June  2015, 8(2): 359-380. doi: 10.3934/krm.2015.8.359

Existence and diffusive limit of a two-species kinetic model of chemotaxis

1. 

Sorbonne Universites, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France

2. 

Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  April 2014 Revised  January 2015 Published  March 2015

In this paper, we propose a kinetic model describing the collective motion by chemotaxis of two species in interaction emitting the same chemoattractant. Such model can be seen as a generalisation to several species of the Othmer-Dunbar-Alt model which takes into account the run-and-tumble process of bacteria. Existence of weak solutions for this two-species kinetic model is studied and the convergence of its diffusive limit towards a macroscopic model of Keller-Segel type is analysed.
Citation: Casimir Emako, Luís Neves de Almeida, Nicolas Vauchelet. Existence and diffusive limit of a two-species kinetic model of chemotaxis. Kinetic & Related Models, 2015, 8 (2) : 359-380. doi: 10.3934/krm.2015.8.359
References:
[1]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multicellular biological growing systems: Hyperbolic limits towards macroscopic description,, Math. Models Methods Appl. Sci., 17 (2007), 1675.  doi: 10.1142/S0218202507002431.  Google Scholar

[2]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1179.  doi: 10.1142/S0218202510004568.  Google Scholar

[3]

N. Bournaveas, V. Calvez, S. Gutiérrez and B. Perthame, Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates,, Comm. Partial Differential Equations, 33 (2008), 79.  doi: 10.1080/03605300601188474.  Google Scholar

[4]

H. Brezis, Analyse fonctionnelle,, Théorie et Applications [Theory and Applications], (1983).   Google Scholar

[5]

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

[6]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinet. Relat. Models, 5 (2012), 51.  doi: 10.3934/krm.2012.5.51.  Google Scholar

[7]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European J. Appl. Math., 24 (2013), 297.  doi: 10.1017/S0956792511000258.  Google Scholar

[8]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species,, Nonlinearity, 26 (2013), 2777.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[9]

C. Di Russo, R. Natalini and M. Ribot, Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis,, Commun. Appl. Ind. Math., 1 (2010), 92.   Google Scholar

[10]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595.  doi: 10.1007/s00285-005-0334-6.  Google Scholar

[11]

B. Elena and B. Howard, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 6 (1995).   Google Scholar

[12]

E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European J. Appl. Math., 24 (2013), 297.  doi: 10.1017/S0956792512000411.  Google Scholar

[13]

E. E. Espejo Arenas, A. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis,, Analysis (Munich), 29 (2009), 317.  doi: 10.1524/anly.2009.1029.  Google Scholar

[14]

L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[15]

A. Fasano, A. Mancini and M. Primicerio, Equilibruim of two populations subject to chemotaxis,, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503.  doi: 10.1142/S0218202504003337.  Google Scholar

[16]

F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[17]

J. Greenberg and W. Alt, Stability results for a diffusion equation with functional drift approximating a chemotaxis model,, Trans. Amer. Math. Soc., 300 (1987), 235.  doi: 10.1090/S0002-9947-1987-0871674-4.  Google Scholar

[18]

T. Hillen, Hyperbolic models for chemosensitive movement,, Math. Models Methods Appl. Sci., 12 (2002), 1007.  doi: 10.1142/S0218202502002008.  Google Scholar

[19]

T. Hillen and A. Stevens, Hyperbolic models for chemotaxis in 1-D,, Nonlinear Anal. Real World Appl., 1 (2000), 409.  doi: 10.1016/S0362-546X(99)00284-9.  Google Scholar

[20]

T. Hofer, J. Sherratt and P. Maini, Dictyostelium discoideum: Cellular self-organization in an excitable biological medium,, Proceedings of the Royal Society of London. Series B: Biological Sciences, 259 (1995), 249.  doi: 10.1098/rspb.1995.0037.  Google Scholar

[21]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species,, J. Nonlinear Sci., 21 (2011), 231.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[22]

B. Howard, E. coli in Motion,, Biological and Medical Physics, (2004).   Google Scholar

[23]

H. J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement,, SIAM J. Math. Anal., 36 (2005), 1177.  doi: 10.1137/S0036141003431888.  Google Scholar

[24]

H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

[25]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101.  doi: 10.1007/s00030-012-0155-4.  Google Scholar

[26]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[27]

E. Keller and L. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[28]

A. Kurganov and M. Lukacova-Medvidova, Numerical study of two-species chemotaxis models,, Discrete and Continuous Dynamical Systems. Series B, 19 (2014), 131.  doi: 10.3934/dcdsb.2014.19.131.  Google Scholar

[29]

T.-C. Lin and Z.-A. Wang, Development of traveling waves in an interacting two-species chemotaxis model,, Discrete Continuous Dynamical Systems Series A, 34 (2014), 2907.  doi: 10.3934/dcds.2014.34.2907.  Google Scholar

[30]

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

[31]

S. Park, P. Wolanin, E. Yuzbashyan, H. Lin, N. Darnton, J. Stock, P. Silberzan and R. Austin, Influence of topology on bacterial social interaction,, Proceedings of the National Academy of Sciences, 100 (2003), 13910.  doi: 10.1073/pnas.1935975100.  Google Scholar

[32]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[33]

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

[34]

J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave,, Proceedings of the National Academy of Sciences, 108 (2011), 16235.  doi: 10.1073/pnas.1101996108.  Google Scholar

[35]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[36]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis,, Kinet. Relat. Models, 3 (2010), 501.  doi: 10.3934/krm.2010.3.501.  Google Scholar

