 Just a few seconds, okay. Welcome everybody for this day of lectures. So we'll start today with the first lecture by Amos Maritan, who I have the pleasure to introduce. Amos Maritan is a professor at the physics department of the University of Padova. And his research interests are very broad and the important contributions in several topics of quantitative biology, from the theory of biopolymers to ecology. And in these lectures, we're gonna talk about scaling in ecology and the relation between metabolic theory and the community practice. So before Amos starts with the presentation, just a few reminders on how to ask questions. So if you are watching from YouTube, you can post your question in the chat. If you are connected in Zoom, you can either write the question in the chat or use the raise-end tool of Zoom. So thank you, Amos, for being with us and please start when you are ready. Thank you, Jacopo, for the introduction. So this is the first lecture of a series of three. The first is delivered by me, by myself. The second will be delivered by Samir Subais and the third by Sandro Zairek. I think that the order is, should be correct. Of course, what I will be telling you is the result of a long collaboration and there is a list, a partial list of, at least of the ones that have contributed more to the ideas that I will tell you today. And in particular, Cheyenne Banova, which is just one of my first collaborators, and then Andrea Rinaldo also that was lecturing in this series of lectures. And Stephen Abel that was introducing us to the ecology of forests that will be, most of this lecture will be devoted to them. So in the last 40 years, there's been a lot of data from bacteria to forest about ecology. And at least there are two mainstream approach to them. One, sorry, there's a mistake here. This is a model dependent approach on the left and model independent approach on the right. So the model dependent approach in the left includes a neutral theory, which something will be said by Sandro Zairek about this in the third lecture and the resource competition model, which will be the subject of the second lecture. Then the Dr. Volterra equation, kind of equation that I think that Stefano Alessina has illuminated you on this issue. And then on the right, there is the model independent approach, which includes the scaling, scaling approach that could be illustrated in this lecture. And all of these should converge to some model that contains the scaling embedded in them in order to be able to reproduce the known results in all the most important ecosystem of interest. And of course to be able to be also predictive and not just the descriptive. Okay, so the first thing that I need to be sure that you are aware of is what are power law and how they are deviations from them in order not to be confused about them. So one example that most of you should know it is the problem of diffusion. For example, in 1D, that's why you have a particle that diffuse in 1D can move in a symmetric way both to the right and to the left. And so the probability to be at some position x at time t is given by Gaussian with the variance that is proportional to the time. And so if you set the x equals zero and you ask what is the probability to be at the origin at time t, this is just a power law. And so in 1D, it is a t to the minus one half and then add you to dimensionary zone there is also the diffusion constant that appear in the denominator to the power one half, which gives the correct dimension to the probability which is one over a length because it is a probability density in 1D. So if we plot this function in a linear plot, we get the plot in the left which is like a similar to an imperbola. But in order to visualize better, the power law it is convenient to plot the logarithm of p versus the logarithm of t. And so if we plot the log of p we get a straight line with low of minus one half. So this is the one on the right is the best way to visualize when a power law is presenting our problem. So typically what we find is not pure power law like this, but power law that are a little bit, let's say dirty or truncated and the truncation can, the reason for the truncation can be at least there are two kinds of truncation. One it is because in the problem it has been introduced a new timescale. In the previous there was no timescale. You see that the only timescale is just the time and there's no characteristic timescale in the problem. And this is why we see pure power law. If for example in the Rano Woker, the diffusion problem there is an external field that tries to push the particle more to the right than to the left. Then there is a timescale in the problem that enters through a sort of average velocity which is tau which is 4D, the diffusion constant time v square where v square is a sort of average velocity. And so this is the right dimension of time. And so the previous law is corrected by an exponential. So for time much less than tau, we see the previous power law with lower mind of one another which is just the prefactor. Whereas for time of order of tau or much larger than tau, we see that the power law is ruined due to a faster decay towards zero. And where the crossover from pure power law to a truncated power law core is just the characteristic timescale that has been introduced to the external field. So other reason for having truncated power law is that the system is finite, for example. So if we have that the particle is contained in a fixed region, in a compact region then there are corrections due to the finite size of the system. And these are the ones that we'll be encountering in this lecture. So there is a problem about the previous two slides where you are encouraged to solve it. It starts from just to describe the movement of a particle in terms of discrete time and spatial steps. And in order to be led to the diffusion equation in one D which is 0.5 and then a solution to the Fourier transformer which is the last but one line where you see that if we set x equals 0 we get exactly what I said in the previous slide about the truncated power law. So even without knowing anything about the exact solution of the diffusion problem in one D you could have argued something based on scaling argument. The scaling argument works as these slides show. So if we have to calculate the probability density to be at position x at time T given that there is a diffusion constant D and there's no bias, no external field at the moment. The diffusion constant has a dimension of length squared divided by time because it describes how the variance of the position varies with time. It is D times time. So D is the dimension of length squared divided by time. So since P x of T in the dimension must have the dimension of one over a volume one way to construct the dimension of volume is just multiplying the time by D and taking the power minus D half. And this put the correct dimension of P and then we must have a function which has to be dimensionless. And so the only dimension and constant it is just, sorry, the only dimension and constant is just x squared divided by DT. And so this is the dimensionless ratio. So without knowing the solution of the problem we just, just due to scaling argument we can predict that the solution must have this form where f is some suitable function that guarantees of course that the probability is also normalized. So in particular if you set the x equals zero f becomes a constant in that case and we get the answer that we asked before. The probability to be at the origin of time D scale like a T to the minus D half. So in one D it is minus one half like we were saying before. In the case that we have an external field in the problem and so meaning that we have another velocity entering the problem then we have another time scale through the velocity and the diffusion constant that I showed you before. And so the new time scale is introduced tau and so this P of T now can have an extra function an extra dependence on time through a function and the dimensionless ratio which is f over T. So in the case of the diffusion that I showed before this function f is an exponential. The scaling argument typically is unable to determine what the small f is. Okay, and so you must have some other information on the system in order to calculate it. The only thing that the scaling argument can tell you is about how the scaling function has to behave for smaller or larger argument. So for small argument meaning when tau is much larger than T then we want that there is a pure power low scaling and so the function has to tend to a constant. Otherwise when T is much larger than tau we want that this is a truncated power low and so it goes to zero. Any question at this point? This is just the preamble. There is no question in the chat but if anyone wants to ask a question please either type it or raise hand using the zoom feature. I think you can move on. I'll tell you if there is any question. Okay, an important concept that will pervade the rest of this lecture is the buzzer metabolic rate B. So we are mammals and we are in thermal engine that we are using energy just through the food, the air that we breathe and so on. And we dissipate energy. And so the dissipated energy is proportional to the energy that we consume. And so this is called the buzzer metabolic rate that has to be distinguished but what it is called for example field metabolic rate which is what happens when you are running or doing some movement. Whereas the buzzer one is more or less when you are almost sleeping. For plants it is the amount of water that is evaporated per day for example. So a plant of 20 meters height it can evaporate about 200 liters per day in good condition. And so the amount of water evaporated for better day is just a measure of the metabolic rate of a plant. So the metabolic rate in principle can depends on many details on the physiology on the specific physiology of the organism be it an insect, a mammal, a bird, a fish or whatever or a plant. So it comes like a surprise that the main trend of the metabolic rate doesn't depend too much on physiological details apart from some particular case but it seems to depend mostly on the body mass of the organism being it a plant or an animal or a bird or whatever. So for example, if we concentrate for a moment in mammals we dissipate the energy through the skin and so you might think that the dissipated heat then is proportional to the surface area of the skin meaning that since we are more or less a complex object the surface scale like the volume