Traveling wave solutions for two species competitive chemotaxis systems

TL;DR

This paper investigates traveling wave solutions in a two-species chemotaxis system with competition, establishing conditions for their existence, non-existence at certain speeds, and the influence of chemotaxis sensitivity on wave speed.

Contribution

It provides new conditions for the existence and non-existence of traveling waves in chemotaxis systems, including explicit ranges for parameters where wave speed is unaffected by chemotaxis.

Findings

Existence of traveling wave solutions for speeds above a critical threshold.

Non-existence of such solutions below a certain critical speed, independent of chemotaxis.

Explicit parameter ranges where chemotaxis does not influence wave speed.

Abstract

In this paper, we consider two species chemotaxis systems with Lotka-Volterra competition reaction terms. Under appropriate conditions on the parameters in such a system, we establish the existence of traveling wave solutions of the system connecting two spatially homogeneous equilibrium solutions with wave speed greater than some critical number c*. We also show the non-existence of such traveling waves with speed less than some critical number c*_0, which is independent of the chemotaxis. Moreover, under suitable hypotheses on the coefficients of the reaction terms, we obtain explicit range for the chemotaxis sensitivity coefficients ensuring c*= c*_0, which implies that the minimum wave speed exists and is not affected by the chemoattractant.