not a river network of course it's not but it allows you to calculate certain quantities exactly for instance if you count the number suppose this is a directed graph and you can count and if you suppose you have a seed point like here for instance or in this case it would be here if you assume that the thickness of the line is proportional to the total contributing area it's a tree so in any place you can calculate the number of nodes you have upstream and what is interesting is that you can calculate exactly the number of sites equally distant from the outlet distance being calculated in what is called normally chemical distance that is along the structure of a network itself which is hydrologically is very important quantity because if you suppose that it rains a given quantity in every single node of that the distance if a system behaves in a dynamically constant way which means the arrival time at the outlet so essentially the response of the system will be related to this quantity. And in the case of piano this is an interesting thing because this is a binomial multiplicative process solved exactly. It's an exact multifractal that has been studied by many including men of itself. In this case what is specific of that is that even if by looking at this you know that this is not a river network however the topological properties which are non-distinctive in many respects almost statistically inevitable the topological structure of this is exactly indistinguishable like bifurcation numbers, lengths, et cetera for the ones exhibited by river networks. So the message I'm conveying here is that in this case you can study exactly the behavior of a traveling wave along this backbone or propagation for instance because you can calculate exactly the number of bifurcation that the system will experience in its traveling downstream. So if you solve this numerically this is let me show it directly what you see it's a diffusion plus a reaction term a logistic reaction term happening in the sense what you're seeing here and this is how the system propagates. What is interesting is that if you take the longitudinal direction what matters for instance from going from source to outlet I put it for clarity horizontally but it could be even any source term would have exactly the same behavior. What happens is that if you do this so this is how the paw propagates into the system depending of different features in this case you have like no drift I'll later on examine the case with have a significant hydrodynamic drift and what happens here is that you're going to have a system with the jagged part here is due to the heterogeneity is generated by the structure itself but it generates a traveling wave and what is phenomenal is that a walker that is at site on the backbone can spare the certain time of a secondary branch if you go here and you get sidetracked in this direction you spend some time around the system here before you can eventually plug back and continue your travel towards the outlet so the main point is that if you go through a system which is one-dimensional but with significant delays because of the sidetracking of the structure essentially produces a slowing of the speed of a traveling front so if you are only interested in the behavior of the system along the backbone you can interpret the secondary branches as essentially a producer of the delay between sides of the backbone so delays could be chemical, physical, biological reactions but generates essentially a lifetime distribution because that's what I'm interested in into your system and you can do that by a technique called continuous time around the walk I'm just leaving a few slides in here but you have the density of the point is something which you take into account the probability that you have of staying you have a distribution of the delay time in a node a might of a delay time in a node at this point is the time you spend if you get kicked out in one of the side structures how long would you spend in the side pockets before you jump into the mainstream and that's a distribution of a lifetime in here and then you have also a term which describes the jump at every side which in this case we call it reactive random walk because you assume that every time step you have a with probability one half you go back and forth you have the system you have to know bifurcation or if you have z nearest neighbors you have a probability one over z of jumping in one of the other ones you have so a long story short this can be treated by the plus transforms it's called Hamilton Jacobi mechanism and you can calculate in fact fairly well yeah the plus transforms how the system behaves so you can calculate you can do the Kolmogorov Fisher system for the homogeneous and dimensional point in particular one dimensional system the speed of propagation of a soliton that we generated by the diffusion plus a logistic reaction term in this particular case and in the case of Piano Network in particular you can do that exactly so a long story short if anybody's interested I told you that in the book that Marino Gatto, Ignacio Rodriguez and myself just published there is a complete derivation of that that speed in fact turns out to be something which is if you look at the size of a jump which you assume that the system would have in discrete space and continuous time would be delta square divided by tau the mean waiting time of system you can see that this is the equivalent of the speed of the other and this is the equivalent of a coefficient so it's twice the square root of r times or a times d but divided by coefficient which is essentially the convolution of the waiting times you have outside the backbone over which you propagate which is larger than one so a theoretical prediction on an abstract model again on a particular structure is telling you that in principle you