Stability and Bifurcation Analysis of a Discrete Gompertz Model with Time Delay
Authors: Yingguo Li
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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1084658Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1569
 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.
 M.J. Piotrowska, U. Forys, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl. 382 (2011) 180-203.
 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.
 L. Dong, L. Chen, L. Sun, Optimal harvesting policies for periodic Gompertz systems, Nonlinear Analysis: Real World Applications 8 (2007) 572-578.
 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.
 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.
 H. Su, X. Ding, Dynamics of a nonstandard finite-difference scheme for Mackey-Glass system, J. Math. Anal. Appl. 344 (2008) 932-941.
 H. Su, X. Ding, Dynamics of a Discretization Physiological Control System, Discrete Dynamics in Nature and Society Article ID 51406 (2007) 16 pages.
 J.F. Neville, V. Wulf, Numerical Hopf bifurcation for a class of delay differential equations, J. Comput. Appl. Math. 115 (2000) 601-616.
 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.
 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.
 Y.A. Kuznetsov, Elements of applied bifurcation theory, New York, Spring-Verlag, 1995.
 B.D. Hassard, N.D. Kazarinoff, Y.H. Wa, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
 S. Wiggins, Introduction to Applied Nonlinear Dynamical System and Chaos, Springer-Verlag, New York, 1990.