Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions.

Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions Short title: Gierer-Meinhardt system with Robin boundary conditions

Bifurcation of Spike Equilibria in the Near-Shadow Gierer-Meinhardt Model

In the limit of small activator diffusivity ε, and in a bounded domain in R with N = 1 or N = 2 under homogeneous Neumann boundary conditions, the bifurcation behavior of an equilibrium onespike solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity D. When D = ∞, and under certain conditions on the exponents in the nonlinear te...

Existence and Stability Analysis of Spiky Solutions for the Gierer-meinhardt System with Large Reaction Rates

We study the Gierer-Meinhardt system in one dimension in the limit of large reaction rates. First we construct three types of solutions: (i) an interior spike; (ii) a boundary spike and (iii) two boundary spikes. Second we prove results on their stability. It is found that an interior spike is always unstable; a boundary spike is always stable. The two boundary spike configuration can be either...

Stable Boundary Spike Clusters for the Two-dimensional Gierer-meinhardt System

We consider the Gierer-Meinhardt system with small inhibitor diffusivity and very small activator diffusivity in a bounded and smooth two-dimensional domain. For any given positive integer k we construct a spike cluster consisting of k boundary spikes which all approach the same nondegenerate local maximum point of the boundary curvature. We show that this spike cluster is linearly stable. The ...

Spikes for the Gierer-meinhardt System in Two Dimensions: the Strong Coupling Case

Numerical computations often show that the Gierer-Meinhardt system has stable solutions which display patterns of multiple interior peaks (often also called spots). These patterns are also frequently observed in natural biological systems. It is assumed that the diffusion rate of the activator is very small and the diffusion rate of the inhibitor is finite (this is the so-called strong-coupling...