  We are interested in the dynamics of a population structured by a phenotypic
trait. Individuals reproduce sexually, which is represented by a non-linear
integral operator. This operator is combined to a multiplicative operator
representing selection. When the strength of selection is small, we show that
the dynamics of the population is governed by a simple macroscopic differential
equation, and that solutions converge exponentially to steady-states that are
locally unique. The analysis is based on Wasserstein distance inequalities
using a uniform lower bound on distributions. These inequalities are coupled to
tail estimates to show the stability of the steady-states.
