The dynamics of boundary spikes for a nonlocal reaction-diffusion model

An asymptotic reduction of the Gierer-Meinhardt activator-inhibitor system in the limit of large inhibitor diffusivity and small activator diffusivity epsilon leads to a singularly perturbed nonlocal reaction-diffusion equation for the activator concentration. In the limit epsilon --> 0, this nonlocal problem for the activator concentration has localized spike-type solutions. In this limit, we analyze the motion of a spike that is confined to the smooth boundary of a two or three-dimensional domain. By deriving asymptotic differential equations for the spike motion, it is shown that the spike moves towards a local maximum of the curvature in two dimensions and a local maximum of the mean curvature in three dimensions. The motion of a spike on a flat segment of a two-dimensional domain is also analyzed, and this motion is found to be metastable. The critical feature that allows for the slow boundary spike motion is the presence of the nonlocal term in the underlying reaction-diffusion equation.