Confinement by biased velocity jumps: aggregation of Escherichia coli

TL;DR

This paper analyzes a kinetic model of bacterial chemotaxis showing that biased velocity jumps lead to exponential confinement and aggregation of Escherichia coli, with proven convergence to a positive equilibrium.

Contribution

It establishes the existence of a non-trivial equilibrium and proves exponential convergence, highlighting the role of inhomogeneous bias in bacterial aggregation.

Findings

Existence of a positive equilibrium with exponential decay

Exponential convergence to equilibrium proven

Bias in velocity jumps causes bacterial confinement

Abstract

We investigate a linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in the presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.