Stochastic Persistence

Abstract

Let $(X_t{)}_{t\ge 0}$ be a continuous time Markov process on some metric space $M,$ leaving invariant a closed subset $M_0\subset M,$ called the {\em extinction set}. We give general conditions ensuring either "Stochastic persistence" (Part I) : Limit points of the occupation measure are invariant probabilities over $M_{+}=M\setminus M_0;$ or "Extinction" (Part II) : $X_t\to M_0$ a.s. In the persistence case we also discuss conditions ensuring the a.s convergence (respectively exponential convergence in total variation) of the occupation measure (respectively the distribution) of $(X_t)$ toward a unique probability on $M_{+}.$ These results extend and generalize previous results obtained for various stochastic models in population dynamics, given by stochastic differential equations, random differential equations, or pure jump processes.

References in corpus (4)

• Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching
• Quantitative ergodicity for some switched dynamical systems
• Smooth invariant densities for random switching on the torus
• Random Switching between Vector Fields Having a Common Zero

Cited by in corpus (3)

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• Quasi-stationary distributions of multi-dimensional diffusion processes
• Stochastic Lotka-Volterra Competitive Reaction-Diffusion Systems Perturbed by Space-Time White Noise: Modeling and Analysis