Vanishing-spreading dichotomy in a two-species chemotaxis competition system with a free boundary

TL;DR

This paper analyzes a two-species chemotaxis model with a free boundary, establishing conditions for solution existence and exploring the vanishing-spreading dichotomy based on interspecies competition strength.

Contribution

It introduces a novel free boundary chemotaxis model with Lotka-Volterra competition and characterizes the vanishing-spreading dichotomy scenarios.

Findings

Vanishing-spreading dichotomy always occurs in the model.

Strong sense vanishing-spreading occurs under weak-weak competition.

Weak sense vanishing-spreading occurs under weak-strong competition.

Abstract

Predicting the evolution of expanding population is critical to control biological threats such as invasive species and virus explosion. In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms and a free boundary. Such a model with a free boundary describes the spreading of new or invasive species subject to the influence of some chemical substances in an environment with a free boundary representing the spreading front. We first find conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we investigate vanishing-spreading dichotomy scenarios for positive solutions. It is shown that the vanishing-spreading dichotomy in the generalized sense always occurs; that the vanishing spreading dichotomy in the strong sense occurs when the competition between two species is weak-weak competition; and that the vanishing spreading dichotomy in the weak sense occurs when the competition between two species is weak-strong competition.