Stability and HOPF Bifurcation Analysis in a Stage-structured Predator-prey system with Two Time Delays

A stage-structured predator-prey system with two time delays is considered. By analyzing the corresponding characteristic equation, the local stability of a positive equilibrium is investigated and the existence of Hopf bifurcations is established. Formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results. Based on the global Hopf bifurcation theorem for general functional differential equations, the global existence of periodic solutions is established.

• Yongkun Li
• Meng Hu

• Predator-prey system
• stage structure
• time delay
• HOPF bifurcation
• periodic solution
• stability.

[1] Z. Jing, J. Yang, Bifurcation and chaos in discrete-time predator-prey
system, Chaos, Solitons & Fractals 27(2006) 259-277.
[2] H. Huo, W. Li, Periodic solutions of a periodic Lotka-Volterra system
with delays, Appl. Math. Comput. 156 (2004) 787-803.
[3] S. Gao, L. Chen, Z. Teng, Hopf bifurcation and global stability for a
delayed predator-prey system with stage structure for predator, Appl.
Math. Comput. 202(2008) 721-729.
[4] S. Li, X. Liao, C. Li, Hopf bifurcation in a Volterra prey-predator model
with strong kernel, Chaos, Solitons & Fractals 22(2004) 713-722.
[5] J. Wei, M. Li, Hopf bifurcation analysis in a delayed Nicholson blowflies
equation, Nonlinear Anal. 60(2005) 1351-1367.
[6] R. Xu, F. Hao, L. Chen, A stage-structured predator-prey model with time
delays, Acta Mathematica Scientia , 26A(3)(2006) 387-395(in Chinese).
[7] R. Xu, Z. Ma, The effect of stage-structure on the permanence of a
predator-prey system with time delay, Appl. Math. Comput. 189(2007)
1164-1177.
[8] K. Wang, W. Wang, H. Pang, X. Liu, Complex dynamic behavior in a
viral model with delayed immune response, Phys. D 226(2007) 197-208.
[9] K. Cooke, Z. Grossman, Discrete delay, distributed delay and stability
switches, Journal of Mathematical Analysis and Applications, 86(1982)
592-627.
[10] L. Wang, W. Li, Existence and global stability of positive periodic
solutions of a predator-prey system with delays, Appl. Math. Comput.
146(2003) 167-185.
[11] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations
of Population Dynamics. Kluwer Academic, Dordrecht Norwell, MA
(1992).
[12] J.K. Hale, S.V. Lunel, Introduction to Functional Differential Equations.
NewYork: Springer-Verlag(1993).
[13] B. Hassard, N. Kazarinoff, Y.H. Wan, Theory and Applications of
Hopf Bifurcation, London Math Soc. Lect. Notes, Series, 41.Cambridge:
Cambridge Univ. Press(1981).
[14] Y. Kuang, Delay Differential Equations with Applications in Population
Dynamics, Academic Press, New York(1993).
[15] J. Wu, Symmetric functional differential equations and neural networks
with memory, Trans. Amer. Math. Soc. 350(1998) 4799-4838.
[16] D. Mehdi, N. Mostafa, R. Razvan, Global stability of a deterministic
model for HIV infection in vivo. Chaos, Solitons & Fractals 34(4)(2007)
1225-1238.
[17] D. Greenhalgh, Q.J.A. Khan, F.I. Lewis, Recurrent epidemic cycles in an
infectious disease model with a time delay in loss of vaccine immunity,
Nonlinear Anal. 63(2005) 779-788.
[18] B.S. Goh, Global stability in two species interactions, Journal of
Mathematical Biology, 3(1976) 313-318.
[19] A. Hastings, Global stability of two-species systems, J. Math. Biology,
5(1977/78) 399-403.
[20] X. He, Stability and delays in a predator-prey system, J. Math. Anal.
Appl. 198(1996) 355-370.