Title: 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.