Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

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

Pages: 1033 - 1064, Volume 4, Issue 4, November 2004

doi:10.3934/dcdsb.2004.4.1033       Abstract        Full Text (499.7K)       Related Articles

Theodore Kolokolnikov - Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2, Canada (email)
Michael J. Ward - Department of Mathematics, University of British Columbia, Vancouver, Canada V6T 1Z2, Canada (email)

Abstract: 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.

Keywords:  Gierer-Meinhardt model, reaction-diffusion equations, Green's function, pattern formation.
Mathematics Subject Classification:  35K57, 34B27, 92C15.

Received: March 2003;      Revised: January 2004;      Published: August 2004.