About reaction-diffusion systems involving the Holling-type II and the Beddington-DeAngelis functional responses for predator-prey models

TL;DR

This paper derives a reaction-cross diffusion predator-prey model with Holling-type II and Beddington-DeAngelis responses from a microscopic reaction-diffusion system, analyzing its stability and Turing patterns.

Contribution

It introduces a new derivation of predator-prey models with complex functional responses from microscopic dynamics, and compares their stability properties.

Findings

Solutions converge to reaction-cross diffusion systems as a parameter tends to zero.

The Turing instability domain is characterized for the derived models.

Comparison shows differences in instability domains between cross and standard diffusion.

Abstract

We consider in this paper a microscopic model (that is, a system of three reaction-diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to $0$ towards the solutions of a reaction-cross diffusion system of predator-prey type involving a Holling-type II or Beddington-DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington-DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.