Asymptotic stability of equilibria of selection-mutation equations.

We study local stability of equilibria of selection-mutation equations when mutations are either very small in size or occur with very low probability. The main mathematical tools are the linearized stability principle and the fact that, when the environment (the nonlinearity) is finite dimensional, the linearized operator at the steady state turns out to be a degenerate perturbation of a known operator with spectral bound equal to 0. An example is considered where the results on stability are applied.