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 interval$$\begin{equation*} \left\{ \begin{array}{ll} A_t=\epsilon^2 A''- \mu(x) A+\frac{A^2}{H}, &x\in(-1, 1),\,t>0,\\[3mm] \tau H_t=D H'' -H+ A^2, & x\in (-1, 1),\,t>0,\\[3mm] A' (-1)= A' (1)= H' (-1) = H' (1) =0, \end{array} \right. \end{equation*}$$$$\begin{equation*}\mbox{where } \quad 0<\epsilon \ll\sqrt{D}\ll 1, \quad \end{equation*}$$$$\begin{equation*} \tau\geq 0 \mbox{ and $\tau$ is independent of $\epsilon$. } \end{equation*}$$Aspike clusteris 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 ofNspikes near a non-degenerate local minimum pointt0of the smooth positive inhomogeneity μ(x), i.e. we assume that μ′(t0) = 0, μ″(t0) > 0 and we have μ(t0) > 0. Here,Nis an arbitrary positive integer. Further, we show that this solution is linearly stable. We explicitly compute all eigenvalues, both large (of orderO(1)) and small (of ordero(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: theO(1) scale of the precursor inhomogeneity μ(x), the$O(\sqrt{D})$scale of the inhibitor diffusivity and theO(ε) scale of the activator diffusivity; (iii) the expressions can be made explicit and often have a particularly simple form.