 The next lecture is by Justin Eken, so Justin, thanks for being with us, and please feel free to share the screen. Okay, great. Thank you. You can hear me okay? Yes, perfect. Okay. Thanks again for having me back. Today I want to switch gears a little bit and think more specifically about the energetic dynamics of consumers on landscapes and think of some of these processes and maybe a little bit of a different way and then aim to leverage these ideas to examine a really really interesting transition in the history of mammalian systems, and this is effectively the evolution of grasslands and I know you've already had some great talks about the evolution or about grasslands and Savannah systems, where we are grass specialists and everything, almost everything we eat is grass, you know, beer is grass juice, bread is grass, rice is grass, we rely a lot on grasses. And so the evolutionary transition when grasslands evolved as a as a unique biome was a really significant, a significant event in mammalian evolution so so that's one of the, that's where I'll end today. Okay, let's see. And really what I want to do is look at the dynamics of starvation and recovery and think of these processes from an allometric perspective as a function of body size and I want to try to make the argument that this time scale of these processes really structures a lot. Some of the constraints of mammals that terrestrial mammals experience. And so again I'm kind of isolating my, my thought processes towards mammals and more specifically terrestrial mammals. The rules change a little bit when when you don't have to support your own weight, like aquatic mammals do, don't have to. So, starting very simply right all organisms as we all know, must find food acquire food and process food. They have to get the energy required for reproduction to pass on their genes to the next generation. And organisms do this in very different ways. Some organisms consume a lower quality food that's more evenly distributed. Some organisms consume higher quality food that's somewhat more clumped over space, and other organisms, such as this hyena in the lower left consume very very energetic energetically rich foods that are very clumped in space, and also happen to move around space. And these are all unique challenges depending on the different foraging strategy organisms are following and they all function to distribute risk. So, so dealing with risk is part of the game it's part of being alive right and organisms evolve different strategies to cope with the risks that they experience. So when we think about just finding food finding resources, we can think about the availability of foods. So organisms must, depending on the foods that they're trying to acquire they must search for foods on a landscape, and whether that landscape has a lot of food or a little food really has an impact on their strategy that's in part due to the natural conditions of the landscape the abiotic conditions. It's also due to the biotic conditions like competition. There's a lot of dispersion how patchily distributed food is across the landscape. Sometimes encounter rates are very short and some sometimes encounter rates are very long are very large. They have to deal with energy densities differing energy densities of foods a lot of foods that are homogenously distributed tend to be poor nutritional quality. Other more clumped in space tend to be higher nutritional quality. You know think on one end of a grazer consuming grass that's relatively poor nutritional quality, compared to a carnivore on the other end of the spectrum, who's searching for these large packets of food. If we can think of herbivores as packets of food that are incredibly energy dense but it takes a lot of effort and skill to define them and track them down and ultimately acquire them. In addition to these food related risks organisms at the same time of course have to balance the, you know, the skill of not dying. So they have to avoid predation. And that is going to change across landscapes it's going to change across is going to influence the behavior of organisms. And I would say the first approximation many of these strategies to deal with risk is can be captured by body size. The body size really alters the risk landscape that organisms experience within a system. And I'll get to that in more detail. I started talking about that a little bit during my last seminar. And we've seen this graph before. What's really important is structures, a lot of things in mammalian systems here it's structuring the, the, the mortality due to predation. So I'm not going to go through this, explaining this too long except to say that here we see herbivores at a rate as a function of their mass on the x axis and in the y axis is the percent mortality due to predation and there's a sharp cut off at 310 kilograms if you're smaller than that you you are consumed. So we know that body size influences interactions, and that is pretty obvious on ecological time scales, however, if we zoom out and think more across evolutionary time scales. I think what is also obvious is that interactions influence body size. So here's this dynamic feedback over evolutionary time between the ecology of an organism as a function of its body size, and the selective forces operating on that organism that's going to perhaps shift his body size over evolutionary time. So, this is one of this is another one of my favorite, like science figures. In a paper by John Alroy 1998. I'm looking in the over here so the upper the upper left called copes rule in the dynamics of body mass evolution North American fossil animals where John Alroy compiled this enormous compendium of, of body sizes of different North American animals over the last 80 million years, as well as the dates of their occurrence the best estimates of their current states and so that's represented in that in this upper left figure. Each played is represented by a line that represents how long it was around and you know back here this is around 66 million years ago this is the when the asteroid hits hits earth and dinosaurs go extinct. And we see before that point and these are just again mammals. Before that point mammals are relatively small. This is a log scale right this is body mass and grams. And after the asteroid impact dinosaurs are wiped out at least non avian dinosaurs are wiped out and mammals fill the space. There's actually been some really nice diffusion models applied towards understanding how mammalian body size fills this empty space