of the organism to the power to third exactly like a sphere as a surface that scale like the radius square and the volume like the radius cube so it means that the surface of the sphere scale like the volume to the two-third. So since the mass of a mammal is proportional to the volume through the density of the body and the density is more or less the one of the water the mass is proportional to the volume. So it means then that if we take that B is proportional to the surface, the mass is proportional to the volume and the surface related to the volume to the power to third we deduce that we should deduce it that the metabolic rate scale like the mass to the two-third. However, at the end of 40s, 1940s Mr. Kleiber found that the two-third should be three-quarter and indeed in his original plot you see the nice picture of Kleiber, Max Kleiber and on the right, he was plotting the heat production per day in kilo calories again in the long plot as I was teaching you before versus the log of the master. So if he's right, it should find that the log of the master versus the log of the body mass, the log of the metabolic rate versus the log of the body mass should be a straight line of a slope, a three-quarter which is the red line. And you see that here a rabbit, dog, woman, I don't know why not men, but it does matter, cow, elephant, whale, the whale is just the green point on the top and the red line as the slope which is three-quarter, the upper dash line, it is just the weight. So if he plotted the weight versus the weight it should be a line with a slope one. So the upper line as a slope one, the lower dash line is the surface and the surface in this scale like two-third. So according to him, the red line is in the middle and it is well-approximated by two-quarter. Anyway, after 71 here, Brown did the one in the first reference on the right bottom, they were calculating again the masses and the metabolic rate corrected with the temperature, so there's some temperature correction that has to enter in this slope in order to make it a little bit better. You see that the log on, it is log of B like before, it is not very visible. It is the log of B versus the log of mass. Doesn't matter if the mass is in gram or in kilograms because that is just a constant that is added to the log. So it is independent on which unit you are using. The power load does depend on the units. And the log is in base, in natural base, otherwise here there are too many order. Anyway, log 10, it is about 18 order, one eight. And this slope is well-approximated by three-quarter. And here there are, you see plants, mammal, bird, bat, zooplankton, insect, fish. And you see that apart from some local deviation that you can see sometime, but overall the trend, it is well-captured by the three-quarter, like I was saying. So one might ask, why this three-quarter is important? What is the impact of this law on our life? So this is something that I will, it is again a small problem, but it is done in detail. So the metabolic rate of an organ can be attributed to the maintenance of the cell. So if you have in your body NC cell at the time t, each cell is consuming BC as unit of energy. So the metabolic rate, the energy input in our body, there is a part that is used by cell and the number of cells is NC. And the part is used to change to the number of cells, meaning doing a new cell, which is the last part. In order to do a new cell, you need the EC of energy. So the net balance is that the maintenance, pure, the use for doing a new cell, overall, it is the total input of energy. So the number of cells is proportional to the mass, okay? MC is the mass of the cell. The mass of the cell doesn't change with age, it is the same at all ages. And so we can transform this equation in an equation for M and using that the metabolic rate scale with the mass to some power. And we set the constant in such a way that it has the correct unit. So B naught and MC, MC is the mass of the cell. So we scale the mass, M over MC is the number of cells. B naught is some constant in order to fix the right dimension. And alpha can be taken three-quarter if we believe in the Clybert law. So this equation in transforming this one, where the first term comes from the Monday, from the Monday, from, sorry, the second term, it is from the maintenance. And the second term, the first term, it is from M, alpha, it is the Clybert law. And this is a DM over the T. So we get this equation. So if we make a small change of variables and we introduce a dimensionless variable, which is M over capital M, where M, capital M is the adult mass, we can transform the equation for the growth in an equation one. And so after we change the variable in terms of dimensionless variables and in terms of a characteristic time, which is this, okay, the red part, we get an equation for X, which is very simple to integrate. And this gives equation three. So then at the end, we find that the mass of an organism should increase like equation four is indicating. Well, you see that, I think that there is a minus sign in the exponential, otherwise it becomes close to infinity. It's not right. So there is a minus sign in front of the exponent. And there is E to the minus T over tau critical, tau characteristic. And the tau characteristic, it is the mass to the minus one minus alpha. So if alpha is three quarters, it is M to the one quarter. So solving the equation for growth of an organism, automatically tell us that there is a characteristic time in our, in our, in our organism. That scale like the muscle to the one quarter. So not only that, but what we get, it is how the organism should increase their mass as time increases. So if you plot everything in terms of, I'm not sorry, in terms of time that is scaled in this way, the time is T tau, sorry, T minus tau critical characteristic divided by this log and we take into account the mass at birth. So all organism plotted the mass of the organism, with respect to the mass of the other, in terms of this scale time, okay, in this dimensional time, they should collapse in a single curve. So here, for example, we have the fit of how the mass of a cow should grow. Here, how the mass of a cow should, and you see that the data that are the green light, the green point, I think, they get the green and the empirical are well fitted by the curve, by the prediction, both with alpha to turn and also with alpha to quarter. The difference is very, is very light. And so each organism has its own ontogenic growth, but if you plot everything in terms of the dimensionless time, this time here that I've written here, and the ratio of the mass, the actual mass divided the mass when they are added, you see that all curves collapse in a single curve. So you see that here they are point from many animals, pigs, shrew, rabbit, cow, rat, and so on, and that they all follow a universal curve. So we are a member of the same family, meaning that we're not too different from a lizard or from a pig in terms of growth if we choose a suitable scale time and the scale mass, meaning we measure the mass with respect to the adult mass and the time has to be scaled according to the mass to the one quarter. For example, the lifespan grows like mass to the one quarter. So small animals like a rat are much less than an elephant. Also the health beta scale like the mass to the minus one quarter. So as a rabbit has a frequency which is much higher than one of an elephant. And so there are many consequences of this and then you can find them in the original paper by us and by West, Brown, and Anglister. So here now we concentrate on the origin of this one quarter because there is this M to the one quarter, the metabolic rate scale like the mass to the three quarter. And so it is called the quarter power loss scale. Are there questions at this point before I go on? So not in the chat, but I'm not there. Yeah, there is a question. Where in this plot, so in these plots, where in this plot falls species in which individual is hard to define. This is uncommon for animals, but very common in plants. Sorry, but this is the question. I don't know if Miguel wants to ask the question. Yes, yes, of course. So here it's fundamental to define an individual. So you can measure mass, the growth of that individual. So where would the organisms where individual is hard to define like this, I don't know, large prairies of grass, where all the grass blades are connected or? I'm not sure that I understand the question. This individual, I think they're grown in some, all in the same condition. Meaning that they are, I think they are feeded regularly. And the point on this graph, I think it's averages over many individuals of the same kind. Right, but the definition of the rescaling of the mass depends on the mass of the adult. Oh, yes, yes, exactly. The average mass of an adult. Exactly, yes. If we have something like in plants, for example, where they can basically continue to grow indefinitely. Oh, yes. So that is a question of how to define the mass of a plant. We can record. I think I will come back later to that. For plants, you should distinguish between the dead mass and the living mass. You're right. Okay. So this, here, I don't know if you can read that there is no plant in this plot. Right. But you're right, okay. Okay, thank you. Okay, so now we will concentrate on the possible origin of this water power law. And so many years ago, we had some inclusion on why or where this trick water comes from. And so we were hypothesizing that this could be the result of an optimization principle. And so we said that for a given metabolic rate, evolution has been such that it selected animals or organism, whose mass is the minimum as possible for a given metabolic rate. Or if you want for a given mass, you want the maximum metabolic rate as possible, which is like to say that if the metabolic rate of a restaurant is the number of meals served per day, and the mass is followed by the number of waiters, you want to have the minimum number of waiters for the given number of meals that you serve every day. So this is, for example, the problem of serving a village or a town with water, and you ask how much water do we want to have in the pipes that are able to fulfill the need of the town. So if you have pipes that goes around in a way like showing the figure like in a spiral, for example, for these nine houses, each house is consumed one unit of water per day. So from the main source, we must have that nine unit of water has to leave the main source every day. The pipe enter in the first