Confinement by biased velocity jumps: aggregation of \textit{Escherichia coli}
type: Item
instance of: scholarly article
title: Confinement by biased velocity jumps: aggregation of \textit{Escherichia coli}
zbMATH Open document ID: 1337.35155
author: Vincent Calvez
author: Gaël Raoul
author: Christian Schmeiser
published in: Kinetic and Related Models
publication date: 2015-08-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1404.0643
review text: The authors investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. The stochastic simulations give the kinetic model \[ \partial_tf(x,v,t)+v\cdot\nabla_xf(x,v,t)=\int_V(K(x,v')f(x,v',t)-K(x,v)f(x,v,t))dv',\eqno{(1)} \] where \(f(t,x,v)\) is the phase (position (\(x\)) -- velocity (\(v\))) space distribution of microorganisms at time \(t\geq 0\), with \(x\in\mathbb R^d\) and \(v\in V\in\mathbb R^d\). The velocity set \(V\) is supposed to be bounded and rotationally symmetric. The right hand side of (1) is the turning operator. It describes the velocity changes due to tumbling. The turning kernel \(K(x,v)\) is the rate of changing from velocity \(v\) to a different velocity \(v'\) at position \(x\). The \(x\)-dependence contains the influence of the chemo-attractant. The fact that the turning kernel only depends on the incoming (pre-tumbling) velocity means a complete randomization of velocity at tumbling events. The main deficiency of (1) as a model for the experiments is the lack of a nonlinear coupling with an equation for the chemo-attractant, describing production by the cells, diffusion, and decay. The restriction to the linear problem has purely mathematical reasons. It is shown that proving the existence and stability of aggregated stationary solutions already pose significant difficulties, even for a simple one-dimensional version of (1). The authors restricted themselves to a one-dimensional model \((d=1)\) with the velocity set \(V=[-1/2,1/2]\), \(|V|=1\). A typical example for the choice of the turning kernel is given by \[ K(x,v)=1+\chi\operatorname{sign}(xv),\quad 0<\chi<1.\eqno{(2)} \] The turning rate takes the larger value \(1+\chi\) for cells moving away from \(x=0\), and the smaller value \(1-\chi\) for cells moving towards \(x=0\). Equilibrium distributions of the turning operator, i.e. , multiples of \(1/K\), have a…
Mathematics Subject Classification ID: 35Q92
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 6469716
zbMATH Keywords: kinetic equations
zbMATH Keywords: velocity-jump processes
zbMATH Keywords: chemotaxis
zbMATH Keywords: equilibrium
zbMATH Keywords: hypocoercivity
reviewed by: Boris V. Loginov
describes a project that uses: Chemotaxis
MaRDI profile type: Publication
Wikidata QID: Q62568221
OpenAlex ID: W2963492180
arXiv ID: 1404.0643
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DOI: 10.3934/KRM.2015.8.651
description: scientific article