[37]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts,, European J. Appl. Math., 13 (2002), 641.  doi: 10.1017/S0956792501004843.  Google Scholar

show all references

References:
[1]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multicellular biological growing systems: Hyperbolic limits towards macroscopic description,, Math. Models Methods Appl. Sci., 17 (2007), 1675.  doi: 10.1142/S0218202507002431.  Google Scholar

[2]

N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1179.  doi: 10.1142/S0218202510004568.  Google Scholar

[3]

N. Bournaveas, V. Calvez, S. Gutiérrez and B. Perthame, Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates,, Comm. Partial Differential Equations, 33 (2008), 79.  doi: 10.1080/03605300601188474.  Google Scholar

[4]

H. Brezis, Analyse fonctionnelle,, Théorie et Applications [Theory and Applications], (1983).   Google Scholar

[5]

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

[6]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinet. Relat. Models, 5 (2012), 51.  doi: 10.3934/krm.2012.5.51.  Google Scholar

[7]

C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European J. Appl. Math., 24 (2013), 297.  doi: 10.1017/S0956792511000258.  Google Scholar

[8]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species,, Nonlinearity, 26 (2013), 2777.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[9]

C. Di Russo, R. Natalini and M. Ribot, Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis,, Commun. Appl. Ind. Math., 1 (2010), 92.   Google Scholar

[10]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms,, J. Math. Biol., 51 (2005), 595.  doi: 10.1007/s00285-005-0334-6.  Google Scholar

[11]

B. Elena and B. Howard, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 6 (1995).   Google Scholar

[12]

E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$,, European J. Appl. Math., 24 (2013), 297.  doi: 10.1017/S0956792512000411.  Google Scholar

[13]

E. E. Espejo Arenas, A. Stevens and J. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis,, Analysis (Munich), 29 (2009), 317.  doi: 10.1524/anly.2009.1029.  Google Scholar

[14]

L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[15]

A. Fasano, A. Mancini and M. Primicerio, Equilibruim of two populations subject to chemotaxis,, Mathematical Models and Methods in Applied Sciences, 14 (2004), 503.  doi: 10.1142/S0218202504003337.  Google Scholar

[16]

F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189.  doi: 10.1007/s00285-004-0286-2.  Google Scholar

[17]

J. Greenberg and W. Alt, Stability results for a diffusion equation with functional drift approximating a chemotaxis model,, Trans. Amer. Math. Soc., 300 (1987), 235.  doi: 10.1090/S0002-9947-1987-0871674-4.  Google Scholar

[18]

T. Hillen, Hyperbolic models for chemosensitive movement,, Math. Models Methods Appl. Sci., 12 (2002), 1007.  doi: 10.1142/S0218202502002008.  Google Scholar

[19]

T. Hillen and A. Stevens, Hyperbolic models for chemotaxis in 1-D,, Nonlinear Anal. Real World Appl., 1 (2000), 409.  doi: 10.1016/S0362-546X(99)00284-9.  Google Scholar

[20]

T. Hofer, J. Sherratt and P. Maini, Dictyostelium discoideum: Cellular self-organization in an excitable biological medium,, Proceedings of the Royal Society of London. Series B: Biological Sciences, 259 (1995), 249.  doi: 10.1098/rspb.1995.0037.  Google Scholar

[21]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species,, J. Nonlinear Sci., 21 (2011), 231.  doi: 10.1007/s00332-010-9082-x.  Google Scholar

[22]

B. Howard, E. coli in Motion,, Biological and Medical Physics, (2004).   Google Scholar

[23]

H. J. Hwang, K. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement,, SIAM J. Math. Anal., 36 (2005), 1177.  doi: 10.1137/S0036141003431888.  Google Scholar

[24]

H. J. Hwang, K. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319.  doi: 10.3934/dcdsb.2005.5.319.  Google Scholar

[25]

F. James and N. Vauchelet, Chemotaxis: from kinetic equations to aggregate dynamics,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101.  doi: 10.1007/s00030-012-0155-4.  Google Scholar

[26]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[27]

E. Keller and L. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[28]

A. Kurganov and M. Lukacova-Medvidova, Numerical study of two-species chemotaxis models,, Discrete and Continuous Dynamical Systems. Series B, 19 (2014), 131.  doi: 10.3934/dcdsb.2014.19.131.  Google Scholar

[29]

T.-C. Lin and Z.-A. Wang, Development of traveling waves in an interacting two-species chemotaxis model,, Discrete Continuous Dynamical Systems Series A, 34 (2014), 2907.  doi: 10.3934/dcds.2014.34.2907.  Google Scholar

[30]

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

[31]

S. Park, P. Wolanin, E. Yuzbashyan, H. Lin, N. Darnton, J. Stock, P. Silberzan and R. Austin, Influence of topology on bacterial social interaction,, Proceedings of the National Academy of Sciences, 100 (2003), 13910.  doi: 10.1073/pnas.1935975100.  Google Scholar

[32]

B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[33]

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

[34]

J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave,, Proceedings of the National Academy of Sciences, 108 (2011), 16235.  doi: 10.1073/pnas.1101996108.  Google Scholar

[35]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[36]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis,, Kinet. Relat. Models, 3 (2010), 501.  doi: 10.3934/krm.2010.3.501.  Google Scholar

[37]

G. Wolansky, Multi-components chemotactic system in the absence of conflicts,, European J. Appl. Math., 13 (2002), 641.  doi: 10.1017/S0956792501004843.  Google Scholar

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