 Okay. So I'll assume that you can see my screen. Thank you. It's a pleasure to be back. Today, I would like to focus on the role of humans in ecosystems and how we can understand that from reconstructing past systems. And so I'm going to spend part of the talk today focusing in general, really, well, more specifically to the Pleistocene with respect to the Pleistocene extinctions and then focus in on a particular system that I've worked on in Egypt. So that's kind of the roadmap for today. Okay. So the effects of humans on ecosystems. We've been around for a while. The human species for 150, 200, 250,000 years, depending on where you make that break point. And of course, our relatives, we've have been around much longer. I like to think of the process of our involvement in systems as a continuum, because it has been and our role in ecosystems has likely changed over the last, you know, five million years or so, that we began to diversify into different species. And then, of course, throughout our history as homo sapiens, I'm really going to, even though I'm showing our ancestors or close cousins, I'm really going to be focusing on humans specifically. So let's start more recently than what is shown on the last slide and think about the Pleistocene extinctions. This is a period of time where we've come to understand the role of humans much better in the last number of years. But this is a period of time where we had a large influence on the global diversity, especially of mammals. And I'm going to focus a lot on mammals today, specifically larger bodied mammals, greater than four kilograms. So this graphic, let me walk you through it a little bit. It shows a lot of information. And it might be a little outdated. It's from 2006. And there's been a lot of work. This is a really quickly developing field. But what we're showing, of course, are different continents and the numbers and pictographs of species relate to the number of extinctions that these different continents experienced as humans colonized them. The black bar represents when humans kind of showed up on the scene. Of course, in Africa, we've been on the scene for a very long time before we were Homo sapiens. And as we colonized the planet, we see a series of extinctions that we collectively call the Pleistocene extinctions. Some of these are very well timed with human arrival. And some of them are less well understood. So in Europe, for example, there's fewer extinctions. And our arrival is likely much earlier than the extinction than we have evidence for the extinctions being. The circles represent evidence for extinctions. Yellow is provisional evidence. Green is robust evidence. And we have really robust evidence around 14,000 years before present. Humans showed up a lot earlier before that. So it's not as clear in Europe what's going on. In Australia, it's very, very, well, I guess I could say clear or maybe to be a little more careful correlated where humans arrive right as we find evidence for mass extinctions in Australia. And that's around 72,000 to 44,000 years before present. In North America, it's also very correlated to use a more diplomatic term where there's a large number of extinctions that we have evidence for right around the time that humans arrive on the scene. In South America, there's also a large number of extinctions, but it's less well understood. So humans appear to, to our understanding, humans appear to have played a large role in the Pleistocene extinctions across the planet. So what can we learn when we get down to business and we want to reconstruct how these ecosystems function before and after human arrival? And so that's really the goal today to reconstruct patterns of interaction to understand the role of humans on community dynamics over this period of time. What I'm showing on the right is an American lion skull. Lions are an interesting example of what occurred at the Pleistocene boundary. Lions were one of maybe the most widespread large mammalian carnivore on land. They were successful in many, many different continents. They were all over North America. The North American lion was actually had larger body size than the modern African lion, which is interesting to imagine. And now, of course, their range is restricted to Sub-Saharan Africa and a small population in India. And I want to focus a little bit more on how we go about reconstructing interactions in this particular way, because I'm a little more involved in this style of research. And really, the idea here is to take advantage of modern contemporary systems to understand the rules of interaction. So we'll use the Serengeti rules to kind of quote the documentary film and book on ecosystem processes and use body size from contemporary systems as a window into what likely occurred in past communities, how past communities were arranged. I'm showing one of my two favorite graphics here, figures from a paper published, it's covered up by the window here, in the 2000s by Anthony Sinclair and Justin Brasheres and Simon Maduma, on patterns of predation in a diverse predator prey system. So this is in the Serengeti and what you're looking at is on the y-axis is mammalian predators arranged by their body size. So the smaller bodied are on the top and the larger bodied is on the bottom. Of course, lions at 150 kilograms are the largest terrestrial carnivores in Sub-Saharan Africa. If we throw in crocodiles, they might not be, but we'll just focus on the grasslands in the savanna. And what we see is on the x-axis is the prey weight range in kilograms. So this is their food. And of course, as we go from the smaller carnivores to larger carnivores, they're eating larger food, they're eating bigger food, right? They're eating herbivores that are larger and that makes sense. But the other key thing to see here is the range. And that's what's highlighted by that black bar. So as we get to larger carnivores, they're eating larger ranges in addition to the larger body size, the mean. And it would be nice to see distributions on these bars, but I guess you want to plan your publication strategy. You don't want to throw it all into one go because I think the data are out there. So lions have this huge range of food that they eat. You can see one of the interesting things here is wild dogs. They have this like shifted range relative to their body size. So they're eating above their weight. And that's because they're such efficient pack hunters. So that's a pack hunting signal. But it's a nested relationship. So the diets of smaller predators are subsets of the diets of larger predators. Now this means that if you're small, if you're a small herbivore and you live in the savanna, you're going to be eaten. And that's shown on the bottom graph. You're somewhat doomed. That's your future is to be food. And only if you're larger do you escape predation. So in the bottom graph, which is really striking, is the percent of mortality due to predation on the y-axis and on