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Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
Wavefront of an angiogenesis model
1.  Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 
References:
[1] 
A. R. A. Anderson and M. A. J. Chaplain, Modelling the growth and form of capillary networks, in "On Growth and Form: SpatioTemporal Pattern Formation in Biology" (eds. J. C. McLachlan M. A. J. Chaplain and G. G. Singh), Wiley, (1999), 225249. 
[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), 413452. 
[3] 
H. Byren and M. A. J. Chaplain, Mathematical models for tumour angiogenesisnumerical simulations and nonlinearwave equations, Bull. Math. Biol., 57 (1995), 461485. 
[4] 
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399439. 
[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), 149168. 
[6] 
L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141146. 
[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), 128. 
[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), 13301355. 
[9] 
A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138163. 
[10] 
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235248. 
[11] 
J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in ReactionDiffusion Theory," Springer Monographs in Mathematics, SpringerVerlag London, Ltd., London, 2004. 
[12] 
H. A. Levine, B. D. Sleeman and M. NilsenHamilton, 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), 77115. 
[13] 
D. Li, T. Li and K. Zhao, On a hyperbolicparabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 16311650. 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 hyperbolicparabolic 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 hyperbolicparabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 19671998. 
[17] 
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 13101333. 
[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), 123131. 
[20] 
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739761. 
[21] 
B. P. Marchant, J. Norbury and J. A. Sherratt, Traveling wave solutions to a haptotaxisdominated model of malignant invasion, Nonlinearity, 14 (2001), 16531671. 
[22] 
J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, SpringerVerlag, New York, 2002. 
[23] 
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169184. 
[24] 
A. NovickCohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria, J. Math. Biol., 19 (1984), 125132. 
[25] 
G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243247. 
[26] 
B. Perthame, PDE models for chemotactic movement: Parabolic, hyperbolic and kinetic, Applications of Mathematics, 49 (2004), 539564. 
[27] 
G. Rosen, On the propogation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185189. 
[28] 
G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations, J. Theor. Biol., 59 (1976), 243246. 
[29] 
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476478. 
[30] 
Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 4570. 
[31] 
J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161230. 
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: SpatioTemporal Pattern Formation in Biology" (eds. J. C. McLachlan M. A. J. Chaplain and G. G. Singh), Wiley, (1999), 225249. 
[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), 413452. 
[3] 
H. Byren and M. A. J. Chaplain, Mathematical models for tumour angiogenesisnumerical simulations and nonlinearwave equations, Bull. Math. Biol., 57 (1995), 461485. 
[4] 
M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med., 1 (2006), 399439. 
[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), 149168. 
[6] 
L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141146. 
[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), 128. 
[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), 13301355. 
[9] 
A. Friedman and J. I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks, J. Math. Anal. Appl., 272 (2002), 138163. 
[10] 
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235248. 
[11] 
J. A. Leach and D. J. Needham, "Matched Asymptotic Expansions in ReactionDiffusion Theory," Springer Monographs in Mathematics, SpringerVerlag London, Ltd., London, 2004. 
[12] 
H. A. Levine, B. D. Sleeman and M. NilsenHamilton, 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), 77115. 
[13] 
D. Li, T. Li and K. Zhao, On a hyperbolicparabolic system modeling chemotaxis, Math. Models Methods Appl. Sci., 21 (2011), 16311650. 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 hyperbolicparabolic 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 hyperbolicparabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 19671998. 
[17] 
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 13101333. 
[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), 123131. 
[20] 
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739761. 
[21] 
B. P. Marchant, J. Norbury and J. A. Sherratt, Traveling wave solutions to a haptotaxisdominated model of malignant invasion, Nonlinearity, 14 (2001), 16531671. 
[22] 
J. D. Murray, "Mathematical Biology. I. An Introduction," Third edition, Interdisciplinary Applied Mathematics, 17, SpringerVerlag, New York, 2002. 
[23] 
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169184. 
[24] 
A. NovickCohen and L. A. Segel, A gradually slowing traveling band of chemotactic bacteria, J. Math. Biol., 19 (1984), 125132. 
[25] 
G. M. Odell and E. F. Keller, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243247. 
[26] 
B. Perthame, PDE models for chemotactic movement: Parabolic, hyperbolic and kinetic, Applications of Mathematics, 49 (2004), 539564. 
[27] 
G. Rosen, On the propogation theory for bands of chemotactic bacteria, Math. Biosci., 20 (1974), 185189. 
[28] 
G. Rosen, Existence and nature of band solutions to generic chemotactic transport equations, J. Theor. Biol., 59 (1976), 243246. 
[29] 
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476478. 
[30] 
Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 4570. 
[31] 
J. Xin, Front propagation in heterogeneous media, SIAM Review, 42 (2000), 161230. 
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