house, one unit is consumed out, eight units are going out from the first house and so on. So overall, in the pipes, you have eight plus seven plus six. And overall, you have that while the metabolic rate is nine, which is nine unit of water per day, the total mass is the total mass present in the pipes. And so if you sum nine plus eight plus off, what you get, it is approximately L to the four. If L by L is the size of the town, L squared is the metabolic rate, the number of unit of water consumed per day. What are L to the four is the total water present on the pipes or the number of waiters that are going around in the tables. And so if you put the two things together, what you have is that the mass scale like the metabolic rate to the square. So we can do better if for example, we choose that the distribution network is ramified, it is a tree. A direct tree meaning that the arrow are not going back, but they are only going in the direction away from the source. So again, if you count how much water is in the pipe, instead of the L to the four, you now you find L to the three. And so the exponent relating the mass to the metabolic rate is three half. So it means that in 3D, we get three four because if we do again the same trick as before, what you find is that the metabolic rate scale like the mass to the power D, the dimension of the system divide D plus one. So in 3D it's three quarter. So this seems a very nice a magic theorem that can be proved exactly that not everything is understood perfectly, but at least it gives an idea that the three quarter may be the result of an optimization principle. For plants, the things is even, it is even simple. So for us that it's for sure, it is one of the most exciting ecosystem because due to their complexity. And the structure of the forest is also related to the forest functionality. By forest structure, I mean the distribution of sizes of three in a forest. So in the last 40 years, I've been a several measurement, several sensors of forest around the world. And typically what they do is that they measure, which is the, which is the easier things, the diameter of the trunk at some height. And then they measure the diameter through just this ribbon like meter. And the distribution of the diameter that the probability to find the fraction of trees with a given diameter is plotted here. This is again a log, log plot. This is the log of individuals, meaning plants, having a given radius, trunk radius at the breast height. So in the horizontal axis, you have the log of the diameter in the vertical axis. You have the log of the number of individuals. Of course, this has freedom of being. And you see that more or less in this one decade, a little bit more than what they did. There is a power law with an exponent, which is around minus two. And you see that this is true for various kind of forest here, we are just showing four later on, which show more of them. And there is some truncation, meaning that you can find trees with sizes that are too big. You can also measure another trait of the plant beside the trunk radius. You can also measure the volume of the crown. So again, if we plot the log of the crown volume versus the log of the height, okay, we see that there is this cloud of point that more or less there's not been no attempt to make an average and just plot also the error on the average and so on. Just these are the raw data, plot in the log log plot as before. What you see is that the crown volume grows like approximately the cube of the height of the tree. So you might think that you would expect that it is more or less something like a very compact cube and a spherical object. But this is not true because as you, so just I need another slide. You can define an exponent that says in principle how the crown radius scale with the height of the tree. And so this is typically described by an exponent that is called the half exponent. So this half exponent for this slide, it is one as I will show in a moment. But before showing that, let me show what happens at various latitudes. This exponent that around the tropics is around one, as you go toward the more northern latitudes, it decay toward the point five, okay. So it's not true that the plants are always like spherical object like in this picture. But as you go to more and more, more to the north, they become shorter in height, but also much thinner. So here are the tropics, it is almost an isotropic scaling of the width and the height. Whereas as you move to the northern, the height scale much faster than the width of the crown. This just means that they are taller and it are shorter in height, but the width of the crown scale with an exponent which is smaller than one. So now let's see what are the impact of this scaling. And before that, we can derive how the metabolic rate of a plant scale with the plant mass. And so we can come to what is meant by mass of a plant, at least for me. So we have that plants are taking water from underground and through silencer that are not cylindrical, but almost cylindrical conduit. They are transported to the leaves and through stone they are evaporated in the air. And so if the leaf density of the crown is more or less constant and if the leaf is an invariant unit, meaning that they are not changing with the age of the tree, which is true. If I give you a leaf of a tree, you can't say if the tree is one year old or a hundred or a thousand year old. It is always the same. Based on that, if the tree volume, if the crown scale like the height to the capital H, then you can calculate what is the volume of the crown. So it is H times H to the power, H, sorry, the R crown square, which means H to the two way capital H, meaning that the crown volume scale like the height of the tree to an exponent, which is one plus two H. So as I said before at the capital H is about one around the tropics. And so it makes the crown of the volume crown to be to scale like more or less tree around the tropics. And then at the most northern latitude, the scale like two because capital H becomes about one half. So we go from H cube near the tropics to H square to the north. And if the leaf density is more or less constant, we can deduce that since the above all transpiration is proportional to the surface of the leaves, the number of leaves is proportional to the volume. So we have that the metabolic rate of the tree is related to the volume of the crown, which is the height of the tree to the power one plus two H, capital H. Now you can also measure the mass of the water inside the tree, okay? And the mass of the water inside the tree is proportional to the, what is called the sub-wood, the living part of the wood, which is just the say the external part of the trunk. Meaning the water is flowing only through the external part of the trunk. Not this is not strictly true for old plant. For example, at the tropics, all the wood is more or less living. And so all trunk is conducting water. Whereas where there is the seasons and our latitude is only the external part that is more conductive. The internal one is dead. And so we have the water is flowing only near the surface of the trunk. And so what we mean by mass of a tree is the water present in the tree. Or if you want the living wood, not the dead part. So you can do a short, a simple calculation to see that the amount of water present, it is the metabolic creator times age. Just calculating how much water is contained in the xylem given that the surface of the tip of the xylem can be identified with the stomach of the lip, which we have already calculated the surface. And so you can calculate that the total mass, the total water present on a plant is proportional to height of the tree to the power two plus two H. So if we use that B is proportional to H to the one plus two capital H and the mass is as an extra power of H, you can put everything together and you find the generalization of the climate flow in terms of the asked exponent. And so you see that when capital H is one, you find the three quarter. So for plants near the tropics, we should have that the metabolic rate scale like the living part of the wood or the total water present in the plant to the power three quarter. Whereas away from the tropics, this exponent becomes smaller till it reaches two-thirds toward the north. So you can also express the metabolic rate if all the cross-section of the trunk transport water or a fraction of it at least, then we can also express the metabolic rate as the water flow times the cross-section, which is approximately pi R squared. And the water density of course does depend on the size of the tree. At the end, what you find is the metabolic rate is also proportional to the cross-section of the trunk of the base, okay? So if we use that, at the end, you can get this nice table where you can express the metabolic rate as the height to the one plus two capital H or two R squared. Or if you use these two relation, you can express the radius of the trunk in terms of the height of the tree. So you see that when H is one, you get that the radius of the trunk grows like super linearly with the height. You might observe that the trunk radius is much smaller than the height of the tree. That is true. I'm not saying the opposite, but it means that here in front, there must be an amplitude that is very smaller. So the amplitude is not the concern of the scaling theory at least at this level, but it is an important issue because that is related to what is the maximum height of the tree, okay? Because we don't see trees that are half a kilometer height, most of them are 100 meter height and that is an open issue. Nobody's know why trees stop to grow in height, but they can grow in mass as one of your colleagues was saying before. The mass of the tree is always growing because there is a part that is dead, that is just a sustained development. It's just the physical support of the plant. The ranges of the crown scale like in this way, okay, just using the scaling of h and r, the mass of the tree, if it is more or less concentrated on trunk and branches scale like h times r square, meaning the more or less the mass of the trunk. And so you can find that the mass of a tree scale like at the tropics scale like h to the four. So the volume scale like h cube, as I said before, the mass scale like h to the four. So then it means that the density of the wood scale like h cube, sorry, h four divide h cube, meaning that the density of the wood or the plant grows like h. So you can imagine that this density cannot grow forever because