Hello, my name is Gabrielle de Michele, and I'm going to be presenting some joint work with Pierre-Yngaudry and Cécile Pierrot on discrete logarithm algorithms in pairing relevant finite fields. So the discrete logarithm problem is one of the two major mathematical problems on which is based the security of asymmetric cryptography protocols, the other one being factorization. So the discrete logarithm problem can be found for key exchange protocols such as Diffie-Hellman, El-Gamal, or some signature protocols such as ECDSA, DSA, etc. So the definition goes as follows. Given a finite secular group G, a generator G of this group, and a target element H of the group G, we want to find the exponent X such that G to the X is equal to the target element H. So one question that arises from this definition is what finite secular group G can I choose? So for cryptographic purposes, we want the group G to be chosen in such a way that DLP is as hard as possible. So commonly used groups are prime finite fields, finite fields, or elliptic curves over finite fields, for example. DLP is used in protocols that are widely deployed such as, for example, the Ephraim or Diffie-Hellman protocol as mentioned before, which is present here, for example, in this TLS handshake. Another interesting example which we're going to focus on is pairing-based protocols. Let me go back to definitions. So what is a cryptographic pairing? A pairing is a map between the product of two groups, two additive groups, into a target multiplicative group, GT. The map has a few properties that one needs to satisfy. So first it has to be billionaire, non-degenerate, and also for practicality reasons we want E to be efficiently computable. In cryptography, the groups that are often chosen for pairings are G1 and G2 are often subgroups of elliptic curves over prime fields or finite fields, and the target group GT is often a subgroup of a finite field, FPN. Pairing has been used a lot in older protocols and in more recent protocols such as ZK-SNARX, which is an applicable use for blockchain, Zcash, etc. So it is an interesting example, not only because it is used in many protocols, but also because it uses the discrete logarithm problems on both elliptic curves and also on finite fields. So one question that we try to answer in this paper that we look at in depth is how to construct a secure pairing-based protocol. And the simple answer, which turns out to be not so simple, is to look at DLP algorithms on both the elliptic curve side and on the finite field side. So let us start with the discrete logarithm problems on elliptic curves. The best known algorithm is Polaro, which has a complexity and square root of the size of the subgroup that is being considered. And not much is said on this slide because there has not been many major gain in the complexity since a few decades regarding DLP on elliptic curves. However, on the finite field side, the discrete logarithm algorithms have a lot evolved a lot in the past decades. There are many of them and their complexities often depend on the relation that there exists between the characteristic p and the extension degree n of a finite field, which we define as fpn throughout this talk. So as mentioned before, the complexity of many algorithms for DLP on finite fields depends on this relation between p and n. And one useful way to express this relation is to use this l notation. So this l notation is this very ugly formula in the green box, which depends on two parameters, lp and c. And it is defined as the exponential of some constant c. So the second parameter is this c in front of a log pn raised to the power lp and the log log pn raised to the power 1 minus lp. So what is interesting regarding the complexities of these algorithms that are expressed using this lp and n notation is that if one tends, makes lp tend to zero, then this whole formula goes to some log of pn, which corresponds to a polynomial time algorithm. On the other hand, if lp tends to one, then we end up with pn, which corresponds to some exponential time algorithm. So this lp that varies between zero and n defines, allows us throughout this l notation to define what is called a sub exponential time algorithm, a complexity, which is not as bad as exponential, but not as good as polynomial. So all the complexities that we are going to see, most complexities that we're going to see, I should be careful with the quasi polynomial time algorithms, are expressed using this l notation, where this l is a constant between zero and one. This l notation also allows us to define three families of finite fields, where the characteristic p is expressed using, again, this l notation. So depending on this relation between p and n, we have finite fields that are said to be a small characteristic, medium characteristic, and large characteristic. And what is interesting with the algorithms that solve the lp on a finite field is that depending on the area, the zones where the finite fields are, these algorithms are going to be different and are not going to perform as well. So the complexities are also going