Quasi-stationary Distributions for Randomly Perturbed Dynamical Systems by Mathieu Faure

We analyze quasi-stationary distributions {μ}ε>0 of a family of Markov chains {X}ε>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 με 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.