Stable spike clusters for the precursor Gierer–Meinhardt system in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ma
Stable spike clusters for the precursor Gierer–Meinhardt system in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ma
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DOI：
10.1007/s00526-017-1233-6
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发表时间：
2017-05
影响因子：
2.1
通讯作者：
Juncheng Wei;M. Winter;Wen Yang
中科院分区：
文献类型：
--
作者：
Juncheng Wei;M. Winter;Wen Yang
我们考虑的Gierer-Meinhardt系统具有小的抑制剂扩散系数，非常小的激活剂扩散系数和前体的不均匀性。对于任意给定的正整数k，我们构造一个由k个尖峰组成的尖峰簇，它们都接近于前体不均匀性的同一个非退化局部极小点。我们表明，这个尖峰集群可以是线性稳定的。特别是，我们表明存在的尖峰集群位于顶点的多边形有或没有中心。此外，没有中心的集群对于多达三个尖峰是稳定的，而具有中心的集群对于多达六个尖峰是稳定的。支撑这些稳定的尖峰集群的主要思想如下：由于抑制剂扩散率小，尖峰之间的相互作用是排斥的，并且尖峰被吸引向前体不均匀性的局部最小点。结合这两种效应可以导致簇内尖峰位置的平衡，使得簇线性稳定。
We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integerkwe construct a spike cluster consisting ofkspikes 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.