Nonlinear subdiffusive fractional equations and aggregation phenomenon

TL;DR

This paper develops nonlinear fractional equations for subdiffusive particles with density-dependent characteristics, revealing a transition from subdiffusive to normal diffusion and highlighting the impact of anomalous exponents on aggregation.

Contribution

It introduces a new nonlinear subdiffusive fractional model with density-dependent parameters and analyzes the transition to normal diffusion, extending previous linear models.

Findings

Transition from subdiffusive to normal diffusion regimes

Nonuniform anomalous exponents significantly influence aggregation

Model applicable to biological cell and population dynamics

Abstract

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean field density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.