 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 Padua. 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 and 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 Subeys 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, Ciaon Banawa, 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 in 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 in the resource competition model, which will be the subject of the second lecture. Then there are a lot of alteration 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 approach that would 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 the internet. 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 there 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 it can move in a symmetric way both to the right and to the left. And so the probability to be at some position x of 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 of time t, this is just a power law. And so in 1D, it is a t to the minus one half. And then due to dimensionary reason, 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 versus log of t we get a straight line with log of minus one half. So this is the one on the right. It is the best way to visualize when a power law is presenting our problem. So typically what we find is not a pure power law like this but that 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 time scale. In the previous there was no time scale. You see that the only time scale is just the time and there's no characteristic time scale in the problem. And this is why we see a pure power law. If for example in the Rano Woker the diffusion problem there is an external field that try to push the particle more to the left, to the right, then to the left. Then there is a time scale 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 which is lower than minus one alpha which is just the pre factor. Whereas for time of order of tau 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 is just the characteristic time scale that has been introduced then to the external field. So other reason for having truncated the 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 the correction 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 describing 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 the solution to the Fourier transformer which is the last but one line where you see that if we set x equals zero 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 you have to calculate the probability density to be at position x of 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 the 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 puts 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. In particular if you set that x equals zero f becomes a constant in that case and we get the answer that we asked before. The probability is to be at the origin of time d scale like 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 a 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. 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. 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 per day is just a measure of the metabolic rate of a plant. So the metabolic rate in principle can depends on many details of 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 if 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 that if we take that B is proportional to the surface, the mass is proportional to the volume and the surface is related to the volume to the power to third we deduce that we should deduce 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 it 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 mass. So if he's right, it should find that the log of the mass versus the log of the body mass, that's all the log of the metabolic rate versus the log of the body mass should be a straight line of a slope three-quarter which is the red line. And you see that here the 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 indeed 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 also the some temperature correction that has to enter in this law 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. Does 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 doesn't depend on the units. And the log is 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 some time, 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 of the energy that they input out, 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 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? NC is the mass of the cell, the mass of the cell doesn't change with age, it's 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 Clibbert law. So this equation in transforming this one, where the first term comes from the second term, it is from the maintenance. And the second term, the first term, it is from M-alpha, it is the Clibbert law. And this is the M 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 the 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 organism that scale like the mass to the one quarter. So not only that, but what we get 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 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 the scale time, 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 guppy should, and you see that the data that are the green light, the green point, I think, 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 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 pointed 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 are 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 earth beat scale like the mass of the minus one quarter. So as a rabbit has a frequency, a frequency which is much higher than one over 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 don't know. I can't hear you. 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, what is the question? Well, I mean, I don't know if Miguel wants to ask him, 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 are grown in some only in the same condition. I mean, I think they are feed 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. 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. You're right. For plants, you should want to distinguish between the dead master and the living master. You're right. Okay. So this here, I don't know if you can read that there is no plants in this plot. Right. But you're right, okay. Okay, thank you. Okay. So now we will concentrate on the possible origin of this quarter power law. And so many years ago, we had some intuition on why or where this three quarter comes from. And so we were hypothesizing that this could be the result of an optimization principle. And so we said that 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 as to leave the main source every day. The pipe enter in the first house, cause 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 six, what you get, it is approximately L to the four. If L by L is the size of the town, L square 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 row, the arrow are not going back, 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 L to the four, you find L to the three. And so the exponent relating the mass, 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 very nice, a magic theorem that can be proved exactly. 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 are, it is even simple. So forest, 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 they measure, 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 prison 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 one decade, 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 a 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 point five, okay? So it's not true that 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 north, the height scale much faster than the width of the crown. This just means that they are taller and these 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 that, to what is meant by mass of a plant, at least for me. So we have that plants are taking water from underground and throw silencer that are not cylindrical, but almost cylindrical conduit. They are transported to the lips 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, the capital H is about one around the tropics. And so it makes the crown of the volume crown not to be to scale like more or less three around the tropics. And then at the most northern latitude 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 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 what is called the sub-wood, the living part of the wood, which is just say the external part of the trunk, meaning the water is flowing only through the external part of the trunk, 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 the water is flowing only near the surface of the trunk. And so what we mean by mass of the 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 rate times H. 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 Earth's exponent. And so you see that when capital H is one, you find the tree 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 tree 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 square. And the water density of course does depend on the size of the tree. At the end, what you find with 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 to R square. 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 age 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 small. 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, but most 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 path that is dead that is just a sustained development. It's just the physical support of the plant. The radius 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 squared, meaning the more or less the mass of the trunk. So you can find that the mass of a tree scale like are 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 of 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 trees, for leaves. So there's no more room for the bubble of transpiration. So this constraint is telling you that at a certain point it must stop, okay? In order to evaluate the where this can occur, we should know what are the constant in front of this power node 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. We can go on and then collect the question. Okay, there's a question. Yeah, perhaps there's a question. Yeah, I have a question. This is very interesting. I would like to ask usual 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, frank radius, scale like R to the minus two. Yeah, my question is based on, because there is also the Damos rule. There is 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 fourth. 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 that can stay on a given area, but trees of the same size, then you are right. Because if each trees, 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 square. Okay? So you have to take the maximum number of trees of the same size times R square, 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 the 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. 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 support that is capital N. Okay. So N times P naught of H prime times D H 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 D H. 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 filling almost uniformly all the room at this point. And so we're 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 differentiate 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 minus one plus two capital H. The density of trees, this picture is taken from the book by Hubble, 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 R to the minus seven over three, and then the truncated power law with some characteristic radius that is related to the availability 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 that. That was always true, but it's not answering to this question or the distribution of the sizes. So the exponent is 2.3. And indeed, we may, from the data we saw that the exponent is near to seven over three originally West and Brown did a mistake and they were saying that this was minus two and they also related to it down below, but it's not the down below. Of course, as we go up in the latitude, this exponent changes. So in order to understand why the finiteness of the resource availability 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, so if this is a 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 the tree. 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 record is this scaling answers. So if we use the scaling answers with a 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, we use what we found in the previous slide. We use this scaling answers, which is the number two which is a generalization of the deterministic answers. And then you do some calculation, you find what is the distribution of the radius which is this power law, one plus six capital H divide 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 disposed of. 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 one. On the other hand, if the resources are very tiny then RC is very small. And so the power law is masked by these corrections here. So we take this, this is a constant by the way it is that 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. We just 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 that 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 the star way 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 there is a forest. And so you can quantify the degrees of where it is. So this is a graph again for various forest and the measurement was done by Selang 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 then 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 is some echo that I feel. So the capital H, how it changing with latitude is an open problem. And in the beginning West, sorry, Anquist and Nicholas was saying that forest away from tropics are exceptional, meaning that our exception 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. 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 the 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 plant our forest is an easy community because the plant 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 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 with not as good statistics as we have for forests. But surely it's something that should be done in the future. The same level of accuracy. Great. So thank you very much again for giving this lecture. And now we are going to.