March  2017, 10(1): 313-327. doi: 10.3934/krm.2017013

Aggregated steady states of a kinetic model for chemotaxis

1. 

Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France

2. 

University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

*

Received  January 2016 Revised  July 2016 Published  November 2016

Fund Project: The second author is supported by the Austrian Science Fund grant no. W1245.

A kinetic chemotaxis model with attractive interaction by quasistationary chemical signalling is considered. The special choice of the turning operator, with velocity jumps biased towards the chemical concentration gradient, permits closed ODE systems for moments of the distribution function of arbitrary order. The system for second order moments exhibits a critical mass phenomeneon. The main result is existence of an aggregated steady state for supercritical mass.

Citation: Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic & Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013
References:
[1]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, J. Diff. Equ., 44 (2006), 32pp.  Google Scholar

[2]

N. Bournaveas and V. Calvez, Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, Ann. de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 1871-1895.  doi: 10.1016/j.anihpc.2009.02.001.  Google Scholar

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V. CalvezG. Raoul and C. Schmeiser, Confinement by biased velocity jumps: Aggregation of Escheria coli, Kinetic and Related Models, 8 (2015), 651-666.  doi: 10.3934/krm.2015.8.651.  Google Scholar

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F. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

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J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS, 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

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J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, DCDS-A, 25 (2009), 109-121.  doi: 10.3934/dcds.2009.25.109.  Google Scholar

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H.J. HwangK. Kang and A. Stevens, Global existence of classical solutions for a hyperbolic chemotaxis model and its parabolic limit, Indiana Univ. Math. J., 55 (2006), 289-316.  doi: 10.1512/iumj.2006.55.2677.  Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

N. MittalE. O. BudreneM. P. Brenner and A. van Oudenaarden, Motility of Escheria coli cells in clusters formed by chemotactic aggregation, PNAS, 100 (2003), 13259-13263.   Google Scholar

[10]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[11]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[12]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses PLoS Comput. Biol., 6 (2010), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.  Google Scholar

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J. J. L. Velazquez, Point dynamics in a singular limit of the Keller-Segel model 1: Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223.  doi: 10.1137/S0036139903433888.  Google Scholar

show all references

References:
[1]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, J. Diff. Equ., 44 (2006), 32pp.  Google Scholar

[2]

N. Bournaveas and V. Calvez, Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, Ann. de l'Institut Henri Poincare (C) Non Linear Analysis, 26 (2009), 1871-1895.  doi: 10.1016/j.anihpc.2009.02.001.  Google Scholar

[3]

V. CalvezG. Raoul and C. Schmeiser, Confinement by biased velocity jumps: Aggregation of Escheria coli, Kinetic and Related Models, 8 (2015), 651-666.  doi: 10.3934/krm.2015.8.651.  Google Scholar

[4]

F. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.  Google Scholar

[5]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS, 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.  Google Scholar

[6]

J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, DCDS-A, 25 (2009), 109-121.  doi: 10.3934/dcds.2009.25.109.  Google Scholar

[7]

H.J. HwangK. Kang and A. Stevens, Global existence of classical solutions for a hyperbolic chemotaxis model and its parabolic limit, Indiana Univ. Math. J., 55 (2006), 289-316.  doi: 10.1512/iumj.2006.55.2677.  Google Scholar

[8]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[9]

N. MittalE. O. BudreneM. P. Brenner and A. van Oudenaarden, Motility of Escheria coli cells in clusters formed by chemotactic aggregation, PNAS, 100 (2003), 13259-13263.   Google Scholar

[10]

H. G. OthmerS. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.  Google Scholar

[11]

C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[12]

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses PLoS Comput. Biol., 6 (2010), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.  Google Scholar

[13]

J. J. L. Velazquez, Point dynamics in a singular limit of the Keller-Segel model 1: Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223.  doi: 10.1137/S0036139903433888.  Google Scholar

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