[Paper Review] Stable spike clusters for the precursor Gierer-Meinhardt system in R2

Juncheng Wei, Matthias Winter+1 more|arXiv (Cornell University)|May 23, 2017

Nonlinear Dynamics and Pattern Formation21 references3 citations

TL;DR

This paper constructs stable spike clusters in the Gierer-Meinhardt reaction-diffusion system on ℝ² with small activator and inhibitor diffusivities and a precursor inhomogeneity. By balancing repulsive spike interactions (due to small inhibitor diffusivity) and attractive forces toward a local minimum of the precursor, the authors prove the existence of linearly stable clusters of k spikes—forming polygons with or without a center—stable for up to 3 (without center) or 6 (with center) spikes.

ABSTRACT

We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of $k$ spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.

Motivation & Objective

• To establish the existence of stable spike clusters in the Gierer-Meinhardt system with a precursor inhomogeneity in two-dimensional space.
• To analyze the interplay between repulsive spike interactions (from small inhibitor diffusivity) and attractive forces toward local minima of the precursor.
• To determine the maximum number of spikes that can form a linearly stable cluster, depending on geometric configuration (with or without a central spike).
• To rigorously construct and validate stable k-spike clusters near a nondegenerate local minimum of the precursor function.

Proposed method

• Use of matched asymptotic expansions to construct multi-spike solutions near a local minimum of the precursor inhomogeneity.
• Derivation of a reduced eigenvalue problem for stability analysis by projecting onto the kernel of the linearized operator.
• Construction of a $2k \times 2k$ matrix $M_\mu(\mathbf{q})$ encoding interactions between spike positions and the precursor's curvature.
• Analysis of the matrix $M_\mu(\mathbf{q})$ to determine linear stability via its eigenvalues, with contributions from spike repulsion and precursor attraction.
• Use of the profile $w(z)$ of the single-spike solution to approximate spike shapes and derive leading-order terms in the asymptotic expansion.
• Incorporation of the inhibitor field $H_\varepsilon$ and its gradient into the stability analysis through integral identities and perturbation expansions.

Experimental results

Research questions

• RQ1 Can stable clusters of k spikes form in the Gierer-Meinhardt system on ℝ² when the inhibitor diffusivity is small and the activator diffusivity is negligible?
• RQ2 What geometric configurations of spike clusters (e.g., regular polygons with or without a center) are linearly stable under the combined effects of spike repulsion and precursor attraction?
• RQ3 What is the maximum number of spikes that can form a stable cluster, and how does this depend on the cluster's symmetry and the precursor's curvature at the minimum point?
• RQ4 How does the curvature of the precursor function at its local minimum influence the stability of spike clusters?

Key findings

• Stable k-spike clusters exist for any positive integer k, all clustering near a nondegenerate local minimum of the precursor inhomogeneity.
• The cluster without a central spike is linearly stable for up to three spikes, while the cluster with a central spike is stable for up to six spikes.
• The stability arises from a balance between repulsive interactions (due to small inhibitor diffusivity) and attractive forces toward the precursor minimum.
• The linear stability of the cluster is determined by the eigenvalues of a $2k \times 2k$ matrix $M_\mu(\mathbf{q})$, which combines spike-spike interaction and precursor curvature effects.
• The leading-order stability condition depends on the second derivative of the precursor function $\mu$ at the minimum point and the repulsion strength encoded in the constant $c_1$.
• The analysis confirms that spike clusters can be stable even when spikes are closely packed, provided the precursor inhomogeneity and diffusivity parameters are appropriately tuned.