Séminaire du LPTMS : Alia Abbara (EPFL)

In this talk, we first give a broad introduction to the field of population genetics, that was introduced by probabilists (J.B.S. Haldane, S. Wright, R. Fisher) to model genetic variations between individuals. We focus on the problem of a single mutant individual that has a fitness advantage with respect to all of the other identical individuals, that we call the wild-types. Our goal is to characterize the spread of the mutant type within evolutionary trajectories. We briefly present a few fundamental models (Moran and Wright-Fisher) that rely on Markovian updates of the state of the population and their analytical description within a diffusion approximation. We then present the problem of structured populations, and introduce a new graph-based model, where subpopulations are placed on the nodes of a given graph, and stochastic migrations can happen along the edges. We analyze this model with a branching process approximation, and notably obtain the fixation probabilities of a single mutant, i.e. how likely it is to take over the whole graph.