Okay, so welcome everyone. Today we have Dr. Gaël Raoul from the Centre Mathematica de

l'École Polytechnique de Paris and he will be speaking about the dynamics of kinetic

models from evolutionary biology. Please, Gaël.

Okay, great. Thank you very much. Thank you. So I would like to speak, as said, about some

work we have been doing on evolutionary models. So my presentation will be in four parts.

First, I will introduce the biological concepts and the model. Then heuristic analysis. This

is typically what the biologists do to consider the kind of biology questions we have in mind.

And then I will proceed to mathematical analysis of this approximation that was proposed by the

biologists and finish by the convergence of a certain solution of a certain model to a state.

And this will be like, say, the core of the study. So I first define what is a phenotypic

trait. So we consider a population of E. coli bacteria and we consider clones. So they are all

the same and we grow them in different temperatures. Okay, what we see is that this temperature

affects the way they grow. We have a maximal growth rate

that is rich for a certain temperature T0. And we say that T0 is the temperature to which

the bacteria is best adapted to. And we choose as a phenotypic trait this temperature

for the genotype of that bacteria, the temperature to which the bacteria is best adapted to.

Okay, so this is what we call a phenotypic trait. There are lots of other examples,

but this is an example and we see that if we modify the temperature of the environment of

a bacteria, we can wonder, is the bacteria going to evolve to match so that its phenotype

matches the temperature you grow the bacteria in. So this kind of idea is what is called

as evolutionary rescue. Evolutionary rescue is you consider here a certain fitness landscape,

so a certain environmental conditions. So fitness landscape.

Where we have a growth rate, right? This is a growth rate

as a function of the phenotype, the phenotypic trait.

Okay, so this is a fitness landscape for the times that are negatives. And when T reaches

zero, we change suddenly the environment. So this is a fitness landscape.

For T positive. So initially the population is adapted to this landscape. So typically

their phenotypes will be something like that, right? Centered around the maximum of this one.

And when we change the environment, well, this population will collapse.

And the phenotype will collapse. So this is a fitness landscape.

And when we change the environment, well, this population will collapse. And the idea is that

maybe you will have a few mutants that will be there or that will be created through mutation

that will succeed to survive. And that will start growing. Okay, so this is a dynamics that we would

have in this situation. And that's what we observe in this experiment. Initially, we have a

population that is ill-adapted to the environment it's introduced in. So there is some initial phase

that is not really clear, but at some point there is a collapse of the population, right?

And we see that the population succeeds to develop new mutants, new strains. And these new mutants

actually succeed in growing in this new medium. And the population reaches again a size that is

of order one, let's say. So this is something that can be done in a lab. You see that it takes

300 hours, so it's not too long. And we can repeat this experiment, see if it's repeatable or not,

and follow the mutation. So we see here the mutations that play a role in this adaptation

to the new environment. Now what is going to happen in a situation where we have two

optima like that, right? Where we have a first optima here, Z bar one, and the second one Z bar two.

Well, let's start with an initial population

that is centered, let's say, in this trait here. Well, so this is a t equals zero.

At a larger time, what's going to happen? Well, it will tend to evolve towards this optimum here.

You have an evolution like that, and the population will tend to

a concentration around this local optimum. It will stay there for a while because it's

a local optimum. The small mutation is the instrumental, but after a while, you will have

some mutants that will reach this higher peak here. All right, so this will take longer,