Invasion Fronts and Pattern Formation in a Model of Chemotaxis in One and Two Dimensions.

The purpose of this paper is to explore spatio-temporal pattern formation via invasion fronts in the one and two dimensional Keller-Segel chemotaxis model. In the one-dimensional case, simulations show that solutions that begin near an unstable equilibrium evolve into periodic patterns. These in turn evolve into new patterns through a process known as coarsening. In the two-dimensional case, we encounter only periodic patterns in the wake of the initial front. Transverse patterning only arises as a result of a transverse instability of these periodic patterns from the leading invasion front