Periodic Oscillations in a Delay Population Model

In this paper, a nonlinear delay population model is investigated. Choosing the delay as a bifurcation parameter, we demonstrate that Hopf bifurcation will occur when the delay exceeds a critical value. Global existence of bifurcating periodic solutions is established. Numerical simulations supporting the theoretical findings are included.

• Changjin Xu
• Peiluan Li

• Population model
• Stability
• Hopf bifurcation
• Delay
• Global Hopf bifurcation.

[1] V. G. Nazarenko, Influence of delay on auto-oscillation in cell populations. Biofisika 21(1976)352-356.
[2] I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay
differential equations of population dynamics. Math. Comput. Modelling
35(3-4)(2002)295-301.
[3] S. H. Saker and S. Agarwal, Oscillation and global attractivity in a
nonlinear delay periodic model of population dynamics. Appl. Anal.
81(4)(2002)787-799.
[4] S. H. Saker, Periodic solutions, oscillation and attractivity of discrete
nonlinear delay population model. Math. Comput. Modelling 47(3)
(2008)278-297.
[5] Y. L. Song and Y. H. Peng, Periodic solution of a nonautonomous
periodic model with continuous and discrete time. J. Comput. Anal. Math.
188(2)(2006)256-264.
[6] K. J. Zhang and Z. H. Wen, Periodic solution for a delayed population
model on time scales. Int. J. Comput. Math. Sci. 4(3)(2010)166-168.
[7] Y. Kuang, Delay Differential Equations with Applications in Population
Dynamics. Academic Press, INC, 1993.
[8] J. Hale,Theory of Functional Differential Equations. Springer-Verlag,
1977.
[9] Jianhong Wu, Symmetric functional differential equations and neural
networks with memory. Transcations of AMS, 350(12)(1998)4799-4838.