Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Area of Science:

• Mathematical Modeling
• Physical Chemistry
• Biophysics

Background:

• Reaction-diffusion equations model diverse physical, chemical, and biological systems.
• Evolving domains, common in biology (e.g., tumor growth), present modeling challenges.
• Existing models often lack analytic solutions for complex geometries.

Purpose of the Study:

• Derive reaction-diffusion equations for transport with reactions on evolving 1D domains.
• Incorporate subdiffusive transport and inhomogeneous domain dynamics.
• Develop analytic methods for short-time moments and validate against simulations.

Main Methods:

• Generalized continuous time random walks to derive model equations.
• Analytic expression construction for short-time particle position moments.
• Comparison with random walk simulations and numerical integration of reaction transport equations.

Main Results:

• Model equations successfully incorporate subdiffusion and inhomogeneous domain evolution.
• Analytic method for short-time moments shows favorable agreement with simulations.
• Initial conditions significantly impact short-time dynamics, introducing drift and diffusion terms.

Conclusions:

• The derived reaction-diffusion equations offer analytic insights into systems with evolving geometries.
• Findings address the scarcity of analytic results for non-uniformly growing domains.
• The approach is applicable to population spreading on evolving interfaces and future first-passage problems.