Spectral Stability of Travelling Wave Solutions in a Keller-Segel Model

TL;DR

This paper analyzes the spectral stability of travelling wave solutions in a Keller-Segel chemotaxis model, revealing conditions under which these solutions are absolutely or transiently unstable based on chemotactic strength.

Contribution

It provides new spectral analysis results for Keller-Segel travelling waves, especially regarding eigenvalues and stability conditions related to chemotactic parameters.

Findings

Eigenvalue at the origin due to translation invariance

Absolute instability when chemotactic coefficient exceeds a critical value

Transient instability below the critical chemotactic coefficient

Abstract

We investigate the point spectrum associated with travelling wave solutions in a Keller-Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise.