Reduced wave Green's functions and their effect on the dynamics of a spike for the Gierer–Meinhardt model

Abstract

In the limit of small activator diusivit y a formal asymptotic analysis is used to derive a dieren tial equation for the motion of a one-spike solution to a simplied form of the Gierer-Meinhardt activator-inhibitor model in a two-dimensional domain. The analysis, which is valid for any nite value of the inhibitor diusivit y D with D 2 , is delicate in that two disparate scales and 1= ln must be treated. This spike motion is found to depend on the regular part of a reduced-wave Green’s function and its gradient. Limiting cases of the dynamics are analyzed. For D small with 2 D 1, the spike motion is metastable. For D 1, the motion now depends on the gradient of a modied Green’s function for the Laplacian. The eect of the shape of the domain and of the value of D on the possible equilibrium positions of a one-spike solution is also analyzed. For D 1, stable spike-layer locations correspond asymptotically to the centers of the largest radii disks that can be inserted into the domain. Thus, for a dumbbell-shaped domain when D 1, there are two stable equilibrium positions near the centers of the lobes of the dumbbell. In contrast, for the range D 1 a complex function method is used to derive an explicit formula for the gradient of the modied Green’s function. For a specic dumbbell-shaped domain, this formula is used to show that there is only one equilibrium spike-layer location when D 1, and it is located in the neck of the dumbbell. Numerical results for other non-convex domains computed from a boundary integral method lead to a similar conclusion regarding the uniqueness of the equilibrium spike location when D 1. This leads to the conjecture that, when D 1, there is only one equilibrium spike-layer location for any convex or non-convex simply connected domain. Finally, the asymptotic results for the spike dynamics are compared with corresponding full numerical results computed using a moving nite element method.