Open Access
April 2014 Quasi-stationary distributions for randomly perturbed dynamical systems
Mathieu Faure, Sebastian J. Schreiber
Ann. Appl. Probab. 24(2): 553-598 (April 2014). DOI: 10.1214/13-AAP923

Abstract

We analyze quasi-stationary distributions {με}ε>0 of a family of Markov chains {Xε}ε>0 that are random perturbations of a bounded, continuous map F:MM, where M is a closed subset of Rk. Consistent with many models in biology, these Markov chains have a closed absorbing set M0M such that F(M0)=M0 and F(MM0)=MM0. 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 MM0), 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.

Citation

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Mathieu Faure. Sebastian J. Schreiber. "Quasi-stationary distributions for randomly perturbed dynamical systems." Ann. Appl. Probab. 24 (2) 553 - 598, April 2014. https://doi.org/10.1214/13-AAP923

Information

Published: April 2014
First available in Project Euclid: 10 March 2014

zbMATH: 1334.60137
MathSciNet: MR3178491
Digital Object Identifier: 10.1214/13-AAP923

Subjects:
Primary: 34F05 , 60J10
Secondary: 60F10 , 60J80 , 92D25

Keywords: large deviations , nonlinear branching process , Quasi-stationary distributions , Random perturbations

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.24 • No. 2 • April 2014
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