Base state of growing reaction-dilution systems exhibiting Turing patterns

Aldo Ledesma-Durán; Consuelo García-Alcántara; Iván Santamaría-Holek; Aldo Ledesma-Durán; Consuelo García-Alcántara; Iván Santamaría-Holek

doi:10.3934/mbe.2026005

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 1: 97-123. doi: 10.3934/mbe.2026005

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Base state of growing reaction-dilution systems exhibiting Turing patterns

Unidad Multidisciplinaria de Docencia e Investigación, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Juriquilla, 76230, Querétaro, Mexico

Received: 29 July 2025 Revised: 29 October 2025 Accepted: 21 November 2025 Published: 28 November 2025

Turing pattern formation in growing domains depends on a steady state that balances reaction rates and local volume changes, leading to more complex patterning conditions than in fixed domains. We analyzed the effects of domain growth and shrinkage on spatially homogeneous concentrations and their stability, demonstrating that long-term behavior depends on the growth type: exponential growth causes asymptotic deviations, linear and quadratic growth enable gradual recovery of the fixed-domain state, and oscillatory growth induces concentration oscillations. Using a linear approximation for the base state, we derived an analytic expression that accurately predicts these effects for slow domain variations. Our theoretical model shows that dilution-induced steady states evolve proportionally to the chemical fixed-point concentration, a result validated through extensive numerical simulations of the Brusselator and BVAM reactions. Additionally, we proposed an approximate framework for evaluating the stability of spatially homogeneous perturbations, interpreting it as a balance between reaction rates and dilution. This yielded an analytical criterion for determining stability in the absence of diffusion, offering an alternative to previously exclusive numerical approaches for identifying the first Turing condition for pattern formation.
- Turing pattern,
- growing domain,
- Lyapunov function,
- base state,
- homogeneous
Citation: Aldo Ledesma-Durán, Consuelo García-Alcántara, Iván Santamaría-Holek. Base state of growing reaction-dilution systems exhibiting Turing patterns[J]. Mathematical Biosciences and Engineering, 2026, 23(1): 97-123. doi: 10.3934/mbe.2026005

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Abstract

Turing pattern formation in growing domains depends on a steady state that balances reaction rates and local volume changes, leading to more complex patterning conditions than in fixed domains. We analyzed the effects of domain growth and shrinkage on spatially homogeneous concentrations and their stability, demonstrating that long-term behavior depends on the growth type: exponential growth causes asymptotic deviations, linear and quadratic growth enable gradual recovery of the fixed-domain state, and oscillatory growth induces concentration oscillations. Using a linear approximation for the base state, we derived an analytic expression that accurately predicts these effects for slow domain variations. Our theoretical model shows that dilution-induced steady states evolve proportionally to the chemical fixed-point concentration, a result validated through extensive numerical simulations of the Brusselator and BVAM reactions. Additionally, we proposed an approximate framework for evaluating the stability of spatially homogeneous perturbations, interpreting it as a balance between reaction rates and dilution. This yielded an analytical criterion for determining stability in the absence of diffusion, offering an alternative to previously exclusive numerical approaches for identifying the first Turing condition for pattern formation.

References

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