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Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
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Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model
1. | Centre for Mathematical Biology, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G2G1, Canada |
2. | Department of Mathematics and Maxwell Institute for Mathematical Sciences, School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom |
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
V. Andasari, A. Gerisch, G. Lolas, A. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171.
doi: 10.1007/s00285-010-0369-1. |
[3] |
S. Banerjee, A. P. Misra and L. Rondoni, Spatiotemporal evolution in a (2+ 1)-dimensional chemotaxis model, Physica A: Statistical Mechanics and its Applications, 391 (2012), 4061-4062.
doi: 10.1016/j.physa.2011.12.054. |
[4] |
D. Barkley, Simplifying the complexity of pipe flow, Physical Review E, 84 (2011), 016309 [8 pages].
doi: 10.1103/PhysRevE.84.016309. |
[5] |
R. Baronas and R. Šimkus, Modelling the bacterial self-organization in circular container along the contact line as detected by bioluminescence imaging,, Nonlinear Anal. Model. Control, 16 (): 270.
|
[6] |
J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds, Princeton University Press, 2008. |
[7] |
E. O. Budrene and H. C. Berg, et al, Complex patterns formed by motile cells of Escherichia coli, letters to Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[8] |
G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps, Phys. Rev. E, 64, 051914, 2001.
doi: 10.1103/PhysRevE.64.051914. |
[9] |
Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170.
doi: 10.1007/s00285-002-0173-7. |
[10] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[11] |
T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[12] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV, 105 (2003), 103-165. |
[13] |
K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[14] |
J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations, Trans. AMS, 261 (1980), 589-604.
doi: 10.1090/S0002-9947-1980-0580905-3. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[17] |
T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, submitted, 2013. |
[18] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[19] |
H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation, In W. Alt, A. Deutsch, and G. Dunn, editors, Dynamics of Cell and Tissue Motion, Birkhäuser, 1996. |
[20] |
H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Appl. Math., 70 (2009), 133-167.
doi: 10.1137/070711505. |
[21] |
M. R. Owen and J. A. Sherratt, Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions, J. Theor. Biol., 189 (1997), 63-80.
doi: 10.1006/jtbi.1997.0494. |
[22] |
K. J. 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. |
[23] |
I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.
doi: 10.1007/s00285-007-0088-4. |
[24] |
R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli, Luminescence, 26 (2011), 716-721. |
[25] |
Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[26] |
D. D. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. |
show all references
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
V. Andasari, A. Gerisch, G. Lolas, A. South and M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: Biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141-171.
doi: 10.1007/s00285-010-0369-1. |
[3] |
S. Banerjee, A. P. Misra and L. Rondoni, Spatiotemporal evolution in a (2+ 1)-dimensional chemotaxis model, Physica A: Statistical Mechanics and its Applications, 391 (2012), 4061-4062.
doi: 10.1016/j.physa.2011.12.054. |
[4] |
D. Barkley, Simplifying the complexity of pipe flow, Physical Review E, 84 (2011), 016309 [8 pages].
doi: 10.1103/PhysRevE.84.016309. |
[5] |
R. Baronas and R. Šimkus, Modelling the bacterial self-organization in circular container along the contact line as detected by bioluminescence imaging,, Nonlinear Anal. Model. Control, 16 (): 270.
|
[6] |
J. T. Bonner, The Social Amoebae: The Biology of Cellular Slime Molds, Princeton University Press, 2008. |
[7] |
E. O. Budrene and H. C. Berg, et al, Complex patterns formed by motile cells of Escherichia coli, letters to Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[8] |
G. de Vries, Bursting as an emergent phenomenon in coupled chaotic maps, Phys. Rev. E, 64, 051914, 2001.
doi: 10.1103/PhysRevE.64.051914. |
[9] |
Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153-170.
doi: 10.1007/s00285-002-0173-7. |
[10] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[11] |
T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Mod. Meth. Appl. Sci., 23 (2013), 165-198.
doi: 10.1142/S0218202512500480. |
[12] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresberichte der DMV, 105 (2003), 103-165. |
[13] |
K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162.
doi: 10.1093/imamat/hxl028. |
[14] |
J. P. Keener, Chaotic Behavior in Piecewise Continuous Difference Equations, Trans. AMS, 261 (1980), 589-604.
doi: 10.1090/S0002-9947-1980-0580905-3. |
[15] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[16] |
E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[17] |
T. Kolokolnikov and J. Wei, Basic mechanisms driving complex spike dynamics in a chemotaxis model with logistic growth, submitted, 2013. |
[18] |
K. Kuto, K. Osaki, T. Sakurai and T. Tsujikawa, Spatial pattern formation in a chemotaxis-diffusion-growth model, Physica D: Nonlinear Phenomena, 241 (2012), 1629-1639.
doi: 10.1016/j.physd.2012.06.009. |
[19] |
H. G. Othmer and J. Dallon, Models of Dictyostelium aggregation, In W. Alt, A. Deutsch, and G. Dunn, editors, Dynamics of Cell and Tissue Motion, Birkhäuser, 1996. |
[20] |
H. G. Othmer and C. Xue, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Appl. Math., 70 (2009), 133-167.
doi: 10.1137/070711505. |
[21] |
M. R. Owen and J. A. Sherratt, Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions, J. Theor. Biol., 189 (1997), 63-80.
doi: 10.1006/jtbi.1997.0494. |
[22] |
K. J. 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. |
[23] |
I. G. Pearce, M. A. J. Chaplain, P. G. Schofield, A. R. A. Anderson and S. F. Hubbard, Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365-388.
doi: 10.1007/s00285-007-0088-4. |
[24] |
R. Šimkus and R. Baronas, Metabolic self-organization of bioluminescent escherichia coli, Luminescence, 26 (2011), 716-721. |
[25] |
Z. A. Wang and T. Hillen, Pattern formation for a chemotaxis model with volume filling, Chaos, 17 (2007), 037108, 13 pp.
doi: 10.1063/1.2766864. |
[26] |
D. D. Woodward, R. Tyson, M. R. Myerscough, J. D. Murray, E. Budrene and H. C. Berg, Spatio-temporal patterns generated by Salmonella typhimurium, Biophys. J., 68 (1995), 2181-2189. |
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