  We consider model arising from evolutionary biology, where the population is
structured by two variable: a spatial variable, and a phenotypic trait. The
model combines a parabolic operator on the space variable to a kinetic operator
on the trait variable (this operator, that represents the effect of sexual
reproduction, is similar to the Boltzmann inelastic operator). Thanks to the
Tanaka inequality, the operator on the trait variable implies a contraction for
the Wasserstein distance. We combine this contraction argument to parabolic
estimates controling the spatial regularity of solutions to derive a
macroscopic limit of this kinetic equation. More precisely when a parameter
$\gamma>0$, representing the rate of reproduction, is large, the moments of
solutions of the kinetic model converge to the solutions of the
Kirkpatrick-Barton model, a widely used model in ecology and evolutionary
biology. Note that in this study, we consider only the case where the spatial
variable is in a compact set, namely the $d-$dimentional torus.
