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Mathematics of traveling waves in chemotaxis --Review paper--
1. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. |
[3] |
S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth, preprint, 2012. |
[4] |
W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722.
doi: 10.1007/BF00275511. |
[5] |
S. Balasuriya and G. A. Gottwald, Wavespeed in reaction diffusion systems, with applications to chemotaxis and population pressure, J. Math. Biol., 61 (2010), 377-399.
doi: 10.1007/s00285-009-0305-4. |
[6] |
D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73. |
[7] |
R. Bellman, "Stability Theory of Differential Equations," McGraw-Hill Book Company, New York-Toronto-London, 1953. |
[8] |
R. D. Benguria, M. C. Depassier and V. Mendez, Minimum speed of fronts of reaction-convenction-diffusion equations, Phys. Rev. E (3), 69 (2004), 031106, 7 pp.
doi: 10.1103/PhysRevE.69.031106. |
[9] |
F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulses 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. |
[10] |
M. P. Brenner, L. S. Levitor and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacterial, Biophys. J., 74 (1988), 1677-1693. |
[11] |
E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. |
[12] |
E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. |
[13] |
S. Childress, Chemotactic collapse in two dimensions, in "Modelling of Patterns in Space and Time" (Heidelberg, 1983), Lect. Notes in Biomath., 55, Springer, Berlin, (1984), 61-66.
doi: 10.1007/978-3-642-45589-66. |
[14] |
W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations," D. C. Heath and Co., Boston, 1965. |
[15] |
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. |
[16] |
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. |
[17] |
F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123. |
[18] |
Y. Ebihara, Y. Furusho and T. Nagai, Singular solution of traveling waves in a chemotactic model, Bull. Kyushu Inst. Tech. Math. Natur. Sci., 39 (1992), 29-38,. |
[19] |
N. Fenichel, Geometric singular perturbation theory of ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[20] |
R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. |
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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. |
[22] |
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. |
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R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778. |
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J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rat. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
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S. Gueron and N. Liron, A model of herd grazing as a traveling wave: Chemotaxis and stability, J. Math. Biol., 27 (1989), 595-608.
doi: 10.1007/BF00288436. |
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J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629-641.
doi: 10.1016/S0252-9602(09)60059-X. |
[27] |
M. Holz and S. H. Chen, Spatio-temporal structure of migrating chemotactic band of escherichia coli, Biophys. J., 26 (1979), 243-262. |
[28] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[30] |
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. |
[31] |
C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. J. Russell) (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995.
doi: 10.1007/BFb0095239. |
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E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1975), 309-317. |
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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[34] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. |
[35] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248. |
[36] |
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1016/S0092-8240(80)80057-7. |
[37] |
I. T. Kiguradse, On the non-negative non-increasing solutions of non-linear second order differential equations, Ann. Mat. Pura ed Appl. (4), 81 (1969), 169-192. |
[38] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. |
[39] |
M. Kot, "Elementals of Mathematical Biology," Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511608520. |
[40] |
K. A. Landman, M. J. Simpson, J. L Slater and D. F. Newgreen, Diffusive and chemotactic celluar migration: Smooth and disconcinuous traveling wave solutions, SIAM J. Appl. Math., 65 (2005), 1420-1442.
doi: 10.1137/040604066. |
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I. R. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13. |
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D. Lauffenburger, C. R. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40. |
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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. |
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T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM. J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
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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.
