Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model
The authors investigate the stability of symmetric \(k\)-spike equilibrium solutions to the Gierer-Meinhardt reaction-diffusion system in one space dimension in the limit of small activator diffusity and for various ranges of the reaction-time constant \(\tau\) and diffusitivity \(D\) of the inhibitor field dynamics. A nonlocal eigenvalue problem is derived that determines the stability on an \(O(1)\) time scale of these \(k\)-spike equilibrium patterns. For \(k=1\) and for various exponent sets of the nonlinear terms, they show that a one-spike solution is stable only when \(0 \leq \tau < \tau_0(D)\) and at \(\tau_0(D)\) a pair of complex conjugate eigenvalues crosses the imaginary axis. For \(k > 1\) positive real eigenvalues occur which lead to an instability of competition type; also synchronous oscillatory instability in the amplitudes of the spikes can occur for positive values of \(\tau\).
scientific article; zbMATH DE number 1971876