The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

Abstract

In a singularly perturbed limit of small diffusivity ε of one of the two chemical species, equilibrium spike solutions to the Gray-Scott (GS) model on a bounded one-dimensional domain are constructed asymptotically using the method of matched asymptotic expansions. The equilibria that are constructed are symmetric k-spike patterns where the spikes have equal heights. Two distinguished limits in terms of a dimensionless parameter in the reaction-diffusion system are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime, the solution branches of k-spike equilibria are found to have a saddle-node bifurcation structure. The stability properties of these branches of solutions are analyzed with respect to the large eigenvalues λ in the spectrum of the linearization. These eigenvalues, which have the property that λ = O(1) as ε → 0, govern the stability of the solution on an O(1) time scale. Precise conditions, in terms of the nondimensional parameters, for the stability of symmetric k-spike equilibrium solutions with respect to this class of eigenvalues are obtained. In the low feed-rate regime, it is shown that a large eigenvalue instability leads either to a competition instability, whereby certain spikes in a sequence are annihilated, or to an oscillatory instability (typically synchronous) of the spike amplitudes as a result of a Hopf bifurcation. In the intermediate regime, it is shown that only oscillatory instabilities are possible, and a scaling-law determining the onset and 9600 Garsington Road, Oxford, OX4 2DQ, UK. 22T. Kolokolnikov et al. of such instabilities is derived. Detailed numerical simulations are performed to confirm the results of the stability theory. It is also shown that there is an equivalence principle between spectral properties of the GS model in the low feed-rate regime and the Gierer-Meinhardt model of morphogenesis. Finally, our results are compared with previous analytical work on the GS model.

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“…The assumptions (4) of stability against homogeneous perturbations A1 and being of inhibitor-activator type A2 must now be checked for an interval of parameter values. The shape of the set of points (a, m) satisfying A1 given by (34) implies that if we pick a ¼ p 1 and m ¼ p 2 for which (w þ , n þ ) is stable against homogeneous perturbations, then this remains true if a is increased or m is decreased, see Figure 4. The inhibitor-activator assumption was already found to …”

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

“…The dimensionless GS model is (cf. [30], [21]) v t = v xx − v + Auv 2 , −1 < x < 1, t > 0; v x (±1, t) = 0, (1.1a) τ u t = Du xx + (1 − u) − uv 2 , −1 < x < 1, t > 0; u x (±1, t) = 0.…”

“…[37]); the existence and stability of equilibrium solutions in the semi-strong interaction regime (cf. [11], [7], [8], [28], [29], [30], [21], [22]), and the dynamics of solutions in the semi-strong interaction regime (cf. [5], [6], [43]).…”

“…In [21] symmetric k-spike equilibrium solutions to (1.3), where the spikes have a common amplitude, were constructed asymptotically in the low feed-rate regime. A construction of similar solutions using the geometric theory of singular perturbations is given in [11] for a special scaling of the GS model.…”

“…A construction of similar solutions using the geometric theory of singular perturbations is given in [11] for a special scaling of the GS model. In [21] the stability of k-spike patterns on an O(1) time-scale was analyzed by first formally deriving a nonlocal eigenvalue problem (NLEP). From a rigorous analysis of this NLEP, two fast instability mechanisms were identified: synchronous oscillatory instabilities of the spike amplitudes, and competition, or overcrowding, instabilities leading to the spike annihilation events.…”