to vary depending on which area we're looking at. So we are interested in a particular area, which is what we call the first boundary case, which is exactly the area between small and medium characteristic. So it is the area where the characteristic p is defined to be exactly of the form lp n of one third. The cp, which is the second constant in the l notation, is often ignored, if not important, depending on the context. So why do we look at this area for two main reasons. The first one is concerning pairings. This is the area where pairings take their values. And since one of our goals is to look at the security of pairing values protocols, then we're interested in, of course, the area that concerns pairings. The other reason why we look at this area is because it is an area where a lot of algorithms overlap. We have algorithms coming from the small characteristic world, from the medium characteristic world, which both apply to the boundary case, but it is not entirely clear which one performs best and which one are actually applicable up to where. So this area required some studying. So going back to the first reason, the first motivation for studying this first boundary case, we go back to the security of pairings. And in order to have a secure pairing, we want the dlp, so the discrete logarithm problem, to be as hard on the elliptic curve side and on the finite field side. So one can look at the known complexity of these algorithms for the finite field side. And the first thing one can do is to ignore the small characteristic area because we know there are some plausible and unknown algorithms with much better complexities. This leads us with lpn of one third complexity for all the other known algorithms in medium and high characteristic. Then the idea is to balance these complexities. And we know that for the elliptic curve side, we have the square root of p, which comes from a polar row. So when we balance all these complexities, we end up with a characteristic p of the form lpn of one third. And this corresponds exactly to this boundary case, which we discussed before. So now what do we focus on on this paper? Well, the idea is to study the behavior of all these algorithms that exist in this area in order to then draw some conclusions for the security of pairing-raised protocols. So let us now discuss about these algorithms that exist to solve DLP in finite fields. So most of them come from a family of algorithms called index calculus algorithms. And they all follow the same steps. We consider a finite field fpn and a factor basis, f, which consists of a small element.  The three main steps of any index calculus algorithms are a relation collection step when we find relations between these elements of our factor basis. A linear algebra step where we solve the linear equations. So if we have enough of these relations that we've collected in step one, we can form a system of equations where the unknowns are the discrete logarithms of the elements of our factor basis. If we have enough equations, as many as we have unknowns, we can solve the system and then we end up with the discrete logarithms of all the elements of our factor basis. This allows us to proceed to the last step, which is the individual logarithm step. We're also called a descent step, where the goal is to solve a DLP for a target element h. And we do so by using the discrete logarithms of the elements of f, which were computed in the step above in the linear algebra step. So one of the most well-known algorithms from this index calculus algorithms family is the number field sieve. The number field sieve is often illustrated by this commutative diagram that is given on the slide. And it starts by choosing two polynomials f1 and f2 in such a way that this diagram commutes. So f1 and f2 allows to define two number fields, one on the left, one on the right. And this is where the name of the algorithm comes from. And now the idea of this diagram is to define, to compute some relations where relations come from algebraic norms that are factored in both number fields. And if they are b-smooths, b-smooths means that for a given constant b that is defined, they factor into elements that are smaller than b. So if they are b-smooths on both sides, then this results in a relation. So we have an equality because the diagram commutes between this product of elements smaller than b. This allows, this creates what we call a relation. And as mentioned before, if we have enough of those, then we can solve our system and have the discrete logarithms of the elements of the factor basis. So all the technical details are in the paper or in general in the literature. There's been a lot of work on the number field set. And in particular, throughout the years, many variants have been put forward to improve on the complexity of the number field set. One of these variants is called the multiple NFS. So the multiple NFS simply considers more number fields than just two, as in the classical NFS setup. So we have, again, a commutative diagram where instead of considering two number fields, we have v-ones, which are defined by polynomials. Again, two are chosen at the very beginning of the algorithm just as an NFS and then the other ones are just linear combinations of these two initial polynomials. So this results in some trade-offs in the complexity, but overall the complexity is lowered using MNFS. Another variant is the tower number field set, where, again, the setup is very similar to NFS. The steps of the algorithms are the same as for any index calculus algorithm, but here we just simply add extra algebraic structure. Again, in order to lower the complexity of NFS, of the variant. Another variant of NFS is the special number field set, where here the difference comes from the characteristic key, which is defined as the evaluation of a polynomial of some degree lambda with some small coefficients. So in this case, p has this special form, which gives the name to the algorithm. And again, in this particular setup with p of this particular form, then we have an algorithm with lower complexity than the classical NFS. So how do we evaluate the complexity of NFS and also all of its variants? So NFS or any index calculus algorithm have these three steps. And so the overall complexity of the algorithm is simply the sum of the three costs of each of these steps. So in order to optimize the overall complexity, we want to optimize the maximum of these three costs. So this is rather complicated for several reasons. First of all, NFS and its variants have a lot of parameters. Some are discrete, some are continuous. These parameters vary depending on the variance we're considering, depending on the polynomial selection that we're using to select the first two polynomials f1 and f2. There are some boundary issues. So this whole optimization problem for which we use Lagrange multipliers becomes all the more complex because of the number of parameters that are being considered. When we solve the polynomial system, we use Grubner basis algorithms. And on top of all of this, a lot of analytic number theory results must be considered. So in the end in this work, we have looked at the complexity of all these algorithms along with all their variants, both for the algorithms and the polynomial selection. And this resulted in these two plots where we have the complexities as a function of CP, which is the second constant in the L notation for all the variants and all the polynomial selections. So a surprising fact that we've noticed when computing these complexities is that not all the variants are applicable at the boundary case. For example, if we want to use the special number filter, so SNFS, with the tower setup, then this results in norms that are much larger than expected and thus a much higher complexity. So some variants cannot be combined precisely in this boundary case. Now, we can also look at algorithms that comes from the small characteristic area. Since we are precisely at the boundary case, we are interested in the ones that come from the medium characteristics, so NFS and all these variants. But we should also look at how applicable and how well these algorithms from the small characteristic area perform at the boundary case. So one of these algorithms is called the function field SIF. And the function field SIF is very similar to NFS, except you should think of function fields instead of number fields and look at it as a special variance. So one result of our work was actually to reduce to lower the complexity of FFS by working in some shifted finite fields. So this is a very similar argument as is done for the tower setup, where if we consider a finite field FPN, instead of actually working in FPN, we work in a translated field F prime eta, where P prime, for example, is P times kappa, where N is this composite kappa times eta. So we were working in a translated finite field, which allows us to lower the complexity. All the technical details are given in the peak. And finally, from the small characteristic world, we should mention the quasi-pollinomial algorithms, which a lot of work has been done in the past decade, up to very recently, in 2019, we're finding that this looks key proved this complexity given by the theorem just stated below. So in the end, what do we have? We have this little image that shows us which are the best algorithms, depending on the area in which you are located. So in the small characteristic on the very left, we have these QP, these quasi-pollinomial algorithms. On the right, in the medium characteristic, we have all these variants of NFS. These variants, of course, are not always applicable. It depends on the considerations that are made on N, whether P is special or not, whether N is composite or not. And then we focus on this boundary case, where not only do we have all the variants of NFS, but we also have the function field sieve, which is applicable in this area. So one part of our work has been to precisely identify the crossover points where FFS stops being the best algorithm, and the variants of NFS starts outperforming FFS. And similarly, we have a crossover point between FFS and the quasi-pollinomial algorithms in