should have but in Peano you can calculate exactly when you can do it numerically in any other case the heterogeneity generated by the bifurcation structure that you will face propagating in one direction slows the front and indeed that's what happens so if you take a two-dimensional system this is what you have in the two-dimensional system of this kind like this is the you inject into a particular place oops and you see how these propagates in space isotropically and the speed is twice the square root of a times d and almost regardless of a particular network structure for Peano you have an exact solution for OCNs which you have seen the other time you have something of the same kind you have systematically a slower speed of propagation you may wonder who cares and to do that I resort to something which I find absolutely fascinating a group of spanish physicists wrote on theoretical population biology in 2006 that started our interest in the subject studied a quantitative model for the American colonization of the race towards the west in the 19th century as seen through the lenses of transport of a fractal network so the idea is that the landscape heterogeneities and the need of water forced the settlers about through their courses that is any settler will move along in a fractal in fact because we move along a river and only crossing divides a small time so when they cross the device like a baroque or whatever why do they need the through your courses because they needed the water resources cost for drinking energy transportation whatever and the idea that the exact reaction diffusion model seems like a sensible way about a river network and a say which you would have that the population growth would be a logistic with a rate parameter a for population growth at a single node what is remarkable that the archives of a U.S. Congress allows you to evaluate in fact how the colonization came about and what these guys had is that you had the yearly value of the relative frequency of them that is the distribution of the relative frequency of the distance crossed the year by year by the colonizers in different directions and thereby one of the key parameters that you have to calculate that the system would have been the five distribution of the distribution of the jumps and was actually given and didn't need to be calibrated at all and what is totally remarkable if you take that so you had essentially the growth rate from the archives of the population growth you have the distribution of them the features of them of a random walk in space for distribution of the jumps and the settlers would go about over in an isotropic march towards the west and what you had that the feature bridge which is then you had the parameters twice the square root of a times t will be of the order of 40 kilometers per year what is totally remarkable if you take any simulation any kind of assumptions etc you go immediately to speeds of the order of 14 kilometers per year and mind you that the actual observed speed of the expansion was of the order of 1.3 13.5 so the idea is that the data suggests that indeed even in this case the very nature of the substrate in this case for the population for population dynamics was the one dominated by the features of the river network itself because of the constraints that we have imposed in the system this can be so this is again the picture so the constraints that we have imposed on the backbone of propagation of the tabling waves facing the bifurcation that I will once where you will see inevitably the topological properties that is the properties of the bifurcations that you would face along your path on any particular backbone would be the same in this structure which is statistically identical to a two river network have a replicator replicable etc like we have seen the case of ocean and the piano network if I behave very much in the same manner so yet another indication that the substrate plays a major role in the ecology of the system so like I did like I showed you the other time the first thing that we need to say okay it's a strong suggestion and can we study in the laboratory the what happened so we had developed the the ability in my lab to work with protists and so you have these essentially protisto drones if you want so we made these running places and we see how we can measure it by more or less progressively more refined tools etc how and inject the population one end can invade and the speed at which it invades the system so this looks like a Kandinsky picture but is essentially the trace taken with microscopy done in my lab of individual trajectories of every single protist moving in this direction and mind you that one of the two dimensions were much smaller than the longitudinal one thereby making the one dimensional assumption is so I am interested in how heterogeneity slows the front and one would be to test how the demographic stochasticity pitches in with the respect to the physical Mogorov model of invasion so essentially what we took the model of this kind and we decoupled the sense of space from the intrinsic noise which are having the system mind you that different realizations of the protist run through the system are things of this kind so our application it was absolutely needed to see how the system behaves but the point is that if you do these and you replicate these normally which had a 95% range you see that still in a context in which you have there's nothing neutral in here there's nothing but demographics stochasticity induced by the biology of the system and still what is phenomenal is that the physical Mogorov system proves able to reproduce with a remarkable accuracy what happens in the laboratory so interesting enough we would see okay now can we generate other kinds of heterogeneity in the system and how can you do that when one idea was used photo taxes in biological invasions and and the idea is to use the generalized receptor law because you have a system in which every node essentially you have a reaction in the node and you have transport among nodes so the idea is to use the photo taxes in a system of this kind this is my the runner the drone the stadium for these properties to run the injection point the collection point and what we have in a photo tactic system would be to use leads to generate resources heterogeneously distributed in space which hadn't been done before and slowing invasion by heterogeneous environments is one of the tenets in which I've been trying to get you interested in and as you're seeing so the idea is that for instance heterogeneity could be special heterogeneity in which for instance you put some drift in a system so you put a bias in the system not simple diffusion but you add advection which makes it even more heterogeneous of course because advection in this case you assume constant of a system and you have a probability in any node which you have is a probability which is an even probability of splitting in the case of diffusion and it's not even if you have drift of any kind so essentially what you have in the system is the parameter that unifies the drift what you get is that this is the wavefront speed in one case so essentially in the case in which you have like zero drift in the square root of AT like you have it in the color of system up to a system which is completely dominated by advection which is important because I'm interested in studying the persistence of species in a river net or season ecological corridor so they may or may not be important depending on ecologically viable quantities like the nature of the organs we are discussing so anyways what we know is that the Hamilton Jacobi method we have used and the numerical simulations match fairly well and if you instead put drifts you start having negative and positive positive fish waves and you can have approximations which are important in this case we shall briefly talk about the so-called telegraph equations peano then stays in the system with features that are calculated exactly of course I won't test you with the details they are quite involved but but again straightforward and analytically solvable and again significant because peano topological is indistinguishable from the real of your network although if you look carefully you see that other metrics simply do not match the telegraph equation is also something interesting of course now for instance this is what happens if you look at how the system propagates having chosen three different spots for colonizing the system so this is how in fact the system propagation behaves and there's a certain degree of commonality there's a degree of commonality in the system here now why the reaction telegraph model against the diffusion when this all happens if you put a drift what happens is that you have to assume that if you use a reaction random work model you essentially unless you do some provisions for it you would assume that the speed of propagation of the jump will be at infinite velocity which is in fact it's a complicated thing to digest long story short if you put the proper model which overcomes the inconsistency well known inconsistencies of a diffusion model in the case of bias in this case long story short in the computation nothing happens so the reaction random work describe what happens in the system even in the case of the drift fairly well so back to the system which heterogeneity is imposed by me by choosing a certain distribution of resources with a certain correlation length for instance you have like silence resources silence you see it's a prosaum process the one we modeled it with in the system here so we had the ports at the bottom of the channel for changing the distance from the lead the angle of light propagation of the light to use something which is called the Keller Siegel framework for calculating how the resource acts like a sort of an advection in the system and I won't get into details of course if you're interested I'd be delighted to see what what happens but what what is interesting in this case is that we do have a framework so you turn off a light the system behaves and that's very clear from the spectral behavior of the system behaves like a linear e to the minus the wave number square times dt so so it turns out to be a normal diffusion if you turn off the light and it comes up into something different if you generate a light field which aggregates and acts as a drift in the system thereby disturbing the other behavior of the system so individual trajectories here are much larger and as you see in the different realizations and different kinds and yet the coherent picture appears so this is light intensity profile we're using the experiment and you see the one spread experiment was done for each landscape and and thereby you have a a significant way of studying the system what is the punchline of these are relevant to our case is that the mean front propagation that we can compute theoretically through the Keller Segel framework and the experimental one in fact they show that the speed has a new actor it is the autocorrelation length of the resource field so heterogeneity appears in the system and affects the propagation of the speed of propagation of the traveling wave and these autocorrelation does slow the speed as you see here in the case you're having there is which have a mean invasion speed computer directly without any light field included the square root of 2D system of that which is the slope of the system here so heterogeneity whether given intrinsic to the structure of the river network whose architecture recurrent characters were long studying and which I have introduced in my first class have an effect and to jump into something connected like believe it or not I'm going to a giant of the field the late Ilka Hansky craft food price few years back sovereign died too early and whom whom we miss sorely in fact had invented a very important quantity that was key to metapropilation analysis called the metapropilation capacity of a fragmented landscape that is you found out that an ecological measure super sound and quite well established experimentally from fieldwork and from theoretical work that characterizes the suitability of the substrate to effect of heterogeneity in his case was a fragmentation of the landscape and there was a main motivation from the ecological view so technically metapropilation capacity is the leading eigenvalue of an appropriate landscape matrix which I'll explain in a minute and essentially the punchline was that a species is created to persist in a landscape which is the matrix for ecological interaction in my case I would put this in in the other in the case of the other network is larger than some threshold value which is divided by properties of a species and which essentially the capability to disperse and the capability to to to the dispersion capability and the you know whatever the mortality and reproduction rate so metapropilation capacity is a number that can be conveniently be used to rank landscapes or for ecological interactions in terms of their capacity to support meta populations metapropilation of course is another one of those bold statements not unlike the neutral model of biodiversity in that metapropilation essentially ignore interspecific interactions among species that means there's no predator pre-relations etc so essentially it is an intrinsic capacity of one species that determines its ability to survive on this I shall not spend issues time because of the limited time I have but we have discussed that at length for the implications so what happens is that the key to that is essentially a focus on the probability to be at point i in space in time of a focus species the one whose ability to survive is studying which is the probability of being present in patch i at time t which is a balance of colonization extension forces and we just kill out the interspecific interactions over which ecologists spent lifetimes in the past that was a greatness of one of the most in fact gifted field ecologists of our times Ilka Hanskin said so the point is that following the rate of change of this probability in time is essentially a balance of the probability of being colonized by different places at distance at places site j at distance di j measured along the system and and that multiplies the colonization rate and then you have a the extension rate which is birth minus death ratios in the two so the ratio e over c is what determines the teacher that you're having there now the main result is that what is lambda and the the the eigenvalue of the system well it's a landscape it's a the maximum eigenvalue of a matrix which is the landscape matrix which essentially this is the dispersal ability of a species and this is the distance which you have in the system itself so it's a it's a positive matrix it's a reducible matrix in fact so ground for various theorem applies and say and that's what he found out and if it turns out that if the maximum eigenvalue of this particular landscape matrix which is very easy to calculate for a landscape like ours like I show in a minute the landscape matrix for us it's easier and more constraining than in say a savanna ecosystem like we had seen the other day so what happens is the following if you take a path along which an open channel network develops so we started from an any particular system and we say essentially streamline a simplified network by assuming that the system would tends to minimize its energy dissipation approximating the structures that and the statistics that the river network inevitably has but like it or not it's some sort of an unintended consequences with the decreasing energy the second line of energy you have as a bonus in fact the best better viability in abstract terms generated by the river network so the ecosystem in a sense benefits from the physical process that determines the substrate that we are talking about and and then well there are other issues but I realized it's I've been chatting too much so I'll be skipping fields but anyways this is a significant result in terms of the probability and if you take several realizations of the same network etc you have a band a narrow band and if you have like different kind of mean field versus OCN thing you see that this is a significant result from the statistical view so I'm I've the last 10 minutes of my talk will be devoted to can we use the same constant to study river networks and biodiversity and I will leave to the third and last class how this in fact is key to evaluate how pathogens spread along river networks psychological corridors thereby propagating deadly disease deadly or chronic disease so what am I saying is that in a river network is a set of is an oriented graph my my notes and edges edges are physical edges obtained offline by digital terrain maps that we see last time you do have reactions which would be physical chemical or or biological reactions in this case biological ecological reactions and essentially the links are what act as transport models between notes and this applies to individuals this applies to species or to populations so essentially we can talk about rather than make a relation make the communities and the first example I want to show it's a paper which we have Marino and I care very much for Lorenzo Mai is the first author on the metapopulation persistence in in river networks what happens is a following so the to study metapopulation dynamics can be done by if by looking at the system in which you may have like species that live on a substrate of this kind which eventually might if I get off a move off of the network itself so the species we are talking about the metapopulation and that somewhat forces what's going to happen with more complex compartmental models for a number of different phenomena will be like you have like in every note you have two ecologically distinct developmental stages so we split the population living in the node which is a reach if you please and in the young non reproductive individuals why and the adult reproductive individuals movement from one node to the other can occur for different pathways but along the structure of the network either along the same network if you are a fish or even overland if you are an amphibian for instance in certain stages of the life you take into account local demographic processes birth growth and death and the dispersal dynamics in each node of a river network which again using the full constraining power of of the system of this kind now this is important to explain what's called the drift paradox because to explain the long-term dispersal in system in which the drift could be very significant effect there's there's something which ecologically puzzled ecology for a long time for instance to explain the long-term persistence of populations for instance empirical documented examples came in for instance for Scandinavian freshwater ecosystems insect species that compensate the larval drift that is transported passively along the river network with upstream directed flight of adults prior to OV position and it's called the Mueller's colonization cycle otherwise an excess production hypothesis has been forth in which drifting organisms would be exceeding the balance of demography of the local scale which implicitly assumed that the drift essentially represents an extra mortality terms of these kinds but anyways all I'm saying is that what is new here is not the ecology which is well known but it's the fact that you're using a subset of this kind and to give an example of how coupled are these in this case two for every node at node I you have the young the number of young no reproductive individuals and the number of adult reproductive individuals they have a certain mortality term possibly density dependent you have them in what you have the most important you have the probability of dispersing from any j site so you may have adults at site I can be taken because of the connectivity matrix for any side j by simple proportion and transport that you're having in the system whereas in the case of the adults in addition to that you have reproduction terms it's not important actually how you do that of the specificity of that which is very well known system that we're having in this place here but what I'm showing you is the fact that how do you study those systems is two equations per end nodes of a river network so you can actually assume that those matrices dispersal matrices are probability which depends on the connectivity structure something which you assign and give offline and which you can study by studying the stability of the the metaprolation persistence is actually related to its stability of the population that is if a state x0 is stable the population cannot persist in any river network nodes if it is unstable juveniles and adult abundances grow and they grow the multivalation persistence so you can study the global equilibrium of those matrices which is essentially it's a super Jacobian in which you have two n by two n matrices that can be done exactly and calculate the survival and the non-survival of the different ranges of the system and what for instance one particular case which was particularly interesting was the study of salamanders for which you had the critical data from I think it was in Virginia some place in which you can relate the conditions from manipulation to persist and manipulation not to persist counting on the fact that the juvenile and the adults have different ability to jump off the river network itself I am almost done regretfully so I will compensate next thing but give me three minutes to complete with another example the invasion of the zebra muscle which paves the way of something which I'll be showing you Friday that is what you're seeing here it was the zebra muscle was by accident introduced in the Mississippi Missouri river basin by introduction it was taken from the veligus without the the larvee that were introduced in ballast water of cargo ships that came in from Europe where it was native and it started spreading through the system reaching the DTAs even embedded hydrologic units in the system have been growing dramatically over the years so up to a point what is totally remarkable about this is the fact that at a certain point it didn't even simply diffuse or disperse downstream all of a sudden you started having places where you started having flare-ups of invasions in completely unrelated places far away from it that introduces the fact that we can study river networks fairly well in those systems and essentially the local reaction equation is the local local age growth model in four stages which have lava production transport of lardy is a passive larvae are small the veligas are essentially passive scalars so they diffuse and they are vetted to the river network for the river network but most important you have to introduce what is called more than a multiplex network why because in those recreational ponds in which you put some boats what happens is that if a guy picks up a boat doesn't empty its ballast water properly put it on the trailer and brings a thousand kilometers away then in as much as COVID-19 spread through human mobility so did the spread of this system here so the long story short on which I shall stop in time regretting that I have dedicated too much time to the early phases but I will be saying this what I miss from here