Bifurcation of spike equilibria in the nearshadow GiererMeinhardt model
Theodore Kolokolnikov  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 onespike solution to the GiererMeinhardt activatorinhibitor system is analyzed for different ranges of the inhibitor diffusivity $D$. When $D=\infty$, it is wellknown that a onespike solution for the resulting shadow GiererMeinhardt 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 oneparameter family of dumbbellshaped domains. This motivates a further analysis of the existence of certain nearboundary spikes. Their location and stability is given in terms of the modified Green's function. Finally, for the dumbbellshaped domain, an intricate bifurcation structure is observed numerically as $D$ is decreased below some $O(1)$ critical value.
Keywords: GiererMeinhardt model, reactiondiffusion equations, Green's function, pattern formation.
Received: March 2003; Revised: January 2004; Available Online: August 2004. 
2014 5year IF.957