that that the asteroid leaves. So they fill this niche space and very quickly attain very large body sizes. You'll also notice there's more structure in this to there's kind of a gap here that begins to form around 4035 million years ago or so, which I'm not really going to touch on today but there's a lot of interesting structure in the body size over time. And we call this copes rule the evolution of larger body size within clades over evolutionary time and this is observed in terrestrial systems it's also observed, not just for mammals but across across different clades it's it's it's observed in marine realms as well as well. So copes rule seems to be a very common and large scale macro evolutionary phenomenon. Another thing that John Alroy did, which was really interesting is he documented. He took this data and looked at the change in mass over a given period of time, relative to the mass of the older species. So he created this kind of a stability diagram right where where if if the trend is in on this side of the graph and I've kind of rotated it just so the masses line up on the y axis of both figures which is kind of a funny way of looking at it. But if the trend is on this side that means there there's growth towards larger body size and if the trend is on the negative side that means there's decline in body size within that temporal window of the clade. And what he found is well it looks like there's an evolutionary attractor and body size at a lower size but there's also an attractor at a larger size and this would describe very well this idea of copes rule where there's this slow tracking towards some kind of attractor at the larger body size limit. More recent work I think this is a 2010 paper by Felicia Smith at all. And they were looking at maximum body size of different groups so they looked at carnivores and herbivores they looked in across different continents and they saw the same pattern, where after the asteroid impact at around 66 million years before present. And they have this kind of space filling dynamic of mammalian groups. Okay, and this on the bottom I'm just giving you a sense of of this, this large body size space filling in extreme extreme on my guess. Modern African elephant. And these are two of the largest mammalian species that have lived on land, again, dinothea in the minus in the myosin and endric a theory in the legacy. So, and your theory is about two and a half times a modern African elephant so that's pretty big. It's not sauropod size but it's pretty big. So today I want to really talk about two things I want to answer address two sets of questions. First, what is the influence of starvation recovery on the dynamics of populations, and can it provide insight into processes driving body size evolution. The second theme, or the second set of questions is, I want to ask how do individuals balance starvation risk against reproductive investment when resource acquisition is uncertain. And what are the evolutionary consequences. And so you'll see how these two different perspectives are really linked together as we go along. So let's start very simply and something with with a model that we're all, I'm sure, quite familiar with, or at least I've seen before, but the lack of Altair consumption model right so the the classic predator prey model, except that here I've included, you know, saturation of the resource. The resource density declines so, oh, sorry, I'm getting ahead of myself. So, so the lack of Altair model really captures starvation implicitly. Okay, because as the resource runs out, the growth of the consumer declines, because we have this function here the growth of the function of the resource density. So as the resource density becomes lower, the growth of the consumer also is is hit. Okay, so I'm illustrating this lack of Altair model on the bottom with this cartoon, where the reproductive capacity of the consumer is really tied to the resource density, which I'm just using with these bonsai trees I guess supplied by by keynote. Okay, so that's the lack of Altair model it's capturing implicitly this idea of starvation. But, could we perhaps think of ways to better capture the starvation process explicitly capturing starvation and could that give us any insight into the dynamics of populations that are contained by resource limitation. And the way that we went about doing this is was to think really carefully about the time scales of growth and recovery, following a starvation process. And so we started with an ontogenetic growth curve. And so now we're not thinking about the population but we're thinking about the dynamics in the mass of an individual. The organism is born, so it has some birth mass, and then it grows along this this this kind of sigmoid sigmoidal curve until it reaches this asymptotic body mass. Now it will never actually because it's asymptotic it never actually reaches the asymptotic mass. So we supply a cut off. So say at 95% the asymptotic mass that is what we characterize as an adult size, where the adult is reproductively active. Now so the adults, it's business it's moving around the landscape looking for food. If it doesn't find food. We have states that the adult can be in. So once the organism hits the adult stage. If it doesn't find food it stars and it reaches the starve state. So it stars along a different trajectory. And it stars to some point along this ontogenetic growth curve. Now once it's in a starve state it can then recover if it finds food again. So we and it follows another trajectory outlined here this recovery trajectory back to its full size. So really what we have here is a is a two state consumer model. The consumer can exist in a full state or starve state. Now, when it reproduces it takes some time for it to reach its full state. And then once it's in its full state it can cycle back and forth between a starve state and a full state. So given the time scales of that process of moving back and forth between a starve state and a full state. We considered the amount of body mass that mammal that mammals typically have as fat. And there is a fairly well known allometric relationship for the amount of fat mass a mammal has and again this is on average so this is taking the average trend across many different mammalian animals. And I'm showing this, I'm moving my windows around here so I can see it too. But I can see this, or we can see this on the bottom so so we have the amount of fat mass illustrated here by the simple allometric equation where there's f not times m to the