The genealogy of a solvable population model under selection with dynamics related to directed polymers

Abstract

We consider a stochastic model describing a constant size $N$ population that may be seen as a directed polymer in random medium with $N$ sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable conditions the generations are independent and the model is characterized by an extended Wright–Fisher model, in which the individual $i$ has a random fitness $\eta_{i}$ and the joint distribution of offspring $(\nu_{1},\ldots,\nu_{N})$ is given by a multinomial law with $N$ trials and probability outcomes $\eta_{i}$’s. We then show that the average coalescence times scales like $\log N$ and that the limit genealogical trees are governed by the Bolthausen–Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright–Fisher model, and show that, under certain conditions on $\eta_{i}$, the limit may be Kingman’s coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.