Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

TL;DR

This paper rigorously derives a cross-diffusion model from a reaction-diffusion system describing competing species with different dietary diversities, analyzing stability and supporting findings with numerical simulations.

Contribution

It provides a rigorous mathematical derivation of a starvation-driven cross-diffusion system as a limit of a reaction-diffusion model, and studies its stability properties.

Findings

The derived cross-diffusion system is stable with no Turing instability.

Numerical simulations confirm the theoretical stability analysis.

The model captures the impact of dietary diversity on species competition.

Abstract

We rigorously prove the passage from a Lotka-Volterra reaction-diffusion system towards a cross-diffusion system at the fast reaction limit. The system models a competition of two species, where one species has a more diverse diet than the other. The resulting limit gives a cross-diffusion system of a starvation driven type. We investigate the linear stability of homogeneous equilibria of those systems and rule out the possibility of Turing instability. Numerical simulations are included which are compatible with the theoretical results.