Wavefront of an angiogenesis model

Discrete and Continuous Dynamical Systems - Series B

Volumn 17, Issue 8, 2012, Pages 2849-2860

  Wang, Zhi An ^a  

^a HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

Angiogenesis; Chemotaxis; DiFusion limits; Fresher equation; Traveling waves; Wave speed

Indexed keywords

EID: 84867055448     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2012.17.2849     Document Type: Conference Paper

References (31)

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.
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 (2009/10), 1522-1541.
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. Difierential 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 haptotaxisdominated 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.