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We are interested in the dynamics of a population structured by a phenotypictrait. Individuals reproduce sexually, which is represented by a non-linearintegral operator. This operator is combined to a multiplicative operatorrepresenting selection. When the strength of selection is small, we show thatthe dynamics of the population is governed by a simple macroscopic differentialequation, and that solutions converge exponentially to steady-states that arelocally unique. The analysis is based on Wasserstein distance inequalitiesusing a uniform lower bound on distributions. These inequalities are coupled totail estimates to show the stability of the steady-states.

Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection