Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T01:05:29.849Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

Published online by Cambridge University Press:  03 June 2015

Jing-Jun Zhao ,
Jing-Yu Xiao  and
Yang Xu
Jing-Jun Zhao*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China
Jing-Yu Xiao
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China
Yang Xu
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China
*
*Corresponding author. Email: [email protected]
Get access

Abstract

This paper is concerned with the Hopf bifurcation analysis of tumor-immune system competition model with two delays. First, we discuss the stability of state points with different kinds of delays. Then, a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points. Furthermore, under this condition, the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory. Finally, a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically. At last, some numerical experiments are given to match well with the main conclusion of this paper.

Keywords

34K1837G1037G1537N25Hopf bifurcationdelaytumor-immunedynamical systemperiodic solution
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 2 , April 2013 , pp. 146 - 162
Copyright
Copyright © Global-Science Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adam, J. and Bellomo, N., A Survey of Models on Tumor Immune Systems Dynamics, Birkhäuser, Boston, 1996.Google Scholar
[2]Bürger, R., The Mathematical Theory of Selection, Recombination and Mutation, John Wiley, New York, 2000.Google Scholar
[3]Dieckmann, O. and Heesterbeek, J. P., Mathematical Epidemiology of Infectious Diseases, John Wiley, New York, 2000.Google Scholar
[4]May, R. M. and Nowak, M. A., Virus Dynamics (Mathematical Principles of Immunology and Virology), Oxford University Press, Oxford, 2000.Google Scholar
[5]Albert, A., Freedman, M. and Perelson, A. S., Tumors and the immune system: the effects of a tumor growth modulator, Math. Biosci., 50 (1980), pp. 25–58.Google Scholar
[6]Thorn, R. M. and Henney, C. S., Kinetic analysis of target cell destruction by effector T cell, J. Immunol., 117 (1976), pp. 2213–2219.CrossRefGoogle Scholar
[7]Thoma, J. A., Thoma, G. J. and Clark, W., The efficiency and linearity of the radiochromium release assay for cell-mediated cytotoxicity, Cell. Immunol., 40 (1978), pp. 404–418.Google Scholar
[8]Lefever, R., Hiernaux, J., Urbain, J. and Meyers, P., On the kinetics and optimal specificity of cytotoxic reactions mediated by T-lymphocyte clones, Bull. Math. Biol., 54 (1992), pp. 839–873.Google Scholar
[9]Galach, M., Dynamics of the tumor-immune system competition the effect of time delay, Int. J. Appl. Comput. Sci., 13 (2003), pp. 395–406.Google Scholar
[10]Yafia, R., Hopf bifurcation in differential equation with delay for tumor-immune system competition model, SIAM J. Appl. Math., 67 (2007), pp. 1693–1703.Google Scholar
[11]Kuznetsov, V.A. and Taylor, M. A., Nolinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis, Math. Biol., 56 (1994), pp. 295–321.CrossRefGoogle Scholar
[12]Hassard, B., Kazarinoff, N. and Wan, Y., Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.Google Scholar
[13]Hale, J., Theory of Functional Differential Equations, Springer, New York, 1977.Google Scholar
[14]In’t Hout, K.J., A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT, 32 (1992), pp. 634–649.Google Scholar
[15]Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations, John Wiley, Chichester, 1987.Google Scholar
[16]Wang, Q., Li, D. S. and Liu, M. Z., Numerical Hopf bifurcation of Runge-Kutta methods for a class of delay differential equations, Chaos Soliton Fract., 42 (2009), pp. 3087–3099.Google Scholar