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 Gray-Scott model in the limiting case when the small parameter $\epsilon$, related to the reaction rate of the reverse reactions, approaches zero. We numerically study the physically more complete Gray-Scott model with various $\epsilon$ in one dimension. By decreasing $\epsilon$, we observed that the stationary pattern in the classical Gray-Scott model can be stabilized as the transient state in the variational model for a significantly small $\epsilon$. Additionally, the variational model admits oscillated and traveling-wave-like pattern for small $\epsilon$. The persistent time of these patterns is on the order of $O(\epsilon^{-1})$. We also analyze the stability of two uniform steady states in the variational Gary-Scott model for fixed $\epsilon$. 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.

翻译：本文探讨了一个四物种变分Gray-Scott模型中的斑图形成问题。该模型包含了所有逆反应，并引入虚拟物种来描述经典Gray-Scott模型中的生死过程。这一修正将经典Gray-Scott模型转化为热力学一致的封闭系统。经典的双物种Gray-Scott模型可视为变分Gray-Scott模型在极限情况下的子系统，此时与逆反应速率相关的小参数$\epsilon$趋近于零。我们对具有不同$\epsilon$值的、物理上更完整的Gray-Scott模型进行了一维数值研究。通过减小$\epsilon$，我们观察到经典Gray-Scott模型中的稳态斑图可以在$\epsilon$极小时作为变分模型的暂态被稳定下来。此外，对于较小的$\epsilon$，变分模型允许振荡型和类行波斑图的存在。这些斑图的持续时间为$O(\epsilon^{-1})$量级。我们还分析了固定$\epsilon$下变分Gray-Scott模型中两个均匀稳态的稳定性。尽管这两个状态在某种意义下都是稳定的，但变分模型的梯度流型动力学表现出基于初始条件的选择效应：仅当初始条件不收敛于边界稳态（对应于经典Gray-Scott模型中的平凡均匀稳态）时，才会发生斑图形成。