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Chemical reactions often occur in a stepwise fashion, involving two or more distinct reactions taking place in a sequence. A balanced equation indicates the reacting species and the product species, but it reveals no details about how the reaction occurs at the molecular level. The reaction mechanism (or reaction path) provides details regarding the precise, step-by-step process by which a reaction occurs.
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Base state of growing reaction-dilution systems exhibiting Turing patterns.

Aldo Ledesma-Durán1, Consuelo García-Alcántara1, Iván Santamaría-Holek1

  • 1Unidad Multidisciplinaria de Docencia e Investigación, Universidad Nacional Autónoma de México, Boulevard Juriquilla 3001, Juriquilla, 76230, Querétaro, Mexico.

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Turing pattern formation in growing domains is complex. Study shows domain growth type dictates pattern stability and concentration dynamics, offering new analytical criteria for pattern formation.

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Lyapunov functionTuring patternbase stategrowing domainhomogeneous

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Area of Science:

  • Chemical kinetics and reaction-diffusion systems
  • Mathematical modeling of biological and chemical pattern formation

Background:

  • Turing pattern formation typically studied in fixed domains.
  • Growing or shrinking domains introduce dynamic volume changes affecting reaction-diffusion systems.
  • Understanding pattern stability in dynamic environments is crucial for biological morphogenesis.

Purpose of the Study:

  • Analyze the impact of domain growth and shrinkage on Turing pattern formation.
  • Investigate the stability of spatially homogeneous concentrations under varying domain dynamics.
  • Develop analytical models to predict pattern behavior in growing domains.

Main Methods:

  • Linear approximation of the base state in dynamic domains.
  • Derivation of analytic expressions for concentration stability.
  • Numerical simulations of the Brusselator and BVAM reaction models.

Main Results:

  • Domain growth type significantly influences long-term concentration behavior (exponential, linear, quadratic, oscillatory).
  • Dilution-induced steady states are proportional to the chemical fixed-point concentration.
  • An analytical criterion for stability in the absence of diffusion was derived.

Conclusions:

  • Domain growth dynamics introduce complex patterning conditions compared to fixed domains.
  • The derived analytical framework accurately predicts pattern behavior in slow domain variations.
  • The study provides a novel analytical approach to identify Turing instability conditions.