next Friday so EGAS is the computer that simulated values in like in this case that is a measure and is a computed as you're seeing here only by our ability to introduce those long range flights generated in the injection by a different network which is not related to the topology of a river network but they're related to the road infrastructure in a certain probability of making a distance in the system you are able to calculate one of the most devastating because zebra mussel grew at the level of density that generated significant damage not only ecological because it displaced every other native species of that kind but it also generated clogging of hydropower plants production and the likes and this is an example of how in fact that biological invasions reach the place yeah this is a good point to stop in here so the point line is that and now we've been taking on the next class to see how this in fact becomes the essence for studying through the same environmental matrix and possibly multiplexed network in which pathogens disperse waterborne I mean through the network and human mobility spreads the disease how in fact they can use oriented graphs and other reactions and hydrologic analogic transport to tackle individuals to make up relations or make up relations now I'm ready for your questions great thanks a lot Andrea for the very very nice lecture so we have time for questions so please if you want to ask one you can use the arrays and feature or you can type it in the in the chat yes please Monday can we emi then Monday can we emi yes I do barely but yeah okay yeah can you hear me sir I I do okay I'm okay so I I'm I'm looking at the last the last point you presented in this slide is it or do I join the class not to at the point to start it is it that the the network of of of of the the model that you brought was but it but it did before you presented the result of the data or is it it it directs imputes without validation but it's a good question actually Monday what happens is the following okay there are networks and networks in this case if you're talking about the transport of you see the main point right you have nodes in which have physical chemical biological reactions right and then and you have transport along the network structure now if you're talking about the dispersal of the organism in the in the larval form for instance they are small seeds they behave like a passive scholar so they diffuse and they are affected and diffuse a lot of the river right but if you get a bunch of them and you take them from a node and you transport that completely artificial into another node like you do if they with the veligas they get trapped in the ballast water uncleaned of the boat then is the mechanism that can explain why you can actually have a flare up of the of a proliferation of a zebra mussel hundreds of kilometers away without any in between why this is interesting because it's like receding the infection if you want or the density of a population somewhere else now in one case the network is essentially the physical substrate which we know directly right because that's unavoidable digital terrain maps allow you to extract those connectivity matrices that is how I know to connect to any other node in the system and you can actually distinguish whether you for instance you can also go you have a preferential transport downstream but you may even go upstream with a probability which is different so you have a bias transport I'll be talking about that Friday but in the case of a of a mobility network generated by the other have time something else so in that case what we did was to evaluate the kernel of dispersion but it was calibrated on the data actually which is the displacement of commercial boats into the place through some proxy which is a data on the number of boats parked in different positions when Mississippi Missouri were basing so when whereas in the case of a river network it's a given the connectivity matrix is given once and for all because it's extracted objectively it's added remotely and objectively manipulated if you have a case that you may have to make assumptions when we'll be talking about human mobility for instance to spread disease which is not unlike these you will have two possibilities one assume one of the models for like normally used like gravity models radiation models and then or you may use cellular phones and track individual mobility of large numbers now you can now in that case you may wonder whether the use of a telephone in certain places where color spreads is socially biased or not if you're looking only a certain segment of a population but mind you high experiences that's not the case that's actually socially neutral the ownership of a phone even in the most dilapidated place on earth and I've seen it in my eyes and I'll tell you next week okay thank you so much for the explanation I will appeal if there's maybe a kind of a reference material then I will like to to read more into it I'd be delighted well actually if I'm not I'm not selling anything because all the money we go to Cambridge University Press but the book that I just published that with Marino Gatto and Ignacio Rodriguez-Turbez called the River Networks as Ecological Corridors Species Population Pathogens had everything where we have done the past 15 years on the subject and I think hopefully a good review of a literature as well thank you Mande great thanks a lot for the question and the answer so we have time for more questions if any also on YouTube you can type the question if you want otherwise I think we are actually on time so otherwise thanks again very much Andrea for the for the lectures and looking forward for the next one if you want to listen to Marino can I stay on here absolutely yes so what we're going to do now is that we are taking a break before the before Marino