Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation

In this paper we derive a constrained Hamilton-Jacobi equation with obstacle from a discrete non-linear integro-differential model of population dynamics, with exponentially decaying mutation kernel. The exponential decay of the kernel leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models in \cite{BMP}. We consider a population composed of individuals characterized by a quantitative trait, subject to selection, mutation and competition. In a regime of small mutations, small spatial discretization step and large time we prove that the WKB transformation of the density converges to a viscosity solution of a constrained Hamilton-Jacobi equation with obstacle.