November  2004, 4(4): 1033-1064. doi: 10.3934/dcdsb.2004.4.1033

Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model

1. 

Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2, Canada, Canada

Received  March 2003 Revised  January 2004 Published  August 2004

In the limit of small activator diffusivity $\varepsilon$, and in a bounded domain in $\mathbb{R}^{N}$ with $N=1$ or $N=2$ under homogeneous Neumann boundary conditions, the bifurcation behavior of an equilibrium one-spike solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity $D$. When $D=\infty$, it is well-known that a one-spike solution for the resulting shadow Gierer-Meinhardt system is unstable, and the locations of unstable equilibria coincide with the points in the domain that are furthest away from the boundary. For a unit disk domain it is shown that as $D$ is decreased below a critical bifurcation value $D_{c}$, with $D_{c}=O(\varepsilon^2 e^{2/\varepsilon})$, the spike at the origin becomes stable, and unstable spike solutions bifurcate from the origin. The locations of these bifurcating spikes tend to the boundary of the domain as $D$ is decreased further. Similar bifurcation behavior is studied in a one-parameter family of dumbbell-shaped domains. This motivates a further analysis of the existence of certain near-boundary spikes. Their location and stability is given in terms of the modified Green's function. Finally, for the dumbbell-shaped domain, an intricate bifurcation structure is observed numerically as $D$ is decreased below some $O(1)$ critical value.
Citation: Theodore Kolokolnikov, Michael J. Ward. Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1033-1064. doi: 10.3934/dcdsb.2004.4.1033
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