Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion

TL;DR

This paper derives fractional Patlak-Keller-Segel equations from a microscopic model to describe chemotactic superdiffusion, capturing long-range organism movement influenced by Levy distributions.

Contribution

It introduces a novel derivation of nonlocal fractional chemotactic equations from microscopic run-time distributions, linking individual movement to macroscopic models.

Findings

Derived fractional chemotactic equations from kinetic models.

Connected Levy-distributed runs to nonlocal PDEs.

Applicable to various biological diffusion processes.

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.