the x-axis is the log prey, log herbivore weight. I have to move around my little zoom bar. That's in my way. And what we see is this very sharp sigmoidal kind of switch here, where below 310 kilograms, you're doomed. And above 310 kilograms, you've effectively escaped predation for the most part. And the dividing line appears to be zebra and buffalo. So the Z stands for zebra, wildebeest. They're eaten. Buffalo, in upwards, you escape predation. So buffalo, giraffe, rhinos, hippos, elephants, a very small percentage of their population is actively killed. We're not counting scavenging here. Okay. So bottom line, body size really matters in these terrestrial systems and these mammalian systems. And we can really think of sub-saharan Africa as being a remnant of the Pleistocene. It's one of the last surviving Pleistocene systems for us to draw inference from and correlate our understanding of these systems to the past, especially a relatively recent past like the Pleistocene. Organisms, it's not like we're going back to the dinosaur world and trying to reconstruct which dinosaurs ate whom. This is a little closer in time, so we can more confidently make relationships between what we see today and what we saw back then. Oh, yeah. Okay. So this is how we're currently doing this. And it's based on a method published by Rudolph Rohr, I think in 2010, an American naturalist, where he establishes a logit function where the log of the ratio of the probability that there is an interaction between a predator and prey divided by the probability that there is not an interaction between predator and prey is a function of the mass ratio between predators and prey. So here m sub i is the mass of a predator and m sub j is mass of a potential prey. And you can see that there's three different terms here, one of which is squared and one is linear and one is constant. And this allows and by fitting alpha, beta and gamma, which are unknown parameters that need to be fit to a particular system, by fitting these parameters, we can establish a Gaussian like distribution and log space that defines the probability of a link existing between a predator and a prey as a function of the predator and prey's body mass in this case. So for example, by fitting, and this is, you know, fitting these three parameters could be done by, you know, any kind of, you know, simulated annealing or an elder meat, you know, algorithm. So by fitting these parameters, for example, in the Serengeti, in the known Serengeti system, we can accurately predict 74% of link presence as well as link absence. And what I'm showing on the bottom here is not the Serengeti. It's the one that I had on hand when I was making my talk, because I'm working on some ocean systems right now, but this is the Benguela system. This is from the famous paper by Peter Yodzis. And it shows predators and prey or species arrayed by their body size. And the known trophic interactions being the darker colors, darker squares in this adjacency matrix, and the predicted probability of an interaction overlaying on top in the color. So by feeding in a known adjacency matrix, we can fit these unknown parameters, alpha, beta, gamma, and derive a equation that tells us the most likely, you know, the probability of an interaction given body size ratios. So by looking at the Serengeti, which I'm showing on the bottom, and for some strange reason, I've put large herbivores on the left and small herbivores on the right. My apologies. I tried to switch it around, but it wouldn't let me. It's a detail. And what we see is by looking at the probability of an interaction or an established interaction in modern systems, we can equate that to the likely interaction structure for past systems, assuming body size rules stand. Okay. And there's not really a whole lot of evidence to suggest that past systems wouldn't be following the same alimentary principles as modern systems, especially if we're within the same kind of families of organisms. But that's an assumption. Okay. So let me see if I can pull up my chat window just in case Okay. All right. So not only do we get mean probabilities, well, this is just kind of showing a different cross section, right? So here on the bottom left, now I'm looking at, we have the probability of an interaction on the y-axis, and this is the mass ratio on the x-axis. And so you can see this isn't logged. The x-axis isn't logged, so it doesn't look Gaussian, but it is, if you log it. And this just shows that for a given predator in prey, let's say if we keep the predator mass constant and we're moving the prey mass, there's a prey mass where the prey is too small for the predator. So it's energetically inefficient for a lion to run around trying to catch mice. But then it's also energetically inefficient and risks death to go after the largest animals as well. So there's these boundaries on either side. And this is what this is capturing, where those boundaries change as a function of what the organism size is, the consumer size is. Okay. So not to belabe, oh yeah. And then obviously not to belabor the point, then I belabor it. But then when we expand this, looking at prey mass by predator mass, we can picture this in two dimensions and you can see how the probability of a link is changing and captures that nested relationship that we see in the Serengeti. So this is one of the important things that this is capturing. Now the original Rural model also allows for latent traits. So traits not associated with body size. So thinking about marine systems, sperm whales, or not sperm whales as a bad example, but humpback whales are not going to obey this relationship because they're going after really small things and they happen to be really, really big. And so there are other latent traits that are not body size related that you could account for using this type of approach as well that I'm not going to go into. Okay. So the questions that we want to address here is how do changes in structure translate to dynamics? So if we take modern Serengeti systems as kind of the baseline and use it to reconstruct Pleistocene systems to get the structure of interactions and really these are probabilities of interactions. So when I say reconstruct a Pleistocene system, I'm really talking about a large number of potential systems that fit those probabilities. Are there differences from one Pleistocene system to another? So this is really awesome work led by Machias Perez in 2015 and Proceedings of the Rural Society B, I really like this paper, where they implement this method, they reconstruct structure, and using body mass ratios and then just look at the structure of the system. They look at modularity by nestedness to see if there are differences between continents. Pleistocene systems from North America, South America, and Africa. And what they find is that there are large differences from one system to another, but it doesn't always divide out by continent. So for example, in South