American Institute of Mathematical Sciences

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November  2012, 17(8): 2849-2860. doi: 10.3934/dcdsb.2012.17.2849

Wavefront of an angiogenesis model

Zhi-An Wang 1,

1. 

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

Received  March 2011 Revised  June 2011 Published  July 2012

In this paper, we show the existence of traveling wave solutions to a chemotaxis model describing the initiation of angiogenesis. By a change of dependent variable, we transform the wave equation of the angiogenesis model to a Fisher type wave equation. Then we make use of the methods of analyzing the Fisher wave equation to obtain the existence of traveling wave solutions to the angiogenesis model. In virtue of the asymptotic behavior of the traveling wave solution at infinity, we find the explicit wave speed for cases of both zero and nonzero chemical diffusion. Finally based on the fact that the wave speed is convergent with respect to the chemical diffusion, we rigorously establish the zero chemical diffusion limit of traveling wave solutions by the energy estimates.
Keywords: fisher equation, angiogenesis, wave speed, diffusion limits., traveling waves, Chemotaxis.
Mathematics Subject Classification: Primary: 35C07, 35K55; Secondary: 46N60, 62P10, 92C1.
Citation: Zhi-An Wang. Wavefront of an angiogenesis model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2849-2860. doi: 10.3934/dcdsb.2012.17.2849
References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Modelling the growth and form of capillary networks, in "On Growth and Form: Spatio-Temporal Pattern Formation in Biology" (eds. J. C. McLachlan M. A. J. Chaplain and G. G. Singh), Wiley, (1999), 225-249.

[2]

N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Modelling, 32 (2000), 413-452.

[3]

H. Byren and M. A. J. Chaplain, Mathematical models for tumour angiogenesis-numerical simulations and nonlinear-wave equations, Bull. Math. Biol., 57 (1995), 461-485.

[4]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.

[5]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168.

[6]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146.

[7]

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.

[8]

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

[9]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.

[10]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248.

[11]

J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in Reaction-Diffusion Theory," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2004.

[12]

H. A. Levine, B. D. 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), 77-115.

[13]

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.

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data, preprint, 2011.

[15]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM. J. Appl. Math., 70 (): 1522. 

[16]

T. Li and Z. A. 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.

[17]

T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.

[18]

T. Li and K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, preprint, 2011.

[19]

Y. Li, The existence of traveling waves in a biological model for chemotaxis, Acta Mathematicae Applicatae Sinica (in Chinese), 27 (2004), 123-131.

[20]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.

[21]

B. P. Marchant, J. Norbury and J. A. Sherratt, Traveling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.

[22]

J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.

[23]

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

[24]

A. Novick-Cohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria, J. Math. Biol., 19 (1984), 125-132.

[25]

G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247.

[26]

B. Perthame, PDE models for chemotactic movement: Parabolic, hyperbolic and kinetic, Applications of Mathematics, 49 (2004), 539-564.

[27]

G. Rosen, On the propogation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185-189.

[28]

G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations, J. Theor. Biol., 59 (1976), 243-246.

[29]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.

[30]

Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.

[31]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.

show all references

References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Modelling the growth and form of capillary networks, in "On Growth and Form: Spatio-Temporal Pattern Formation in Biology" (eds. J. C. McLachlan M. A. J. Chaplain and G. G. Singh), Wiley, (1999), 225-249.

[2]

N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Modelling, 32 (2000), 413-452.

[3]

H. Byren and M. A. J. Chaplain, Mathematical models for tumour angiogenesis-numerical simulations and nonlinear-wave equations, Bull. Math. Biol., 57 (1995), 461-485.

[4]

M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399-439.

[5]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168.

[6]

L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146.

[7]

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.

[8]

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

[9]

A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138-163.

[10]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248.

[11]

J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in Reaction-Diffusion Theory," Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2004.

[12]

H. A. Levine, B. D. 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), 77-115.

[13]

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.

[14]

T. Li, R. H. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data, preprint, 2011.

[15]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM. J. Appl. Math., 70 (): 1522. 

[16]

T. Li and Z. A. 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.

[17]

T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.

[18]

T. Li and K. Zhao, Quantitative decay of a hybrid type chemotaxis model with large data, preprint, 2011.

[19]

Y. Li, The existence of traveling waves in a biological model for chemotaxis, Acta Mathematicae Applicatae Sinica (in Chinese), 27 (2004), 123-131.

[20]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.

[21]

B. P. Marchant, J. Norbury and J. A. Sherratt, Traveling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity, 14 (2001), 1653-1671.

[22]

J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.

[23]

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

[24]

A. Novick-Cohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria, J. Math. Biol., 19 (1984), 125-132.

[25]

G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247.

[26]

B. Perthame, PDE models for chemotactic movement: Parabolic, hyperbolic and kinetic, Applications of Mathematics, 49 (2004), 539-564.

[27]

G. Rosen, On the propogation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185-189.

[28]

G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations, J. Theor. Biol., 59 (1976), 243-246.

[29]

H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.

[30]

Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.

[31]

J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161-230.

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