Volume 5, Issue 2
Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

Adv. Appl. Math. Mech., 5 (2013), pp. 146-162.

Published online: 2013-05

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• 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

Hopf bifurcation delay tumor-immune dynamical system periodic solution

34K18 37G10 37G15 37N25

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@Article{AAMM-5-146, author = {Jing-Jun Zhao, Jing-Yu Xiao and Yang Xu}, title = {Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2013}, volume = {5}, number = {2}, pages = {146--162}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-m1224}, url = {http://global-sci.org/intro/article_detail/aamm/62.html} }
TY - JOUR T1 - Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays AU - Jing-Jun Zhao, Jing-Yu Xiao & Yang Xu JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 146 EP - 162 PY - 2013 DA - 2013/05 SN - 5 DO - http://dor.org/10.4208/aamm.12-m1224 UR - https://global-sci.org/intro/article_detail/aamm/62.html KW - Hopf bifurcation KW - delay KW - tumor-immune KW - dynamical system KW - periodic solution AB -

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.

Jing-Jun Zhao, Jing-Yu Xiao & Yang Xu. (1970). Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays. Advances in Applied Mathematics and Mechanics. 5 (2). 146-162. doi:10.4208/aamm.12-m1224
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