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Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems.

Anotida Madzvamuse1, Andy H W Chung1, Chandrasekhar Venkataraman1

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Summary

New models for coupled bulk-surface reaction-diffusion systems reveal that bulk dynamics can drive surface patterning. Surface dynamics alone cannot pattern the entire bulk, highlighting the crucial role of boundary conditions.

Keywords:
Robin-type boundary conditionsTuring diffusively driven instabilitybulk-surface finite-elementsbulk-surface reaction–diffusion equationslinear stabilitypattern formation

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Area of Science:

  • Mathematical Biology
  • Chemical Engineering
  • Physical Chemistry

Background:

  • Reaction-diffusion systems are fundamental to understanding pattern formation in biological and chemical systems.
  • Coupling bulk and surface dynamics introduces complex behaviors and analytical challenges.
  • Understanding diffusion-driven instability is key to predicting pattern formation.

Purpose of the Study:

  • To develop and analyze new mathematical models for coupled bulk-surface reaction-diffusion systems.
  • To investigate the conditions for diffusion-driven instability in these coupled systems.
  • To explore the influence of boundary conditions on pattern formation.

Main Methods:

  • Formulation of novel mathematical models for coupled bulk-surface reaction-diffusion equations.
  • Analytical investigation of necessary conditions for diffusion-driven instability.
  • Decoupling of stability analysis for bulk and surface dynamics.
  • Numerical simulations to validate theoretical findings.

Main Results:

  • The bulk reaction-diffusion system can induce surface patterning independently of surface dynamics.
  • Surface reaction-diffusion dynamics alone cannot generate bulk patterns universally; patterns are localized near the surface.
  • Robin-type boundary conditions create a boundary layer that couples bulk and surface dynamics.

Conclusions:

  • The interplay between bulk and surface dynamics is crucial for pattern formation.
  • Boundary conditions significantly influence pattern localization and stability.
  • The developed models provide a framework for studying complex spatio-temporal patterns in coupled systems.