Modeling Subdiffusion Using Reaction Diffusion Systems

Abstract

In many types of media, particularly within living cells or within their membranes, diffusing species do not follow Fick's laws but instead show transient subdiffusive behavior. Formulating spatiotemporal models that take this behavior into account is a delicate matter, as one is faced with the choice of resorting either to fractional calculus or to microscopic descriptions. In this article, we provide an alternative designed to be easier to handle analytically and numerically than the existing approaches. Specifically, starting from the continuous time random walk model, we construct linear reaction diffusion systems that can be used as components within such a model and which capture the defining properties of subdiffusion. We show how to impose physically relevant parameters, and prove stability and mass conservation. While applications to cellular biology are our main motivation, our approach is abstract and should thus be applicable to any situation where anomalous subdiffusion is observed.