A piecewise deterministic model for a prey-predator community

TL;DR

This paper introduces a piecewise deterministic model for prey-predator communities with fast predator dynamics, proving ergodicity and deriving an explicit invariant measure, with numerical simulations confirming convergence properties.

Contribution

It develops a novel piecewise deterministic framework for prey-predator systems with fast predator evolution and provides explicit invariant measures and convergence analysis.

Findings

Unique invariant probability measure established

Exponential ergodicity proven for the process

Numerical simulations confirm convergence of measures

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.