Abstract
In this talk, we will study the dynamics of a population structured by a phenotypic trait. We will assume that individuals reproduce sexually, which can be represented by a nonlinear and non-local operator, analogous to an inelastic collision operator in statistical mechanics. This operator is combined with a multiplicative operator representing the effect of natural selection. When the strength of selection is weak (a classical assumption in population genetics), we show that population dynamics can be described by a differential equation. It is possible to develop this idea to show that in long time, the solution converges exponentially to a unique stationary state. This result is obtained by combining Wasserstein-type estimates from the reproduction operator with estimates of the macroscopic quantities described by the differential equation. Finally, we discuss the addition of a spatial variable to this model.
