Citation: |
[1] |
J. Al-Omari and S. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Analysis: Real World Applications 6 (2005), 13-33.doi: 10.1016/j.nonrwa.2004.04.002. |
[2] |
B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, "Molecular Biology of the Cell," 5th edition, Taylor & Francis Ltd., New York, 2007. |
[3] |
A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations," Clarendon Press, New York, 2003. |
[4] |
S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, The dynamics of population models with distributed maturation periods, Theoretical Population Biology, 25 (1984), 289-311.doi: 10.1016/0040-5809(84)90011-X. |
[5] |
G. Bocharov and K. P. Hadeler, Structured population models, conservation laws, and delay equations, Journal of Differential Equations, 168 (2000), 212-237.doi: 10.1006/jdeq.2000.3885. |
[6] |
M. Chaplain and A. Matzavinos, Mathematical modeling of spatio-temporal phenomena in tumor immunology, in "Tutorials in Mathematical Biosciences. III," Lecuture Notes in Math., 1872, Berlin, (2006), 131-183. |
[7] |
K. L. Cooke and P. van den Driessche, On zeros of some transcendental equations, Funkcialaj Ekvacioj, 29 (1986), 77-90. |
[8] |
G. M. Cooper and R. E. Hausman, "The Cell: A Molecular Approach,'' ASM Press, Washington, 1997. |
[9] |
A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences, Physica D, 208 (2005), 220-235.doi: 10.1016/j.physd.2005.06.032. |
[10] |
A. d'Onofrio, Tumor-immune system interaction: Modelling the tumor-stimulated proliferation of effectors and immunotherapy, Mathematical Models and Methods in Applied Science, 16 (2006), 1375-1401.doi: 10.1142/S0218202506001571. |
[11] |
A. d'Onofrio, F. Gatti, P. Cerrai and L. Freschi, Delay-induced oscillatory dynamics of tumour-immune system interaction, Mathematical and Computer Modelling, 51 (2010), 572-591. |
[12] |
J. Dyson, R. Villella-Bressan and G. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Deterministic and Stochastic Modeling of Biointeraction (West Lafayette, IN, 2000), Mathematical Biosciences, 177/178 (2002), 73-83.doi: 10.1016/S0025-5564(01)00097-9. |
[13] |
J. Dyson, R. Villella-Bressan and G. Webb, A spatial model of tumor growth with cell age, cell size, and mutation of cell phenotypes, Mathematical Modelling of Natural Phenomena, 2 (2007), 69-100.doi: 10.1051/mmnp:2007004. |
[14] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.doi: 10.1038/287017a0. |
[15] |
Y. Kuang, "Delay Differential Equations: With Applications in Population Dynamics,'' Academic Press, New York, 2003. |
[16] |
W. Liu, T. Hillen and H. Freedman, A mathematical model for $M$-phase specific chemotherapy including the $G_0$-phase and immunoresponse, Mathematical Bioscience and Engineering, 4 (2007), 239-259.doi: 10.3934/mbe.2007.4.239. |
[17] |
H. Lodish et al., "Molecular Cell Biology,'' 3rd Ed. Scientific American Books , New York, 1995. |
[18] |
N. MacDonald, "Biological Delay Systems: Linear Stability Theory,'' Cambridge University Press, 1989. |
[19] |
R. Nisbet and W. Gurney, The systematic formulation of population models for insects with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135.doi: 10.1016/0040-5809(83)90008-4. |
[20] |
T. Roose, S. J. Chapman and P. K. Maini, Mathematical models of avascular tumor growth, SIAM Review, 49 (2007), 179-208.doi: 10.1137/S0036144504446291. |
[21] |
R. A. Santiago-Mozos, I. G. Khan and M. Madden, Revealing the origin and nature of drug resistance of dynamic tumour systems, International Journal of Knowledge Discovery in Bioinformatics, 1 (2010), 26-53. |
[22] |
F. R. Sharpe and A. J. Lotka, A problem in age distribution, Philosophical Magazine Series 6, 21 (1911), 435-438.doi: 10.1080/14786440408637050. |
[23] |
H. Smith, "Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems," Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island, 1995. |
[24] |
H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Springer, New York, 2011.doi: 10.1007/978-1-4419-7646-8. |
[25] |
U. Veronesi and G. Quaranta, "Un Male Curabile,'' Mondadori Editore, Milano, 1986. |
[26] |
M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, Journal of Mathematical Biology, 47 (2003), 270-294.doi: 10.1007/s00285-003-0211-0. |
[27] |
G. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 1985. |