American system, there's a large range of modularity predicted by this approach and a relatively constrained amount of nestedness that's predicted by this approach as well in all places. In North America, there's more modularity than in some of the Pleistocene systems than is seen in some of the Pleistocene systems from South America. And then Africa kind of falls in the middle here, it's in the mid-range and that's represented by this, I'm pointing at my screen like you can see that, but I should use my pointer I guess. So African systems fall here somewhere. But Machias and his group went one step further and they asked, well, since we can estimate the structure of these Pleistocene systems, can we assume some very simple dynamics and put those dynamics on that structure? So they used a lot of Altaira framework with allometric vital rates. So I'm just showing the very basic framework that they used, Arsabae. So this is the change in abundance for species eye over time. And Arsabae is positive if it's an herbivore, negative if it's a carnivore to capture that difference in growth and mortality. And in addition to the consumption of different prey species. And R is assumed to be an allometric, so it's scaled allometrically to mass to the negative one quarter. And this term B sub ij is extrapolated from these probabilities of a link existing between species. And the system, it's a high-dimensional ODE system is stable if the leading eigenvalue is less than zero. And what they experimented with is simulating the system, the Pleistocene system as is, and with the presence of an added apex predator. So they did it without humans and then they included humans and they assumed humans had a similar hunting behavior or hunting focus as other apex predators in the system. So as for example, a large lion or a saber-toothed cat. And there is a lot of evidence to suggest that humans were consuming at the top of the food web in the Pleistocene. And by looking at the system without humans and with the added effect of humans, they could assess the destabilizing effect of humans. Okay, so this is what they found. So on the top, I'm just showing the structure as before. On the bottom, the y-axis is the destabilizing effect of humans. And we have the different systems across continents according to the colors coded to the key on the left. And what we see is that in North America and South America, humans had a much larger destabilizing effect on the system. Whereas in Africa, they had a much smaller destabilizing effect. Okay. And this is suggestive that because what the idea here is that humans evolved in Africa and they involved, they co-evolved with African systems. And it's possible that African systems, as we know them, are to some extent a product of those interactions. So the the organisms that are present in African systems and the interactions that they have within themselves are a consequence of interacting with humans, in particular over the last 200,000 years or so. And this may explain, to some extent, why there have been so few documented Pleistocene extinctions in Africa relative to North America or South America, where humans were brand new on the scene. So those communities had no preparation for the arrival of humans. And that could explain the destabilizing effects that we observe when applying this type of theory to those communities. Just a very brief mention. So we're applying some of these same approaches to understand the change in the Northwest Atlantic over time. We're specifically focusing on the Nova Scotian shelf here. And that's circled in the map and thinking about how that system might have changed over the course of the Holocene. Of course, that system began as a subarctic kind of icy environment and is now not. In addition, human fishing has been a huge stressor on that system for hundreds of years. And we wanted to understand how that system may have changed as species were lost. And we're using the same types of approaches here. I'm just showing the same Benguela food web adjacency matrix that we're using as one of the food webs that we're characterizing the probability of lengths occurring between species. One of the challenges we've been having is many of the species in the systems that we're trying to parameterize the contemporary systems that we're using as a baseline for this. It's very hard to find mass estimates. If you're trying to do a complete system, you know, like what's a mass of a jellyfish. So we think so far it looks like these estimates work for length as well. So we can use, and that makes sense because it's a ratio anyway. So it's really whatever the organism is queuing off of to make its foraging decisions is what's important for this type of procedure. And just to give you a sense, this is what the Northwest Atlantic looks like today. We have some very charismatic species like the Greenland shark, which is one of the oldest organisms possible animals in the world. We have poor beagle sharks, blue sharks, haddock shrimp, Kaplan, all these interesting organisms. And this is what it looked like. Historically in the 1700, it was very different. I circled the organisms that are no longer large players in the system or extinct locally. We have Atlantic system, coastal sharks, white sharks, killer whales, walrus, cod, all of whose populations have either collapsed or gone away in the system that we're looking at. And so there's large structural difference between these systems and we expect that there's going to be large dynamic differences as well. What's interesting is all of the fishing and managerial decisions that we make in the modern system here at the Northwest Atlantic are basically working under the assumption that this is the system as it should exist. Whereas very recently it was very different. So the management decisions are contingent on our historical understanding of the system. And we think the historical understanding of the system isn't very good. So we're trying to add to that. Okay, I'm moving on now to Egypt. This is a beautiful palette at the Ashmolean Museum of that's dated to the pre-dynastic era in Egypt. And what it shows is among some mythical organisms, there's many, many modern species that are no longer in Egypt. So it's a portrait of the past where the past is very different than the present. And Egypt in the pre-dynastic era was a very different place. And our goal here is to reconstruct the patterns of extinction to understand what Egypt looked like before and use that to inform how Egypt operates today. And this is common. We have beard witnesses. Sorry, as I hit the table. We have beard witness to these changes. That's humans and we've documented these changes in many different places. This is a cave in Spain in a pictograph of a bison. Of course, there's cave paintings all over Europe and many other parts of the world documenting organisms that no longer live where they live. Sometimes organisms that are extinct. And that's in the