论文标题
与前体场的Gierer-Meinhardt模型的稳定的不对称尖峰平衡
Stable Asymmetric Spike Equilibria for the Gierer-Meinhardt Model with a Precursor Field
论文作者
论文摘要
反应扩散系统中的前体梯度在反应运动中是空间变化的系数。此类梯度已用于各种应用，例如Hydra中的头部形成，以模拟预模式的效果，并在各个空间区域中定位模式。对于1-D Gierer-Meinhardt（GM）模型，我们表明，激活剂衰减速率中的简单前体梯度可以导致存在稳定的，不对称的，两种尖峰模式，对应于不同高度激活因子中的局部峰。对于GM模型来说，这是一种质性的新现象，因为在没有前体场的情况下，不对称的尖峰模式都不稳定。通过确定两种尖峰稳态模式的全局分叉图，我们表明，沿对称性两尖峰分支的超临界对称性分叉从前体场中的参数中出现了不对称的模式。通过一种组合的分析数字方法，我们分析了GM模型在两尖峰稳态周围线性化的光谱，以确定不对称溶液分支的一部分是线性稳定的。在此线性稳定性分析中，得出和分析了新的一类新的矢量值非局部特征值问题（NLEP）。
Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction-kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1-D Gierer-Meinhardt (GM) model we show that a simple precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric, two-spike patterns, corresponding to localized peaks in the activator of different heights. This is a qualitatively new phenomena for the GM model, in that asymmetric spike patterns are all unstable in the absence of the precursor field. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach we analyze the spectrum of the linearization of the GM model around the two-spike steady-state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis a new class of vector-valued nonlocal eigenvalue problem (NLEP) is derived and analyzed.
