QUASI-STATIONARY DISTRIBUTIONS FOR RANDOMLY PERTURBED DYNAMICAL SYSTEMS

• 成果类型: Article
• 署名作者: Faure, Mathieu; Schreiber, Sebastian J.
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Aix-Marseille Universite; Aix-Marseille Universite; University of California System; University of California Davis
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP923
• 发表日期: 2014
• 页码: 553-598
• 关键词: small random perturbations; markov-chains; persistence; EXISTENCE; game; permanence; dispersal; EVOLUTION; coherence

摘要

We analyze quasi-stationary distributions {mu(epsilon)}(epsilon>0) of a family of Markov chains {X-epsilon}(epsilon>0) that are random perturbations of a bounded, continuous map F : M -> M, where M is a closed subset of R-k. Consistent with many models in biology, these Markov chains have a closed absorbing set M-0 subset of M such that F(M-0) = M-0 and F (M \ M-0) = M \ M-0. 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 \ M-0, then the weak* limit points of mu(epsilon) 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.