our paper. So the motivation for having all this precise analysis about these algorithms is the security of parents. So we would like to answer the following question asymptotically. What finite fields at the end should be considered in order to achieve the highest level of security when constructing a parent? So the goal is to precisely find P and N that answers the question above. So this question can be answered now that we have this entire analysis of all the algorithms at this boundary case, which is the area where parents take their values. And so the idea here is to look for the value of CP. So again, CP is the second constant in the L notation that maximizes the minimum between the complexity on the elliptic curve side and the complexity on the finite field side. So from this whole analysis, we've seen that the complexities for the finite field side for DLP are decreasing functions of CP. On the other hand, for elliptic curves, we have polar rho, which is an increasing function. And the complexity depends on this value of rho, which characterizes the size of the subgroup considered. So here in this plot, we take rho equals 1 to 2. And again, details about rho are given in the paper. And so the optimal CP that has to be chosen in order to balance two complexities is given by the intersection point between the complexity of polar rho and the complexity of the best algorithm for DLP. So as one can see, FFS is the plot in yellow or in yellow, which is far on the left. So it is not considered in this analysis for pairings as its complexity becomes much higher for values of CP that are considered. So this plot gives the intersection points for all the variants of NFS. So not only MNFS, but also MXTNFS when we have a composite N, when we have the tower set up, and also curves for the special number fields. So one can then identify precisely, depending on the considerations you want to make on the characteristic P and the extension degree N throughout this plot, can precisely identify the crossover points. This is summarized in the following table. So for example, if we take N prime, non-composite, and a normal P, so no special form, then one can see that the best algorithm to solve DLP for finite field is the multiple number field. So A is the polynomial selection that is being used. And so the crossover point is for CP equals 4.45. This allows one to define a precise P and then the corresponding N in order to define the finite field that has to be used to construct a secure pairing. So there were some surprising facts that we've noticed when considering the security for pairings, one of which is the fact that using a special form for P does not always make the pairing less secure, as one could think, because SNFS would tend to lower the complexity. Indeed, if we look at the first line of the table, we see that if we take a special prime and a lambda, which is the degree of the polynomial used for defining P, and this lambda is equal to 20, then the curve of SNFS is above the curve of MNFS. So SNFS does not perform better for this particular value of lambda. So one can choose values of lambda for which MNFS will outperform SNF. So this was one surprising fact that we've noticed when studying the security of pairings. So this concludes my talk. Thank you for listening. And I will be happy to answer questions during the session.


We are having these battles again and again and again. What does long-term resistance look like? One, you absolutely do not adhere to the rules of politeness when it comes to challenging racism, either in terms of saying like, oh you know, don't get me wrong, I love a good curry, but right, so you challenge like your racist aunt Mabel, you see someone reading the sun, you ask them why you're reading that and you start normalising these conversations because in silence white supremacy can thrive.   There are liberals who will tell you, oh that makes us just as bad as them. It's like, no, because we don't want a genocide, right? So, no, it doesn't make us as bad as them. Yeah, there is a line in that line is maybe actually murdering people, like it's not hard guys. Three, is that when it comes to movements of racial emancipation, like Black Lives Matter or whether it's organising against the far right in this country, or whether it's migrant solidarity movements in Europe, as white people or as indeed people of colour with relative amounts of social or economic privilege, you loudly, noisily proclaim your support for it and you put your body to it. I was really struck by a tweet that I saw from a friend of mine who was saying that, you know, I'm thinking of activists in Charlottesville putting their bodies on the line without free healthcare, which wouldn't really think about that as an image of solidarity.  No, none of that. Because we're in an unacceptable situation where people are saying, oh well, if it wasn't for Black Lives Matter, maybe Charlottesville wouldn't be happening and we're like, oh, if you didn't call the Nazis, maybe they wouldn't be Nazis. No, what we need to do is normalise this course of racial emancipation.