gamma. Now gamma in this case, as measured from empirical data is 1.19 so it turns out that the amount fat, the amount of fat and organism can hold for mammals for terrestrial mammals is super linear. An organism that's a mammal that's larger can hold relatively more fat than you'd expect, based on its mass compared to a smaller organism. So the, if we think of this ontogenetic ontogenetic growth curve, in terms of the proportion of the adult mass that anywhere along this curve represents, we can actually identify those proportions as a proportion of the allometric relationships of fat mass and muscle mass. So for example, the proportion of adult mass. When you lose all of your fat is given by epsilon sub sigma so it's one minus. This is the proportion of fat mass that an organism is expected to be able to hold given its mass. So that for muscle and fat mass as well. And so we call this epsilon sub mu. And so this is just one minus this summation between the amount of fat mass and the amount of muscle mass which has a different allometric scaling coefficient exponent, divided by the adult mass of the organism. So here we've identified some some key points along this ontogenetic growth curve that's going that are going to be helpful for us in determining both what the different states of the of the organism can be identified as as well as the time scales required to move from one state or the other. So we are going to define the full state as epsilon sub lambda so this is 95% of the adult mass. So we're going to identify the starved state as epsilon sub sigma and it's this is exaggerated here is just a cartoon. So epsilon sub sigma is the mass of an adult minus its fat. So we're assuming, and this is kind of a, you know, an extreme, an extreme assumption but it serves our ends and being relatively simple to calculate. Essentially the adult mass without the fat mass so once you run out of fat you're in a starved state. But if you run out of fat and muscle, because when organisms run out of fat they begin catabolizing their muscle. And that usually means that's usually the death now for the organism. But once they burn all of their fat and their muscle, we consider them dead. So the consumer can exist in the full state starved state or dead state. Okay. So we have an explicit change in mass that the organism experiences whether they're full starved or dead and it doesn't really matter what their masses if they're dead. So now that we have these geometric relationships for these different rates that characterize how long it takes the time scale of starvation, the time scale of mortality, which is the process of moving from the adult state through the starved state to the dead and again this is mortality entirely due to starvation. And, and another relationship for the growth of the consumer the reproductive capacity of the consumer as a function of body mass, we can look at how these rates change as a function of body mass. So here on the body mass of consumers, mammalian consumers. What we've seen terrestrial mammals obtain is outlined in yellow on the along the bottom of the x axis here. And this is the rate. So this is the average rate of starvation and how it changes as a function of body size mortality recovery and consumer growth. And I just want to point out a couple of fun little features of these rates. And that is these these strong upper bounds these asymptotic kind of mathematical weird things that happen at very very large body size. So for example, we can solve for the point where an organism is 100% fat. Okay, so given the elementary relationship between the amount of fat mammalian organism can hold. We can solve for the point at which it's equal to one, the point at which, at what size would the mammal be expected to be if it was 100% fat. In other words, starvation time at this point is infinitely long. And this is why we have this, you know, this, this, this decline here this sharp decline. Okay, so if we solve for the point where the organism is 100% fat. We have mass is equal to eight times 10 to the 8.3 times 10 to the eighth grams. And so this is an organism the size of about 140 African elephants, which would be very large indeed. In other words, it would be about five times the largest blue whale ever recorded. So that's that's large and it's not. It's fun, but it's not biologically meaningful. So asymptotes are by definition unrealistic bounds. And of course if we're thinking about body size evolution. We really require thinking about within lineage selective driver that's also mechanistic. And that's what we want to work towards here. Okay, so we've talked about how these different rates. How we identify different states that a consumer could be in. We've thought about how those rates and time scales change with body size. Now could we put these into a simple Latke-Valterra type model, but make starvation explicit. So on the top I have a cartoon that you've already seen that illustrates the classic Latke-Valterra predator prey model where the, where the reproduction of the consumers tie to the density of resources. So I've illustrated a little bit of a more complicated model where we have two states for the consumer. The consumer can be full or the consumer can be hungry. And it moves between the states by starvation, which is illustrated by this red line, red arrow, and then recovery illustrated by this. Wow. Thank you. And, and, and there's also rates determining the maintenance of the biological tissues that's being supported by the full and hungry consumers and they're a little bit different. And so that's illustrated by these orange arrows. The blue arrow here is the mortality. So, so when you lose all of your fat and all of your muscle you're kicked out. Now the other assumption that we make in this model and I think this is a key assumption is when you're full you're full. And so you're thinking about reproduction. So reproduction is turned on when the organism is full and reproduction occurs at a constant rate. So reproduction occurs at a constant rate when you're full, but then as the organism transitions to being hungry, reproduction is turned off. And here's a switch. It doesn't scale smoothly with resource density. What does scale smoothly with resource density is the rate of starvation. Okay, so starvation is tied to resource density as the resources become less dense starvation picks up. And so, in a in a in a poor resource landscape there's going to be more hungry individuals so reproduction is going to be tuned down. So in a rich resource landscape, there's going to be more full individuals that are reproducing, and which which then occur at a constant rate. Okay, so those are the assumptions of the model. I already kind of explained everything on the slide but I just want to show you the non dimensional version of this explicit starvation model, where we have the dynamics of the full consumer the dynamics of the hungry consumer. The dynamics of the resource, the resources. So we have consumer growth being constant, only occurring for full individuals. We have starvation which moves individuals from the full to hungry class. We have recovery which moves individuals from the hungry to full class. And then we have mortality of those individuals in the hungry class if they continue to starve. The resources change they they they have carrying capacity here normalized to one. And they're consumed to both replenish hungry consumers that replenish full consumers will consume resource at a particular rate, but also hungry and full consumers, consumer resources just to maintain their own tissues. And that's what's outlined in orange. Okay. So we we called it the nutritional state structured model. And again all of the time skills are based on this on the genetic curve. All right. So, now that we've discussed the different rates and how they skill allometrically, we've discussed this model that incorporates those various rates that are allometric. We can combine them, and we can ask, given allometrics starvation recovery rates can we predict observed mammalian densities does this model make any sense can we make any prediction to verify that the model is telling us something interesting about the real world. We think that we can so the results of this starvation explicit nutritional state structured model map really really well on to known densities of different mammalian systems. Okay, so, so the relationship between the steady state densities of mammalian populations as a function of body mass is known as damage law. I'm sure, many of you are familiar with it. We're showing the data here as the blue points. And this is the steady state this is actually I think it's individuals per meter squared, but that's that's fixed in the paper. We have body mass in grams on the x axis. The steady states of the nutritional state structured model are shown in green and orange. We have a steady straight for the steady state for the full population and a steady state for the hungry component of the population. If we add these two together that's the total population, and it doesn't look any different than the green line because again this is log scale and so so the orange trajectory here is much much smaller than the green trajectory. But the important thing is that the green trajectory maps really well on top of the data. And so, we can modify the intercept with by changing alpha in our model, which is the resource growth rate. To something that resembles grass. We get this intercept here. Okay. And of course that's going to vary from place to place and area to area and that that might explain, you know, if we were to subdivide these points into different regions and different NPP, you know, or types of areas then, then we might find some finer structure here. But really the important take home is that the slope is purely a function of the dynamics and rate equations. And here's just the equation for the steady state down here. Now as I mentioned, the, the only source of mortality that we have in this model is starvation. If we were to add an additional external source of mortality, we would modify the steady state trajectories illustrated here. And I'm just going to include I'm just going to compare that now I'm going to now add in an additional external mortality source and see how that modifies f star. What we find is that additional mortality in this model really affects larger species. And you can see that the red line starts veering away from the green line at around 10 to the fourth grams. And so it's these larger species that are really impacted by external sources of mortality. And that coincides pretty well with what we understand about natural systems when we look at large scale extinction events. It's often the large organisms that go extinct. The steady state population sizes are more sensitive to external sources of mortality. As we see in this kind of, you know, cartoon view of the world, then, then that might in part contribute to increased extinction risk. However, there, there, there is a lot of ambiguity in the fossil record about whether or not large organisms are more prone to extinction. And one of this is a data problem. There is some evidence that that is the case that larger organisms have shorter persistence times in the fossil record, compared to smaller organisms but it's it's really hard to get at, and based on resolution limitations of the fossil record. Well, our previous finding was by setting the starvation recovery rate to known allometric relationships. However, we can get additional insight into extinction risk by allowing the starvation rate and the recovery rate to vary. So here we're not constraining the starvation rate and recovery rate. We have starvation rate on the x axis and recovery rate on the y axis. And what, and these are a simulation results now so, so what we're doing is simulating the nutritional state model, state structured model over time, and applying some perturbations to it just some simulation induced perturbations and seeing when the system falls below some critical threshold. And we do that many times and we turn that in. If it falls below a critical threshold we deem it extinct. If it stays above the critical threshold, then we assume that it's that it's not extinct and so we can look at the probability of extinction. That's the function of starvation rate and recovery rate. And what we find is, is there's two regimes of extinction so so blue is a low probability of extinction, and yellow is a high probability of extinction and there's really too large extinction regimes. This is actually a half bifurcation, which we can solve for analytically. And so as you move towards this area you get, you know, cycles and cyclic dynamics and that promotes extinction. So here, the steady state of the consumer is being lowered too far. So, up here in this a regime we have death by oscillations and, and in this B regime we have death by scarcity and in the middle we have this nice little window where where extinction risk is low. So what I'm showing in the white dot, and this is for 