at a certain point, branches started to collapse one on the other and there's no more room for the leaves. So there's no more room for the bubble transpiration. So this constraint is telling you that at a certain point, it must stop, okay? In order to evaluate them where this can occur, we should know what are the constant in front of this power law that has to be there also for dimensional reason. So now we can also use this geometrical trace to understand what happens for an assembly of plant or let's say for forest. Any question at this point? There are not a question in the chat. Let's go on. You can go on and then collect the question again. Okay, there is a question? Yeah, perhaps there is a question. Yeah, I have a question. This is very interesting. I would like to ask, you showed before that there is this paper that they have found that the relationship of the body mass with the number of the individuals, they found the scaling exponent of minus two. Oh, yes, yes, yes. Around minus two, yes. The number of individuals of plant with a given radius, frankly, a scale like R to the minus two. Yeah, my question is based on, because there is also the Damos rule, which also D. Damos rule, who also- Damos law, okay? Damos law, yeah, that he also predicts that the relationship of the body size and the abundance should scale with the minus three-fourths. Yes, that is another problem. Damos was one of, we have collaborated. So I know what it was meant. So if you take, if you are interested in cultivating trees and you want to know how many trees can stay on a given area, but trees of the same size, then you are right. Because if each trees, that they're all the same size, so they are all consuming the same amount of energy, and we know that each of them is consuming R squared. Okay? So you have to take the maximum number of trees of the same size times R squared, which is the size, as to be equal to the amount of resources that are avoidable. So if the amount of resources that are avoidable is fixed, the total number of trees is R to the minus two. Exactly. But this exponent is related, what you are saying is related to trees that are more or less of the same size. I'm not talking of that. We are talking of the distribution of sizes, like it is observing forests like this. I will come back to that problem in a moment. Okay. So in order to understand exactly this problem, how the sizes of trees are distributed on a forest, we must introduce another optimization principle, which is that we assume that plants are trying to fill all possible room that is free. Of course, a fraction of it, a finite fraction, it's not just an infinitesimal fraction, but a fraction of it. So if we assume that we must have that at a given, at any given height, we must have that the room, that the space is occupied by leaves. So if we say that the probability to have plants of a given height, P naught of H, the P naught is introduced because I want to call it ideal distribution, and we know that the volume of a crown scale like small H to the power of one plus two capital H. The distribution of height is P naught of H. So the total number of trees, suppose that is capital N, okay. So N times P naught of H prime times DH prime is, it is just the fourth line. It is the total number of trees with height in the range between H and H plus DH. If we multiply this by the volume, which is H prime to the power of one plus two H, and we integrate between zero and capital and the small H, we require that this is proportional to the volume of the forest of area A, meaning this is, it is feeling almost uniformly or the room at this point. And so we are saying that the total volume occupied by the leaves is proportional to A, the area of the forest, times H. This has to be true for all H in the range between, let's say zero and some maximum height that I've called H critical. If I define H with respect to H, then what you derive from here is that the ideal distribution, it is some constant, the area of the plot divided by N, which is the inverse density of plants, times H to the power of H. Minus one plus two capital H. The density of trees, this picture is taken from the book by Habbel, the Neutral Theory of Biogeography. And you see that the density is constant. The density of plants as the number, as the plot area increases, you see that there is a perfect straight line with slope one. And so it means that this is just constant. And so it is just the ideal distribution of height, it is just H to the exponent of the crown volume. So now we take into account that there is a constraint that the plot has to satisfy that the resource availability is fixed. So if we take into account the resource availability, the ideal distribution has to be modified by taking into account that we cannot have as many plants as we want. So if we take into account the limitation in the resource availability, we find that after we change the variable from height to radius of the trunk, we find that the distribution at the tropics, just for simplicity, it is H to the minus seven over three and then the truncated power law with some characteristic radius that is related to the avoidability of the resources. So the minus seven over three is just the ideal distribution without taking into account the finiteness of the resources. Sorry, this is my dog. And so the exponent is not a minus two, like a simple application of the downflow is true, but it's not answering to this question or the distribution of the sizes. So the exponent is 2.3. And indeed, from the data, we saw that the exponent is nearer to seven over three than two. Originally, West Anquist and Brown did a mistake and they were saying that this was minus two and they also related to the downflow, but it's not the downflow. Of course, as we go up in the latitude, this exponent changes. So in order to understand why the finiteness of the resource avoidability impact on the ideal distribution one has to use a minimum principle. So we say that the true distribution with this P of H has to be as near as possible to the ideal one taking into account that the total amount of resources has to be finite. So if we impose that the total number of resources has to be finite, and this is a question two, and we want to minimize the distance of the real distribution from the ideal one after some calculation, we find that the distribution of height is the ideal one, which is the red here, times the exponential. So if we now do a change of variables with some assumption that we know that the radius of the trunk scale like the height of the tree in this way, we saw this in the table that I was showing before. So if we do the, if this is deterministic, meaning that given the radius of the trunk, we automatically say what is the height of the tree, then we should say that it is a delta of theta. This is not true because there are two random variables. So in principle, what we have to see is that the probability that we have a given radius of the trunk, given that the height of the tree is h, a generalization of a delta of the delta is this scaling answer. So if we use the scaling answer with the distribution of the height, the known ideal one that we have just found in the previous slide that you can go over carefully by yourself, you will use what we found in the previous slide. We use this scaling answer, which is the number two, which is a generalization of the deterministic answer. And then you do some calculation, you find what is the distribution of the radius, which is the, this power law, one plus six capital H divided one plus two H. When capital H is one, you find seven over three corrected by this exponential, which is a Gaussian in this case, which is R divided over RC. RC is just related to the amount of resources that are at disposal. So if the resources are very, very large, RC is very large. And so the exponential comes into play only for large radius, and you can see a beautiful power line. On the other hand, if the resources are very tiny, then RC is very small. And so the power law is masked by this correction here. So we take this, this is a constant, by the way. It can be fixed by normalization. So if we take the distribution for granted, and we just have only one parameter in our problem, which is RC, because we don't have a measure of the source avoidability, and we measure capital H from the crown geometry, then we plug it there, and we can make prediction of the distribution of radius of trunk in various forests around the world. This is just a sample of nine. We have about 50 of them. So the red point are the prediction based on the previous slide, with just a fit in just one parameter, which is RC, and the black dots are the data. And you see that the agreement with the data is quite satisfactory. The data are not perfect because there are sensors. Sensors are done every five years. The statistic is good, but it's not exceptional. And so you can find anyway that it is quite reasonable. So I think that I should finish, let me just finish. These are just a problem for you, okay? These are two problems for you. This I can skip. And based on the ideal distribution that I saw, that I was showing, this ideal distribution, you can go to forest and measure how far you are from the ideal distribution, from the prediction. So if we have a forest, where there's some disturbance, like a road or something that is going on some part, nearer you are to the disturbance, which is part B. If you go and measure the distribution of sizes in part B and in part A, part A is undisturbed if it's far away and the touch green line, it is the prediction and the black line, it is the data. And you see that the prediction agree with the data. On the other hand, given that the crown radius is not a disturbance in the presence of the road, we can measure the crown radius. We can measure the capital H exponent. We can predict what is the power law of the ideal distribution. And if we compare the ideal prediction, which is the touch line for both A and B, the touch line is the prediction. You see that in B, the distribution of trees differ from the ideal one. Whereas in A, which is far away from the disturbance, agree with the prediction. So the scaling approach can tell us, can quantify the degrees of disturbance in a forest just going and measuring the distribution of sizes. The distribution of sizes deviated from the prediction. Based on that, you can say how well is a forest. And so you can quantify the degrees of wilderness. So this is a graph, again, for various forests. And the measurement was done