Dynamics of a boundary spike for the shadow Gierer-Meinhardt system

Abstract

The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of [3] showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.

Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

“…On the other hand, when we consider the instability of a bounded stationary solution with multiple spots in (6), we do not need to rewrite (6) into a new system like (8), that is, we may treat (1) with d = 0. Thus, the two examples motivate us to consider the system (1) with the factor 1/ε d for 0 ≤ d ≤ n instead of (5).…”