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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[11]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[11]

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

[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.

[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.

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