Spatial pattern formation in reaction-diffusion models: a computational approach
Spatial pattern formation in reaction-diffusion models: a computational approach
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DOI：
10.1007/s00285-019-01462-0
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发表时间：
2020-01-01
影响因子：
1.9
通讯作者：
Xue, Chuan
中科院分区：
文献类型：
--
作者：
Hao, Wenrui;Xue, Chuan
反应扩散方程已被广泛用于描述生物模式的形成。反应扩散模型的非均匀稳态对应于这些模型支持的平稳空间模式。通常，这些稳态并不是唯一的，并且对应于生物学中观察到的各种空间模式。传统上，时间推进方法或稳态求解器的基础上牛顿的方法来计算这样的解决方案。然而，这些方法收敛到的解高度依赖于初始条件或猜测。本文提出了一种计算反应扩散模型多个非一致定态的系统方法，并确定了它们对模型参数的依赖性。该方法是基于同伦连续技术，并涉及网格细化，这显着降低了计算成本。该方法生成的单参数稳态分岔图，可能包含多个不连通的组件，以及两个参数的解决方案地图，根据稳态的数量划分成不同的区域的参数空间。我们将该方法应用于两个经典的反应扩散模型，并将我们的结果与文献中的理论分析进行了比较。第一个是Schnakenberg模型，它已被用来描述生物模式的形成，由于扩散驱动的不稳定性。第二个是20世纪80年代提出的Gray-Scott模型，用于描述自催化糖酵解反应。在每种情况下，该方法揭示了许多，如果不是所有的，非均匀的稳态和它们的稳定性。
Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction-diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.