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

Wavefront of an angiogenesis model

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.
Citation: Zhi-An Wang. Wavefront of an angiogenesis model. Discrete & 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, (1999), 225. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in Reaction-Diffusion Theory,", Springer Monographs in Mathematics, (2004). Google Scholar

[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. 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,, preprint, (2011). 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 (): 1522. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[22]

J. D. Murray, "Mathematical Biology. I. An Introduction,", Third edition, 17 (2002). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

show all references

References:
[1]

A. R. A. Anderson and M. A. J. Chaplain, Modelling the growth and form of capillary networks,, in, (1999), 225. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[11]

J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in Reaction-Diffusion Theory,", Springer Monographs in Mathematics, (2004). Google Scholar

[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. 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,, preprint, (2011). 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 (): 1522. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[20]

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

[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. Google Scholar

[22]

J. D. Murray, "Mathematical Biology. I. An Introduction,", Third edition, 17 (2002). Google Scholar

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

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