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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), 341-356. doi: 10.1146/annurev.bi.44.070175.002013.

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. doi: 10.1126/science.166.3913.1588.

[3]

S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1-21. doi: 10.3934/dcdsb.2015.20.1.

[4]

J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical, Dicrete Contin. Dyn. Syst., 34 (2014), 5165-5179. doi: 10.3934/dcds.2014.34.5165.

[5]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.

[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), 1-28. doi: 10.1007/s00032-003-0026-x.

[7]

M. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046.

[8]

M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245. doi: 10.4171/IFB/141.

[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), 193-219. doi: 10.1016/j.jde.2013.04.002.

[10]

Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448. doi: 10.1016/j.bpj.2008.10.027.

[11]

E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[12]

C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. doi: 10.1007/BF02460793.

[13]

I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13. doi: 10.1016/S0006-3495(78)85466-6.

[14]

D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40.

[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), 71-115. doi: 10.1016/S0025-5564(00)00034-1.

[16]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519.

[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), 2819-2849. doi: 10.1142/S0218202514500389.

[18]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X.

[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), 1967-1998. doi: 10.1142/S0218202510004830.

[20]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333. doi: 10.1016/j.jde.2010.09.020.

[21]

R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0.

[22]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538. doi: 10.4171/IFB/200.

[23]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334.

[24]

R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406. doi: 10.1016/0025-5564(72)90058-2.

[25]

G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotaxis bacteria, J. Theor. Biol., 49 (1975), 311-321.

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674. doi: 10.1007/BF02460738.

[27]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279. doi: 10.1016/0025-5564(75)90080-2.

[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), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.

[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), 16235-16240. doi: 10.1073/pnas.1101996108.

[30]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478. doi: 10.1002/pamm.200310508.

[31]

C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713. doi: 10.1137/060655122.

[32]

Z. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849-2860. doi: 10.3934/dcdsb.2012.17.2849.

[33]

Z. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601.

[34]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 44 (1975), 341-356. doi: 10.1146/annurev.bi.44.070175.002013.

[2]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. doi: 10.1126/science.166.3913.1588.

[3]

S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1-21. doi: 10.3934/dcdsb.2015.20.1.

[4]

J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical, Dicrete Contin. Dyn. Syst., 34 (2014), 5165-5179. doi: 10.3934/dcds.2014.34.5165.

[5]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0.

[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), 1-28. doi: 10.1007/s00032-003-0026-x.

[7]

M. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046.

[8]

M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245. doi: 10.4171/IFB/141.

[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), 193-219. doi: 10.1016/j.jde.2013.04.002.

[10]

Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448. doi: 10.1016/j.bpj.2008.10.027.

[11]

E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[12]

C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. doi: 10.1007/BF02460793.

[13]

I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13. doi: 10.1016/S0006-3495(78)85466-6.

[14]

D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40.

[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), 71-115. doi: 10.1016/S0025-5564(00)00034-1.

[16]

D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 1631-1650. doi: 10.1142/S0218202511005519.

[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), 2819-2849. doi: 10.1142/S0218202514500389.

[18]

T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X.

[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), 1967-1998. doi: 10.1142/S0218202510004830.

[20]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333. doi: 10.1016/j.jde.2010.09.020.

[21]

R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0.

[22]

G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538. doi: 10.4171/IFB/200.

[23]

T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334.

[24]

R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406. doi: 10.1016/0025-5564(72)90058-2.

[25]

G. Rosen, Analytical solution to the initial-value problem for traveling bands of chemotaxis bacteria, J. Theor. Biol., 49 (1975), 311-321.

[26]

G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674. doi: 10.1007/BF02460738.

[27]

G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279. doi: 10.1016/0025-5564(75)90080-2.

[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), e1000890, 12pp. doi: 10.1371/journal.pcbi.1000890.

[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), 16235-16240. doi: 10.1073/pnas.1101996108.

[30]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478. doi: 10.1002/pamm.200310508.

[31]

C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713. doi: 10.1137/060655122.

[32]

Z. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849-2860. doi: 10.3934/dcdsb.2012.17.2849.

[33]

Z. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601.

[34]

Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70. doi: 10.1002/mma.898.

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