Chemotactic waves of bacteria at the mesoscale

Abstract

The existence of travelling waves for a model of concentration waves of bacteria is investigated. The model consists of 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. Serious mathematical difficulties arise in comparison with the diffusive regime which was studied in a previous work. The cornerstone of the proof is establishing monotonicity properties of the spatial density of cells. Travelling waves exist under certain conditions on the parameters. Counterexamples to both existence and uniqueness are found numerically after a careful analysis of the discrete velocity problem.