100 gram organism, I'm showing the white dot illustrates the expected value of the starvation rate and recovery rate for an organ organism of 100 grams. Okay, so it falls within this extinction wind or so this this low extinction risk window. So if we move up to 10 to the fourth grams, we find that the windows changing shape a little bit it's actually getting smaller. One important thing to notice is that the, the, the scales on the axes are changing. And if we move to 10 to the six grams, the windows a lot smaller but the point is still the expected value of starvation recovery rate is still within that window. So it doesn't look a whole lot smaller than this one but because the axes are changing it's actually about 30 times smaller. So that also that's another way to assess, I suppose, the extinction risk of organisms as they increase in size the window in which their recovery rates and starvation rates operate in a way that shouldn't produce large oscillations in their population sizes is lower. Now, can we say anything about the evolution of large body size. To do so, we ran a bit, we ran an experiment, a competition experiment. So the competitive advantage of body size among closely related species is what we wanted to evaluate we wanted to evaluate if to very closely related species that very in body mass by a small amount. So competing for the same resource who would win and we used, you know, classic our star theory to determine the winner, whichever consumer pushed its resource to a lower steady state would be declared the winner. So we imagined a competition. A competition between a resident consumer with that has mass M and a competitor which has mass imprime which is just modified a little bit. So it's M times one plus Kai, where Kai is the percent change in mass. So the competitor could be a little fatter if Kai is greater than zero, or it could be a little leaner if Kai is less than zero and there is a limit because you can only lose so much fat before you dip below that. The amount of fat an organism has at which point it would be assumed to be to starve to continue so there is a lower threshold to how lean an organism can be as a function of its body mass. So we essentially are searching across different masses of the resident and different values of Kai the percent change in mass of its competitor to see who pushes its resource to a lower steady state and see if we can reconstruct that. What Alroy was was seeing in his data back in 1998 this idea of an attractor at a large body mass. And this is what we found so let me walk you through this figure so on the x axis we have the resident body mass this is the mass of the resident consumer, and on the y axis we have Kai. So above Kai, the fatter competitor wins, for lack of a better term, and below Kai the leaner competitor wins. And we're searching across body mass M of the resident and the competitor. So what we find is that for lower body masses, the fatter competitor always wins and that's illustrated by this blue region up here. At very, very, very large body sizes the leaner competitor wins. Okay, so it switches. And the switching point occurs right here this is what we call M opt. Okay, so let's see. All right. So this switch occurs at M opt, and if we calculate the value of M opt the optimum mass so this again would function as an attractor because it's the larger the fatter organisms that are winning at lower body size which means that given all of the things being equal the larger organism would be selected for given this pure resource competition situation, and then at larger masses it would be the smaller that were selected for. And the value of that switch point is 1.748 times 10 to the seventh grams so this is purely a function of the simulation that and the rates that we put into the model. And the fossil record to see it, the mass of the largest organism to see how well that maps on to this optimum mass. We find that the observed values the best estimates for the mass of the largest mammalian organisms are for and drink 1.5 times 10 to the seventh grams and for dinotherium actually 1.74 times 10 to the seventh grams so very accurate relative to our simulation so accurate that we checked in double check to make sure we weren't in some tautological space that and it made us sure for a while but but we weren't. So, we were able with this explicit starvation model to reconstruct the upper bound of mammalian body size, thereby, you know, putting a putting an expectation putting a specific model to alroy's insight from the North American fossil data. And we don't have any sign of a lower bound but our, you know, state structured model is pretty minimal. We don't have any higher terrific effects, we don't have anything other than starvation so we might not expect to see too much from this one simple model. Okay, so I want to change gears now. In the time that I have remaining to the second question how do individuals balance starvation risk against reproductive investment when resource acquisition is uncertain what are the evolutionary consequences. All right, so so far we have been assuming that only the mean resource density impacts starvation. But of course we know, as I mentioned in the beginning that resources can be heterogeneous in space. They can be patchy, and this patchiness depends a lot on, you know, resource type. It also depends on the consumer body size and the area over which it forages. And we can capture these relationships by the scaling of the variability of the resource and the area over which is over which it forages with body size. And I should mention that this this work was led by the postdoctoral Jedi, who Tom bot, who's who's unfortunately no longer in my lab, but he's at UC Santa Cruz. And what I mean by this idea that patchiness scales with body mass of the consumer is, you know, let's consider the Savannah landscape. If we look really closely at the Savannah mouse. It seems it sees a very different type of landscape in terms of its resource patchiness, then does an elephant that's moving across this broader plane. Okay, so so in this case the Savannah mouse sees a very patchy distributed landscape in terms of its resources whereas the elephant would see a much more homogenously distributed landscape. And we capture this idea that patchiness scales with body size with this parameter zeta. Okay, and this tells us essentially how the coefficient of variation of resources changes with body mass. So when zeta is equal to one, we have a very uniform landscape and that uniformity