Hopf Bifurcations and Oscillatory Instabilities of Spike Solutions for the One-Dimensional Gierer-Meinhardt Model

2006

Interfaces Free Bound.

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.

Section: In the Intermediate Regime O(1)mentioning

confidence: 99%

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

Kolokolnikov

Ward

2006

Interfaces Free Bound.

“…where √ 1 + τα denotes the principal branch of the square root function, compare [35]. Now, we have Proof Lemma 13 follows from Theorem 19 by a regular perturbation argument.…”

Section: Theorem 12mentioning

confidence: 85%

“…Stability in the case where τ is large has been investigated in [35] for single spikes and those results on Hopf bifurcation are expected to carry over to the case of an N-spike cluster considered here. We remark that stability in the case of large τ for the shadow system has been studied in [6,34].…”

Section: Remarkmentioning

confidence: 87%

See 1 more Smart Citation

Stable spike clusters for the one-dimensional Gierer–Meinhardt system

Winter

2016

Eur. J. Appl. Math

We consider the Gierer-Meinhardt system with precursor inhomogeneity and two small diffusivities in an intervalτ > 0 and τ is independent of ε.A spike cluster is the combination of several spikes which all approach the same point in the singular limit. We rigorously prove the existence of a steady-state spike cluster consisting of N spikes near a non-degenerate local minimum point t 0 of the smooth positive inhomogeneity μ(x), i.e. we assume that μ (t 0 ) = 0, μ (t 0 ) > 0 and we have μ(t 0 ) > 0. Here, N is an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of order O (1)) and small (of order o (1)). The main features of studying the Gierer-Meinhardt system in this setting are as follows: (i) it is biologically relevant since it models a hierarchical process (pattern formation of small-scale structures induced by a pre-existing large-scale inhomogeneity); (ii) it contains three different spatial scales two of which are small: the O(1) scale of the precursor inhomogeneity μ(x), the O( √ D) scale of the inhibitor diffusivity and the O(ε) scale of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly simple form.

“…Existence and stability of spiky steady states have been studied for 1-D in [13] and their instabilities have been investigated in [27]. For 2-D the existence and stability of multiple spikes has been investigated in [31], [32], [33].…”

Section: Introductionmentioning

confidence: 99%

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

Winter

2009

J Nonlinear Sci

Abstract. We rigorously prove results on spiky patterns for the Gierer-Meinhardt system [10] with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini andFirstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity.This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper.To derive these new effects in a mathematically rigorous way, we use analytical tools like LiapunovSchmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work [33].Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.1991 Mathematics Subject Classification. Primary 35B35, 76E30; Secondary 35B40, 76E06.