Reduction to a single closed equation for 2 by 2 reaction-diffusion systems of Lotka-Volterra type

TL;DR

This paper derives a simplified single closed-form reaction-diffusion equation for 2x2 Lotka-Volterra systems modeling interacting species, enabling easier analysis of population dynamics in large, competitive environments.

Contribution

It introduces a novel reduction method that simplifies complex coupled systems into a single equation using singular limit and compactness techniques.

Findings

Reduction to a single reaction-diffusion equation for large populations

Application to Wolbachia spread model in arthropods

Validation of the approach through classical bistable equations

Abstract

We consider general models of coupled reaction-diffusion systems for interacting variants of the same species. When the total population becomes large with intensive competition, we prove that the frequencies (i.e. proportions) of the variants can be approached by the solution of a simpler reaction-diffusion system, through a singular limit method and a relative compactness argument. As an example of application, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals.