Spatial pattern formation in the Keller-Segel Model with a logistic source

This paper deals with a Neumann boundary value problem in a d-dimensional box T-d = (0, pi)(d) (d = 1, 2, 3) for the chemotaxis-diffusion-growth model {U-t = del(D-u del U - chi U del V) + rU(1 - U K), V-t = D-nu del V-7 + alpha U - beta V, (star) which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that given any general perturbation of magnitude delta, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order In 1/delta. Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical characterization for the early-stage pattern formation in the Keller-Segel model (star). (C) 2013 Elsevier Ltd. All rights reserved.