Hi. In this lecture we're again going to talk about replicator dynamics. And Remember replicator dynamics have this idea in them that the proportion of people playing a particular action at time T plus one depends on the proportion paid at a time T And then the payoff for that action at time T. Now, in the previous lecture, we talked about replicator dynamics in the context of people playing actions or strategies, so populations of individuals. In this lecture, we're gonna talk about it in an ecological context. So we're thinking about, we're thinking about different phenotypes of a species, and those phenotypes having different fitness?s. And think about replicator dynamics as a way to capture the dynamics of that population, that, of a species. Let me explain what I mean. Remember, in replicator dynamics, there's a set of types. But now, instead of assuming a payoff to each type, I'm gonna assume there's a fitness to each type. That's sort of how fit the species is. That, remember the species is two if particular [inaudible]. A logical match and I'm also going to assume there's some proportion of each type. Well, now what we can do is we can think of the exact same logic. How many of each type are gonna get reproduced in the next population? Well, it's gonna depend on the fitness of each type, and the proportion. Because the more birds there are of a particular phenotype, the more offspring they're gonna have. But it's also due to the more fit a particular type of a species is, the more offspring it's gonna have. So the fitness and the proportion are gonna determine how many there'll be in the next population. Now how I want to think of this is a fitness wheel. So you can think of is, when you're choosing a mate, that there's this, this giant wheel. And you sort of spin this wheel of fortune. And so there's a diff, bunch of different types. There's type 1s, type 2s and type 3s. And you spin the wheel and it stops on type two. Now the property [inaudible] on the type 2's depend on two things, the number of 2's. So there's only two of them And the thickness of 2's. The reason we call this the fitness wheel and not just a wheel is the size of the pie here you can think of as being proportional to the fitness. So the more fit you are the bigger your slices. So 2's are fit so they get really big slices. 1's are not very fit so they get small slices. But then it's also the case, the more of who they are, the more slices you get. So there's lots of ones, there's four ones, so they get more slices. Well this fitness wheel [inaudible] metaphor is the same thing as replicator dynamics. You can think of the size of the slice. As being proportional to the fitness, is you're gonna get a number of slices, representing the number of species of that type, and this will give you exactly replicator dynamics. And what you can think of, and sometimes it's bunching those all together, so putting all the 1's in one big slice, all the 2's in one big slice, and all the 3's in one big slice, and then spin the wheel that way. And that's another way to think of these replicator dynamics. Now, I'm going to use replicator dynamics and maybe of the fitness field implicitly, to explain something called Fisher's fundamental theorem. Fisher's theorem is going to be really cool, because it is going to allow us to combine a bunch of models that we have already used. So remember we had the model that said there is no cardinal that meant there is a lot of variation within a species. Second, we had that model of rugged landscapes, the idea being that like, when you encode a function, you could think of it as a rugged landscape, that you are trying to climb hills. And then third, we've got these model of replicator dynamics. What Fisher's fundamental theorem is going to do, is it's going to combine all. All of these models into one, and give us an insight about the role that variation plays in adaptation. Okay, so hang on, a lot going on here. So remember our, there is no cardinal. That meant that there's a population of things that we call cardinal. Ther e's genetic and phenotypic variation in the population of cardinals. And remember we also had the rugged landscape model, saying that if you think of a cardinal, it could be, have a fitness, which is sort of, maybe, somewhere here. This one here will have a fitness of this height. One down here is a lower fitness. This has a fairly high fitness, and this has the highest fitness of all. So we can place different cardinals on the landscape, and different cardinals are gonna have different fitness?s. So we could think of, then. Replicator dynamics is saying, what's gonna happen? You're gonna copy the [inaudible] fit, and you're also gonna copy the people who exist in higher proportion. So we can use, we can place all of those diverse cardinals on one landscape. And then we can imagine that replic-, that replicator dynamics are gonna help us choose the ones that are sort of higher up on the landscape. So here's Fisher's theorem, the idea anyway, that higher variances, If you have more variation, then you should be able to adapt faster. You should be able to climb the landscape faster. Let's see why that's probably true. So suppose there's low variation. There's very low variation. And now I apply some selective pressure. I can only climb up a little bit. But if there's high variation, then I can climb a lot faster. So the fast, the more variation, the more people I've got to copy, the more likely there is to be someone good, the better I'm gonna do. So let's do an example and see why this is the case. So let's start with the population that has one third of people at fitness three, one third of the fitness four and one third of the fitness five. So note that the average fitness here is just gonna equal four. Well let's look at the weights, let's use [inaudible] dynamics and let's figure out the weights for each of these different strategies. The weight on strategy one Is going to be one third times three, which is one. The wait on strategy two is one third times four, which is four thirds. And the wait on strategy t hree is one third times five, Which is five thirds. Well now, let's compute the proportion we are gonna have in the next period of each type is of type one. Proportion of type one is just gonna equal one over one plus four thirds plues five thirds, which is gonna be. Three over twelve, to proportion of type two, is gonna be four thirds over one, plus four thirds, plus five thirds which is gonna be four over twelve. And a proportion of type three, is just gonna be five thirds over one, plus four thirds, plus five thirds, which is five over twelve. So we're gonna have 3/12's fitness three, 4/12's fitness four and 5/12's fitness five. Now if we figure out what's are new average fitness gonna be, that's gonna be three Times three 12's, plus four, times four 12's, plus five, Times five 12's. So what we're going to get is nine plus sixteen which is 25, [inaudible] 50 over twelve. >> Which if we divide here is gonna be four and a sixth. So what we get is we started on an average fitness of four, we end up with an average fitness of a sixth. Let's now do a case where we've got medium variants, so before the paths were three four, and five, now the variances, the fitness?s are two, four, and six. Let's do the same thing. So what's the weight on strategy one, that's gonna be one third times two, Which is 2/3's. The weight on strategy two is going to be one-third times four which is 4/3's and the weight on strategy three is going to be one-third times six, Which is six thirds. And again, the average fitness here, as before, was equal to four, So now if we want to complete the probability that someone?s going to be of type one in the next period. That's just going two-thirds over two-thirds + 4/3 + 6/3, so that's going to be two over twelve. The probability of someone?s, of type two Is gonna be, and notice we can get rid of all the thirds here. So that's just gonna be four over two plus four plus six, so that's 4/12. And the probability [inaudible] of type three is gonna be just six over two plus four plus six, whic h is 6/12. So now, if we wanna [inaudible] the new average fitness in this new population, 'cause before, it went from one-third, one-third, one-third, to 2/12, 4/12, 6/12, We're gonna get that it's two. X(2x12)+ 4x(4x12) + 6x(6x12). So that's gonna be four+16 which is twenty+36 which is 56/12. All right, And so that's gonna to be Four, and 4/6ths, So before we had four and six. Now we're going to get four and 4/6ths. Last let's do a population where we have really high variance. So 1/3rd of a population of zero, 1/3rd of a fitness of four, and 1/3rd of a fitness of eight. Again let's do all the math. Here's what we get. For Wayern strategy one, this will be easy. It's going to be zero, because the fitness is zero. Latent strategy two, is gonna be four thirds. And latent strategy three is gonna be eight thirds. So the probability of someone's strategy one next time is gonna be zero. The probability of someone's strategy two is just gonna be four thirds over four thirds plus eight thirds. So that's a third, which means a probability of someone's a strategy three is gonna be two thirds. So when we compute the new average fitness, number will be four there. The average fitness was four. The new average fitness is gonna be one third Times four plus two-thirds times eight. So that's 4/3 plus sixteen, which is 20/3, right? So what we get is we get the average fitness is then gonna be six and two-thirds. So in the first case, what we get is we got a gain. We need to finish this where at three, four, and five, fitness increased by one-sixth. In the second case, remember we had a little bit more variation, the fitness increased by 4/6. And in the third case when there had been greater variation, the fitness increased by two and 4/6. Remember because the average came up from four to six and two-thirds. So what we see, we see the, the amount of gain seems to be increasing in the variation. So the more variation, the faster the population can adapt. Well, let's compute the variation in each of these populations. Remember va riation is just the difference from the mean. So in the first case, the variation will be three minus four, squared, plus four minus four squared, plus five minus four squared, so that's just gonna be two. And now just to the last, in the last case you're gonna get zero minus four squared which is sixteen, plus four minus four squared which is zero. Plus eight minus four squared, which is also sixteen, we're gonna get 32. So this gives us the variation within each population. So what we had before was the gain, and if we put this in terms of six, the gains are one sixth, four sixths, and sixteen sixths. And if we look at the variation, it's two, eight, and 32. We'll notice, this is gis one goes to two, this is gis times two, this is gis times two, and this is gis times two. So the gain is exactly half the variation, in each case. In effect, this is Fisher's fundamental theorem. The change in average fitness due to selection, if we have replicator dynamics, is gonna proportional to the variation. So more variation, More adaptation and they're proportional, And we saw this by combining three models. There is no cardinal, there's a rugged landscape in replicator dynamics. And we get this really interesting result. That the change in average fitness due to selection, due to replicator dynamics, is gonna be proportional to the variation. And again, we got it by combining three different models. So this one of the powers of being a many model thinker, is you can then combine them to ask much more deeper scientific questions. But there's a rub here, and this is what we're gonna come to in the next lecture. We just got this idea that says more variation is better. But this one's counter to something we learned very early on in the course, which is that you wanna reduce variation because of six sigma. So what I wanna do in the last lecture mission is contrast these two models. Because when we think about becoming model thinkers, what you'd like to do is have lots of models in your head, and use those to adjudicate differ ent intuitions. To figure out which logic applies in which situation. So that's where we'll go next. Alright, thank you.