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show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. |
[2] |
J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. |
[3] |
S. B. Ai, W. Z. Huang and Z. A. Wang, Traveling wave solutions to a chemotaxis system with logistic growth, preprint, 2012. |
[4] |
W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722.
doi: 10.1007/BF00275511. |
[5] |
S. Balasuriya and G. A. Gottwald, Wavespeed in reaction diffusion systems, with applications to chemotaxis and population pressure, J. Math. Biol., 61 (2010), 377-399.
doi: 10.1007/s00285-009-0305-4. |
[6] |
D. Balding and D. L. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73. |
[7] |
R. Bellman, "Stability Theory of Differential Equations," McGraw-Hill Book Company, New York-Toronto-London, 1953. |
[8] |
R. D. Benguria, M. C. Depassier and V. Mendez, Minimum speed of fronts of reaction-convenction-diffusion equations, Phys. Rev. E (3), 69 (2004), 031106, 7 pp.
doi: 10.1103/PhysRevE.69.031106. |
[9] |
F. S. Berezovskaya, A. S. Novozhilov and G. P. Karev, Families of traveling impulses 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. |
[10] |
M. P. Brenner, L. S. Levitor and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacterial, Biophys. J., 74 (1988), 1677-1693. |
[11] |
E. Budrene and H. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633. |
[12] |
E. Budrene and H. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. |
[13] |
S. Childress, Chemotactic collapse in two dimensions, in "Modelling of Patterns in Space and Time" (Heidelberg, 1983), Lect. Notes in Biomath., 55, Springer, Berlin, (1984), 61-66.
doi: 10.1007/978-3-642-45589-66. |
[14] |
W. A. Coppel, "Stability and Asymptotic Behavior of Differential Equations," D. C. Heath and Co., Boston, 1965. |
[15] |
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. |
[16] |
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. |
[17] |
F. W. Dahlquist, P. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123. |
[18] |
Y. Ebihara, Y. Furusho and T. Nagai, Singular solution of traveling waves in a chemotactic model, Bull. Kyushu Inst. Tech. Math. Natur. Sci., 39 (1992), 29-38,. |
[19] |
N. Fenichel, Geometric singular perturbation theory of ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[20] |
R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369. |
[21] |
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. |
[22] |
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. |
[23] |
R. E. Goldstein, Traveling-wave chemotaxis, Phys. Rev. Lett., 77 (1996), 775-778. |
[24] |
J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rat. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
[25] |
S. Gueron and N. Liron, A model of herd grazing as a traveling wave: Chemotaxis and stability, J. Math. Biol., 27 (1989), 595-608.
doi: 10.1007/BF00288436. |
[26] |
J. Guo, J. X. Xiao, H. J. Zhao and C. J. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629-641.
doi: 10.1016/S0252-9602(09)60059-X. |
[27] |
M. Holz and S. H. Chen, Spatio-temporal structure of migrating chemotactic band of escherichia coli, Biophys. J., 26 (1979), 243-262. |
[28] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. |
[29] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69. |
[30] |
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. |
[31] |
C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. J. Russell) (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995.
doi: 10.1007/BFb0095239. |
[32] |
E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1975), 309-317. |
[33] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. |
[34] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. |
[35] |
E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 26 (1971), 235-248. |
[36] |
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1016/S0092-8240(80)80057-7. |
[37] |
I. T. Kiguradse, On the non-negative non-increasing solutions of non-linear second order differential equations, Ann. Mat. Pura ed Appl. (4), 81 (1969), 169-192. |
[38] |
A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. |
[39] |
M. Kot, "Elementals of Mathematical Biology," Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511608520. |
[40] |
K. A. Landman, M. J. Simpson, J. L Slater and D. F. Newgreen, Diffusive and chemotactic celluar migration: Smooth and disconcinuous traveling wave solutions, SIAM J. Appl. Math., 65 (2005), 1420-1442.
doi: 10.1137/040604066. |
[41] |
I. R. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13. |
[42] |
D. Lauffenburger, C. R. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40. |
[43] |
K. J. Lee, E. C. Cox and R. E. Goldstein, Competing patterns of signaling activity in dictyostelium discoideum, Phys. Rev. Lett., 76 (1996), 1174-1177. |
[44] |
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. |
[45] |
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. |
[46] |
T. Li, R. H. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM. J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[47] |
T. Li and Z.-A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (): 1522.
doi: 10.1137/09075161X. |
[48] |
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.
doi: 10.1142/S0218202510004830. |
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