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 terms, it is well-known that a one-spike solution for the resulting shadow Gierer-Meinhardt system is unstable but with an asymptotically exponentially small growth rate as ε → 0. The asymptotic location of this unstable spike is determined by critical points of the distance function. Hence, on a one-dimensional interval of length two this unstable equilibrium spike is located at the midpoint of the interval. Similarly, when D = ∞, an unstable spike is located at the center of a circular cylindrical domain of radius one. For these two types of symmetrical domains, it is shown that as D is decreased below a critical bifurcation value Dc, with Dc = O(ε e), 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. For these domains, when D = ∞, an unstable spike can be located either in the neck or in one of the two lobes of the dumbbell. It is shown that when D is exponentially large as ε → 0, the bifurcation behavior is such that the spike in the neck of the dumbbell becomes stable and the spikes in the lobes of the dumbbell tend to the boundary of the domain. This motivates a further analysis of the existence and stability of certain near-boundary spikes. The location and stability of these spikes depend on the behavior of the regular part of the Green’s function near the boundary of the domain. Finally, for the oneparameter family of dumbbell-shaped domains, it is shown numerically that as D is decreased below some O(1) critical value there is an additional pitchfork bifurcation through which the spike in the neck of the dumbbell loses its stability to stable spike solutions that migrate towards the two lobes of the dumbbell as D is decreased.