Stability and Bifurcation Analysis of a Discrete Gompertz Model with Time Delay
In this paper, we consider a discrete Gompertz model with time delay. Firstly, the stability of the equilibrium of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark- Sacker bifurcations occur when the delay passes a sequence of critical values. The direction and stability of the Neimark-Sacker are determined by using normal forms and centre manifold theory. Finally, some numerical simulations are given to verify the theoretical analysis.
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Volterra model, Advances in Differential Equations, vol. 1, no. 3, (1996)
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of angiogenesis models, J. Math. Anal. Appl. 382 (2011) 180-203.
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[6] C. Zhang, Y. Zu, B. Zheng, Stability and bifurcation of a discrete red
blood cell survival model, Chaos, Solitons and Fractals 28 (2006) 386-
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[7] H. Su, X. Ding, Dynamics of a nonstandard finite-difference scheme for
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[8] H. Su, X. Ding, Dynamics of a Discretization Physiological Control
System, Discrete Dynamics in Nature and Society Article ID 51406
(2007) 16 pages.
[9] J.F. Neville, V. Wulf, Numerical Hopf bifurcation for a class of delay
differential equations, J. Comput. Appl. Math. 115 (2000) 601-616.
[10] Z. He, X. Lai, A Hou, Stability and Neimark-Sacker bifurcation of
numerical discretization of delay differential equations, Chaos, Solitons
and Fractals 41 (2009) 2010-2017.
[11] H. Shu, J. Wei, Bifurcation analysis in a discrete BAM network model
with delays, Journal of Difference Equations and Applications 17:1 69-84
(2011) 69-84.
[12] Y.A. Kuznetsov, Elements of applied bifurcation theory, New York,
Spring-Verlag, 1995.
[13] B.D. Hassard, N.D. Kazarinoff, Y.H. Wa, Theory and Applications of
Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
[14] S. Wiggins, Introduction to Applied Nonlinear Dynamical System and
Chaos, Springer-Verlag, New York, 1990.
[1] J. Lopez-Gomez, R. Ortega, A. Tineo, The periodic predator-prey Lotka-
Volterra model, Advances in Differential Equations, vol. 1, no. 3, (1996)
403-423.
[2] M.J. Piotrowska, U. Forys, Analysis of the Hopf bifurcation for the family
of angiogenesis models, J. Math. Anal. Appl. 382 (2011) 180-203.
[3] J. Jia, C. Li, A Predator-Prey Gompertz Model with Time Delay and
Impulsive Perturbations on the Prey. Discrete Dynamics in Nature and
Society Article ID 256195 (200) 15 pages.
[4] L. Dong, L. Chen, L. Sun, Optimal harvesting policies for periodic
Gompertz systems, Nonlinear Analysis: Real World Applications 8 (2007)
572-578.
[5] Q. Wang, D. Li, M.Z. Liu, Numerical Hopf bifurcation of Runge-Kutta
methods for a class of delay differential equations, Chaos, Solitons and
Fractals 42 (2009) 3087-3099.
[6] C. Zhang, Y. Zu, B. Zheng, Stability and bifurcation of a discrete red
blood cell survival model, Chaos, Solitons and Fractals 28 (2006) 386-
394.
[7] H. Su, X. Ding, Dynamics of a nonstandard finite-difference scheme for
Mackey-Glass system, J. Math. Anal. Appl. 344 (2008) 932-941.
[8] H. Su, X. Ding, Dynamics of a Discretization Physiological Control
System, Discrete Dynamics in Nature and Society Article ID 51406
(2007) 16 pages.
[9] J.F. Neville, V. Wulf, Numerical Hopf bifurcation for a class of delay
differential equations, J. Comput. Appl. Math. 115 (2000) 601-616.
[10] Z. He, X. Lai, A Hou, Stability and Neimark-Sacker bifurcation of
numerical discretization of delay differential equations, Chaos, Solitons
and Fractals 41 (2009) 2010-2017.
[11] H. Shu, J. Wei, Bifurcation analysis in a discrete BAM network model
with delays, Journal of Difference Equations and Applications 17:1 69-84
(2011) 69-84.
[12] Y.A. Kuznetsov, Elements of applied bifurcation theory, New York,
Spring-Verlag, 1995.
[13] B.D. Hassard, N.D. Kazarinoff, Y.H. Wa, Theory and Applications of
Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
[14] S. Wiggins, Introduction to Applied Nonlinear Dynamical System and
Chaos, Springer-Verlag, New York, 1990.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64236", author = "Yingguo Li", title = "Stability and Bifurcation Analysis of a Discrete Gompertz Model with Time Delay", abstract = "In this paper, we consider a discrete Gompertz model with time delay. Firstly, the stability of the equilibrium of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark- Sacker bifurcations occur when the delay passes a sequence of critical values. The direction and stability of the Neimark-Sacker are determined by using normal forms and centre manifold theory. Finally, some numerical simulations are given to verify the theoretical analysis.
", keywords = "Gompertz system, Neimark-Sacker bifurcation, stability, time delay.", volume = "5", number = "8", pages = "1427-4", }