by Sloan, one of our collaborators. And you see that when you are far away from the disturbance, the touch line is the ideal. And as you move towards the height disturbance, you see that the disturbance is quite visible. And you see a large deviation from the ideal one. So this is just a cartoon. So these are the main conclusions. So thank you for your attention. And sorry, I'm almost at the end of the hour. Thank you, Amos, for the very nice lecture. So if you have a question, please raise your hand. So in the meanwhile, there is a question actually from the chat by Jordi, who is asking a clarification on the harvest exponent. So is the its value obtained experimentally? Or there are any physical arguments or biological arguments for its value dependence on the latitude? On the latitude. Yes, so that is our problem. This is what we haven't understood. There's some echo that I feel. So the capital H, how it's changing with latitude is an open problem. And in the beginning, west, sorry, anguished in Nicholas was saying that forests away from tropics are exceptional, meaning that they are exceptional to the power law with an ideal exponent, let's say 3, for the count volume. Whereas we are saying that they are being a different scale. But we're not able to present an argument to see how the exponent capital H change with latitude. It's just an empirical observation based on that. Great. If there is any other question, please, this is the time to ask. Everything was over. So the lecture will be available on YouTube. And there is a question by Martina. How do you measure RC? Can you repeat? How do you measure RC? RC is a fit parameter. We have to fit the data because RC, as I said, is related to where the truncation of power law occurred. And in the derivation that I did, I was relating RC to the amount of avoidable resources. So if the resources of the avoidable resources are huge, RC becomes very high. Otherwise, if the resources are very tiny, RC become very small. But since we don't have access to the avoidable resources, we have to fit this parameter. So there is another question from YouTube, actually, asking, can you apply the same scaling concept to other communities that are not plant communities? Yes, we do. Plants are, forest is an easy community because the plants are not moving. And you can measure sizes very easy. But we don't have such a facility for other communities. But we are trying to do it for bacteria's community. I think that Rinaldo, I don't know which lecture he has done. But we were doing measurement of bacteria. And we are seeing similar things also for bacteria. But it was not as good statistics as we have for forests. But surely, it's something that should be done in the future. At the same level of accuracy. Great. So thank you very much again for giving this lecture. And now we are going to take a 12-minute break before James O'Dwyer Q&A session. And now we are going to be randomly assigned to break out rooms. So see you in about 10 minutes. Thank you. Thank you again. Can I close the breakout rooms? Yes, please. After one minute. Great. So if anyone is following from YouTube, we are going to start again in about one minute when people are going to join back to the main Zoom meeting room. So if you want to ask questions, as usual, you can use the YouTube chat to do that. And I await the question for you. So we are waiting for the participants to join back. I think everyone should be back in the main meeting room. So before we start with the next lecture, I'd like to say a few reminder about the information about the school. So please remind to check frequently the program on the ICTP website, especially the program of next week is changing rapidly. And next week, beyond other lectures, we are going to also have round tables with discussions among speakers that can also be participated by you. And the other point I wanted to make is that next Thursday, the 16, there is going to be a colloquium, an ICTP colloquium by Professor Ned Wingring, which will be livestreamed on YouTube and can be followed on Zoom. But for that, if you want to follow on Zoom and ask question, since it's a separate event, you have to register. So you find all the information on the ICTP website. Great. So the next slot is a Q&A session, like the one we had earlier. And in this respect, it's my pleasure to introduce the next lecturer, James O'Dwyer. James is a professor in the plant biology department at the University of Illinois Interfunctional Pain. And his research is focused on modeling and analyzing complex ecological community, combining data and the theory. So he pre-recorded two lectures on cooperation, stability, and resilience. So what I'm going to do now is to leave the floor to James and leave the floor to you as well to ask questions. So please, if you have any question about the pre-record lectures, don't hesitate to type it in the chat or raise hand. OK. So we have a question by Washington, please. Yeah. Hi, James. I enjoyed your lectures quite a bit. Very interesting stuff. I got a bunch of questions. Maybe I'll just throw a couple of questions at you that were things you commented on that I wasn't sure I fully understood what you meant. One was you said that in your models, you weren't really depending on or it didn't really matter whether you chose a random distribution for the interactions. I mean, you had the luck of Volterra, and then you also had the separate resource business. Yeah. So the first question is, can you say a little bit more about whether you use random interactions or what you use there? And I'll just ask the second question you can use. And the second question is, you made a comment about being able to integrate out the resources and get a generalized luck of Volterra model with some extra terms for the other things. And the second is, whether the resources actually have independent trajectories. So those would be hidden variables and some time dependencies, or whether it's something simpler than that. Yeah, super interesting questions. Thank you very much for the questions, Washington. So let me, the first one, as you probably know, and I know other lectures have talked about it, many of those classic luck of Volterra results do rely on results for the eigenvalue spectra of random matrices. So that I'm not disputing. My comment there was, but it's a restriction, because that may not, it's certainly not telling you about every possible luck of Volterra system, that's for sure. If you fine-tune the interactions, you could get not any spectrum you want, but you could certainly violate those general rules for stability. So my comment was about the consumer resource models in contrast to that. And so there are matrices obviously involved there that you saw in the lectures, like in that. And I think with the additional layer of mechanism, there comes more choices to make. So there's many different versions of those consumer resource models, which is one issue. But in the version I showed in the lecture where you have substitutable resources with some preferences for those resources and you have some production or returning of resources to the common pool, there are two matrices there, right? The C matrix and the P matrix as I was telling them. And our results about stability, there are definitely some assumptions along the way to get the strongest analytical results that we have to assume things like equal abundances for the different taxa, but we can still get a range of results weakening those kinds of assumptions. But what we don't ever have to assume is that the C and P matrices are typical of a random draw. So they can be literally anything you want so that for the results that we can prove. So the C matrix could be just diagonal, which is one of the cases I looked at in slightly more detail. So you'd specialize on one resource or it could be random, could be some mixture of specialism and then some off diagonal elements might be random. So you have some additional ability to use other resources, but it really doesn't matter for the results that we looked at. So that, what I was trying to say was that it's not the whole story and there are some things that we don't have a handle on in that consumer resource framework, but one nice thing is that we don't have to make that assumption about the random matrices to get to the results, that the results about getting out, understanding that the structure, the whole spectrum would be another question. We don't know that, but just saying whether it's stable or not, we can say without making those kinds of assumptions. Just to clarify, I mean, not saying it's true for every possible CMP set of matrices, right? Every possible set of CMP matrices within some, so there are some constraints, like for example, that'd be positive. The P matrix, the way I formulated that model was such that the diagonal was zero. So you weren't recycling into, well, in the case of the specialist matrix, I actually think maybe what generally, even the diagonal could be non-zero, but there are some constraints on the P and C matrix for it to make biological sense. Other than that, no. I see, very interesting. So you're saying it's a very general result, independent? Yeah, in that respect, yes. That now I won't sort of say there aren't a lot of assumptions going in there because even building the structure of that model is making, if you like, more assumptions than Locke-Volterra. And I can give you one flavor of where even though, exactly as you say, that the results I presented are nicely very general. I'll give you an example of a flavor of something which is sort of not exactly covered up, but something that's not maybe immediately obvious. So one assumption I made, and I did state it in the lecture, but the benefit I derive from a resource in the model I showed you is proportional to the rate at which I deplete that same resource. So if I'm eating something, I'm growing in proportion to that. But of course, that doesn't have to be the case. I could degrade resources without caring about them. So I could take up resources and just dump them in some form which was unusable to me or anyone else without my population growing. And that's not implausible in the sense that there are examples of cases in real ecological systems where that is to some extent the case, maybe not as extreme as that. And it also makes intuitive sense in that I could gain a competitive advantage if I just degrade the resource you use. It doesn't necessarily matter to me that I can grow with it. So making that generalization, you have a different