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

This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a d-dimensional box Td=(0,π)d  (d=1,2,3). It is proved that given any general perturbation of magnitude δ, 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⁡(1/δ). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns