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Modeling contact inhibition of growth: Traveling waves
1. | Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, University of Rome Tor Vergata, Via dei Taurini 19, 00185 Rome, Italy |
2. | Meiji Institute of Advanced Mathematical Sciences, Meiji University, 1-1-1, Higashi-mita, Tama-ku, Kawasaki, 214-8571 |
3. | Department of Basic Sciences, Kyushu Institute of Technology, 1-1, Sensui-cho, Tobata, Kitakyuchu, 804-8550, Japan |
References:
[1] |
D. G. Aronson, Density-dependent interaction systems, in "Dynamics and Modelling of Reactive Systems" (eds. W. H. Steward, W. R .Ray and C. C Conley), Academic Press, New York-London, (1980), 161-176. |
[2] |
M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250.
doi: 10.4171/IFB/233. |
[3] |
M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157.
doi: 10.7153/dea-04-09. |
[4] |
M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Singular limit problem of a nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013).
doi: 10.7153/dea-04-09. |
[5] |
M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013). |
[6] |
M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, in preparation (2013). |
[7] |
Z. Biró, Stability of traveling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371. |
[8] |
M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour deveropment, Math. Med. Bio., 23 (2006), 197-229.
doi: 10.1093/imammb/dql009. |
[9] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York-Toronto-London, 1955. |
[10] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[11] |
G. Garcia-Ramos, F. Sanches-Garduño and P. K. Maini, Dispersal can sharpen parapatric boundaries on a spatially varying environment, Ecology, 81 (2000), 749-760. |
[12] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
[13] |
B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7964-4. |
[14] |
D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion, J. Differential Equations, 244 (2008), 2870-2889.
doi: 10.1016/j.jde.2008.02.018. |
[15] |
S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Ser. 9 Mat. Appl., 15 (2004), 271-280. |
[16] |
N. S. Kolmogorov, N. Petrovsky and I. G. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière e son application a un problème biologique, (Russian), Bull. Univ. État Moscou, Série internationale A, 1 (1937), 1-26. |
[17] |
G. S. Medvedev, K. Ono and P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovsky-Piskunov equation, European J. Appl. Math., 14 (2003), 343-367.
doi: 10.1017/S0956792503005102. |
[18] |
J. A. Sherratt, Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, Proc. R. Soc. Lond. A, 456 (2000), 2365-2386.
doi: 10.1098/rspa.2000.0616. |
[19] |
F. Sanches-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.
doi: 10.1006/jdeq.1995.1055. |
[20] |
Y. Tsukatani, K. Suzuki and K. Takahashi, Loss of density-dependent growth inhibition and dissociation of $\alpha$-catenin from E-cadherin, J. Cell. Physiol., 173 (1997), 54-63. |
show all references
References:
[1] |
D. G. Aronson, Density-dependent interaction systems, in "Dynamics and Modelling of Reactive Systems" (eds. W. H. Steward, W. R .Ray and C. C Conley), Academic Press, New York-London, (1980), 161-176. |
[2] |
M. Bertsch, R. Dal Passo and M. Mimura, A free boundary problem arising in a simplified tumour growth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010), 235-250.
doi: 10.4171/IFB/233. |
[3] |
M. Bertsch, D. Hilhorst, H. Izuhara and M. Mimura, A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, Differ. Equ. Appl., 4 (2012), 137-157.
doi: 10.7153/dea-04-09. |
[4] |
M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Singular limit problem of a nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013).
doi: 10.7153/dea-04-09. |
[5] |
M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura and T. Wakasa, Traveling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth, in preparation (2013). |
[6] |
M. Bertsch, H. Izuhara, M. Mimura and T. Wakasa, in preparation (2013). |
[7] |
Z. Biró, Stability of traveling waves for degenerate reaction-diffusion equations of KPP-type, Adv. Nonlinear Stud., 2 (2002), 357-371. |
[8] |
M. A. J. Chaplain, L. Graziano and L. Preziosi, Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour deveropment, Math. Med. Bio., 23 (2006), 197-229.
doi: 10.1093/imammb/dql009. |
[9] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, New York-Toronto-London, 1955. |
[10] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[11] |
G. Garcia-Ramos, F. Sanches-Garduño and P. K. Maini, Dispersal can sharpen parapatric boundaries on a spatially varying environment, Ecology, 81 (2000), 749-760. |
[12] |
R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. |
[13] |
B. Gilding and R. Kersner, "Travelling Waves in Nonlinear Diffusion-Convection Reaction," Birkhäuser Verlag, Basel, 2004.
doi: 10.1007/978-3-0348-7964-4. |
[14] |
D. Hilhorst, R. Kersner, E. Logak and M. Mimura, Interface dynamics of the Fisher equation with degenerate diffusion, J. Differential Equations, 244 (2008), 2870-2889.
doi: 10.1016/j.jde.2008.02.018. |
[15] |
S. Kamin and P. Rosenau, Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Ser. 9 Mat. Appl., 15 (2004), 271-280. |
[16] |
N. S. Kolmogorov, N. Petrovsky and I. G. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière e son application a un problème biologique, (Russian), Bull. Univ. État Moscou, Série internationale A, 1 (1937), 1-26. |
[17] |
G. S. Medvedev, K. Ono and P. Holmes, Traveling wave solutions of the degenerate Kolmogorov-Petrovsky-Piskunov equation, European J. Appl. Math., 14 (2003), 343-367.
doi: 10.1017/S0956792503005102. |
[18] |
J. A. Sherratt, Wave front propagation in a competition equation with a new motility term modelling contact inhibition between cell populations, Proc. R. Soc. Lond. A, 456 (2000), 2365-2386.
doi: 10.1098/rspa.2000.0616. |
[19] |
F. Sanches-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319.
doi: 10.1006/jdeq.1995.1055. |
[20] |
Y. Tsukatani, K. Suzuki and K. Takahashi, Loss of density-dependent growth inhibition and dissociation of $\alpha$-catenin from E-cadherin, J. Cell. Physiol., 173 (1997), 54-63. |
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