Nonlinear subdiffusive fractional equations and aggregation phenomenon

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.

10 pages

References in corpus (4)

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• Generalized Elastic Model: Fractional Langevin Description, Fluctuation Relation, and Linear Response
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• Nonlinear degradation enhanced transport of morphogens performing subdiffusion
• A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk
• Anomalous metapopulation dynamics on scale-free networks
• Self-organized anomalous aggregation of particles performing nonlinear and non-Markovian random walk
• A non-linear subdiffusion model for a cell-cell adhesion in chemotaxis
• Density-Dependence Subdiffusion in Chemotaxis
• Anomalous stochastic transport of particles with self-reinforcement and Mittag-Leffler distributed rest times