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

Wenrui Hao1, Chun Liu2, Yiwei Wang3

  • 1Department of Mathematics, Pennsylvania State University, University Park, 16802, PA, United States.

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|May 2, 2025
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Summary
This summary is machine-generated.

This study introduces a four-species variational Gray-Scott model, enhancing pattern formation analysis. The model stabilizes classical patterns and reveals new oscillating and traveling-wave dynamics for small reaction rates.

Keywords:
Energetic variational approachPattern formationReaction-diffusion systemsTransient patternVariational Gray-Scott models

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Area of Science:

  • Chemical kinetics
  • Pattern formation
  • Computational modeling

Background:

  • The classical Gray-Scott model is a widely studied reaction-diffusion system known for generating complex spatial patterns.
  • Existing models often lack thermodynamic consistency and do not fully account for birth-death processes.
  • A need exists for a more complete and thermodynamically sound framework to study pattern formation.

Purpose of the Study:

  • To introduce and analyze a four-species variational Gray-Scott model that incorporates reverse reactions and a virtual species.
  • To investigate pattern formation in this enhanced model, particularly how it relates to the classical model.
  • To explore the stability and dynamics of steady states and emergent patterns within the variational framework.

Main Methods:

  • Development of a four-species variational Gray-Scott model, ensuring thermodynamic consistency.
  • Numerical exploration of pattern formation in one spatial dimension.
  • Analysis of the influence of a small parameter ϵ (related to reverse reaction rates) on pattern stability and dynamics.
  • Investigation of energy stability for uniform steady states.

Main Results:

  • Stationary patterns from the classical Gray-Scott model can be stabilized as transient states in the variational model for small ϵ.
  • The variational model exhibits novel oscillating and traveling-wave-like patterns for small ϵ, with persistence times of O(ϵ-1).
  • Energy stability analysis reveals that while two uniform steady states are stable, the model exhibits initial-condition-dependent pattern selection.

Conclusions:

  • The four-species variational Gray-Scott model provides a more physically complete and thermodynamically consistent framework for studying reaction-diffusion systems.
  • This enhanced model expands the observable dynamics beyond the classical model, including stable transient patterns and new wave phenomena.
  • The model's gradient flow dynamics demonstrate a selection mechanism influencing pattern emergence based on initial conditions.