A focused mind is a dangerous mind. If you're able to empower the thought, suddenly the power of that is big. They can make you sick just by looking at you. We have simple methods. It's a tremendous possibility. Namaskaram Sadhguru. They say evil eye brings bad luck and can cause harm.  So does that things exist? Yes. People can influence things. Right now, if I just take this water and look at it for a minute, we can prove today the molecular structure of the water will change. So now if I look at this, why shouldn't the molecular structure of the water inside change? It will. It can change in a positive way. It can change in a negative way. Both are possible. A focused mind is a dangerous mind. Dangerous or not is determined by the intent. If you're able to empower the thought, then the thought becomes very powerful. An angry mind, a lustful mind, a very loving mind, all these are powerful minds. The reason is not because of anger or love, simply because they've become one-pointed. If something becomes one-pointed, suddenly the power of that is big. Now if you do this, it won't hurt somebody, you know? You make it like this, it hurts somebody, isn't it? Because you become one-pointed. The same force, but it's one-pointed, suddenly it's creating a bigger impact. So one-pointedness may happen out of sheer awareness, or with sadhana, or because of anger. When you're really angry with this person, your mind is one-pointed. Just see, in love it may wander, but in anger and hatred, mind becomes one-pointed. Because of that one-pointedness, it can influence. So people are in extreme states of anger, hatred, greed. When they really focus on something, things can happen with the other person. I said it can happen, not that it will happen. It also depends on how vulnerable you are. If you are organized in such a way, external influences do not determine what happens within you, doesn't matter who looks at you, you can be free from that.  They can make you sick just by looking at you. It's very much possible. There are people who can cause death just by looking at somebody. If you can cause ecstasy in somebody just by looking at them, you can cause agony if you want, you know? Yesterday, different switch, that's all. It's like your tap. If you turn this way, it's hot water, cold water. Mind is just like that. It can be used in many positive, phenomenal ways. It's a tremendous possibility. But people choose to use it in different ways.  Every day it's happening, technology, you could use it whichever way. This is also just that. This body and this mind can be used in the same way. This is why Indian women used to wear a big and it was supposed to be made of certain material. Turmeric and lime was used so that it doesn't affect. Particularly if a woman was considered very beautiful, then they would make it much bigger because naturally people's eyes will fall on somebody who has a certain physical presence. So it should not affect her health and well-being and whatever. So they made it much bigger. So if you consider yourself beautiful, you must have it.  So don't get into all these things because then you will start thinking right now I am getting as the monkey because she looked at me. I got a headache because he looked at me. Don't get into all that. But this we have simple methods.  Don't know what's fire wash? It's called clay shana shana. That means an act which will take away the impurities that gather around you. Because your body is not just here, your body doesn't stop here. It extends itself. Have you noticed probably women are more sensitive to this than men? You don't have to be touched by somebody. If somebody comes a bit too close, not a psychological thing, that's a different matter. Psychologically and emotional aspects are there. Leaving that, if somebody comes too close, right there itself there is a feeling that you know you want some distance. They haven't touched you just like this. This itself causes some disturbance, isn't it? Because the body doesn't end here. The energy is little beyond that. How much beyond that depends on how exuberant your life energies are. If your life energies are truly exuberant, you can fill this hall with it. Otherwise, it is at least little bit. For most human beings it could be anywhere between three to nine inches outside of their body. So you gather material on your body, you wash it. Similarly you're gathering other kinds of material upon your energetic system. You can wash this water with water. Certain impurities will go with that.  Is it necessarily happening because of others people's look or not? That's a questionable matter. But impact of life around you upon your energy system is definitely happening. If you have a spiritual process within you, that is you're doing some kriya or you're doing some meditative processes, then you don't need all this because you know how to wash yourself from within.


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 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.   And that usually means that's usually the death now for the organism.  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.  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.  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.


Does fair trade really work? The Fair Trade Certification System was created to support and empower marginalized small-scale producers and workers in low-income regions of the world. Fair trade enables farmers and workers to tackle poverty and improve their own financial sustainability by creating a trade model with fair prices and working conditions. But to what degree does it work in practice? How does fair trade overcome limitations of its system for greater impact? Within the fair trade model, certified buyers and traders must pay the fair trade price on any goods they purchase from fair trade cooperatives. The fair trade price is comprised of the minimum price, or market price, whichever is higher, plus a fair trade premium. To support small-scale farmers who often already struggle with poverty, the minimum price acts as a safety net for farmers against volatile market prices that could result in losses. And the premiums are additional funds for cooperatives and workers to reinvest into their communities, like building roads or funding education. Do producers really benefit from fair trade? It's important to point out that benefits for fair trade producers will vary depending on product and region. Different studies, which focus on coffee producers, have demonstrated that fair trade system does indeed improve the lives of producers and their surrounding community. They have higher bargaining power in trade negotiations, they receive higher shares of revenue because of the minimum price model, and the fair trade premium has had a significant impact on the social welfare of the community as a whole. However, other studies have raised questions about the long-term impacts of the fair trade model. Here are four ways in which the fair trade system may be limited in its benefits for small-scale producers. Not enough fair trade sales. Producers only receive the fair trade price if fair trade certified buyers purchase their products. If farmers can't find a fair trade buyer, they will have to sell their crop under normal terms, which means no minimum price and no fair trade premium, and this can lead to lower revenue. So even if the producers abide by fair trade standards, there is no guarantee that they will always sell at fair trade prices. Some products sell better under the fair trade label, while others may not. For example, according to a review by the state of sustainable initiatives, only 6% of total fair trade produced tea was sold to fair trade buyers in 2014. On the other hand, two-thirds of fair trade produced bananas were sold to fair trade buyers in 2018. But what's worth remembering here is that the number of fair trade sales is growing. Fair trade tea sales had more than doubled after UK retailers like Sainsbury's and Coop switched all their tea brands to fair trade, and fair trade banana sales have grown by 114% in the last 10 years. So the more retailers switch to fair trade products and consumer demand increases, the more producers receive the fair trade price. Another point to remember is that while it may not seem like much to us, farmers can at least rely on a portion of their products selling at a fair trade price. A stable income that farmers under conventional trade standards may not always get to enjoy. They also benefit from other non-monetary aspects of fair trade standards, like protection of workers' rights in large plantations. Some have raised the concern that fair trade's auditing and licensing fees may detract from producers' benefits. To be clear, while cooperatives are responsible for auditing fees, licensing fees are paid by companies near the end of the fair trade chain, who want to show that their product is fair trade certified with a label. Like all certification systems, these fees are necessary for operation costs. As fair trade certification is built on ethical standards, auditing is integral to maintain the integrity of the certification. The auditing fees are necessary to ensure that fair trade producers and traders are moving towards better, fairer standards. In my interview with FloCert, the third-party auditing body for fair trade, they explain how auditing fees are determined. The fee depends on the size and setup of the cooperative or company. For example, is it a cooperative with 20 members or with 2,000? And of the number of products it wants to certify. For fair trade cooperatives or organizations that cannot afford to pay their fees due to circumstances, FloCert applies fee adjustments. For example, in the face of the global COVID-19 pandemic, FloCert had offered its customers who got into financial troubles to pay their fees in installments or defer payments. Long-term financial sustainability While fair trade focuses on the long-term financial sustainability of its producers and workers, one potential issue lies in its fundamental philosophy of an open-door system. Fair trade always welcomes producers who want to join the fair trade chain. But what happens when you have a growing number of farmers focused on producing one specific crop? You potentially end up with an oversupply, as a supply exceeds demand. This ultimately impacts the number of products that end up being sold under fair trade terms, meaning farmers may need to sell their products under conventional trade. Nicholas Lambert, who at the time of this interview was CEO of Fair Trade Belgium, and he shared the difficulty in balancing this challenge with their philosophy. What we've done last year is we've put temporary rules to limit the entry of new producer organizations in the system so that we can keep our promises and that people are not disappointed from becoming fair trade certified but end up not selling anything under fair trade terms. Because there are so many people joining the system and while the fair trade demand is growing, it's not as fast. Another aspect Lambert mentions is the impact of this growth on the limited number of fair trade staff who become stretched thin and cannot provide the usual quality guidance and support to farmer cooperatives. Lambert is still hopeful that this limited entry rule will be released in accordance with fair trade's open-door policy. Environmental challenges Growing clusters of monocultures also present an issue from an environmental perspective and this represents another problem. Long-term financial sustainability cannot be achieved without considering environmental sustainability. Fair trade international says it themselves on their website.  Fair trade is now exploring the concept of climate risk insurance for its small-scale producers and is working with small holder farms to increasingly diversify their land. However, while fair trade standards encourage sustainable agricultural practices like responsible water use and waste management, enhanced biodiversity and soil fertility, it is not mandatory. Though it's important to consider that sustainable agricultural practices like organic farming can also have higher costs. The benefit of being part of the fair trade system though is that producers can earn more and reinvest some of their higher revenue into sustainable practices. Are farm workers really protected from exploitation? Perhaps one of the biggest critiques the fair trade system faces is its unapparent impact on farm workers. It's important to distinguish here that farm workers are usually found on larger fair trade-certified farms, plantations or estates, not as much on small-scale farms as they are typically family-owned and worked by family members. Here are two ways in which the fair trade system may be limited in its benefits for workers. Limited wages for workers. One study in the Costa Rican coffee sector showed that fair trade certification successfully increased the income for coffee millers and had benefited their local community overall. But pointed out that unskilled workers, like coffee pickers and farm laborers, saw no increase on their wages. Under fair trade, all laborers are guaranteed the legal minimum wage of their region. However, there's often a gap between minimum wage and living wage. Meaning the cost of having a decent home, decent food, medical care, etc. How does paying a minimum wage then line up with fair trade's aim to empower the most vulnerable actors of the trade chain? Nicholas Lambert again shed light on the complexity of the situation. It's very delicate from an ethical standpoint because the farmer himself is already poor. With a big plantation, it's easy to go to the big boss and say, hey, you have to pay a decent wage to your workers. When you have these very small farms where maybe a few weeks a year they will hire someone to help them and who are already poor, it's difficult to go and tell them you have to pay them a wage that you yourself are not receiving. First, you have to make sure that a small farmer is getting enough pay so that he can then afford to pay his workers correctly. Otherwise, you'll chase the farmer into poverty. But it's a very difficult issue and that's something we have and are trying to regulate and check. So what has been done to improve conditions for workers? On fair trade certified plantations and estates, fair trade premiums are 100% reserved for laborers. The workers manage a joint committee that receives the premiums and there they can collectively agree to use part of the money as supplemental wages, additional to their minimum wage. However, other times as Lambert shared, they have decided to use premiums on buying coats, distributing milk to families, or paying for tuition fees for their kids. The most important aspect of the fair trade system is that it leaves the decision solely up to the workers, allowing them to invest these additional funds in what they feel best fits their needs. Fair trade's auditing system has also come under fire as child labor was allegedly found on a fair trade cocoa plantation in the Ivory Coast by an investigative journalist team from Danwatch in 2020. It's unclear whether the lack of auditing is due to travel restrictions under COVID-19 or systematic flaws. But whenever you hear one case, it makes you wonder how many more cases like this exist, especially considering many other non-fair trade cocoa farmers in West Africa have been discovered to have a widespread use of child labor. How does fair trade truly ensure that its producers are not using illegal forms of labor? Fair trade audits are usually announced in advance, as auditing often takes time to prepare. For example, cooperatives prepare paperwork for review and they also