Estrada-Rodriguez, Gissell; Gimperlein, Heiko; Painter, Kevin J. Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion. (English) Zbl 1390.92024 SIAM J. Appl. Math. 78, No. 2, 1155-1173 (2018). Summary: The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Lévy 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 behavior 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. Cited in 17 Documents MSC: 92C17 Cell movement (chemotaxis, etc.) 35R11 Fractional partial differential equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences Keywords:chemotaxis; Patlak-Keller-Segel equation; velocity jump model; nonlocal diffusion; Lévy walk; cell motility Software:DLMF PDFBibTeX XMLCite \textit{G. Estrada-Rodriguez} et al., SIAM J. Appl. Math. 78, No. 2, 1155--1173 (2018; Zbl 1390.92024) Full Text: DOI arXiv References: [1] W. Alt, {\it Biased random walk models for chemotaxis and related diffusion approximations}, J. Math. Biol., 9 (1980), pp. 147-177. · Zbl 0434.92001 [2] W. Alt, {\it Singular perturbation of differential integral equations describing biased random walks}, J. Reine Angew. Math., 322 (1981), pp. 15-41. · Zbl 0437.60059 [3] G. Ariel, A. Rabani, S. Benisty, J. D. Partridge, R. M. Harshey, and A. Be’Er, {\it Swarming bacteria migrate by Lévy walk}, Nature Commun., 6 (2015). [4] M. D. Baker, P. M. Wolanin, and J. B. 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