Quasi-stationary distributions for randomly perturbed dynamical systems

We analyze quasi-stationary distributions {..}.>0 of a family of Markov chains {X.}.>0 that are random perturbations of a bounded, continuous map F:M.M, where M is a closed subset of Rk. Consistent with many models in biology, these Markov chains have a closed absorbing set M0.M such that F(M0)=M0 and F(M.M0)=M.M0. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for F (i.e., an attractor for F in M.M0), then the weak* limit points of .. are supported by the positive attractors of F. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.