Stochastic eco-evolutionary model of a prey-predator community

TL;DR

This paper develops a stochastic eco-evolutionary model for prey-predator communities, analyzing their phenotypic coevolution through differential equations and jump processes, and illustrating the dynamics with simulations.

Contribution

It introduces a comprehensive individual-based model capturing prey-predator coevolution, extending existing frameworks with new analytical results and simulation insights.

Findings

Existence of a unique globally stable equilibrium under certain conditions.

Extension of the Polymorphic Evolutionary Sequence to a coevolutionary context.

Simulation results showing the impact of prey defense mechanisms on community dynamics.

Abstract

We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the Polymorphic Evolutionary Sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these different limits with an example of prey-predator community that takes into account different prey defense mechanisms. We observe through simulations how these various prey strategies impact the community.