Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth

Abstract

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a $d$-dimensional box ${\mathbb{T}}^d=(0,\pi {)}^d(d=\mathrm{1,2},3)$. 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 $\ln (\mathrm{1}/\delta )$. Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns.