Chemotactic waves of bacteria at the mesoscale

TL;DR

This paper investigates the existence of traveling waves in a bacterial chemotaxis model combining kinetic cell motion with chemical signal dynamics, highlighting mathematical challenges and conditions for wave existence.

Contribution

It introduces a kinetic model for bacterial chemotactic waves, establishing existence conditions and analyzing non-uniqueness through numerical counter-examples.

Findings

Traveling waves exist under specific parameter conditions.

Mathematical difficulties are greater than in diffusive regimes.

Counter-examples show non-uniqueness of solutions.

Abstract

The existence of travelling waves for a model of concentration waves of bacteria is investigated. The model consists in a kinetic equation for the biased motion of cells following a run-and-tumble process, coupled with two reaction-diffusion equations for the chemical signals. Strong mathematical difficulties arise in comparison with the diffusive regime which was studied in a previous work. The cornerstone of the proof consists in establishing monotonicity properties of the spatial density of cells. Travelling waves exist under certain conditions on the parameters. Counter-examples to both existence and uniqueness are found numerically after careful analysis of the discrete velocity problem.