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

Wenrui Hao, Chun Liu, Yiwei Wang, Yahong Yang

TL;DR

The paper addresses pattern formation in a thermodynamically consistent variational Gray-Scott model that augments the classical Gray-Scott system with reversible reactions and a virtual species $Y$, embedding the open system into a closed energetic framework. It derives the model via the energetic variational approach (EnVarA), analyzes energy stability of uniform states, and uses one-dimensional simulations with non-uniform steady states from the classical model as initial data across $\varepsilon$-regimes to reveal stationary patterns, oscillations, and traveling-wave–like dynamics. A key finding is that the interior steady state is a global energy minimizer for all $\varepsilon>0$, while the boundary state is virtually stable for small $\varepsilon$ due to the dominance of $Y$, enabling long-lived non-uniform patterns when $\varepsilon$ is small. The results demonstrate how thermodynamic constraints and the $Y$-driven influx/efflux can sustain complex patterns and quantify pattern persistence via $T=O(\varepsilon^{-1})$, offering a principled lens for pattern formation in open biological and chemical systems with energy exchange.

Abstract

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 pattern in the classical Gray-Scott model can be stabilized as the transient state 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 $O(ε^{-1})$. 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.

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

TL;DR

The paper addresses pattern formation in a thermodynamically consistent variational Gray-Scott model that augments the classical Gray-Scott system with reversible reactions and a virtual species , embedding the open system into a closed energetic framework. It derives the model via the energetic variational approach (EnVarA), analyzes energy stability of uniform states, and uses one-dimensional simulations with non-uniform steady states from the classical model as initial data across -regimes to reveal stationary patterns, oscillations, and traveling-wave–like dynamics. A key finding is that the interior steady state is a global energy minimizer for all , while the boundary state is virtually stable for small due to the dominance of , enabling long-lived non-uniform patterns when is small. The results demonstrate how thermodynamic constraints and the -driven influx/efflux can sustain complex patterns and quantify pattern persistence via , offering a principled lens for pattern formation in open biological and chemical systems with energy exchange.

Abstract

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 , 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 , 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 . Additionally, the variational model admits oscillating and traveling-wave-like patterns for small . The persistent time of these patterns is on the order of . We also analyze the energy stability of two uniform steady states in the variational Gary-Scott model for fixed . 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.
Paper Structure (13 sections, 2 theorems, 55 equations, 13 figures)

This paper contains 13 sections, 2 theorems, 55 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Variational Gray-Scott Model
  3. Derivation of the Variational Gray-Scott Models
  4. Formal limit of the Variational Gray-Scott Model
  5. Uniform steady states in the Variational Gray-Scott Model and their energy stability
  6. Stability of the interior steady-state
  7. Stability of the boundary steady-state
  8. Pattern formulation in Variational Gray-Scott Models
  9. $\epsilon = 10^{-2}$: relatively large $\epsilon$
  10. $\epsilon=10^{-4}$: Intermediate-sized $\epsilon$
  11. $\epsilon = 10^{-6}$: significantly small $\epsilon$
  12. Pattern persistence time v.s. $\epsilon$
  13. Conclusions

Key Result

Proposition 3.1

For any $\epsilon>0$, the interior steady state is a local minimizer of the energy $E$ on the manifold ${\bm{u}}_0 + \mathcal{S}_1$, where ${\bm{u}}_0 = (u_0, v_0, p_0, y_0/\epsilon)$ is the initial condition.

Figures (13)

  • Figure 1: Energy Landscape for $\epsilon = 0.01$ and $\epsilon_3=0.0001$ along the $\bm{s}_i$ directions relative to the interior steady state.
  • Figure 2: Numerical results demonstrating near-stability when Y dominates almost the entire system.
  • Figure 3: Steady states of the classical Gray-Scott model in the domain $\Omega = (0, 1)$, subject to the non-flux boundary condition, with parameter values $D_u = 5 \times 10^{-4}$ and $D_v = 2.5 \times 10^{-4}$, $f = y_0 = 0.04$, and $k = 0.065$, computed in hao2020spatial: (a) Linearly stable steady states ($u$ : solid line, $v$: dashed line). (b) Linearly unstable steady states ($u$ : solid line, $v$: dashed line).
  • Figure 4: (a) Numerical results for the variational Gray-Scott model with $\epsilon = 10^{-2}$ using linearly stable steady states of the classical Gray-Scott model as initial conditions. (b) Numerical results for the variational Gray-Scott model with $\epsilon = 10^{-2}$ using linearly unstable steady states of the classical Gray-Scott model as initial conditions
  • Figure 5: (a) Numerical results for the variational Gray-Scott model with $\epsilon = 10^{-4}$ using linearly stable steady states of the classical Gray-Scott model as initial conditions. (b) Numerical results for the variational Gray-Scott model with $\epsilon = 10^{-4}$ using linearly unstable steady states of the classical Gray-Scott model as initial conditions.
  • ...and 8 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof