A piecewise deterministic model for a prey-predator community

Abstract

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 rescaled processes.

Theorem 2.2 is wrong stated as it is, a corrigendum is available (link provided in the article)

References in corpus (5)

• A microscopic probabilistic description of a locally regulated population and macroscopic approximations
• Evolution of discrete populations and the canonical diffusion of adaptive dynamics
• The emergence of the deterministic Hodgkin--Huxley equations as a limit from the underlying stochastic ion-channel mechanism
• Quantitative speeds of convergence for exposure to food contaminants
• A multi-scale study of a class of hybrid predator-prey models

Cited by in corpus (4)

• Probabilistic and Piecewise Deterministic models in Biology
• Dynamical systems with fast switching and slow diffusion: Hyperbolic equilibria and stable limit cycles
• Analytic solutions for stochastic hybrid models of gene regulatory networks
• Exponential ergodicity of a degenerate age-size piecewise deterministic process