Hopf Bifurcations and Oscillatory Instabilities of Spike Solutions for the One-Dimensional Gierer-Meinhardt Model

In the limit of small activator diffusivity ε, 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 τ ≥ 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 ≤ τ < τ0(D). As τ increases past τ0(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 τ is well beyond τ0(D) is computed numerically and explained qualitatively. For k ≥ 2, we show that a k-spike solution is unstable for any τ ≥ 0 when D > Dk, where Dk > 0 is the well known stability threshold of a multi-spike solution when τ = 0. ForD > Dk and τ ≥ 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 < Dk, we show that a k-spike solution is stable with respect to the O(1) eigenvalues only when 0 ≤ τ < τ0(D; k). When τ increases past τ0(D; k) > 0, a synchronous oscillatory instability in the amplitudes of the spikes is initiated. For certain exponent sets and for k ≥ 2, we show that τ0(D; k) is a decreasing function of D with τ0(D; k) → τ0k > 0 as D → D− k .