The Existence and Stability of Spike Patterns in a Chemotaxis Model

Abstract

In the limit of small chemoattractant diffusivity $\epsilon$, the existence, stability, and dynamics of spiky patterns in a chemotaxis model are studied in a bounded multidimensional domain. In this model, the transition probability density function $\Phi(w)$ is assumed to have a power law form $\Phi(w)=w^p$, and the production of chemoattractant w is assumed to saturate according to a Michaelis--Menten kinetic function. In the limit $\epsilon \to 0$, it is proved that there is a steady-state single boundary spike solution located at the maximum of the mean curvature of the boundary. Moreover, a steady-state interior spike solution is proved to concentrate at a maximum of the distance function. The single interior spike solution is shown to be metastable for certain ranges of p and the dimension N. The stability of a single boundary spike solution is also analyzed in detail. Finally, a formal asymptotic analysis is used to characterize the metastable interior spike dynamics in both a one-dimensional and a multidimensional domain.