  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.
