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 kspike 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.

The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: The pulse-splitting regime

“…For the one-spike case k = 1, (4.37) reduces to the problem (A.10) for the small eigenvalue of a one-spike solution in the intermediate regime. Therefore, Principal Result A.4, established in [21], also applies to the stability of a one-spike solution in the pulse-splitting regime.…”

“…In Section 4 we use an asymptotic matching analysis to analyze the stability of k-spike equilibria for A = O(1) and εA/ √ D 1 with respect to the drift instabilities associated with eigenvalues of order λ = O(ε). This stability analysis is significantly more intricate than that studied in [21] for the drift instability a one-spike solution in the intermediate regime.…”

“…Related results for a one-spike solution for the infinite-line problem were given in [29]. Regarding the small eigenvalues of order λ = O(ε 2 ), it was shown in [21] that k-spike equilibria in the intermediate regime are all stable with respect to this class of eigenvalues when τ = O(1). However, in Section 5 of [21], it was shown that a small eigenvalue can become unstable, leading to a drift instability of a one-spike solution, when τ is asymptotically large as ε → 0.…”

“…Regarding the small eigenvalues of order λ = O(ε 2 ), it was shown in [21] that k-spike equilibria in the intermediate regime are all stable with respect to this class of eigenvalues when τ = O(1). However, in Section 5 of [21], it was shown that a small eigenvalue can become unstable, leading to a drift instability of a one-spike solution, when τ is asymptotically large as ε → 0. This drift instability for the finite-domain problem, which results from a Hopf bifurcation, leads to an oscillatory behavior for the spike-layer location.…”

“…In [20] a rigorous stability analysis of these solutions with respect to the large eigenvalues of order λ = O(1) in the spectrum of the linearization showed that the stability properties depend intricately on A, D, τ, and k. Related work in the low feed-rate regime for a one-spike solution for the infinite-line problem was given in [29]. For the low feed-rate regime, the stability of k-spike equilibria with respect to the small eigenvalues of order λ = O(ε 2 ) in the spectrum of the linearization was studied in [21].…”