Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Abstract

Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k - 1 small eigenvalues, governing the translational stability of a symmetric k-spike pattern, simultaneously cross through zero at precisely the same parameter value at which k - 1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB ... BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory. © European Mathematical Society 2006.