February  2022, 15(1): 27-48. doi: 10.3934/krm.2021043

A kinetic chemotaxis model with internal states and temporal sensing

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received  August 2020 Revised  September 2021 Published  February 2022 Early access  December 2021

Fund Project: The author is supported by the Hong Kong Research Grant Council General Research Fund No. PolyU 153055/18P (Project ID: P0005472)

By employing the Fourier transform to derive key a priori estimates for the temporal gradient of the chemical signal, we establish the existence of global solutions and hydrodynamic limit of a chemotactic kinetic model with internal states and temporal gradient in one dimension, which is a system of two transport equations coupled to a parabolic equation proposed in [4].

Citation: Zhi-An Wang. A kinetic chemotaxis model with internal states and temporal sensing. Kinetic and Related Models, 2022, 15 (1) : 27-48. doi: 10.3934/krm.2021043
References:
[1]

W. Alt, Biased random walk model for chemotaxis and related diffusion approximation, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

S. BlockJ. Segall and H. Berg, Adaption kinetics in bacterial chemotaxis, J. Bacteriology, 154 (1983), 312-323.  doi: 10.1128/jb.154.1.312-323.1983.

[3]

F. A. C. C. 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.

[4]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanics, J. Math. Biol., 51 (2005), 595-615.  doi: 10.1007/s00285-005-0334-6.

[5]

C. EmakoL. Neves De Almeida and N. Vauchelet, Existence and diffusive limit of a two-species kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359-380.  doi: 10.3934/krm.2015.8.359.

[6]

R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1239-1260.  doi: 10.3934/dcdsb.2006.6.1239.

[7]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.  doi: 10.1137/S0036139903433232.

[8]

R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in $E$. coli: A paradigm for multiscale modeling in biology, Multiscale Model. Simul., 3 (2005), 362-394.  doi: 10.1137/040603565.

[9]

R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.  doi: 10.1007/s00285-007-0070-1.

[10]

R. M. Ford and D. A. Lauffenburger, Measurement of bacterial random motility and chemotaxis coefficients: II. Application of single cell based mathematical model, Biotechnol. Bioeng., 37 (1991), 661-672.  doi: 10.1002/bit.260370708.

[11]

T. Hillen, ${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.  doi: 10.1007/s00285-006-0017-y.

[12]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1055-1080.  doi: 10.3934/dcdsb.2010.14.1055.

[13]

H. J. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.

[14]

H. J. HwangK. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM J. Math. Anal., 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.

[15]

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.

[16]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.  doi: 10.1007/s00030-012-0155-4.

[17]

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

[18]

J. Liao, Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit, J. Differential Equations, 259 (2015), 6432-6458.  doi: 10.1016/j.jde.2015.07.025.

[19]

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.

[20]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1122-1250.  doi: 10.1137/S0036139900382772.

[21]

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

[22]

B. PerthameM. Tang and N. Vauchelet, Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway, J. Math. Biol., 73 (2016), 1161-1178.  doi: 10.1007/s00285-016-0985-5.

[23]

B. PerthameN. Vauchelet and Z. Wang, The flux limited {K}eller-{S}egel system; properties and derivation from kinetic equations, Rev. Mat. Iberoam., 36 (2020), 357-386.  doi: 10.4171/rmi/1132.

[24]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372.  doi: 10.1002/zamm.19920720813.

[25] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[26]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.  doi: 10.1073/pnas.1101996108.

[27]

G. SiM. Tang and X. Yang, A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and its hyperbolic and parabolic limits, Multiscale Model. Simul., 12 (2014), 907-926.  doi: 10.1137/130944199.

[28]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28 (1973/74), 303-315.  doi: 10.1007/BF00532948.

[29]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis, Kinet. Relat. Models, 3 (2010), 501-528.  doi: 10.3934/krm.2010.3.501.

