Random Perturbations of Dynamical Systems with Absorbing States

Let F : M → M be a continuous dissipative map of a separable metric space M . Consider a finite collection A of closed F -forward invariant sets that is closed under intersection and that contains M . For all ! > 0, let X be a Markov chain for which the elements of A are absorbing (e.g., extinction boundaries for a population, genotype, or strategy) and such that d(X t+1, F (X ! t )) ≤ ! for all t. Under an additional nondegeneracy condition (i.e., the noise extends locally in all nonabsorbing directions) and a continuity-like condition on the supports of the random perturbations, we show that for sufficiently small values of !, X asymptotically spends all of its time near certain invariant sets of F , so-called absorption preserving chain attractors. Moreover, the weak* limit points of X’s stationary distributions as ! → 0 are F -invariant probability measures whose supports lie in the absorption preserving chain attractors. Applications to the dynamics of structured and unstructured populations, multispecies interactions, and evolutionary games are given.