relatively short history of our being on the planet. The world has changed quite a bit and we've likely played a role in it, not always. And sometimes there's natural climate change reasons as well, but we've certainly played a role. This is another beautiful picture, the lion panel in cave in France, I believe. I've not had the opportunity to see it in person, although many of you might have. But it's just incredible, not only the artistry and the images, but they almost depict motion. And it's actually been thought that multiple pictures like this of lions next to each other were kind of a cartoon used to illustrate motion. Oh yeah, I wanted to delete this. Okay, so I'm just going to skip, go a little faster here. Okay, so the goal here is to use paleontological and historical information to reconstruct the pattern of extinctions in a single community over millennial timescales. And specifically we want to ask what have been the cumulative dynamic effects of climate, urbanization and industrialization on mammalian communities and can this inform our understanding of how communities operate today. Now I want to be really explicit in saying that I'm not trying to say what caused the changes that I'll be presenting. The causation is a different problem. Really, we're looking here at the consequence. Reconstructing what those changes are and looking at the consequences of them. So let me just introduce you very briefly to the area. We're looking at the Nile Valley in Egypt. This is how it looks today. And I just love this description of the area. Desert vegetation can be classified into three basic subdivisions based on how much water those areas receive, perennial, ephemeral, and accidental. It's a very dry area. And water really drives everything. And if we look in Egypt today, these are some of the species we might be lucky enough to see. We have the Blackback Jackal, Canosaurus, or the Golden Jackal, sorry, hyena hyena, striped hyena, caracals, these really cute small cats, foxes. And if we were to go back in time, we would have seen other organisms as well. We would have seen leopards. We would have seen cheetah. These organisms are no longer there. The last individuals were killed relatively recently, although their populations collapsed a long time ago. The last leopard was observed in 1913. The last cheetah was killed in 1974 near Almagra. If we look at herbivore species, we see, of course, a little more diversity. We see the wild ass. We see ibex, gazelle, many multiple species of gazelle, many of which are red listed. And then we look at those that are no longer there. The Heart of Beast was last observed in 1935. The Egyptian boar, the last specimen in Egypt, was British specimen number 2450, 1912 attics, 1957, and oryx, at the bottom, disappeared in the first half of the 19th century. However, if we look farther back, humans, keen observers of the natural world that we are, have been recording our observations for a long period of time. Our goal here is to not only integrate paleontological information of species occurrences, but integrate artistic representations of species occurrences as they have been depicted in Egyptian artwork over time. Because Egyptian antiquity is so well documented and dated, dating artwork and the presence of organisms in artwork is relatively straightforward. And so the idea is to use these representations, many of which are in ecological settings. So pictured here is the enjoyment of hunting. And we see herbivores being hunted by dogs, but herbivores in their natural environment being hunted by Egyptians. And what we're going to capitalize on are these ecological depictions of organisms rather than religious iconography. Here's, again, a beautiful pictograph from the tomb of Amin Amhat in Dynasty 12, picturing many different species of herbivores being hunted with a net, which was a common way of surrounding area with a net and then hunt them at ease. Heart of East, Dorcas gazelle, leopards, oryx, fox, cheetah, roan antelope. So just incredible diversity, most of which are not currently in Egypt today. Again, here's a couple different pictures just to show you the diversity of, in natural history in some of these depictions, we have bird hunting on the upper left and the nets that, or the traps that they used in the upper right. And then the fish diversity on the bottom is also really kind of astounding. I don't know fish very well, but I would bet a lot of these fish could be identified pretty well from these pictures. Now, one of the other important things about Egyptian artistry is that they were very focused on documenting. They documented a lot of details and they clearly distinguished organisms that were domesticated and imports from other places that might have been captured and paid in tribute to a pharaoh or whatever was going on. And this is the symbol on the bottom that symbolized domesticated organism. It's pronounced ran and it means fat fattened. And they use this to distinguish between wild and domesticated stock. So just a really quick and really brief history lesson on the Egyptian past and what I really want to focus on, which is the climate change. So 15,000 years before present, we have the oldest known artwork out in the desert. And I'm showing that picture on the sides of these rock cliffs. Now 5,000 years before present was the end of the African humid period. So before that period in time, Egypt was like East Africa. It was wet. It was a Savannah woodland and it had all of the organisms that we associate with Savannah woodlands, which I'll show in a moment. But at around 5,000 years before present, the monsoon shifted and the rain stopped falling on the area and it began its desertification process. Of course, that had nothing to do with humans. 4,580 years before present, we have the establishment of the old kingdom. 4,140 years before present, we have the establishment of the intermediate period along with a large eridification event, which I'm showing with the orange dots by the way. And this eridification event was an important one. It's noted it's discussed in tablets and pictograms and it's been linked to this eridification event has been linked to large political upheaval and quick successions of rulers. And it's thought that that quick succession of rulers was brought about by large famines from the eridification. 