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We consider a stochastic model describing a constant size $N$ population thatmay be seen as a directed polymer in random medium with $N$ sites in thetransverse direction. The population dynamics is governed by a noisy travelingwave equation describing the evolution of the individual fitnesses. We showthat under suitable conditions the generations are independent and the model ischaracterized 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 andprobability outcomes $\eta_i$'s. We then show that the average coalescencetimes scales like $\log N$ and that the limit genealogical trees are governedby the Bolthausen-Sznitman coalescent, which validates the predictions byBrunet, Derrida, Mueller and Munier for this class of models. We also study theextended Wright-Fisher model, and show that, under certain conditions on$\eta_i$, the limit may be Kingman's coalescent, a coalescent with multiplecollisions, or a coalescent with simultaneous multiple collisions.

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