Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model

In the limit of small activator diffusivity, the stability of a one-spike solution to the shadow Gierer-Meinhardt activator-inhibitor system is studied for various ranges of the reaction-time constant tau associated with the inhibitor field dynamics. By analyzing the spectrum of the eigenvalue problem associated with the linearization around a one-spike solution, it is proved, for a certain parameter regime, that a complex conjugate pair of eigenvalues crosses into the unstable right half-plane Re(lambda) > 0 as tau increases past a critical value tau(0). For this parameter regime, it is proved that there are exactly two eigenvalues in the right half-plane when tau > tau(0) and none when 0 less than or equal to tau < tau(0). It is shown numerically that this critical value of tau represents the onset of an oscillatory instability in the height of the spike. For other parameter regimes, a similar Hopf bifurcation is confirmed numerically. Full numerical solutions to the shadow problem are computed for a spike that is initially centred at the origin of a radially symmetric domain. Different types of large-scale oscillatory motions for the height of a spike are observed numerically for values of tau well beyond tau(0).