3,270 years ago was the establishment of the new kingdom and then 3,000 years before present was another large eridification event that's documented. In the records as well as sediment cores now we can confirm a lot of these eridification events. And then more recent history with the Greco-Roman description of Egypt and of course more recent industrialization. Okay, that was a very quick history of Egypt. Just to show you some early, early Holocene rock art from the Egypt area, we have elephants described, elephants depicted up here, so we don't usually associate elephants with Egypt, they were there. Giraffes were in Egypt. The Damodir was likely a recent migrant from Mesopotamia and just a whole suite of organisms that we don't think about living in Egypt today because they don't. And if you go through and categorize all of the different species that are present both in archaeological deposits or paleontological deposits as well as those represented in ecological settings, this is the list that you have. You have cob, wildebeest, hardebeest, oryx, camels, deer, Damodir, giraffes, hippos, you know just everything we associate with the Serengeti was in Egypt. In terms of the carnivores we had striped hyenas, spotted hyenas, cheetahs, leopards, two distinct species or subspecies of lion, a short main lion and a long main lion. And I'll get to that in a minute and that was distinguished apart from being male and female and there's, well I'll get to it, I'm getting ahead of myself, but if we go through time and document when these organisms disappear from these ecological depictions and from deposits, we can build a pattern of extinctions and we can put error on that too to account for organisms being depicted after they've gone. And so what I'm showing here is the first appearance and for some organisms but primarily the last appearance of these different species which is the black circle and then the red, the color that's overlaying on top is the probability of extinction. So where we have a last appearance we have a gradually declining, sorry, gradually increasing probability of extinction following their last appearance, okay. And so it's a very particular pattern and so that is what we used to infer changes in food web structure over time. Now I mentioned that there's two subspecies of lion, a large-bodied, long main lion, I gotta keep an eye on the time because I get trapped in Egypt stories here, a large-bodied, long main lion and a smaller-bodied, short main lion and these were represented as two distinct subspecies within the artwork and descriptions and this actually correlates with a known lion population that was last seen in the Atlas Mountains. So we think this might represent the Barbary lion which was a larger-bodied, longer-maned lion that existed in North Africa until relatively recently. There's still stories of relict populations surviving in the Atlas Mountains but it really disappeared at the end of the Second Dynasty in terms of its representation in the artwork and then of course the short main lion lasted much longer so it didn't disappear from Egypt until the end of the, until around 3000, 35 years before present, actually correlating with it within a ratification event. Okay, so one way that we can depict structure over time before we even begin to reconstruct interactions is to just simply look at the predator-to-prey ratio. So what we see over time, and I would also draw attention to the fact that these time-bends are not equally sized, okay, they really correspond to the information that becomes present due to different dynastic cycles beginning or ending and antiquity. But if we look at the predator-to-prey ratio over time, we see that it's increasing until about 3000 years before present as herbivores disappear. So herbivore species are disappearing first, increasing the predator-to-prey ratio after which it decreases at around 3000 years before present and then increases again. Of course, as the system becomes, as it loses species, it becomes much more volatile. The predator-to-prey ratio becomes more volatile because smaller changes make larger differences in the ratio. But I'd like to draw your attention to three larger shifts that we see in the predator-to-prey ratio, at least historically, and that's outlined by these stippled lines, and those correspond to the three larger ratification events. So it would seem in a very correlative way that ratification events appear to have impacted how the structure of the community may have been changing over time. And again, I'm not really drawing causation to this. There's many different causes that we could imagine that could feed into this, and this might be a room for theory in the future. It could be driven by bottom-up forcing. It could just be driven by changes to the environment as directed by changes to the climate, which I'm showing on the right. Or humans could play a large role. We could play a role by competing for space. Really, in this area, as it becomes more and more desertified, water is the key. And finding habitat where you have access to water is really the limiting resource in these systems. So as humans begin to expand their agricultural base, they're taking up space where water is available, and that's pushing out wild animals. And then this could certainly have led to many of the extinctions that we observe in the record. Or humans could have actively been hunting these organisms and had a large impact, especially as we have the establishment of a relatively sedentary human civilization for one of the first times where they're also subsisting on crops for food. They could have abnormally large impact on populations of wild organisms. Okay. Now, I'm not going to go through this because I already did, but we use body mass information because all of these species are extant. So we know the sizes of these organisms, and from their body sizes, we can reconstruct the probability of a link existing between them and then build series of food webs that we think best represent these systems. Dynamics. Okay. So this is kind of the fun part. And I know you guys have had awesome talks about dynamics so far. We wanted to not only reconstruct food webs but say something about the stability of those systems. And we wanted to see how stability or measures of stability might have changed over time. To do this, we used an approach called generalized modeling. And I've got a couple of key citations on the bottom. This was really pioneered by Tilo Gross in 2006 and later in 2009. And then we have a description really focused on ecological systems in theoretical ecology in 2011. And I've been meaning, yeah, I'll try to make a reference list for all of these things that I'm citing and not doing a very good job of reporting what journal they're in. But I'll make a reference list for all of the different things that I'm talking about for those of you who are interested that I can make available. Okay, generalized modeling. So this is really useful where we know the architecture of the dynamics of the system, but we don't know the functional relationships that are embedded within that architecture. So for example, let's consider an ecological system where we have the change in biomass over time. So B is biomass. And we have some function of growth. So we have a source, right? So there's biomass is growing by the function S and it's draining where there's a function of mortality as this function D. But we don't know the specific nature of these functional forms. This actually might better represent our knowledge of the system. But, and this is