In this talk, I will present a derivation of a Hamilton-Jacobi equation with obstacle from a discrete integro-differential model in population dynamics. We consider a population parameterized by a scaling parameter $K$ and composed of individuals characterized by a quantitative trait, subject to selection and mutation. The phenotypic density then solves a discrete integro-differntial model. In the regime of large population $K \to +\infty$, small mutations and large time we prove that the WKB transformation of the density converges to the unique viscosity solution of a Hamilton-Jacobi equation with obstacle.
