On the weakly competitive case in a two-species chemotaxis model

TL;DR

This paper studies a two-species chemotaxis model with weak competition, demonstrating global stability of coexistence under small chemotactic strengths using the method of eventual comparison.

Contribution

It introduces a more natural smallness condition for chemotactic strengths ensuring stability, and applies the method of eventual comparison to chemotaxis systems.

Findings

Global asymptotic stability of coexistence steady state

Stability condition is more natural than previous ones

Method of eventual comparison is effective in chemotaxis models

Abstract

In this article we investigate a parabolic-parabolic-elliptic two-species chemotaxis system with weak competition and show global asymptotic stability of the coexistence steady state under a smallness condition on the chemotactic strengths, which seems more natural than the condition previously known. For the proof we rely on the method of eventual comparison, which thereby is shown to be a useful tool even in the presence of chemotactic terms.