The fractional Keller–Segel model

The Keller–Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing-up solutions for large enough initial conditions in dimensions d ≥ 2, but all the solutions are regular in one dimension, a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller–Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyse this fractional Keller–Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller–Segel model.

[1]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[2]  H. Berg,et al.  Complex patterns formed by motile cells of Escherichia coli , 1991, Nature.

[3]  Collapsing bacterial cylinders. , 1999, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  H. Berg,et al.  Dynamics of formation of symmetrical patterns by chemotactic bacteria , 1995, Nature.

[5]  Yin Tintut,et al.  Pattern formation by vascular mesenchymal cells. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[6]  T. Hillen,et al.  The one‐dimensional chemotaxis model: global existence and asymptotic profile , 2004 .

[7]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[8]  F. Bartumeus,et al.  Helical Lévy walks: Adjusting searching statistics to resource availability in microzooplankton , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[9]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[10]  M. Brenner,et al.  Physical mechanisms for chemotactic pattern formation by bacteria. , 1998, Biophysical journal.

[11]  M. Levandowsky,et al.  Random movements of soil amebas , 1997 .

[12]  J. G. Skellam Random dispersal in theoretical populations , 1951, Biometrika.

[13]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[14]  Miguel A. Herrero,et al.  Self-similar blow-up for a reaction-diffusion system , 1998 .

[15]  Zumofen,et al.  Absorbing boundary in one-dimensional anomalous transport. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Leo P. Kadanoff,et al.  Diffusion, attraction and collapse , 1999 .

[17]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[18]  Hans G. Othmer,et al.  Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[19]  Dirk Horstmann,et al.  Boundedness vs. blow-up in a chemotaxis system , 2005 .

[20]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[21]  H. Zaag One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations , 2002 .

[22]  H. Bhadeshia Diffusion , 1995, Theory of Transformations in Steels.

[23]  Katja Lindenberg,et al.  Extinction in population dynamics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Miguel A. Herrero,et al.  Finite-time aggregation into a single point in a reaction - diffusion system , 1997 .

[25]  J. J. L. Velázquez,et al.  Stability of Some Mechanisms of Chemotactic Aggregation , 2002, SIAM J. Appl. Math..

[26]  P. A. Robinson,et al.  LEVY RANDOM WALKS IN FINITE SYSTEMS , 1998 .

[27]  C. Escudero Chemotactic collapse and mesenchymal morphogenesis. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  M. A. Herrero,et al.  Singularity patterns in a chemotaxis model , 1996 .

[29]  J. Klafter,et al.  Microzooplankton Feeding Behavior and the Levy Walk , 1990 .