particularly true for past systems where we really don't expect to know the exact architecture of the system, or at least we don't want to assume that we know. The problem is, of course, we can't simulate a system like this and we can't solve for its fixed point or steady state. So what can we do with it? We can do quite a bit actually. First, the method very simply and generally and quickly is to establish B star as a variable representing all internal equilibria of the system. So these are all non-negative, non-zero equilibria that represent states where all of the organisms persist in our system. But it's unknown because we don't know the functions. We can then define a new set of parameters representing the normalized variables of the generalized system. So here we're defining little b as the biomass divided by the biomass at steady state, which is unknown. A little s of little b is the growth function of the biomass divided by the growth function of biomass at the steady state. And little d of little b is the mortality function divided by the mortality of the biomass at steady state. The important thing about all of these new parameters, these normalized parameters, is that steady state they're all equal to one. And that helps us a lot. This normalization procedure of the generalized system allows us to extrapolate biologically meaningful, and this is the key, biologically meaningful and relatable coefficients, terms, variables, etc. So here we see that when we set the system equal, so when we set the normalized system equal to zero, so assess the system at steady state, little s of b and little d of b, I'm getting covered up here. So this term and this term become one and we have this simple relationship of s star over b star, which is just a constant, is equal to d star over b star, another constant. And we'll just define this as the turnover rate at steady state. So this this gamma here is directly biologically meaningful, it is the biomass turnover rate at steady state, and it's equal to the normalized function of growth minus the normalized function of mortality. Okay, so we've generalized, we've taken this general system and we've normalized it. Okay, now we want, and then we've redefined some of the constants in the system to be biologically meaningful, but now we want to assess the system's stability and what's our, what's our interpretation of, what can be our interpretation of this. So we're going to perturb it and we're going to look at the conduct linear stability analysis as we would to any other ODE system that we might investigate. So we're going to look at the change in d little b over dt as a function of little b with respect to little b and this equals the single eigenvalue of the system and we simply take the the derivatives with respect to little b across the different elements of the system. So we end up with this derivative, for example, for the growth function for the normalized growth function with respect to the normalized biomass is equal to the derivative of the log of the unnormalized growth function divided by the derivative of the log of the unnormalized biomass evaluated at the steady state. And this actually has a direct biological interpretation. It is the percent change in growth divided by the percent change in the argument and in the unnormalized biomass. It's a functional elasticity. So functional elasticities in this system are the logarithmic derivatives of the unnormalized functions relative to the unnormalized argument. And this provides a nonlinear measure for the sensitivity of the function to variations in biomass. So we have in this very simple 1d system, we have the single eigenvalue of the system equal to the biomass turnover rate times the elasticity of growth minus the elasticity of mortality. And we can directly relate this to the conditions under which this single eigenvalue is going to be greater than zero or lesser than zero as a function of the values of the elasticities of growth, the elasticities of mortality and the biomass turnover rate, which doesn't matter so much in terms of determining the positive negative value of the eigenvalue. Okay, so one of the important messages here is that or why are we doing this is that elasticities characterize whole families of functional forms. So consider power law functions. We have a function where the word's constant, a function with a with a linear power, a function with a square term. And if we take the elasticities of these functions, we see that the constant term is simply zero, the linear term is one, and the squared term is two. So power law functions have direct single values associated with their functional elasticities. And this allows us to set realistic bounds to the elasticities in the Jacobian if we're talking about a multi-dimensional ODE system that go into the Jacobian. So you'll have a Jacobian matrix that you derive to look at the linear stability of the system. That's a function of all of these elasticities from the generalized model. But the elasticities themselves have very small bounds associated with them to be biologically meaningful. And that really sets a lot of constraints into the system. So you can more efficiently search across combinations of parameters that are biologically realistic and get a better sense of what the stability of biologically realistic systems is given the architecture fed into the generalized model. More complex functions have elasticities that are functions of the unknown steady state, but also range between small ranges of values. So for example, an elasticity that's more complex that might be a function of the unknown steady state may only realistically be able to vary between zero and one, or between one and two. And this really sets lots of constraints that enables us to more effectively search biologically reasonable space. So just to kind of depict this as a cartoon, if we're doing a random matrix type of approach, we have a large parameter space to search across to get a sense of the stability of the system. However, the generalized modeling approach allows us to subdivide this large parameter space to biologically meaningful units so we can more efficiently search across the system and ignore parameter combinations that are biologically unrealistic. Okay, so now that we've done this, we have, oh yeah, so we have, I'm pointing at my screen again, we have a large system of many interacting species and we have, so change in the abundance of species i over time is equal to the growth of i if it's an herbivore plus the growth of i if it's a carnivore consuming many other herbivores minus the mortality of species i if it's a carnivore or minus the predative loss if it's an herbivore. We normalize that system to a steady state. So this is just kind of the higher dimensional version of what I've already shown you, but all the principles are the same. And then we can drive the Jacobian matrix. So we can derive the on diagonal and off diagonal elements of