A PIECEWISE DETERMINISTIC MODEL FOR A PREY-PREDATOR COMMUNITY

• 成果类型: Article
• 署名作者: Costa, Manon
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Universite de Toulouse; Institut National des Sciences Appliquees de Toulouse; Universite Toulouse III - Paul Sabatier
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1182
• 发表日期: 2016
• 页码: 3491-3530
• 关键词: individual stochastic-processes; Markovian processes; STABILITY; discrete; CONVERGENCE; ergodicity; population; diffusion; criteria; limit

摘要

We are interested in prey-predator communities where the predator population evolves much faster than the prey's (e.g., insect-tree communities). We introduce a piecewise deterministic model for these prey-predator communities that arises as a limit of a microscopic model when the number of predators goes to infinity. We prove that the process has a unique invariant probability measure and that it is exponentially ergodic. Further on, we rescale the predator dynamics in order to model predators of smaller size. This slow-fast system converges to a community process in which the prey dynamics is averaged on the predator equilibria. This averaged process admits an invariant probability measure which can be computed explicitly. We use numerical simulations to study the convergence of the invariant probability measures of the resealed processes.