Reaction-diffusion problems : from front dynamics to genealogies (5)

The fifth lecture began by describing some models of evolution with and without selection. In the neutral case, as in the Wright Fisher model, the stochastic evolution of the proportion of a population carrying an allele is given by the Kimura equation. In the presence of selection, the simplest model wherefitness is independent of time, and in the absence of mutations, it can be shown that the average fecundity of the population increases with time, as predicted by Fisher's natural selection theorem. If we take mutations into account, we arrive at the notion of Eigen's quasi-species around points of locally maximum fecundity, with rare transitions from a maximum to a higher maximum due to mutations. The dynamics of these evolutionary models are very reminiscent of the dynamics of glasses, which get stuck in local free-energy minima before rare fluctuations allow them to find deeper minima. The lecture went on to introduce models of evolution with selection, such as the problem of the propagation of a gene in a population, or N-BBM( N branching Brownian motion ). These models lead to noisy Fisher-KPP-type equations, a subject that has motivated a great deal of work in recent years. After introducing the idea of cut-off to explain the speed of these fronts in the presence of noise, the lecture ended by discussing a simpler selection model than N-BBM, with branching Brownian motion where selection is due to an absorbing wall advancing at a constant speed.