We study two equations of Lotka-Volterra type that describe the Darwinian
evolution of a population density. In the first model a Laplace term represents
the mutations. In the second one we model the mutations by an integral kernel.
In both cases, we use a nonlinear birth-death term that corresponds to the
competition between the traits leading to selection. In the limit of rare or
small mutations, we prove that the solution converges to a sum of moving Dirac
masses. This limit is described by a constrained Hamilton-Jacobi equation. This
was already proved by B. Perthame and G. Barles for the case with a Laplace
term. Here we generalize the assumptions on the initial data and prove the same
result for the integro-differential equation
