Explicitly solvable eigenvalue problem and bifurcation delay in sub-diffusive Gierer-Meinhardt model

A spike solution is constructed on the infinite line for a sub-diffusive version of the Gierer-Meinhardt reaction - diffusion model. A non-local eigenvalue problem governs the spike's stability and is explicitly solvable for a certain choice of the kinetic parameters. Its solution generalises former results for the Gierer-Meinhardt model with regular diffusion, and the normal and anomalous systems' properties are juxtaposed. It is shown that a Hopf bifurcation occurs in the sub-diffusive system for larger values of the time parameter tau(0) as compared to the normal counterpart, rendering the anomalous system more stable. Asymptotic solutions are obtained near important values of the diffusion anomaly index gamma and collectively shown to be valid over most of the applicable range 0 < gamma < 1. A bifurcation delay scenario is described for the sub-diffusive system, and the WKB exponent is computed analytically.