have to arrange time with farmers so the auditors can visit their fields or organize meetings for interviews. But by announcing their audit schedule, how can we be sure that there are no opportunities for cover-ups? Fair trade has criteria to identify high-risk situations and in areas that have shown to have a higher risk of illegal labor, certifiers do conduct unannounced audits. The general spirit of auditing is one of trust, explains Lambert. Sometimes the western point of view is to say, we are going to check these bad guys, because they are the bad guys. But they're not necessarily the bad guys, they're just people like you and me. That's why Lambert stressed that auditing is only a part of the certification process, that it's only a means to an end. Auditing can help improve situations and prevent milk practice from happening, but it doesn't solve the root issue. Lambert further elaborated, take the painful example of child labor. You can say child labor is forbidden, that's it, that's the rule and I'm going to check it. But at the same time, if you don't do anything to make sure that these people are paid better prices, you're not going to tackle the root cause of child labor. Because these people don't want their children to work on farms. They would rather send them to school, but they often can't afford it. Sometimes farmers feel compelled to have their children work on their farms, because they can't afford to hire someone for a few weeks to help them harvest their crops. That's why it doesn't work to just create rules and check that they are followed. Lambert concluded, it helps to have rules because at least things are clear. And it creates a little bit of pressure. But if you're not working on the cause of the issue, then you're not going to solve anything. Fairtrade's approach is one of balance between strictness and encouraging development. Of course, if major certification requirements have been breached, then certifiers will suspend or decertify a producer organization or trader. But Fairtrade's overall focus is to give their producers the opportunity to address problems and improve over time. But what do Fairtrade producers and workers think? It's clear that Fairtrade is not a perfect system, but it's a good start. In 2021, Fairtrade counted 1.7 million farmers and workers in 72 different countries and territories, as well as 35,000 certified products. It might seem like a lot, but Fairtrade producers actually account for less than 1% of the 570 million smallholder farms worldwide. This means that more than 99% of smallholder farms do not work under a trade model that can guarantee fair prices or the respect of basic working rights. However, the number of Fairtrade producers is enough to put criticisms in perspective. While the studies we referred to were conducted to assess the real impact of Fairtrade's model, they're also inevitably limited in their scope and voice. While they study specific product chains and specific cooperatives, they cannot represent the voice and felt impact of all the people in the Fairtrade system. That's why it's important to hear directly from farmers and workers when asking, Did Fairtrade really work for you? Thank you for listening to this episode of our Food Unfolded audio articles. Has listening to this article given you a new perspective on Fairtrade? Let us know on social media linked in the description box. This article was written and read by me, Jane Alice Lu, originally posted on foodunfolded.com. Food Unfolded explores the stories behind the food on our plate, reconnecting us to the origins and sustainability of our food. Co-funded by the EU and powered by EIT Food.


Hello and welcome to the session. In this session we discussed the following question which says find the quadratic polynomial whose 0's are 1 upon 2 and 1 upon 3. We know that a quadratic polynomial whose 0's are alpha and beta is given by px is equal to x square minus alpha plus beta into x plus alpha beta. This is the key idea to be used in this question. Let's move on to the solution now. Now we are given that the 0's of the quadratic polynomial 1 upon 2 and 1 upon 3, we take let alpha be equal to 1 upon 2 and beta be equal to 1 upon 3. That is we have alpha and beta are the 0's of the quadratic polynomial. Then alpha plus beta is equal to 1 upon 2 plus 1 upon 3 and this is equal to, taking LCM we get 6 in the denominator and in the numerator we have 3 plus 2. So we get this is equal to 5 upon 6 that is alpha plus beta is equal to 5 upon 6. Now the product of the 0's that is alpha beta is equal to 1 upon 2 into 1 upon 3. Therefore we get alpha beta is equal to 1 upon 6. Now the required polynomial say px is given by x square minus sum of the 0's of the polynomial into x plus the product of the 0's. So px would be equal to x square minus 5 upon 6 into x plus the product that is 1 upon 6. So this is equal to x square minus 5 upon 6 into x plus 1 upon 6. This is our required polynomial px so this means we get px is equal to 6x square minus 5x plus 1. So this is the required polynomial that is we say that px equal to 6x square minus 5x plus 1 is the quadratic polynomial with 0's 1 upon 2 and 1 upon 3. So this completes the session hope you have understood the solution for this question.