Genealogical transition in the noisy $N$-Branching Random Walk. How stronger selection may promote genetic diversity

Abstract

We consider an extension of the noisy $N$-Branching Random Walk that models the evolution of a population subject to natural selection. We show the existence of a critical value for the noise which separates the limiting genealogical structure into two regimes, which we respectively call the semi-pulled and the fully-pulled regimes. In the fully-pulled regime, the genealogy converges to a discrete time Poisson-Dirichlet coalescent. In the semi-pulled regime, the genealogy converges to the Bolthausen-Sznitman coalescent. We discuss some interesting biological consequences of this result. In particular, our model predicts a non-monotone relation between the selection strength and the effective population size.