does not change with organismal body size so as the organismal gets larger and larger. The landscapes get gets more and more uniform okay so the coefficient of variation is declining. And as I just spoke, the coefficient of variation declines as the organism gets larger. So here, when the organism is small, it sees a more patchy landscape when it's large it seems a more uniform landscape. However, when zeta is equal to two, the granularity of the landscape is preserved as the organism increases in size. So if we consider, you know, a smaller carnivore that sees a very patchy landscape or perhaps a frugivore that's looking that's that's patchily distributed as we get larger and larger body sizes that patchiness is maintained, for example by a large carnivore also searching for patchy food. So the coefficient of variation when zeta is equal to two is preserved as a over consumer body sizes. So we simulated a consumer model on top of these on top of these and especially implicit areas okay so so this whole model is based on this energetic model of consumption, where a consumer has some amount of reserves, and as it finds food, it adds to the foraging in a landscape and when it adds when it finds food it's energetic stores increase and then of course they decrease due to metabolic costs. So the organism spends some amount of energy, which we have as be it's replenished as they find food. And importantly, when it when the organisms energetic state hits a certain point it invests a large amount of energy into reproduction and that energy then is distributed across L offspring and it's litter. Okay, so it's a relatively simple model where the landscape and resources themselves are implicit. And what we're tracking is the energetic state of a given consumer so so it actually replicates a lot of the dynamics that we had in the nutritional state structured model, except that also includes this the ability for the landscape to be heterogeneous right it's not we're not constrained to this mean field type of perspective. So, this is really taking an individual perspective of a system and then expanding it to populations so we can and do treat this as not as an unconstrained optimization problem and explore different life history strategies over resource richness variability and patchiness. We can also assume these life history traits that I illustrated before how much energy the organism spends in different processes, how it's distributed across litters of different sizes. We can assume all of those parameters follow expectations from allometric relationships for terrestrial mammals, and then numerically solve for population study states. And when we do that, we can find how steady state dynamics of the consumer population changes as a function of resource patchiness. And now this is just kind of a cartoon example to illustrate one of our findings, and that is that when the system is not patchy, we find a single steady state. When it is patchy or as it becomes more patchy, we have the appearance of an elite effect. Okay, and the elite effect is represented by this orange line here where, as we get close to zero the change in population size over time is, is, is negative. We can get positive above this critical threshold, I think everyone's probably familiar with Ali effects here. And we can derive a measure of stability of the system by looking at the, the single eigenvalue of the population. Okay, where the eigenvalue is telling us, giving us the steepness of this trajectory at the steady state and star, and the steeper the degree at the steady state, the smaller the time scale of the perturbations of the system, which, which implies a fitter system so we're taking this now as a measure of fitness for the population. We measure this, this fitness metric. Okay, again, the, the leading eigenvalue and where the, we're, we're a larger value implies a smaller time scale of perturbations. So a more stable system, we can then map that across a body mass of the consumer and how clustered the resources, which is given by Zeta. And so again one is a very uniform landscape resource landscape and two is a very patchy landscape. And we can see that lambda or the log of lambda in this case, which is more stable in the blue side so the larger values are more stable and red is less stable changes a lot as a function of body mass and resource clustering. And this gives us essentially a fitness landscape that we can interpret this as a fitness landscape, where more stable regions organisms should be moving towards larger body size, right, and, but it, but it depends on the types of resource and how clustered their resources is, in terms of their trajectories towards these larger body size. And there's a lot to kind of think about with with a figure like this or a prediction like this but one of the things that we wanted to do is try to leverage this prediction against something major in the fossil record where resource distributions really shifted. And these shifts occur if are really obvious if we look at systems from the eosin and then the late myosin. If you look at these two illustrations of what the eosin looked like or likely looked like and what the late myosin likely look like. One of the big differences is grass, grass evolved around 10 million years ago and eosin did not have grasslands there weren't any such thing as grasslands grasslands really exploded in the late myosin. And you might assume then that before the explosion of grasslands food, especially for smaller mammals, and mammals happen to be smaller before the explosion of grasslands tended to be more heterogeneously distributed, whereas in the late myosin they were more uniformly distributed. So can this perspective that we've built with this really simple model, provide any insight into the ecological forces driving the transition to grassland resources. This is the evidence of the explosion of grasslands beginning at around 15 but not really picking up until about 10 million years before present. This is from a paper by Caroline Stromberg, and yellow here is the presence of open grass dominated vegetation in the fossil record. So this is really worth grasslands come on the scene. So this grassland explosion this grassland transition, we see the evolution in mammalian diet. So we see many different mammalian clades begin consuming lots of grass resources whereas before they weren't. These are different mammalian clades illustrated here, and what's on the y axis is Delta C 13. And that is a measure of where they're getting their energy, low values of Delta C 13 mean they're getting their energy from browse resources, and high values of Delta C 13 mean they're getting their energy from grass resources because grasses tropical grasses use a different photosynthetic mechanism and that has a different isotopic signature than than non grass resources. And on the x axis we have age and million years before present so what you see for equids bobbins hippos, gonfitheers, and really all tax if we average them together is this move towards grassland resources is what we see as things transition from C three to C four as grasses become more prevalent. This is in Turkana basin in Africa we see the same shift in North America, and it's a general shift in mammalian communities where grasses become available. So we can make the assumption that grasses are evenly distributed, or in pure grasslands grasses are evenly distributed. And so if you have a Delta C 13 value associated with grasses, you're getting your resources from a more even landscape. Whereas if you have a lower Delta C 13 value that is, you know, that that that is the value of browse resources that you'd be getting your resources from a patchier environment so this is an assumption that we're making. When we look at the fossil data. So let's assume then that a heterogeneous, homogeneous landscape of grass has a zeta equal to one where it's just purely even, which which may be unrealistic but it's going to be close to one. Whereas a landscape that's patchier is going to have a higher zeta value. What is the zeta value of a patchier landscape let's take a Savannah Woodland. It's very very very heterogeneous. We can go and grab pictures of these Savannah Woodlands from Google Earth. We can turn the vegetation into black pixels and the, the non vegetation or the, the non browse vegetation into white pixels, and we can actually measure using a box counting algorithm, the average zeta of the system and it turns out if we do this a lot for a lot of landscapes, the average zeta of these these Woodland resources is 1.71. So now we have this correlation between high Delta C 13 values of grasslands where Zeta should be close to one and a low Delta C 13 value of browsers with an average zeta of 1.71. We can then map extinct organisms into this fitness landscape, where again we have zeta on the x axis, and this is also corresponding to our kind of interpolation of Delta C 13 values so we're using the delta C 13 values of consumers to map them into this zeta space. And then of course they're reconstructed body size. And what we see if we plot a bunch of different organisms so we have suites equids rhinos, let's just focus on the smaller ones at first, across different time periods so the darker values are longer ago at around 10 million years before present and the wider values are towards the present. We see a movement, a trajectory that follows the predicted fitness landscape, as they consume more even foods with the evolution of grasslands they're increasing in body size, following what we would predict based on this fitness landscape. Now things get a little more messy as we get larger and larger and larger and this is effectively because larger organisms have a lot more fat. Again fat mass is super linear with respect to body size, and so they can kind of eat what they want. And so the relationships somewhat fall apart up here, as would be predicted because they have the their bodies can take advantage of many different types of resources whether they're evenly distributed or unevenly distributed. Rhinos are kind of in the middle, they follow the trend, except for black rhinos, which which tend to be choose your consumers they tend to browse for foods, whereas the the white rhinos tend to graze. So we find when we look at these extinct species they tend to follow this fitness landscape as predicted from our very simple model, where we incorporate the energetic demands of consumers, and the patchiness of foods. Okay, so thank you. I know that was a lot to fit into an hour, and I appreciate being able to visit Italy, kind of over the last few days and thanks very much. Yes, I hope you enjoyed the food. Thanks a lot, Justin for the very nice lecture. So we have time for a few questions so please if you want to ask any. You now know how to do it we use the raise and the tool or type it in the chat. Well actually I have one in the about the first part. In the, when you have the model with the starvation and and the grazing so in some sense, it seems to me intuitive that there is a separation of timescale, perhaps I'm interpreting from the equation but that is a separation of time scales between the starvation versus grazing and the population dynamics. I mean what typically happens when you do the separation of timescale is that what emerges like a nonlinear functional response. So I was wondering if there is a relation there. That's, that's an interesting question I. Yeah, you're absolutely right. There, there's a large difference in the time scales. And we haven't really investigated that possibility. In terms of how that might be interpreted. So you're saying effectively that you can capture that process because the time scale starvation recovery is so short. You can capture that process with a with a nonlinear functional response embedded within the pure population equations without the individual dynamics. So if you have like let's say a predator who is in two states like handling and and foraging what you get is a holding type one function of the spawning type two functional response. Right. If you do the separation of timescale so I was wondering if there is a map being there. That's a really interesting question that would be interesting to poke around with I honestly haven't even thought about it. That would be interesting and that would simplify things a lot. I think, you know, what would be really helpful. And I think it's helpful thinking about it in both ways right because kind of complete picture with all of the time scales allows you to kind of think real hard about the time scales of the different processes but then how they might condense into a single functional response. May allow you to kind of travel to a simpler system with those known time scales. Yeah, that would be interesting. We should look into it. So there is a question from Washington. Yeah, so those were really great lectures and I thought that was really interesting stuff. I just have a quick question so you apply your model in the context of mammals and you give this very specific