set of results. That's something we haven't published yet, but I'm working on with Theo, who's one of the grad students I mentioned and an undergrad in my lab. But yeah, so I guess what I'm trying to add then, what I'm trying to add is that with the consumer resource models, there are a lot of choices to make before you even get to those equations. Once you get there, let's see in PB, whatever you want, as long as they are interpretable biologically. Cool. Second question was about integrating out, right? Integrating out resources. So that's a great question. Has sort of bugged me for quite a long time as I started to think about ecological questions many years ago because you had things like, well, you had these two kind of parallel frameworks, right, thinking about competition or interactions more generally, Locke-Volterra and then adding that layer of mechanism. And yeah, it's certainly seen, and I learned more about it as I went on, seeing the statement that those two frameworks can be made equivalent by integrating out these additional degrees of freedom, the resources. So, but there are some subtleties there. So I wrote another paper a couple of years ago, which I didn't talk about in this talk, but trying to get at, when can that be done exactly without, as you said, I think in your earlier information, the question, the resources having their own independent trajectories or being in independent degrees of freedom. And there are some cases where that's true. And you can probably guess that in those cases, there's got to be some kind of conserved quantity. And that, so the dynamics of something involving the consumers and something involving the resources turns out to have no time derivative. And so in some models that is the case. And so the simplest one, the very simplest case would be one consumer and one resource. So think of the resource as space. I like there's some finite space I can occupy and I'm trying to, as I reproduce, my population is growing and filling out that space. So then you could divide up that space into a space which is filled by individuals in my population and then empty space that I'm able to expand into. And you could think of the empty space as an available resource, right? That's sort of something I can take advantage of. Access to light, you could think about it literally as a addictive resource. But there's something conserved, right? Which is the total amount of space is fixed if I'm not expanding it in some way. And so you could write down a consumer resource model for one consumer and the available space, but then write down that the total occupied space which would be the consumer population density, roughly your population size, plus the available space, the resource always is fixed, integrate out the resource that way and you end up with logistic growth. So you could start off with something which looks like it's linear growth for the consumer but multiplied by the amount of available resource that's left because you're gonna grow more slowly if there are less patches left that you can go into. Looks linear, but you integrate out the resource because there is this conserved quantity, total space isn't changing, you get logistic growth where you saturate up to the total, the size of the field, right, whatever it is. So that's a simple case where there's a conserved quantity you can exactly integrate out the resource. Then there are, you could write down multivariable consumer models and multiple resources, but it's certainly not trivial that there's going to be anything like that, any kind of conserved quantity. So that's, but there are cases where that's true and in those cases, you exactly can integrate out the resources, you have an exact description in terms of the consumer population sizes. I mean, it will not in general be log of Ulterra but it'll be something. So that's possible. If you have as many algebraic equations as you have resources, you can basically solve for the resources and just eliminate them from the equation. Exactly, exactly. But that's rare. So that, you know, that system is integrable in some respect, but that's not typical. But the cases which are more talked about or at least when I first started reading about an ecology, the kind of canonical way of thinking about it would be, okay, that's probably not always going to be the case, typically, that there's some exactly conserved, some number of exactly conserved quantities. But there is a thought that maybe resources would be like fast moving would approach their equilibrium values quickly. And so in those, if you buy that idea, then the approach would be basically to set all of the dr by dT's on the left-hand side of the resource equations equal to zero and solve the other sides algebraically. That's not gonna be an exact, that's not gonna be exactly, that won't match the true numerical solution, say, of those ODE's. It may match it certainly close to the, an equilibrium if you linearize the system and say there is a group of eigenvalues which is very large and negative and then a group which is very small and negative. So you'll have some slow and fast directions. And then the full generalization of that result, I guess, would be something that is very hard to get a handle on, at least, I mean, it's certainly not obvious how to get a handle on it, but you might think about the full dynamical system more generally. So you've got some space of consumers and resources. So like living in R to the two N, I guess if there's N consumers and N resources, suppose you're at some arbitrary point in this space, so you're not near equilibrium and you wanna know, could I approximate this by just a model of consumers? I think that's very much in general not gonna be true, but what you could imagine is quite plausible is that maybe quite quickly the dynamics relax to a slow manifold and then kind of cruise in on that lower dimensional manifold to the equilibrium. Now that manifold is in general gonna be some non-linear shape presumably because you're not linearized obviously near, and even if you were linearizing the equilibrium, it could still be a linear combination of consumers and resources, just so happens that for the typical models, people write down most of the fast eigenvalues tend to be overlapping the resource eigenvectors, or sorry, the eigenvectors corresponding to the fast eigenvalues tend to overlap the resource vectors, if you like, the resource directions, but they don't have to. And certainly I think as you get further away from the equilibrium, it's not obvious that slow manifold is just going to be well-described by trajectory of the consumers alone. But I think there's some interesting open scope there to understand, yeah, what happens far away from equilibrium where you simply can't just set dr by dt equal to zero. So yeah, so three ways to answer your question, one is rarely you might have enough algebraic equations to eliminate resources. In that case, and there's one, at least very simple example where that's true and sort of plausible with one consumer, but it's going to be rarer in general. Two, you can kind of say if you're near equilibrium and the spectrum looks the right way, you can more or less ignore the resource dynamics. If you're further away from equilibrium, I think it's plausible that there may be still separations in time scales, but it much harder to get a handle on exactly who is undergoing the interesting slower dynamics. Does that answer your question? Yeah, that's great. Thanks very much and thanks again for the nice talk. No problem. Great, there is a question from Ankit. Hi. Yeah, hi. Very interesting set of lectures. So based on this, what I could gather was that, like in such bacterial communities, since you also have this additional goods production, so to say, that sort of brings down the competition. And like in general Lotka-Waltera, we usually think of interaction matrices and like we directly write down competition terms for like between species, but here like there's no direct sense of competition. Like it's through like mediated through resources and goods production. But like, is there any way of like looking at different levels of competition, like maybe as a mixture of resources and goods production, which could give you like some limits to the stability of the system? Yeah, that's a good question. So yeah, you're absolutely right that I went from a picture where the interactions were pair-wise, i.e. Lotka-Waltera, whether it was competition or you could write down mutualistic interactions by just changing the signs, right? In the inter-specific interactions. And I went from there directly to a system where it was consumption and production of resources. But so a couple of points about that that I would make. One is if you go back, I mean, and not that it's not intuitive anyway, but it's certainly if you even go back to the Lotka-Waltera papers on competition, the interpretation was often written down in terms of resources, right? That these competition coefficients would be large if there was a substantial overlap in the kinds of resources that two species use. So what I'm getting at is, yeah, the competition becomes indirect. There's no direct competition anymore in the consumer resource models. And there's no direct pair-wise mutualistic interaction. It's mediated by I'm producing something that you can use. So it is indirect, but that interpretation was probably always underlying even those pair-wise models. So from a certain point of view, people probably thought about those pair-wise models and still do as an approximation to a more indirect process. That would be one interpretation. But let me answer your question in a different way as well. And that is at least in principle, and probably you can identify cases in practice where competition, well, when I teach competition in a class, you do have different kinds of competition and some. And so the classically you might separate out into some which look more like resource overlap and some which do look more like direct interactions between individuals, right? It could be a territorial interaction or something like this. So now ultimately those are probably for competition for resource. So territory would be an example, right? But nevertheless, it could play out in terms of more direct pair-wise interactions between individuals. So in other words, there is a difference potentially in the dynamics of we're in the same location and I happen to get forage for something before you do. There's a difference between that and me kind of pushing you out, right? So I