The dynamics and pinning of a spike for a reaction-diffusion system

The motion of a one-spike solution to a simplified form of the Gierer--Meinhardt activator-inhibitor model is studied in both a one-dimensional and a two-dimensional domain. The pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator, referred to as precursor gradients, is studied in detail. In the one-dimensional case, we derive a differential equation for the trajectory of the spike in the limit $\epsilon \rightarrow 0$, where $\epsilon$ is the activator diffusivity. A similar differential equation is derived for the two-dimensional problem in the limit for which $\epsilon \ll 1$ and $D \gg 1$, where $D$ is the inhibitor diffusivity. A numerical finite-element method is presented to track the motion of the spike for the full problem in both one and two dimensions. Finally, the numerical results for the spike motion are compared with corresponding asymptotic results for various examples