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

Abstract

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 $0 \leq \tau < \tau_0$. 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 $\tau_0$.