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

Macroscopic limit from a structured population model to the Kirkpatrick-Barton model