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

## TL;DR

This paper introduces a new instability mechanism in chemotactic bacteria due to stiff responses, leading to pattern formation, supported by theoretical proofs and numerical simulations aligning with biological observations.

## Contribution

It presents a novel instability mechanism based on stiff chemotactic responses, with proofs for both microscopic and macroscopic models, and demonstrates pattern formation through simulations.

## Key findings

- Instability occurs in both microscopic and macroscopic models.
- Unstable frequencies are bounded, similar to Turing instability.
- Numerical simulations show formation of periodic patterns.

## Abstract

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