Confinement by biased velocity jumps: aggregation of \textit{Escherichia coli}
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 jump at \(x = 0\) and are therefore not solutions of (1). Stationary solutions have to balance the turning operator with the transport operator \(v\partial_x\). It is proved the existence of a positive equilibrium distribution with an exponential decay at infinity. 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, the results of [\textit{J. Dolbeault} et al., Trans. Am. Math. Soc. 367, No. 6, 3807--3828 (2015; Zbl 1342.82115)] are essentially used.
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