Two-species competition model with chemotaxis: well-posedness, stability and dynamics

TL;DR

This paper analyzes a PDE model of two competing species influenced by diffusion and chemotaxis, establishing well-posedness, stability, and long-term dynamics without smallness constraints on chemotactic effects.

Contribution

It proves global well-posedness of the system and demonstrates species coexistence in the weak competition regime regardless of chemotaxis strength.

Findings

Solutions are globally well-posed without small chemotaxis assumptions.

Species cannot go extinct in the weak competition regime.

Long-term behavior analyzed both analytically and numerically.

Abstract

We study a system of PDEs modeling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviors of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.