Transport Equations for Subdiffusion with Nonlinear Particle Interaction

TL;DR

This paper derives nonlinear, non-Markovian transport equations for subdiffusive cell movement, revealing how volume filling can prevent anomalous aggregation in systems with spatially varying anomalous exponents.

Contribution

It introduces a systematic derivation of nonlinear fractional transport equations incorporating volume filling and adhesion effects in subdiffusive systems.

Findings

Nonlinear equations combine fractional derivatives with nonlinear terms.

Long-time limit equations simplify to non-fractional forms.

Volume filling prevents anomalous aggregation in subdiffusive systems.

Abstract

We show how the nonlinear interaction effects `volume filling' and `adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.