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 $\tau$ is asymptotically large as $\epsilon \to 0$. 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.