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Quasi-stationary distributions for randomly perturbed dynamical systems

Abstract : We analyze quasi-stationary distributions \μ ε \ ε\textgreater0 of a family of Markov chains \X ε \ ε\textgreater0 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 ⊂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 μ ε 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.
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Submitted on : Wednesday, February 22, 2017 - 4:16:28 PM
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Mathieu Faure, Sebastian Schreiber. Quasi-stationary distributions for randomly perturbed dynamical systems. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2014, 24 (2), pp.553--598. ⟨10.1214/13-AAP923⟩. ⟨hal-01474257⟩



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