Absolute instabilities 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 model of bacterial chemotaxis, identifying conditions under which these solutions become absolutely unstable due to spectrum crossing.

Contribution

It provides a detailed spectral analysis of travelling waves in a Keller-Segel model, revealing conditions for absolute instability and spectrum crossing in parameter space.

Findings

All travelling wave solutions have essential spectrum in the right half plane.

For certain parameters, the absolute spectrum is in the left half plane, indicating potential stability.

Critical parameter values are identified where the absolute spectrum crosses into the right half plane, signaling absolute instability.

Abstract

We investigate the spectral stability of travelling wave solutions in a Keller-Segel model of bacterial chemotaxis with a logarithmic chemosensitivity function and a constant, sublinear, and linear consumption rate. Linearising around the travelling wave solutions, we locate the essential and absolute spectrum of the associated linear operators and find that all travelling wave solutions have essential spectrum in the right half plane. However, we show that in the case of constant or sublinear consumption there exists a range of parameters such that the absolute spectrum is contained in the open left half plane and the essential spectrum can thus be weighted into the open left half plane. For the constant and sublinear consumption rate models we also determine critical parameter values for which the absolute spectrum crosses into the right half plane, indicating the onset of an absolute instability of the travelling wave solution. We observe that this crossing always occurs off of the real axis.