Stiff-response-induced instability for chemotactic bacteria and flux-limited Keller–Segel equation

Abstract

Collective motion of chemotactic bacteria as E. Coli relies, at the\nindividual level, on a continuous reorientation by runs and tumbles. It has\nbeen established that the length of run is decided by a stiff response to a\ntemporal sensingof chemical cues along the pathway.We describe a novel\nmechanism for pattern formation stemming from the stiffness of chemotactic\nresponse relying on a kinetic chemotaxis model which includes a recently\ndiscovered formalism for the bacterial chemotaxis. We prove instability both\nfor amicroscopic description in the space-velocity space and for the\nmacroscopic equation, a flux-limited Keller-Segel equation, which has attracted\nmuch attention recently.A remarkable property is that the unstable frequencies\nremain bounded, as it is the case in Turing instability. Numerical\nillustrations based on a powerful Monte Carlo method show that the stationary\nhomogeneous state of population density isdestabilized and periodic patterns\nare generated in realistic ranges of parameters. These theoretical developments\nare in accordance with several biological observations.\n