doi: 10.3934/krm.2021043
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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 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 & Related Models, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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