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

Abstract

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.

Stable spike clusters for the precursor Gierer–Meinhardt system in $$\mathbb {R}^2$$ R 2

“…Whereas in one spatial dimension the spikes in a cluster are aligned with equal distance in leading order (although they differ in higher order) [27], in two spatial dimensions a variety of different spike configurations are possible. In this paper we have considered regular polygons and polygons with a spike in the centre.…”

“…Recently, the first two authors in [27] studied the Gierer-Meinhardt system with precursor in one spatial dimension and proved the existence and stability of a cluster, which consists of N spikes approaching the same limiting point. More precisely, they consider the existence of a steady-state spike cluster consisting of N spikes near a nondegenerate local minimum point y 0 of the inhomogeneity μ(y), i.e., μ (y 0 ) = 0, μ (y 0 ) > 0, where y ∈ (−1, 1), y 0 ∈ (−1, 1).…”

Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

“…Moreover, for the system, which replaces u t , εv t , and the term 1 2 to ε 2 u t , 0, and a non-homogeneous term A(x) in (1.1), respectively, Kolokolnikov and Wei [10] studied the existence and the stability of N -peak stationary solutions. For the one-dimensional Gierer-Meinhardt system with precursors, the existence and stability of N -peak solutions and the stable spike clusters were studied in [15] and [18], respectively. Moreover, the spike density was studied in [11].…”

“…Moreover, the spike density was studied in [11]. The Gray-Scott model is a closely related model (see e.g [14,18] and the references therein.) For spiky patterns in one-dimensional, the existence and stability were studied in [3,4,5,8,9].…”

Pulse Solutions for an Extended Klausmeier Model with Spatially Varying Coefficients

“…However, these studies are usually limited to models with constant coefficients. Some research has focused on the introduction of localized spatial inhomogeneities [44,34,35,48,49,21]; also (often formal) research has been done on reaction-diffusion equations with (less restricted) spatially varying coefficients [9,8,2,47,46,7]. In this article, we aim to expand the knowledge of such systems, by studying a reaction-diffusion system with fairly generic spatially varying coefficients rigorously; motivated by its use in ecology (see Remark 2), we consider the following extended Klausmeier model with spatially varying coefficients [27,4]:…”