Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray-Scott Model

Two different types of instabilities of equilibrium stripe and ring solutions are studied for the singularly perturbed two-component Gray-Scott (GS) model in a two-dimensional domain. The analysis is performed in the semi-strong interaction limit where the ratio O(e -2 ) of the two diffusion coefficients is asymptotically large. For e → 0, an equilibrium stripe solution is constructed where the singularly perturbed component concentrates along the mid-line of a rectangular domain. An equilibrium ring solution occurs when this component concentrates on some circle that lies concentrically within a circular cylindrical domain. For both the stripe and the ring, the spectrum of the linearized problem is studied with respect to transverse (zigzag) and varicose (breakup) instabilities. Zigzag instabilities are associated with eigenvalues that are asymptotically small as e → 0. Breakup instabilities, associated with eigenvalues that are O(1) as e → 0, are shown to lead to the disintegration of a stripe or a ring into spots. For both the stripe and the ring, a combination of asymptotic and numerical methods are used to determine precise instability bands of wavenumbers for both types of instabilities. The instability bands depend on the relative magnitude, with respect to e, of a nondimensional feed-rate parameter A of the GS model. Both the high feed-rate regime A = O(1), where self-replication phenomena occurs, and the intermediate regime O(e 1/2 ) « A « O(1) are studied. In both regimes, it is shown that the instability bands for zigzag and breakup instabilities overlap, but that a zigzag instability is always accompanied by a breakup instability. The stability results are confirmed by full numerical simulations. Finally, in the weak interaction regime, where both components of the GS model are singularly perturbed, it is shown from a numerical computation of an eigenvalue problem that there is a parameter set where a zigzag instability can occur with no breakup instability. From full-scale numerical computations of the GS, it is shown that this instability leads to a large-scale labyrinthine pattern.