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Traveling bands for the Keller-Segel model with population growth
1. | Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899 |
2. | Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 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] |
S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1-21.
doi: 10.3934/dcdsb.2015.20.1. |
[4] |
J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical, Dicrete Contin. Dyn. Syst., 34 (2014), 5165-5179.
doi: 10.3934/dcds.2014.34.5165. |
[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. 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] |
H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[10] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[11] |
E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[12] |
C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1007/BF02460793. |
[13] |
I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13.
doi: 10.1016/S0006-3495(78)85466-6. |
[14] |
D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40. |
[15] |
H. Levine, B. 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), 71-115.
doi: 10.1016/S0025-5564(00)00034-1. |
[16] |
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. |
[17] |
J. Li, T. Li and Z. WAng, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[18] |
T. Li and Z. 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. |
[19] |
T. Li and Z. 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. |
[20] |
T. Li and Z. 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. |
[21] |
R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[22] |
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. |
[23] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[24] |
R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406.
doi: 10.1016/0025-5564(72)90058-2. |
[25] |
G. Rosen, Analytical 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 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. |
[28] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS computational biology, 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[29] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[30] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[31] |
C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713.
doi: 10.1137/060655122. |
[32] |
Z. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[33] |
Z. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[34] |
Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 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] |
S. Ai, W. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Dicrete Contin. Dyn. Syst.-Series B, 20 (2015), 1-21.
doi: 10.3934/dcdsb.2015.20.1. |
[4] |
J. Anh and K. Kang, On a keller-segel system with logarithmic sensitivity and non-diffusive chemical, Dicrete Contin. Dyn. Syst., 34 (2014), 5165-5179.
doi: 10.3934/dcds.2014.34.5165. |
[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. 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] |
H. Jin, J. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[10] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu, Logarithmic sensing in escherichia coli bacterial chemotaxis, Biophysical Journal, 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[11] |
E. Keller and L. Segel, Traveling bands of chemotactic bacteria: A theorectical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[12] |
C. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1007/BF02460793. |
[13] |
I. Lapidus and R. Schiller, A model for traveling bands of chemotactic bacteria, Biophy. J., 22 (1978), 1-13.
doi: 10.1016/S0006-3495(78)85466-6. |
[14] |
D. Lauffenburger, C. Kennedy and R. Aris, Traveling bands of chemotactic bacteria in the context of population growth, Bull. Math. Biol., 46 (1984), 19-40. |
[15] |
H. Levine, B. 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), 71-115.
doi: 10.1016/S0025-5564(00)00034-1. |
[16] |
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. |
[17] |
J. Li, T. Li and Z. WAng, Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[18] |
T. Li and Z. 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. |
[19] |
T. Li and Z. 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. |
[20] |
T. Li and Z. 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. |
[21] |
R. Lui and Z. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[22] |
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. |
[23] |
T. Nagai and T. Ikeda, Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[24] |
R. Nossal, Boundary movement of chemotactic bacterial population, Math. Biosci., 13 (1972), 397-406.
doi: 10.1016/0025-5564(72)90058-2. |
[25] |
G. Rosen, Analytical 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 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. |
[28] |
J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan and B. Perthame, Mathematical description of bacterial traveling pulses, PLoS computational biology, 6 (2010), e1000890, 12pp.
doi: 10.1371/journal.pcbi.1000890. |
[29] |
J. Saragosti, V. Calvez, N. Bournaveas, B. Perthame, A. Buguin and P. Silberzan, Directional persistence of chemotactic bacteria in a traveling concentration wave, PNAS, 108 (2011), 16235-16240.
doi: 10.1073/pnas.1101996108. |
[30] |
H. Schwetlick, Traveling waves for chemotaxis systems, Proc. Appl. Math. Mech., 3 (2003), 476-478.
doi: 10.1002/pamm.200310508. |
[31] |
C. Walker and G. Webb, Global existence of classical solutions for a haptoaxis model, SIAM J. Math. Anal., 38 (2006), 1694-1713.
doi: 10.1137/060655122. |
[32] |
Z. Wang, Wavefront of an angiogenesis model, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2849-2860.
doi: 10.3934/dcdsb.2012.17.2849. |
[33] |
Z. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst.-Series B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[34] |
Z. Wang and T. Hillen, Shock formation in a chemotaxis model, Math. Methods. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
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