Persistence in the Moran Model with Random Switching

The paper is devoted to the study of the asymptotic behaviour of Moran process in random environment, say random selection. In nite population, the Moran process may be degenerate in nite time, thus we will study its limiting process in large population which is a Piecewise Deterministic Markov Process, when the random selection is a Markov jump process. We will then study its long time behaviour via the stochastic persistence theory of Benaïm [9]. It will enable us to show that persistence can occur, i.e. asymptotic coexistence of all species, when there are enough switching possibilities. This is true even if one species has never a predominant selection