Linearized Stability and Hopf Bifurcations for a Nonautonomous Delayed Predator-prey System

Volume 1, Issue 1, October 2016     |     PP. 63-70      |     PDF (242 K)    |     Pub. Date: October 20, 2016
DOI:    503 Downloads     5469 Views  


Li Wang, Assistant Professor, School of Applied Mathematics, Xiamen University of Technology, Xiamen, Fujian Province, Xiamen, 361024 China
Lei Jin, PhD, Faculty of Engineering, University of Regina, Regina, SK S4S 0A2, Canada

In past many years, biomathematics population models are constructed based on plausible explicit and implicit biological assumptions. In the case that not enough analysis is carried out for a well-motivated and plausible model, the result is no or minimum insights gained. In this study, existence of Hopf bifurcations of a nonautonomous delayed predator-prey system with stage-structure for predator is proposed. Furthermore, conditions of linearized stability and Hopf bifurcations for this system are established. Numerical simulations are presented it illustrate the feasibility of our main result.

Hopf bifurcations; stage-structure; positive periodic solation; linearized stability

Cite this paper
Li Wang, Assistant Professor, Lei Jin, PhD, Linearized Stability and Hopf Bifurcations for a Nonautonomous Delayed Predator-prey System , SCIREA Journal of Mathematics. Volume 1, Issue 1, October 2016 | PP. 63-70.


[ 1 ] B.S.Goh, Global stability in two species interactions, J.Math.Biol., 3(1976), 313-318.
[ 2 ] A.Hasbings, Global stability of two species system, J.Math.Biol., 5(1978), 399-402.
[ 3 ] X.Z.He, Stability delay in a predator system, J.Math.Anal.Appl., 198(1996), 355-370.
[ 4 ] Zhengqiu Zhang and Xiawu Zeng, On a periodic stage-structure model, Applied Mathematics Letter, 2003, 16(7), 1053-1061.
[ 5 ] Zhengqiu Zhang, Jun Wu and Zhicheng Wang, Periodic solution of nonautonomous satge-structured cooperative system, Computers Math.Appl., 47(2004), 699-706.
[ 6 ] Yongkun Li,Periodic solution for delay Lotha-Volterra competition system, J.Math.Anal.Appl., 246(2000), 230-240.
[ 7 ] K.Gopalsamy, M.R.Kunlenovic and G.Ladas, Environmental periodicity and time delays in a food-limited population model, J.Math.Anal.Appl., 147(1990).
[ 8 ] B.G.Zhang And K.Gopalsamy, Global attractively and oscillations in a periodic delay logistic equation, J.Math.Anal.Appl., 150(1990), 274-183.
[ 9 ] Wendi Wang, A periodic-prey system with stable-structure for predator, Computers Math.Applic., 33(8),1997,83-91.
[ 10 ] W.G.Aiello, H.I.Fredman, A time delay model of single-species growth stage-structure, Math.Biosic. 101(1990), 139-153.
[ 11 ] W.G.Aiello, H.I.Fredman and J.Wu, Analysis of a model representing stage-structure population growing with state dependent time delay, J.Appl.Math., 52(1992), 855-869.