Hao Wenrui, Liu Chun, Wang Yiwei, Yang Yahong
Department of Mathematics, Pennsylvania State University, University Park, 16802, PA, United States.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, United States.
Math Biosci. 2025 Jul;385:109453. doi: 10.1016/j.mbs.2025.109453. Epub 2025 Apr 30.
In this paper, we explore pattern formation in a four-species variational Gary-Scott model, which includes all reverse reactions and introduces a virtual species to describe the birth-death process in the classical Gray-Scott model. This modification transforms the classical Gray-Scott model into a thermodynamically consistent closed system. The classical two-species Gray-Scott model can be viewed as a subsystem of the variational model in the limiting case when the small parameter ϵ, related to the reaction rate of the reverse reactions, approaches zero. We numerically explore pattern formation in this physically more complete Gray-Scott model in one spatial dimension, using non-uniform steady states of the classical model as initial conditions. By decreasing ϵ, we observed that the stationary patterns in the classical Gray-Scott model can be stabilized as the transient states in the variational model for a significantly small ϵ. Additionally, the variational model admits oscillating and traveling-wave-like patterns for small ϵ. The persistent time of these patterns is on the order of O(ϵ). We also analyze the energy stability of two uniform steady states in the variational Gary-Scott model for fixed ϵ. Although both states are stable in a certain sense, the gradient flow type dynamics of the variational model exhibit a selection effect based on the initial conditions, with pattern formation occurring only if the initial condition does not converge to the boundary steady state, which corresponds to the trivial uniform steady state in the classical Gray-Scott model.
在本文中，我们探索了一个四物种变分格雷 - 斯科特模型中的模式形成，该模型包含所有逆向反应，并引入了一个虚拟物种来描述经典格雷 - 斯科特模型中的生死过程。这种修改将经典格雷 - 斯科特模型转变为一个热力学上一致的封闭系统。在与逆向反应速率相关的小参数ϵ趋近于零的极限情况下，经典的两物种格雷 - 斯科特模型可被视为变分模型的一个子系统。我们在一维空间中对这个物理上更完整的格雷 - 斯科特模型中的模式形成进行了数值探索，使用经典模型的非均匀稳态作为初始条件。通过减小ϵ，我们观察到对于一个足够小的ϵ，经典格雷 - 斯科特模型中的稳态模式可以在变分模型中作为瞬态状态被稳定下来。此外，对于小的ϵ，变分模型允许出现振荡和行波状模式。这些模式的持续时间为O(ϵ)量级。我们还分析了在固定ϵ的情况下变分格雷 - 斯科特模型中两个均匀稳态的能量稳定性。尽管这两个状态在某种意义上都是稳定的，但变分模型的梯度流型动力学基于初始条件表现出一种选择效应，只有当初始条件不收敛到边界稳态时才会发生模式形成，该边界稳态对应于经典格雷 - 斯科特模型中的平凡均匀稳态。
