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The existence of travelling waves for a model of concentration waves ofbacteria is investigated. The model consists in a kinetic equation for thebiased motion of cells following a run-and-tumble process, coupled with tworeaction-diffusion equations for the chemical signals. Strong mathematicaldifficulties arise in comparison with the diffusive regime which was studied ina previous work. The cornerstone of the proof consists in establishingmonotonicity properties of the spatial density of cells. Travelling waves existunder certain conditions on the parameters. Counter-examples to both existenceand uniqueness are found numerically after careful analysis of the discretevelocity problem.

Chemotactic waves of bacteria at the mesoscale