Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability
Stationary peaks in a multivariable reaction–diffusion system: foliated snaking due to subcritical Turing instability
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DOI：
10.1093/imamat/hxab029
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发表时间：
2020-07
影响因子：
1.2
通讯作者：
E. Knobloch;A. Yochelis
中科院分区：
文献类型：
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作者：
E. Knobloch;A. Yochelis
在肺血管和肺发育的背景下使用的侧分支的激活剂-抑制剂-底物模型被认为是基于激活剂的空间局部浓度触发局部侧分支的假设。该模型由四个耦合反应扩散方程组成，因此其稳态局部解服从一维（1D）的八维空间动力系统。模型内的固定局部结构被发现与亚临界图灵不稳定性相关，并组织在独特类型的叶状蛇形分叉结构内。这种行为又与参数空间中交换点的存在相关，在该交换点处，均匀浓度状态的复杂主导空间特征值被一对实特征值取代；该点在该系统中起到别利亚科夫-德瓦尼点的作用。主要的叶状蛇形结构由周期性尖峰或峰值序列组成，具有$N$相同的等距峰值，$N=1,2,\dots\,$，以及由不相同、非等距峰值组成的交叉链接。该结构因大量多脉冲状态而变得复杂，其中一些也经过计算，并且跨越了从主要图灵分岔一直到 $N=1$ 状态折叠的参数范围。这些状态形成了一个复杂的模板，从该模板中局部物理结构在二维横向上发展。
An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $N$ identical equidistant peaks, $N=1,2,\dots \,$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D.