Stochastic Persistence.

By Michel Benaim. 2018

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Let $(X_t)_{t \geq 0}$ 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) : $X_t \rightarrow M_0$ 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.

Subfields: Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/d1acd2dcee624aef50cdcd7c680f2ec5f4027890