Abstract

Consider the following model for the evolution of a population undergoing natural selection. At each generation, individuals give a fixed or random number of offsprings, whose fitness is perturbed by some independent additive noise. Selection maintains the population size fixed by selecting only the fittest individuals. Physicists have recently made some very precise predictions for the genealogy of this population in the limit of inifinite population size : namely, the characteristic timescale is (log N)^3 generations, and when measured in these units, genealogical trees are described by the so-called Bolthausen-Sznitman coalescent. These predictions build on a non-rigorous analysis of stochastic effects in front propagation of KPP equations. We report on some recent progress on this issue (joint work with J. Berestycki and J. Schweinsberg), where this conjecture is shown to hold for a slightly simpler model of branching Brownian motions with a suitable cutoff which keeps the population size approximately fixed.