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2015, 12(4): 717-737. doi: 10.3934/mbe.2015.12.717

Traveling bands for the Keller-Segel model with population growth

Shangbing Ai 1, and  Zhian Wang 2,

1. 

Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899
2. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China

Received  April 2014 Revised  November 2014 Published  April 2015

This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.
Keywords: traveling waves, minimal wave speed., cell kinetics, Keller-Segel model, Chemotaxis.
Mathematics Subject Classification: 35C07, 35K55, 46N60, 62P10, 92C1.
Citation: Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717
References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 44 (1975), 341.  doi: 10.1146/annurev.bi.44.070175.002013.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.  doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation,, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1.  doi: 10.3934/dcdsb.2015.20.1.  Google Scholar

[4]

J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical,, Dicrete Contin. Dyn. Syst., 34 (2014), 5165.  doi: 10.3934/dcds.2014.34.5165.  Google Scholar

[5]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis,, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141.  doi: 10.1016/S1631-073X(02)00008-0.  Google Scholar

[6]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in high space dimensions,, Milan j. Math., 72 (2004), 1.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[7]

M. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis,, SIAM J. Math. Anal., 33 (2002), 1330.  doi: 10.1137/S0036141001385046.  Google Scholar

[8]

M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model,, Interfaces Free Bound., 8 (2006), 223.  doi: 10.4171/IFB/141.  Google Scholar

[9]

H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity,, J. Differential Equations, 255 (2013), 193.  doi: 10.1016/j.jde.2013.04.002.  Google Scholar

[10]

Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis,, Biophysical Journal, 96 (2009), 2439.  doi: 10.1016/j.bpj.2008.10.027.  Google Scholar

[11]

E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[12]

C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397.  doi: 10.1007/BF02460793.  Google Scholar

[13]

I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria,, Biophy. J., 22 (1978), 1.  doi: 10.1016/S0006-3495(78)85466-6.  Google Scholar

[14]

D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth,, Bull. Math. Biol., 46 (1984), 19.   Google Scholar

[15]

H. Levine, B. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. i. the role of protease inhibitors in preventing angiogenesis,, Math. Biosci, 168 (2000), 71.  doi: 10.1016/S0025-5564(00)00034-1.  Google Scholar

[16]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci., 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar

[17]

J. Li, T. Li and Z. WAng, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity,, Math. Models Methods Appl. Sci., 24 (2014), 2819.  doi: 10.1142/S0218202514500389.  Google Scholar

[18]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[19]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967.  doi: 10.1142/S0218202510004830.  Google Scholar

[20]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310.  doi: 10.1016/j.jde.2010.09.020.  Google Scholar

[21]

R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, J. Math. Biol., 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

[22]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms,, Interfaces Free Bound., 10 (2008), 517.  doi: 10.4171/IFB/200.  Google Scholar

[23]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169.  doi: 10.1007/BF00160334.  Google Scholar

[24]

R. Nossal, Boundary movement of chemotactic bacterial population,, Math. Biosci., 13 (1972), 397.  doi: 10.1016/0025-5564(72)90058-2.  Google Scholar

[25]

G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotaxis bacteria,, J. Theor. Biol., 49 (1975), 311.   Google Scholar

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671.  doi: 10.1007/BF02460738.  Google Scholar

[27]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273.  doi: 10.1016/0025-5564(75)90080-2.  Google Scholar

[28]

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

[29]

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

[30]

H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476.  doi: 10.1002/pamm.200310508.  Google Scholar

[31]

C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (2006), 1694.  doi: 10.1137/060655122.  Google Scholar

[32]

Z. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. Wang, Mathematics of traveling waves in chemotaxis,, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

[34]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.  doi: 10.1002/mma.898.  Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Science, 44 (1975), 341.  doi: 10.1146/annurev.bi.44.070175.002013.  Google Scholar

[2]

J. Adler, Chemoreceptors in bacteria,, Science, 166 (1969), 1588.  doi: 10.1126/science.166.3913.1588.  Google Scholar

[3]

S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation,, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1.  doi: 10.3934/dcdsb.2015.20.1.  Google Scholar

[4]

J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical,, Dicrete Contin. Dyn. Syst., 34 (2014), 5165.  doi: 10.3934/dcds.2014.34.5165.  Google Scholar

[5]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis,, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141.  doi: 10.1016/S1631-073X(02)00008-0.  Google Scholar

[6]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis system in high space dimensions,, Milan j. Math., 72 (2004), 1.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[7]

M. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis,, SIAM J. Math. Anal., 33 (2002), 1330.  doi: 10.1137/S0036141001385046.  Google Scholar

[8]

M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model,, Interfaces Free Bound., 8 (2006), 223.  doi: 10.4171/IFB/141.  Google Scholar

[9]

H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity,, J. Differential Equations, 255 (2013), 193.  doi: 10.1016/j.jde.2013.04.002.  Google Scholar

[10]

Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis,, Biophysical Journal, 96 (2009), 2439.  doi: 10.1016/j.bpj.2008.10.027.  Google Scholar

[11]

E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis,, J. Theor. Biol., 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[12]

C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397.  doi: 10.1007/BF02460793.  Google Scholar

[13]

I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria,, Biophy. J., 22 (1978), 1.  doi: 10.1016/S0006-3495(78)85466-6.  Google Scholar

[14]

D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth,, Bull. Math. Biol., 46 (1984), 19.   Google Scholar

[15]

H. Levine, B. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. i. the role of protease inhibitors in preventing angiogenesis,, Math. Biosci, 168 (2000), 71.  doi: 10.1016/S0025-5564(00)00034-1.  Google Scholar

[16]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci., 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar

[17]

J. Li, T. Li and Z. WAng, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity,, Math. Models Methods Appl. Sci., 24 (2014), 2819.  doi: 10.1142/S0218202514500389.  Google Scholar

[18]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar

[19]

T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967.  doi: 10.1142/S0218202510004830.  Google Scholar

[20]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310.  doi: 10.1016/j.jde.2010.09.020.  Google Scholar

[21]

R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, J. Math. Biol., 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

[22]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms,, Interfaces Free Bound., 10 (2008), 517.  doi: 10.4171/IFB/200.  Google Scholar

[23]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model,, J. Math. Biol., 30 (1991), 169.  doi: 10.1007/BF00160334.  Google Scholar

[24]

R. Nossal, Boundary movement of chemotactic bacterial population,, Math. Biosci., 13 (1972), 397.  doi: 10.1016/0025-5564(72)90058-2.  Google Scholar

[25]

G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotaxis bacteria,, J. Theor. Biol., 49 (1975), 311.   Google Scholar

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen,, Bull. Math. Biol., 40 (1978), 671.  doi: 10.1007/BF02460738.  Google Scholar

[27]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria,, Math. Biosci., 24 (1975), 273.  doi: 10.1016/0025-5564(75)90080-2.  Google Scholar

[28]

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

[29]

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

[30]

H. Schwetlick, Traveling waves for chemotaxis systems,, Proc. Appl. Math. Mech., 3 (2003), 476.  doi: 10.1002/pamm.200310508.  Google Scholar

[31]

C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model,, SIAM J. Math. Anal., 38 (2006), 1694.  doi: 10.1137/060655122.  Google Scholar

[32]

Z. Wang, Wavefront of an angiogenesis model,, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. Wang, Mathematics of traveling waves in chemotaxis,, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

[34]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.  doi: 10.1002/mma.898.  Google Scholar

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