the Jacobian matrix, which are now all functions of the functional elasticities, each of which has relatively constrained ranges. And so now we can search across many different potential Jacobians by randomly drawing from within those ranges to determine a percent of those systems that are stable relative to a percent that are unstable. And this allows us to translate the structure that we've reconstructed from the body mass ratios to the likely dynamics of the system and do so for all of the different snapshots that we have of Egypt in the past as it changes over time. So we are really going to look at three different measures here. We're going to look at food web stability, which is just the percentage of drawn food webs that are stable out of the large number that we simulate. I think we simulate around 10 to the seventh. And so this gives us a sense, again, of just the general stability of the entire system is the system stable. And that's just what percentage of those systems, those drawn systems, those Jacobians that we draw from the functional elasticities have a leading eigenvalue that's less than zero. Secondly, we wanted to assess the species specific roles and sensitivities to change. Okay, so let's see. Yeah. So we did two things, which I'll describe in the next slide. So one is related to the stability of the system. And that is once we have the stability of the system, then we can start pulling out species and determining whether or not that stability increases or decreases with the absence of a given species. So for example, if 90% of our food webs that we draw are stable, and then I pull out lions, and now 95% are, well, then that means that lions are a destabilizing force in that system. Okay, if we have a system that's 90% stable and I pull out gazelles, and suddenly there's 10% that are stable, then that would mean that gazelles are an incredibly stabilizing force in that system. So that's something that we're going to look at. We're going to look at the change in the percent of stable food webs relative to different species being present or absent in the system. The third thing that we're going to look at, I don't even know if I have a three on here, I should. Do I? No, I don't. Okay, the third thing that we're going to look at is the sensitivity of species to a perturbation, to a different type of perturbation. So we're going to use this idea of a press perturbation where we push the system to a different equilibrium, to a different steady state, and examine how each of those species that are in the system respond to that push. And this is given by the equation in the bottom right, and it's a function of both the eigenvectors and the eigenvalues in the system where we sum across the modes to get the stability, or sorry, the sensitivity of each individual species in the system. Sorry, I'm going through this really quickly. We explain it more slowly in the paper. I guess as slow as you'd like to read it. So what are our results? What were the dynamic consequences of community change over time? So on the y-axis is the proportion of stable food webs with 100% at the top. And on the x-axis we have our time bins, and so we look at each system and simulate many different iterations of its structure as well as the potential dynamics that can go onto that structure. And what we see very straightforwardly is over time the system becomes less stable. And this is true whether we apply tons of uncertainty into the disappearance of species. So upwards of plus or minus 200 years, plus or minus 500 years to the different disappearance of species from both the artwork and the paleontological information, archaeological information, it doesn't change the overall picture of what occurs as species go extinct. So the Egyptian system at the end of the Pleistocene we can interpret as being much more stable as it is today as it has fallen apart and unraveled over the course of the Holocene. Perhaps a little more interestingly, we can evaluate the stabilizing effects of different species in the system. So again on the x-axis I have the time bins, and here we're trying to determine whether there are key species that contribute to the stability of the system, in which case if they're lost they are contributing destabilizing forces into the food web. Now on that y-axis we have the change in the percentage of stable food webs with or without a given species, and each line and each trajectory represents a comparison of the presence or absence of different species. And I've illustrated some of the species, it turns out all of the herbivores are stabilizing, so and meaning that their trajectories are going to be above zero because they have a net positive contribution to the stability of the food web, whereas carnivores tend to be destabilizing. And what we find is that the most stabilizing species are the smaller-bodied herbivores. And as we've lost the redundancy of smaller-bodied herbivores over time, their stabilizing effect on the system has increased, which means the consequence of losing them has become larger as well. We can see farther back in time losing any species has very small effect on the stability of the system, whereas today because there's so few organisms left in the system, losing any species for the most part has a much larger effect, but it's the smaller-bodied herbivores that have the largest effect. Because they're shared by so many carnivores, because the carnivore species rely on those herbivores, losing them is losing the core of the system. And the last result that I'd like to share with you has to do with the sensitivity. Does sensitivity predict persistence? I mentioned yesterday that one of the keys about paleoecology is that we know the future when we look into the past. We know the future when we look into the past. So we can evaluate whether species that we think might be likely to be important have large effects on the system, or perhaps we can consider whether species that appear to be more sensitive, based on some theoretical argument, actually go extinct. So here we're calculating the sensitivity of species to a perturbation. In other words, we're pushing the system to a new equilibrium and seeing how different species react, how sensitive are they to that disturbance. And we could postulate then that species that are more sensitive to a disturbance may be more likely to go extinct. In other words, species that are more sensitive to a disturbance may persist for a smaller amount of time in the system. And because we know the persistence of species, how long they are in the system after the end of the Pleistocene, that's what we have on the y-axis. We have how long the species are in the system after the Pleistocene. It's logged. And then we have the log sensitivity on the x-axis. And what we find is a negative relationship. So species that are more sensitive have shorter persistence times in the system, whereas species that are less sensitive have longer persistence times