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January  2015, 20(1): 1-21. doi: 10.3934/dcdsb.2015.20.1

Reaction, diffusion and chemotaxis in wave propagation

Shangbing Ai 1, , Wenzhang Huang 1, and  Zhi-An Wang 2,

1. 

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

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

Received  June 2014 Published  November 2014

By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
Keywords: traveling waves, Reaction-diffusion-chemotaxis, minimal wave speed., cell growth.
Mathematics Subject Classification: 35C07, 35K55, 46N60, 62P10, 92C1.
Citation: Shangbing Ai, Wenzhang Huang, Zhi-An Wang. Reaction, diffusion and chemotaxis in wave propagation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 1-21. doi: 10.3934/dcdsb.2015.20.1
References:
[1]

J. Adler, Chemotaxis in bacteria,, Annual Review of Biochemistry, 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]

F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulse and fronts in some models with cross-diffusion,, Nonlinear Analysis: Real World applications, 9 (2008), 1866.  doi: 10.1016/j.nonrwa.2007.06.001.  Google Scholar

[4]

M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumors: The mathamatical modeling of the stages of tumor development,, Math. Comput. Modeling, 23 (1996), 47.   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. A. 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

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D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis,, J. Nonlin. Sci., 14 (2004), 1.  doi: 10.1007/s00332-003-0548-y.  Google Scholar

[10]

E. F. Keller and L. A. 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

[11]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis,, J. Math. Biol., 42 (2001), 195.  doi: 10.1007/s002850000037.  Google Scholar

[12]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar

[13]

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

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar

[15]

T. Li and Z. A. 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

[16]

T. Li and Z. A. 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

[17]

R. Lui and Z. A. 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

[18]

M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31.   Google Scholar

[19]

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

[20]

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

[21]

, National Cancer Institute,, , ().   Google Scholar

[22]

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

[23]

C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model,, SIAM Appl. Dyn. Sys., 8 (2009), 390.  doi: 10.1137/08072797X.  Google Scholar

[24]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[25]

G. Rosen, Analytically 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, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis,, Bull. Math. Biol., 45 (1983), 151.   Google Scholar

[28]

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

[29]

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

[30]

Y. S. Tao, L. H. Wang and Z. A. Wang, Long-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Cont. Dyn. Syst.-Seris B, 18 (2013), 821.  doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[31]

C. Walker and G. F. 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. A. Wang, Wavefront of an angiogenesis model,, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007).  doi: 10.1063/1.2766864.  Google Scholar

[34]

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

[35]

C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695.  doi: 10.1007/s11538-010-9586-4.  Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria,, Annual Review of Biochemistry, 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]

F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulse and fronts in some models with cross-diffusion,, Nonlinear Analysis: Real World applications, 9 (2008), 1866.  doi: 10.1016/j.nonrwa.2007.06.001.  Google Scholar

[4]

M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumors: The mathamatical modeling of the stages of tumor development,, Math. Comput. Modeling, 23 (1996), 47.   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. A. 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]

D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis,, J. Nonlin. Sci., 14 (2004), 1.  doi: 10.1007/s00332-003-0548-y.  Google Scholar

[10]

E. F. Keller and L. A. 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

[11]

H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis,, J. Math. Biol., 42 (2001), 195.  doi: 10.1007/s002850000037.  Google Scholar

[12]

H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar

[13]

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

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar

[15]

T. Li and Z. A. 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

[16]

T. Li and Z. A. 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

[17]

R. Lui and Z. A. 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

[18]

M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model,, Adv. Differential Equations, 16 (2011), 31.   Google Scholar

[19]

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

[20]

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

[21]

, National Cancer Institute,, , ().   Google Scholar

[22]

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

[23]

C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model,, SIAM Appl. Dyn. Sys., 8 (2009), 390.  doi: 10.1137/08072797X.  Google Scholar

[24]

K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model,, Physica D: Nonlinear Phenomena, 240 (2011), 363.  doi: 10.1016/j.physd.2010.09.011.  Google Scholar

[25]

G. Rosen, Analytically 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, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis,, Bull. Math. Biol., 45 (1983), 151.   Google Scholar

[28]

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

[29]

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

[30]

Y. S. Tao, L. H. Wang and Z. A. Wang, Long-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Cont. Dyn. Syst.-Seris B, 18 (2013), 821.  doi: 10.3934/dcdsb.2013.18.821.  Google Scholar

[31]

C. Walker and G. F. 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. A. Wang, Wavefront of an angiogenesis model,, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849.  doi: 10.3934/dcdsb.2012.17.2849.  Google Scholar

[33]

Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007).  doi: 10.1063/1.2766864.  Google Scholar

[34]

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

[35]

C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations,, Bull. Math. Biol., 73 (2011), 1695.  doi: 10.1007/s11538-010-9586-4.  Google Scholar

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