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

Abstract

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.