prediction for the upper bound on mammal body size, which matches remarkably with what you get out of known data. What is the difference if you apply this to say reptiles or dinosaurs I mean what would be different in the model and have you tried trying to make a prediction there as well. Yeah, that's one thing we really want to do the only thing that because because the models written generally without assuming any particular exponent or coefficients etc. We applied those known for mammals, because they were readily available and I tend to think a lot about mammalian systems might my dream to is to you know kind of incorporate a dinosaur perspective. There are some good estimates of, and so basically, it wouldn't really take much except to change in the coefficients, and I don't even think, yeah the exponents wouldn't change, but the coefficients would change because, for example, dinosaurs, some were certainly endothermic, some were certainly ectothermic and a lot looked like they were mesothermic they were falling in between, kind of like great white sharks are mesotherms or tuna. So it's metabolic distinction basically the metabolic metabolism basically metabolic processes will determine different coefficients which would give you exactly yeah yeah so it's really just changing the the coefficients of the system. And then maybe the other thing that would need to be thought about is whether you know just the productivity of the landscape and what types of landscapes different. Again dinosaurs thinking about dinosaurs we're we're forging within. And I think all of those would be really fun kind of thought experiments to run to see, you know, even if we could, because these are analytical, you know, equation, or you know, these are equations. It would be like changing what coefficients starts pushing that upper bound up, and what coefficients start pushing that upper bound down, because of course sauropods pushed the upper bound up. And, you know, the rapcids and other smaller kind of reptile like mammals were had had a lower bound, they didn't get as big so it would be a really interesting kind of thought process to work through. Kind of the flexibility of this space. Cool. Thanks a lot I got one other quick question actually which is you alluded to the higher extinction rate for larger mass organisms. So I guess I'd always kind of assumed that had to do with longer life spans and smaller absolute abundances. But it sounds like you're saying there's more to it than that potentially. Is that correct. Yeah, you know, I think it's, I think that the classic thought and I don't think it's wrong necessarily is that, you know, especially the gestation times of larger mammals makes it really difficult to recover from any big population problem. I think large mammals in it. Yeah, it kind of falls into this general specialist thing that I was I mentioned one of the earlier talks. It's like large mammals, they have more resilience within their lifetime so smaller perturbations to the system within their lifetime. You know if you if I if I don't eat for a week. I'll be unhappy but I'll survive but if a mouse doesn't eat for a day or two it's it's gone. There's less flexibility with those smaller perturbations but then kind of less flexibility with those large perturbations and I think a lot of it has to do with gestation time, and just how to read, you know how quickly it takes to rebuild those populations. Cool, well anyway thanks very much for the great lectures. Yeah, thank you. Great so we have time for one quick question from Monday, Monday. Can you hear me. Thank you. Yes. Okay, thank you very much. For that wonderful lecture, please. I want to look deeply into this grassland interaction with the reproductive pattern of my mouse does this means that one in the population of my mouse can be fast disappearing due to climate change and a genetic factors, because you, you talked about that, that the grassland contributes to how fast they grew into a big and bigger sizes. So I'm looking at situation whereby and also affects their reproduction. So I'm looking at a situation whereby if grassland is fast disappearing due to the impact of climate change and other factors, it means we are going to have a lesser population of this my mouse. Can that apply. Yeah, I mean, thank you. I think there's a lot of mammalian diversity tied to grass resources. And I would certainly expect as you lose those grass resources that would contribute to two extinctions. I guess I would be careful about, you know, the models that I was talking was talking about today. These are very, these are kind of like average models applied to like the average mammal. So I would be really careful about making a specific prediction with respect to any specific species based on these dynamics because they're based on average trends. And each system has idiosyncrasies and a lot more that goes into determining an organism's niche space, etc. And contribute differently to the extinction dynamics. I think, you know, taken across many systems and this is where the model these models are more appropriate where we're competing, where we're comparing average trends across many systems. I think that that would absolutely be true that is, you know, you have organisms that are tied to grasslands as you take those away, or in under really short amounts of time. And we will witness extinctions of those organisms. I think, you know, one of the important things is in the past these transitions from more forested or wooded landscapes to grasslands. They were relatively slow. You know, our grassland indicators in the paleo record, they're over, you know, millions of years, or at the least hundreds of thousands of years so there was time for evolutionary response. There was time for selection to modify what organisms were doing, which we see when we look at the isotopes these they switched to grassland resources. We're making modifications to the landscapes on time scales much shorter than that. And so I would expect that just the differences in time scales of disturbance versus, you know, the evolutionary process is going to increase extinctions, relative to, you know what you might expect during a slow transition. Great. Thank you. Thank you very much. Thank you. Thank you. So thank you very much. Justin for the lectures and answering all the questions. We are now taking a six minute break and we'll be back at six 15pm Italian time with Justin again and other panelists for the last session of the school. Thank you very much to everyone.