in the system. And there's this kind of roof here, the ceiling, because that's the maximum amount of time since we begin making the record. So at the end of the Pleistocene until now, this is kind of the maximum. These species are still present in the system and these species have been lost. So what have we found? We found in Egypt as the system has unraveled in the presence of all of the changes that have gone on in Egypt, both climate, human induced, everything added together, we found that with the unraveling of the food web has come decline in stability. And that there are key species that have played in a larger role than others in determining whether that stability is eroded or not. And finally, the sensitivity of these species, which is again a consequence of their structural relationship within the food web, as well as the elasticities that we attribute to them through this generalized modeling perspective, that the more sensitive species are, the less likely, the less length of time they are expected to remain in the system. So persistence in this case is predictable. Okay, so moving on into the future. We think these community level frameworks are vital for understanding how our ecosystems are expected change in the face of these traumatic events that they witness over long periods of time. And of course, we've looked into the past at mass extinctions in the past, and we've seen large changes in food web structure and function on either side of extinctions. And today, it's important to note, not to end on too sour of a note anyway, but that that we're in the six mass extinction, all evidence points to the fact that we are in the six mass extinction. And so building upon paleontological insight, I think, in my perspective is vital, absolutely vital for having a sense of what to expect in the future. Yeah, so if there's any time, I'm happy to take questions. And then we'll on Thursday, I'll I'll really switch gears and and take a much more a close view of how energetic constraints can give us insight into systems that are that are no longer around that are extinct. So thanks so much. Thank you so much, Justin, for the very nice lecture. So you're sure there is a bit of time for questions. So if anybody has any question, you can either raise your hand or write it in the chat, as usual. I guess everybody's a bit tired after hours of lectures. Anyway, Justin will give another lecture on Thursday, if I'm not wrong. So if you get any questions later, you can always ask them on on Thursday. Great. Sounds great. Well, thank you, everybody. I appreciate it. Oh, actually, there was Oh, sorry, there were raised hand. I didn't see them. Sorry. So there's a question from Flavia. Hey, Justin, how are you? Good. Hi, Flavia. How are you? Very nice to all congratulations. Thank you for that. I'm curious with one result you presented, you showed that the small herdivores are more or less the keystone species. And this is a little bit contradictory. I don't know how to say that, but it's like the opposite of the expected and theory. Usually we think that large mammals or large predators are those that are the keystone species. Is it, is your result a consequence of the dynamics or a consequence of the biomass approach that you use? What is that? Yeah, you know, I think, I think, and I wouldn't necessarily say they're keystone species because that's assuming, you know, some kind of stabilization relative to their known abundances. But yeah, they're key to stability, but they're, but I would say they're only key to stability in the systems that we were looking at recently. So it's only after the systems are unraveled that they really become that important. But if you look at a fully fledged system, so if you go back to the end of the Pleistocene, which is very Serengeti-like, they, you know, there really aren't any species that make a large, they have a large impact on stability when they fall out. And so I guess I would argue that that is an artifact of the unraveled state of the system, and that perhaps only very disturbed systems will observe that importance for the smaller-bodied herbivore. But, you know, even in these large systems, and I think that's because of redundancy, you know, if you look at that, that mortality graph in that Sinclair paper showing that sigmoidal relationship to the probability of predative mortality, the smaller animals are getting hammered, but there's a lot of them. And so if one goes extinct, you know, I can't find a topi, I'll go after Thompson's Gazelle. But if there are, if there's only Thompson's Gazelle, then I think that would be a very different situation. Okay, let me see. Thank you. Okay, there is another question from Mateo. Hi, yes, just a curiosity. Just to what extent do you think the sixth mass extinction is similar to the others, because essentially it looks like there's a factor which is the prevalence of humans, which is might be different from the other extinctions. Yeah, that's a really good question. And I'll be waiting and I guess to answer that, I'll wait into things I don't know a ton about, because a lot of people are working on this, right, just determining whether or not we're actually in a mass extinction, which has a lot to do with figuring out extinction rates and projecting into the future, into a very unknown future. I guess, if anything, I would think that the rate of the extinctions that we're experiencing today is probably a lot faster than the rate of extinctions experienced in the past, which are kind of unknowable. You know, the Permitriassic mass extinction likely occurred over hundreds of millions of years, or sorry, not hundreds of millions of years, millions of years, or hundreds of thousands of years. And because it was this kind of gradual change to the climate brought on by these massive volcanic events, the KT extinction might have been a lot faster, the asteroid impact, but we don't really know how the extinction rates changed for different taxa afterwards. But we've had a huge impact on a relatively short time scale. And I would say if anything's different, it's faster, we're more efficient than volcanoes. So in terms of contributing to extinctions. So that would be my guess, but I know a lot of people are working on trying to understand how the current mass extinction, if it is a mass extinction, and it does seem like it is, compares to prior extinction events. Thanks. Okay, I still see the raised hand of Flavia. I don't know if she has another question or if she didn't unraise her hand. Okay, it was not a question. So I don't see any other raised hand. So if nobody has any other question, or if I am not seeing them like before, I think we can say thank you again to Justin, and hear you again on Thursday. Thursday. Have a nice evening. Thank you. Well, you guys have had a long day, but my day is just beginning. So have a good day to you then, and to all the other people on the other side of the world. That's right. Thank you so much.