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A simple model of carcinogenic mutations with time delay and diffusion
1. | Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw |
2. | College of Inter-faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Zwirki i Wigury 93, 02-089 Warsaw, Poland |
References:
[1] |
J. A. Adam and N. Bellomo, "A Survey of Models for Tumor-imune System Synamics," Birkhäuser, Boston, 1997. |
[2] |
R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, Electron. J. Diff. Eqns., 10 (2003), 33-53. |
[3] |
P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[4] |
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. |
[5] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[6] |
U. Foryś, Comparison of the models for carcinogenesis mutations - one-stage case, in "Proceedings of the Tenth National Conference Application of Mathematics in Biology and Medicine," Świçety Krzy.z, (2004), 13-18. |
[7] |
U. Foryś, Time delays in one-stage models for carcinogenesis mutations, in "Proceedings of the Eleventh National Conference Application of Mathematics in Biology and Medicine", Zawoja, (2005), 13-18. |
[8] |
U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, J. Appl. Anal., 11 (2005), 200-281.
doi: 10.1515/JAA.2005.283. |
[9] |
U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations, Math. Meth. Appl. Sci., 32 (2009), 2287-2308.
doi: 10.1002/mma.1137. |
[10] |
J. K. Hale, "Theory of Functional Differential Equations," Springer, 1977. |
[11] |
J. D. Murray, "Mathematical Biology I: An Introduction," Springer, 2002. |
[12] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer, 2003. |
[13] |
A. S. Perelson and G. Weisbuch, Immunology for physicists, Rev. Mod. Phys., 69 (1997), 1219-1267.
doi: 10.1103/RevModPhys.69.1219. |
show all references
References:
[1] |
J. A. Adam and N. Bellomo, "A Survey of Models for Tumor-imune System Synamics," Birkhäuser, Boston, 1997. |
[2] |
R. Ahangar and X. B. Lin, Multistage evolutionary model for carcinogenesis mutations, Electron. J. Diff. Eqns., 10 (2003), 33-53. |
[3] |
P. K. Brazhnik and J. J. Tyson, On travelling wave solutions of Fisher's equation in two spatial dimensions, SIAM J. Appl. Math., 60 (1999), 371-391.
doi: 10.1137/S0036139997325497. |
[4] |
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. |
[5] |
T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
doi: 10.1006/jmaa.2000.7182. |
[6] |
U. Foryś, Comparison of the models for carcinogenesis mutations - one-stage case, in "Proceedings of the Tenth National Conference Application of Mathematics in Biology and Medicine," Świçety Krzy.z, (2004), 13-18. |
[7] |
U. Foryś, Time delays in one-stage models for carcinogenesis mutations, in "Proceedings of the Eleventh National Conference Application of Mathematics in Biology and Medicine", Zawoja, (2005), 13-18. |
[8] |
U. Foryś, Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations, J. Appl. Anal., 11 (2005), 200-281.
doi: 10.1515/JAA.2005.283. |
[9] |
U. Foryś, Multi-dimensional Lotka-Volterra system for carcinogenesis mutations, Math. Meth. Appl. Sci., 32 (2009), 2287-2308.
doi: 10.1002/mma.1137. |
[10] |
J. K. Hale, "Theory of Functional Differential Equations," Springer, 1977. |
[11] |
J. D. Murray, "Mathematical Biology I: An Introduction," Springer, 2002. |
[12] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer, 2003. |
[13] |
A. S. Perelson and G. Weisbuch, Immunology for physicists, Rev. Mod. Phys., 69 (1997), 1219-1267.
doi: 10.1103/RevModPhys.69.1219. |
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