[30]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

show all references

References:
[1]

W. Alt, Biased random walk model for chemotaxis and related diffusion approximation, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

S. BlockJ. Segall and H. Berg, Adaption kinetics in bacterial chemotaxis, J. Bacteriology, 154 (1983), 312-323.  doi: 10.1128/jb.154.1.312-323.1983.

[3]

F. A. C. C. 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.

[4]

Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanics, J. Math. Biol., 51 (2005), 595-615.  doi: 10.1007/s00285-005-0334-6.

[5]

C. EmakoL. Neves De Almeida and N. Vauchelet, Existence and diffusive limit of a two-species kinetic model of chemotaxis, Kinetic and Related Models, 8 (2015), 359-380.  doi: 10.3934/krm.2015.8.359.

[6]

R. Erban and H. J. Hwang, Global existence results for complex hyperbolic models of bacterial chemotaxis, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1239-1260.  doi: 10.3934/dcdsb.2006.6.1239.

[7]

R. Erban and H. G. Othmer, From individual to collective behavior in bacterial chemotaxis, SIAM J. Appl. Math., 65 (2004), 361-391.  doi: 10.1137/S0036139903433232.

[8]

R. Erban and H. G. Othmer, From signal transduction to spatial pattern formation in $E$. coli: A paradigm for multiscale modeling in biology, Multiscale Model. Simul., 3 (2005), 362-394.  doi: 10.1137/040603565.

[9]

R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.  doi: 10.1007/s00285-007-0070-1.

[10]

R. M. Ford and D. A. Lauffenburger, Measurement of bacterial random motility and chemotaxis coefficients: II. Application of single cell based mathematical model, Biotechnol. Bioeng., 37 (1991), 661-672.  doi: 10.1002/bit.260370708.

[11]

T. Hillen, ${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.  doi: 10.1007/s00285-006-0017-y.

[12]

T. HillenP. Hinow and Z.-A. Wang, Mathematical analysis of a kinetic model for cell movement in network tissues, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1055-1080.  doi: 10.3934/dcdsb.2010.14.1055.

[13]

H. J. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.

[14]

H. J. HwangK. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM J. Math. Anal., 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.

[15]

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.

[16]

F. James and N. Vauchelet, Chemotaxis: From kinetic equations to aggregate dynamics, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 101-127.  doi: 10.1007/s00030-012-0155-4.

[17]

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

[18]

J. Liao, Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit, J. Differential Equations, 259 (2015), 6432-6458.  doi: 10.1016/j.jde.2015.07.025.

[19]

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.

[20]

H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1122-1250.  doi: 10.1137/S0036139900382772.

[21]

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

[22]

B. PerthameM. Tang and N. Vauchelet, Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway, J. Math. Biol., 73 (2016), 1161-1178.  doi: 10.1007/s00285-016-0985-5.

[23]

B. PerthameN. Vauchelet and Z. Wang, The flux limited {K}eller-{S}egel system; properties and derivation from kinetic equations, Rev. Mat. Iberoam., 36 (2020), 357-386.  doi: 10.4171/rmi/1132.

[24]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372.  doi: 10.1002/zamm.19920720813.

[25] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001.  doi: 10.1007/978-94-010-0732-0.
[26]

J. SaragostiV. CalvezN. BournaveasB. PerthameA. Buguin and P Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.  doi: 10.1073/pnas.1101996108.

[27]

G. SiM. Tang and X. Yang, A pathway-based mean-field model for E. coli chemotaxis: Mathematical derivation and its hyperbolic and parabolic limits, Multiscale Model. Simul., 12 (2014), 907-926.  doi: 10.1137/130944199.

[28]

D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 28 (1973/74), 303-315.  doi: 10.1007/BF00532948.

[29]

N. Vauchelet, Numerical simulation of a kinetic model for chemotaxis, Kinet. Relat. Models, 3 (2010), 501-528.  doi: 10.3934/krm.2010.3.501.

[30]

C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM J. Appl. Math., 70 (2009), 133-167.  doi: 10.1137/070711505.

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