On pattern formation in the thermodynamically-consistent variational Gray-Scott model

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.