Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation

Shuonan Wu, Bing Yu, Yuhai Tu, Lei Zhang
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Abstract

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether the Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically map the solution landscape to find all steady-state solutions. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability. Furthermore, by using a generalized action in functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.
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反应-扩散系统的解景观揭示了模式形成的非线性机制和空间稳健性
虽然图灵证明了当均相状态变得线性不稳定时,反应-扩散(RD)系统中会出现空间模式,但图灵机制是否是模式形成的唯一途径仍不清楚。在此,我们开发了一种高效算法,用于系统地绘制解景观图,以找到所有稳态解。通过将我们的方法应用于一般的 RD 模型,我们发现在图灵不稳定性出现之前,稳定的空间模式可以通过鞍节点分岔出现。此外,通过使用基于大偏差理论的函数空间广义作用,我们的方法被扩展用于评估空间模式对抗噪声的稳定性。我们在三物种 RD 模型中应用了这种通用方法,结果表明虽然图灵模式的形成只需要两个化学物种,但在小型生化系统中,第三个物种对于稳定模式抵御强固有噪声至关重要。
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