We provide an analysis of some concentration phenomena which appear in problems related to biology. In particular, we study the adaptive dynamics of a population structured by a phenotypical trait. The mathematical modeling of these problems gives rise to parabolic equations with small diffusion. The presence of a small term leads to multi-scale models. The asymptotic solutions of these equations concentrate on one or several points in the trait space that are evolving in time. Our approach is based on the Hopf-Cole transformation which leads to Hamilton-Jacobi equations with constraint. We also show how this approach can be extended to models with a spatial structure.