Spatial pattern formation in the Keller–Segel model with a logistic source

This paper deals with a Neumann boundary value problem in a dd-dimensional box Td=(0,π)d(d=1,2,3) for the chemotaxis–diffusion–growth model equation(⋆ ){Ut=∇(Du∇U−χU∇V)+rU(1−U/K),Vt=Dv∇2V+αU−βV, 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 δδ, 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 ln1δ. 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 (⋆)(⋆).