Analysis of a delayed mathematical model for tumor growth

Nonlinear Analysis: Real World Applications

Volumn 11, Issue 5, 2010, Pages 4121-4127

  Xu, Shihe ^a,b  

^a ZHAOQING UNIVERSITY   (China)

^b SUN YAT SEN UNIVERSITY   (China)

Author keywords

Dynamical behavior; Retarded differential equations; Stationary solutions; Tumor growth

Indexed keywords

APOPTOSIS; DYNAMICAL BEHAVIORS; MASS CONSERVATION; NON-NEGATIVITY; NONLINEAR REACTION-DIFFUSION EQUATIONS; PROLIFERATION AND APOPTOSIS; RETARDED DIFFERENTIAL EQUATIONS; STABILITY OF STATIONARY SOLUTIONS; STATIONARY SOLUTIONS; TUMOR GROWTH;

CELL DEATH; DIFFERENTIAL EQUATIONS; DYNAMICS; NONLINEAR EQUATIONS; TUMORS;

MATHEMATICAL MODELS;

EID: 77955510611     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2010.04.001     Document Type: Article

References (24)

1. H. Byrne The effect of time delays on the dynamics of avascular tumor growth Math. Biosci. 144 1997 83 117
2. H. Byrne, and M. Chaplain Growth of nonnecrotic tumors in the presence and absence of inhibitors Math. Biosci. 130 1995 151 181
3. H. Byrne, and M. Chaplain Growth of necrotic tumors in the presence and absence of inhibitors Math. Biosci. 135 1996 187 216
4. M. Dorie, R. Kallman, and D. Rapacchietta Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids Exp. Cell Res. 141 1982 201 209
5. H. Greenspan Models for the growth of solid tumor by diffusion Stud. Appl. Math. 51 1972 317 340
6. H. Greenspan On the growth and stability of cell cultures and solid tumors J. Theoret. Biol. 56 1976 229 242
7. K. Thompson, and H. Byrne Modelling the internalisation of labelled cells in tumor spheroids Bull. Math. Biol. 61 1999 601 623
8. J. Ward, and J. King Mathematical modelling of avascular-tumor growth II: modelling growth saturation IMA J. Math. Appl. Med. Biol. 15 1998 1 42
9. M. Bodnar, and U. Fory Time delay in necrotic core formation Math. Biosci. Eng. 2 2005 461 472
10. S. Cui Analysis of a mathematical model for the growth of tumors under the action of external inhibitors J. Math. Biol. 44 2002 395 426
11. S. Cui, and A. Friedman Analysis of a mathematical model of the effect of inhibitors on the growth of tumors Math. Biosci. 164 2000 103 137
12. S. Cui Analysis of a free boundary problem modeling tumor growth Acta. Math. Sinica 21 2005 1071 1082
13. U. Fory, and M. Bodnar Time delays in proliferation process for solid avascular tumour Math. Comput. Modelling 37 2003 1201 1209 (Pubitemid 38517054)
14. U. Fory, and M. Kolev Time delays in proliferation and apoptosis for solid avascular tumour R. Rudnicki, Mathematical Modelling of Population Dynamics vol. 63 2004 Banach Center. Publ. 187 196
15. A. Friedman, and F. Reitich Analysis of a mathematical model for the growth of tumors J. Math. Biol. 38 1999 262 284
16. M.J. Piotrowska A remark on the ODE with two discrete delays J. Math. Anal. Appl. 329 2007 664 676
17. M.J. Piotrowska Hopf bifurcation in a solid avascular tumor growth model with two discrete delays Math. Comput. Modeling 47 2008 597 603
18. R.R. Sarkar, S. Banerjee, A time delay model for control of malignant tumor growth, in: National Conference on Nonlinear Systems and Dynamics, 2006, pp. 14.
19. S. Xu Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays Chaos Solitons Fractals 41 2009 2491 2494
20. S. Xu Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation Nonlinear Anal., B: RWA 11 2010 401 406
21. S. Xu Hopf bifurcation of tumor growth under direct effect of inhibitors with two time delays Int. J. Appl. Math. Comput. 1 2009 97 103
22. S. Cui, and S. Xu Analysis of mathematical models for the growth of tumors with time delays in cell proliferation J. Math. Anal. Appl. 336 2007 523 541
23. J. Smoller Shock Waves and ReactionDiffusion Equations 1994 Springer-Verlag New York Inc.
24. M. Bodnar The nonnegativity of solutions of delay differential equations Appl. Math. Lett. 13 2000 91 95