On persistence and extinction for randomly perturbed dynamical systems

Let $f:\M\to\M$ be a continuous map of a locally compact metric space. Models of interacting populations often have a closed invariant set $\partial \M$ that corresponds to the loss or extinction of one or more populations. The dynamics of $f$ subject to bounded random perturbations for which $\partial \M$ is absorbing are studied. When these random perturbations are sufficiently small, almost sure absorbtion (i.e. extinction) for all initial conditions is shown to occur if and only if $M\setminus \partial M$ contains no attractors for $f$. Applications to evolutionary bimatrix games and uniform persistence are given. In particular, it shown that random perturbations of evolutionary bimatrix game dynamics result in almost sure extinction of one or more strategies.