think it's a great question that you could easily imagine layering on top of the consumer resource models that I wrote down and other people obviously work on as well. You could layer on top of that a direct interaction. There'll be no reason not to. And you're right also that it would like, it would certainly change the dynamics and very probably the stability property. So I think there's no reason not to do that. And there are probably many situations where species are competing both indirectly for resources and maybe directly in terms of direct pair-wise interactions. So I guess what I'm saying is one interpretation block of ulterior competition really is just an approximation to resource acquisition. But another interpretation as well, it may be really accounts for those direct pair-wise cases where two individuals really are interacting directly with each other. And I don't think there's any reason not to put the two together. I have not done that, but it would be kind of interesting to see what the outcome would be. Interesting, thanks. Great, there is a question from Pablo, please. Hi, Pablo. Hello, James. Thank you for your lectures. They were really interesting. So my question is related to the one that Washington had. I'm working with the Mars-Lan model, which you probably are familiar with. And random matrix theory is really interesting, but it has one problem that if you're not able to analytically find the equilibrium, you can't do anything. And this is the problem with the Mars-Lan model. Even if I do timescale separation and I assume that research dynamics are fast, I'm not able to find a stable solution for the resources. And therefore, I'm not able to find an analytical solution for the equilibrium of the populations. So I was wondering if you have faced this problem because I see that you've done random matrix theory with consumer research model where you have cross-feeding. And what are the type of assumptions if you can detail that that you do in order for you to get analytical equilibrium or if you have faced this problem in this particular model, do you have any idea? So how to tackle it? Yeah, well, first of all, yeah, thanks for the question about a couple of points. So to the extent I use random matrix theory in these models related to Washington's question, it's to provide examples rather than a necessary element. In other words, just to give numerical examples in some cases we chose that consumer preferences were drawn from a random distribution, drawn from a distribution. But now let me also point out that a couple of things. One is, so the Rosland and collaborators, of course, that the model that they have developed, which is in, it's sort of very similar to the most general model I wrote down of production in one of my slides. And then I simplified to a different model, which is maybe a little bit easier to analyze in some respects, but the more general model allows for the production of resources by me to depend on the resources that are available to me. And that's very plausible. And it's probably the right way to, well, I'm giving myself a lot of parentheses here to get back to. Let me just say for production of resources as a byproduct of metabolism to depend on the resources around me makes total sense, because if I eat burgers, maybe my byproducts are different than if I eat apples and pears. So that makes total sense. But it adds an extra layer of difficulty in analyzing those models. So the way that we formulated production of resources probably more easily interpreted as a kind of recycling process. So following mortality that there's some characteristic composition of a cell of each taxon and some of it is returned to the common pool. So there are differences. I guess that's my main point in saying describing those details of Rob Marsland's model and what I talked about in detail. But I also totally buy that allowing production to be more generally dependent on the resources around me makes sense. So in terms of analytical solutions of the model I presented, they're relatively straightforward, just involve kind of matrix inversions and nothing overly complicated. That may become more complicated that they're more involved you make the production term for sure. So that there's no guarantee, right? There's no guarantee that you're going to always be held to even. I mean, you have algebraic equations, right? If you're looking for equilibria. So they're certainly simpler than solving the dynamics but there's no guarantee you'll have a nice form or even and certainly there's no guarantee of having a stable equilibrium. So I wonder, I don't know this for sure but certainly in the models that we have looked at and that I talked about, there are certainly regimes in which you won't find the resources settling to an equilibrium. These are precisely the cases where there are instabilities, right? So I don't know if that is related to what you're saying and the stability properties that the Marsland model are different in some respect. So it depends on the details of how you're implementing that but certainly what we find is that there are regimes of resource inflow and obviously depending on the structure of the consumer preferences and the production of resources, there are regimes where there won't be a stable positive equilibrium. And so if you were to solve those equations numerically which we did just to show what it looks like you get some kind of limit cycle and it's not, yeah, that's something which I don't understand fully the properties of what does happen to the genetics when those equilibria become unstable. But yeah, in our models what seems to be key is the level of resource inflow for determining that. And so there are some regimes of resource inflow in coupled to the structure of the preferences and the production matrix. There are some regimes where you won't find stable solutions in some way you will. So I don't know if that's what's maybe happening in the solutions you're looking at that there actually isn't a positive stable equilibrium. It could be that or maybe it's just hard to find your solution in a nice form. And one other point I wanted to make was in the second lecture of mine that is part of the school I talked about what we called metabolically informed community dynamics. And there, what I was really trying to get at I was paid with Mario Mascrella who was in my lab at the time what we were trying to get at was, okay, we do know that the production of resources is going to often depend on the resources I take up. The point I make about the Mars and model being a bit different from what I talked about in lecture one. But what should that look like? You know, I think there's a bit of guesswork involved in formulating these consumer resource models. And that goes back also to pointing up in response to Washington's question. You have many more choices to make. There are these different flavors of consumer resource and production models. And so what I wanted to get at in that second lecture was okay, can we narrow down the possibilities? Can we understand whether there are, what is, what are the most plausible ways for production of resources that depend on the resources I'm taking up? Cause you could write down, you could write down more and less plausible functions, but I mean, there's nothing really stopping you from running out some arbitrary horrible function of resources and different metabolic pathways. And that could be very plausible, even if it's horrific, right? And so that was the idea of that second talk and that paper was to begin to think about what are the most plausible? Can we from something like first principles derive what those production matrices should look like or could look like or constrain what they could look like. So yeah, maybe part of what you're seeing is just that there's many ways to formulate these resource consumption and production models. They're not all guaranteed to have nice closed form solutions for sure for the equilibria. They're not all guaranteed in it, well, in any of the ones we've looked at to have stable positive equilibria and the capital, we don't really know exactly what the right formation of these models is. So there's a lot of question marks there. So I'm really answering your question with a bunch of questions, but hopefully that's at least adjacent to what you're thinking about. Thank you. I think it was a very nice set of questions as an answer. There is a question by Martina. Hi Martina. Hello James. So I have a question that is related to what you were seeing just three seconds ago. So how do you think you, so these metabolic informed models, how do you think they scale when you add more resources that you produce? And whether you can, I don't know, make what you produce changing in time depending on whether you are in the exponential phase, the lag phase. So, I mean. Yeah, yeah. Yeah, basically, does it, yeah. I think I get the question, but if I didn't re-ask it again, if I'm answering totally the wrong thing. So that metabolic informed model, it's a really simple model of what's happening inside a cell, right? That's kind of, I think, what you're getting at. It's just two, yeah. Oh, I guess three resources involved basically in each intracellular process. So two things coming in, an interaction between them and something comes out at the end and then that's excreted by itself. So how does that scale when you have more resources involved? And so let me say back to you how I'm understanding the question and maybe I'll give you a chance just to say if I'm on the right lines. So I think you're asking, well, in any real cell, the processes are more complicated. They will involve discrete changes like maybe processes being switched on and switched off in response to what cells are sensing externally. And so there could be a lag phase or something like this or yeah, as a cell switches between resources. There could also be many different, well, there will be many different resources and other molecules involved in these processes inside the cell. I think you're asking how much of what we see in that really simplified model could possibly carry over in that more general picture. Is that a fair summary? Yeah, actually I was thinking more about what happens in the cell which is related to what happens in the cell, but is, so you start from glucose but you produce 30 other metabolites and more or less the cell excretes, I don't know, 20 of them because you have the metabolic overflow or you have all these molecules that can diffuse passively outside the membrane. So