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Reaction, diffusion and chemotaxis in wave propagation
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 |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Annual Review of Biochemistry, 44 (1975), 341-356.
doi: 10.1146/annurev.bi.44.070175.002013. |
[2] |
J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597.
doi: 10.1126/science.166.3913.1588. |
[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), 1866-1881.
doi: 10.1016/j.nonrwa.2007.06.001. |
[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), 47-87. |
[5] |
L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[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), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[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), 1330-1355.
doi: 10.1137/S0036141001385046. |
[8] |
M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.
doi: 10.4171/IFB/141. |
[9] |
D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[10] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[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), 195-238.
doi: 10.1007/s002850000037. |
[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), 683-730.
doi: 10.1137/S0036139995291106. |
[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, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[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), 1522-1541.
doi: 10.1137/09075161X. |
[16] |
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[17] |
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[18] |
M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model, Adv. Differential Equations, 16 (2011), 31-60. |
[19] |
G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[20] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[21] |
, National Cancer Institute,, , ().
|
[22] |
R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406.
doi: 10.1016/0025-5564(72)90058-2. |
[23] |
C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model, SIAM Appl. Dyn. Sys., 8 (2009), 390-416.
doi: 10.1137/08072797X. |
[24] |
K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[25] |
G. Rosen, Analytically solution to the initial-value problem for traveling bands of chemotaxis bacteria, J. Theor. Biol., 49 (1975), 311-321. |
[26] |
G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.
doi: 10.1007/BF02460738. |
[27] |
G. Rosen, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis, Bull. Math. Biol., 45 (1983), 151-153. |
[28] |
G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279.
doi: 10.1016/0025-5564(75)90080-2. |
[29] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[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), 821-845.
doi: 10.3934/dcdsb.2013.18.821. |
[31] |
C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713.
doi: 10.1137/060655122. |
[32] |
Z. A. Wang, Wavefront of an angiogenesis model, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[33] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[34] |
Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[35] |
C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.
doi: 10.1007/s11538-010-9586-4. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Annual Review of Biochemistry, 44 (1975), 341-356.
doi: 10.1146/annurev.bi.44.070175.002013. |
[2] |
J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597.
doi: 10.1126/science.166.3913.1588. |
[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), 1866-1881.
doi: 10.1016/j.nonrwa.2007.06.001. |
[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), 47-87. |
[5] |
L. Corrias, B. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris. Ser. I., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[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), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[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), 1330-1355.
doi: 10.1137/S0036141001385046. |
[8] |
M. Funaki, M. Mimura and T. Tsujikawa, Travelling front solutions arising in the chemotaxis-growth model, Interfaces Free Bound., 8 (2006), 223-245.
doi: 10.4171/IFB/141. |
[9] |
D. Horstmann and A. Stevens, A constructive approach to traveling waves in chemotaxis, J. Nonlin. Sci., 14 (2004), 1-25.
doi: 10.1007/s00332-003-0548-y. |
[10] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[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), 195-238.
doi: 10.1007/s002850000037. |
[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), 683-730.
doi: 10.1137/S0036139995291106. |
[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, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[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), 1522-1541.
doi: 10.1137/09075161X. |
[16] |
T. Li and Z. A. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[17] |
R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[18] |
M. Meyries, Local well posedness and instability of travelling waves in a chemotaxis model, Adv. Differential Equations, 16 (2011), 31-60. |
[19] |
G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[20] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[21] |
, National Cancer Institute,, , ().
|
[22] |
R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406.
doi: 10.1016/0025-5564(72)90058-2. |
[23] |
C. H. Ou and W. Yuan, Traveling wavefronts in a volume-filling chemotaxis model, SIAM Appl. Dyn. Sys., 8 (2009), 390-416.
doi: 10.1137/08072797X. |
[24] |
K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D: Nonlinear Phenomena, 240 (2011), 363-375.
doi: 10.1016/j.physd.2010.09.011. |
[25] |
G. Rosen, Analytically solution to the initial-value problem for traveling bands of chemotaxis bacteria, J. Theor. Biol., 49 (1975), 311-321. |
[26] |
G. Rosen, Steady-state distribution of bacteria chemotactic toward oxygen, Bull. Math. Biol., 40 (1978), 671-674.
doi: 10.1007/BF02460738. |
[27] |
G. Rosen, Theoretical significance of the condition $\delta=2 \mu$ in bacterical chemotaxis, Bull. Math. Biol., 45 (1983), 151-153. |
[28] |
G. Rosen and S. Baloga, On the stability of steadily propogating bands of chemotactic bacteria, Math. Biosci., 24 (1975), 273-279.
doi: 10.1016/0025-5564(75)90080-2. |
[29] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[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), 821-845.
doi: 10.3934/dcdsb.2013.18.821. |
[31] |
C. Walker and G. F. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713.
doi: 10.1137/060655122. |
[32] |
Z. A. Wang, Wavefront of an angiogenesis model, Discrete Cont. Dyn. Syst.-Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[33] |
Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[34] |
Z. A. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[35] |
C. Xue, H. J. Hwang, K. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.
doi: 10.1007/s11538-010-9586-4. |
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