从一种模式到另一种模式：基于WKBJ的异构领域图灵模式分析
From One Pattern into Another: Analysis of Turing Patterns in Heterogeneous Domains via WKBJ
摘要：
从均匀性形成斑图形成已得到充分研究，但对非均匀介质中的破裂不稳定性知之甚少。将由于固有的空间异质性而观察到的空间图案与由于非线性不稳定性而出现的图案分开是不平凡的。利用WKBJ渐近方法研究了空间非均匀反应扩散系统的图灵不稳定性，得到了经典图灵条件的局部形式的不稳定性条件。我们发现，不稳定模的结构与空间均匀条件下的典型三角函数有很大的不同。不同的增长率模式局限于不同的空间区域。这种局部化有助于解释在异构环境中图灵系统模拟中观察到的常见振幅调制。我们数值证明了这一理论，给出了一个说明性的例子，涌现的不稳定性和惊人的复杂性所产生的空间异质反应扩散系统。我们的研究结果提供了深入了解系统驱动的外源异质性，以及连续的模式形成过程，注意到生物学中的大多数情况下不涉及对称性破缺从同质性，而是由顺序演变的异质状态。这里报告的不稳定性机制准确地捕捉到了这种演变，并将图灵的原始论文扩展到更广泛和更现实的系统类别。
英文摘要：
Pattern formation from homogeneity is well-studied, but less is known concerning symmetry-breaking instabilities in heterogeneous media. It is nontrivial to separate observed spatial patterning due to inherent spatial heterogeneity from emergent patterning due to nonlinear instability. We employ WKBJ asymptotics to investigate Turing instabilities for a spatially heterogeneous reaction-diffusion system, and derive conditions for instability which are local versions of the classical Turing conditions We find that the structure of unstable modes differs substantially from the typical trigonometric functions seen in the spatially homogeneous setting. Modes of different growth rates are localized to different spatial regions. This localization helps explain common amplitude modulations observed in simulations of Turing systems in heterogeneous settings. We numerically demonstrate this theory, giving an illustrative example of the emergent instabilities and the striking complexity arising from spatially heterogeneous reaction-diffusion systems. Our results give insight both into systems driven by exogenous heterogeneity, as well as successive pattern forming processes, noting that most scenarios in biology do not involve symmetry breaking from homogeneity, but instead consist of sequential evolutions of heterogeneous states. The instability mechanism reported here precisely captures such evolution, and extends Turing's original thesis to a far wider and more realistic class of systems.
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