the question was, okay, if I start from glucose, do you think you can scale your processes to account for, I don't know, more metabolites that are produced? Yeah, I don't know, maybe, yeah. That's the... Yeah, that's a great question and that's an easier question to answer because the answer is yes, I think that that side of things is much easier to scale. That they may be different, obviously ratios or proportions of those metabolites produced, but the functional forms will be pretty similar. So scaling on that side is pretty nice if there's many, many outputs and they can diffuse passively across the cell wall and then are put into the common pool. That works nicely in that same framework and will look very similar. But like you say, I know this is relevant to your work as well. That will make a big difference to the community dynamics for sure. And of course, to make full contact with what I talked about in the first lecture where you have many consumers and many resources, that's certainly one way to get there, one plausible way to get there with the metabolic inform model is to have these multiple metabolites produced and that could lead to a really rich set of community dynamics and we didn't really get there in that first paper with Mario. That's an interesting scope for interesting development there for example, to say, if you have those, maybe you keep the input that the essential resources relatively simple for each taxon, but you have a wide range of outputs but following the kinds of functional forms that we talked about, it'll be really interesting to understand what that changes about the dynamics and the stability and the equilibria and so on. I think that would be really interesting as a comparison with all the stuff in the first lecture. I just haven't got there yet. 2020 happened basically, but yeah, I think that's a really interesting question and it is easier to tackle than the other way around. Now, the other way around would be if there are many kind of like essential resources that get involved in some way in the overall set of pathways that lead to those many metabolites. I think that's not an impossible question to answer. That's the scaling up of that side but at least it's harder. And a question to me that I don't really have a good handle on is how to systematically pare down the true complexity of that metabolism and to a point where you can say, these results are robust. I don't think it's implausible that, I mean, look, ecologists have been looking at these relatively simplified dynamics this whole time, right? For decades. So we've been, had this guesswork about how the internals of not just single celled organisms but multi-celled or more complex organisms, how the internals affect ecological dynamics, right? Behavior, it would be maybe underexplored in terms of its impact on population and community dynamics but certainly is something people think about a lot. So guess what I'm saying, it's not implausible to me that those internal dynamics can boil down to something manageable but also we have not at all proved that in that paper. But the other way around, producing many outputs I think is much more doable and not at all uninteresting. It would be very interesting to see how that affects stability and dynamics for larger communities. One more thing I wanted to say, sorry if I'm just taking opportunity to ramble but your question is interesting and it's about this, again, additional layer of mechanism inside the cell. And I think it's just a super interesting question because it's not just about throwing more resources in and having more resources come out. I think it's also about, you know, what is that, you know, there may be other elements of the set of rules but obviously there are other elements to the set of rules by which cells are operating and how do we pair those down to the, you know, to at least a simpler model that we can extract robust results for the community dynamics. So I think that's it. So I've seen a few other, you know, few other approaches to thinking about that. You know, there's like, there's papers from Terry Huar's group which go back many years looking at a portion of resources inside the cell to different categories of process. That's an, in my mind, it's conceptually similar. It's a way of, you know, a simplified model of the internals of the cell, which then can give rise to different community dynamics at this larger scale. And the cell wall of course kind of provides you this somewhat a natural separation of scales which is interesting. So, yeah, I think it's just an answer to your question is, I think some of it can be scaled up. The more general answer in my mind is another question which is how do we systematically show what kinds of community dynamics are robust or the most likely outcomes of whatever is going on in the cell? And that's a harder question but I think it's super interesting. Thank you. Thanks for the answer. So is there any other question? I don't see any hand raised in the list but please, we have time for more question and answers or if you don't want to talk, you can type it in the chat. So it has been pretty intense so far. There is another question from Washington. Yeah, if no one else is asking questions, I'll ask another one. So have you or when you think about resources, I gather you're primarily thinking about like physiological resources, like material resources like phosphate and things like that. Have you thought about how energy as a resource fits into that or are you aware of other work where people have looked at sort of energy flow and systems like this? Yeah, good question. Yeah, there are papers and approaches to thinking about communities or maybe more ecosystem dynamics in terms of energy flows and thermodynamic properties more generally of ecological communities. There is a whole, you know, not field but like approach of thinking about, I don't know if you are familiar with the, in non-equilibrium system mechanics, but people have proposed maximizing entropy as a principle, not proven, but just as a maybe as a guideline. So there's definitely people thinking about whether ecological systems change over time in order to maximize entropy production. And so, you know, obviously that's not just the energy but it's sort of thinking about the system more thermodynamically maybe, which is maybe along the lines of what you're wondering. I don't know if, I mean, for the kinds of things we're looking at here, I mean, the resources, I guess that it, I can't think of a way they would not have an energetic value as such. I think all the things we're thinking about, whether that's light capture or it's eating glucose and kind of using that to derive energy. I mean, they're all, energy I think is inevitably involved, obviously, but... Good, you're saying that some of the resources being passed include energy as a component and others contain other crucial nutrients and things. So it's in some sense implicit, in some sense implicit in what you're doing that energy would be one of the features involved, but... That's right, that's right. And I'm just wondering, you know, I like the work I just referenced, I'm just thinking about these sort of, if you like, coarser, grained pictures of how ecosystems are working and the flows of energy and other thermodynamic properties. So I'm not saying I don't like that stuff and I'm interested in it, but it will be, you know, part of what we're thinking about here is what happens as a level of, for example, as... So you could imagine that the substituted resources may be all forms of organic carbon, you know? And so there are some other resources which are less energetically useful, which may be essential, but we could sort of ignore those and just say, well, carbon is sort of the limiting resource in a given context, maybe. And so then, what we're interested in here comes down to more exactly looking at the differences between those different forms of energy, if you like. So calling those, you know, collapsing down that matrix to just energy could be kind of reductive, right? Because you wouldn't... So for example, many of the... Not many, but several of the talks in the school I've thought about and mine too, in a way, think about coexistence, right? Of many different kinds of species. So not that entirely relies on differentiation of resource use, but it can do, right? And so you'd sort of lose that. Yeah, which is fine, depending on the question, right? Because you actually maybe you wanna lump all heterotrophs into a category, right? And then you're thinking about a much bigger cycle of just, you know, orotrophs, heterotrophs, decomposers, or something like this. And in that case, those kind of coarser flows of energy might be the right language to use. But maybe the right way to answer your question is it probably just depends on the question of interest. And if your question of interest is understanding communities of many different species doing slightly different things, you know, but kind of at the large scale, kind of, maybe they're doing slightly different things in sort of boringly, right? They're not vastly different. Then, you know, the language of these kinds of resources is probably the right language. But if you're interested in those sort of larger scale flows of energy and, you know, you might think about just flows of nutrients like C and N and P rather than specific forms of them, yeah, then that would be a sort of different language to use maybe for different kinds of questions. Cool, thanks. Great, thanks a lot, James. So we have space for more questions if anyone wants to ask. Really good questions, by the way. Thanks, everyone, for watching the lectures and for the great questions. Yeah, I totally agree. I mean, it was very, very interesting. So if there are no more questions, what I would say is that we can move to the next question. Breakout rooms. And James can stay, let's say, another 15 minutes with us. And you're free to chat informally in the breakout rooms. I just as a technical reminder, if you have a Zoom version that is five or higher, you can also change the breakout rooms. So you will be randomly assigned, but you can of course flock in the breakout room where James is, if you want to chat with him. So with that, thanks a lot, James, very much for recording the lectures and staying with us for this Q&A. We'll be back in these main meeting rooms in half an hour with the lecture by Mercedes Pasquale. Awesome, thank you for coming in. Enjoy the lecture by Mercedes. I'm sure it'll be super good. Okay, thank you.