Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model

In the limit of small activator diffusivity E, the stability of symmetric k-spike equilibrium solutions to the Gierer-Meinhardt reaction-diffusion system in a one-dimensional spatial domain is studied for various ranges of the reaction-time constant tau greater than or equal to 0 and the diffusivity D > 0 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. The spectrum of this eigenvalue problem is studied in detail using a combination of rigorous, asymptotic, and numerical methods. For k = 1, and for various exponent, sets of the nonlinear terms, we show that for each D > 0, a one-spike solution is stable only when 0 less than or equal to r less than or equal to tau(0) (D). As tau increases past tau(0) (D), a pair of complex conjugate eigenvalues enters the unstable right half-plane, triggering an oscillatory instability in the amplitudes of the spikes. A large-scale oscillatory motion for the amplitudes of the spikes that occurs when r is well beyond tau(0)(D) is computed numerically and explained qualitatively. For k greater than or equal to 2, we show that a k-spike solution is unstable for any tau greater than or equal to 0 when D > D-k, where D-k > 0 is the well-known stability threshold of a multispike solution when tau = 0. For D > Dk and tau greater than or equal to 0, there are eigenvalues of the linearization that lie on the (unstable) positive real axis of the complex eigenvalue plane. The resulting instability is of competition type whereby spikes are annihilated in finite time. For 0 < D < D-k, we show that a k-spike solution is stable with respect to the O (1) eigenvalues only when 0 less than or equal to tau < tau(0)(D; k). When tau increases past tau(0)(D; k) > 0; a synchronous oscillatory instability in the amplitudes of the spikes is initiated. For certain exponent sets and for k greater than or equal to 2, we show that tau(0)(D; k) is a decreasing function of D with tau(0)(D; k) --> tau(0k) > 0 as D--> D-k(-).