Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion

Abstract

The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy distribution. This article clarifies the form of biologically relevant model equations: We derive Patlak-Keller-Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behaviour of the individuals. A fractional chemotactic equation is obtained in a biologically relevant regime. Apart from chemotaxis, our work has implications for biological diffusion in numerous processes.

20 pages, 4 figures, to appear in SIAM Journal on Applied Mathematics

References in corpus (4)

• Lévy walks
• Persistent Cell Motion in the Absence of External Signals: A Search Strategy for Eukaryotic Cells
• Emergence of Lévy Walks in Systems of Interacting Individuals
• A Fractional Diffusion Equation for an n-Dimensional Correlated Levy Walk

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• The fractional diffusion limit of a kinetic model with biochemical pathway
• Distribution and pressure of active Lévy swimmers under confinement
• Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices