Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems
Localized patterns and semi-strong interaction, a unifying framework for reaction–diffusion systems
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DOI：
10.1093/imamat/hxab036
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发表时间：
2021-08
影响因子：
1.2
通讯作者：
F. Saadi;A. Champneys;N. Verschueren
中科院分区：
文献类型：
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作者：
F. Saadi;A. Champneys;N. Verschueren
使用各种分析和数值方法研究了无限直线上的激活剂-抑制剂反应-扩散方程组。考虑了规范形式，其中包含具有简单立方自催化非线性和任意常数和线性动力学的所有已知模型。此类模型仅限于具有独特均质平衡的模型，包括经典的 Schnakenberg 和 Brusselator 模型，以及文献中提出的用于模拟形态发生的其他系统。众所周知，此类模型具有图灵不稳定性，当激活剂的扩散速度比抑制剂慢时，就会产生稳定的空间周期性模式。相反，在小进给速率的限制下，半强相互作用渐近分析表明存在孤立的尖峰模式。本文描述了连接这两种制度的广泛分歧结构。揭示了某种通用的二参数状态图，其中图灵分岔变为亚临界，导致同宿蛇行的开始。然后，该状态转变为尖峰状态，外层折叠由半强渐近预测。参数和场浓度的重新调整显示了如何独立于扩散速率来研究该状态图。然而，时间动力学被发现强烈依赖于扩散率。霍普夫分岔沿着稳定尖峰的分支发生，这对于小扩散比来说是亚临界的，导致塌陷到均匀状态。随着扩散比的增加，这种分岔通常会变得超临界，并与同宿蛇行以及超临界均匀 Hopf 分岔相互作用，从而导致复杂的时空动力学。使用弱非线性分析、半强渐近和不同数值连续算法的混合来计算适合该理论的许多不同模型的细节。
Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.