Base state of growing reaction-dilution systems exhibiting Turing patterns

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.