Abstract
In this paper, we consider the differential-algebraic predator–prey model with predator harvesting and two delays. By using the new normal form of differential-algebraic systems, center manifold theorem and bifurcation theory, we analyze the stability and the Hopf bifurcation of the proposed system. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of the delayed differential-algebraic systems. Finally, numerical simulations are given to show the consistency with theoretical analysis obtained here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Leslie, P.H., Gower, J.C.: The properties of a stochastic model for the predator–prey type of interaction between two species. Biometrika 47, 219–234 (1960)
Ma, Y.F.: Global Hopf bifurcation in the Leslie–Gower predator–prey model with two delays. Nonlinear Anal., Real World Appl. 13, 370–375 (2012)
Chen, F.D., Chen, L.J., Xie, X.D.: On a Leslie–Gower predator–prey model incorporating a prey refuge. Nonlinear Anal., Real World Appl. 10, 2905–2908 (2009)
Li, N., Yuan, H.Q., Sun, H.Y., Zhang, Q.L.: An impulsive multi-delayed feedback control method for stabilizing discrete chaotic systems. Nonlinear Dyn. (2012). doi:10.1007/s11071-012-0434-y
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)
Song, Y.L., Yuan, S.L.: Bifurcation analysis in a predator–prey system with time delay. Nonlinear Anal., Real World Appl. 7, 265–284 (2006)
Ji, L.L., Wu, C.Q.: Qualitative analysis of a predator–prey model with constant rate prey harvesting incorporating a constant prey refuge. Nonlinear Anal., Real World Appl. 11, 2285–2295 (2010)
Song, Y.L., Peng, Y.H., Wei, J.J.: Bifurcations for a predator–prey system with two delays. J. Math. Anal. Appl. 337, 466–479 (2008)
Wang, J.L., Wu, H.N.: Synchronization criteria for impulsive complex dynamical networks with time-varying delay. Nonlinear Dyn. 70, 13–24 (2012)
Kar, T.K., Pahari, U.K.: Non-selective harvesting in prey-predator models with delay. Commun. Nonlinear Sci. Numer. Simul. 11, 499–509 (2006)
Wei, J.J., Ruan, S.G.: Stability and bifurcation in a neural network model with two delays. Physica D 130, 253–272 (1999)
Zhang, G.D., Shen, Y., Wang, L.M.: Global anti-synchronization of a class of chaotic memristive neural networks with time-varying delays. Neural Netw. 46, 1–8 (2013)
Xiao, D.M., Li, W.X., Han, M.A.: Dynamics in a ratio-dependent predator–prey model with predator harvesting. J. Math. Anal. Appl. 324, 14–29 (2006)
Zhang, G.D., Shen, Y., Chen, B.S.: Positive periodic solutions in a non-selective harvesting predator–prey model with multiple delays. J. Math. Anal. Appl. 395, 298–306 (2012)
Kar, T.K., Ghorai, A.: Dynamic behaviour of a delayed predator–prey model with harvesting. Appl. Math. Comput. 217, 9085–9104 (2011)
Gordon, H.S.: Economic theory of a common property resource: the fishery. J. Polit. Econ. 62, 124–142 (1954)
Zhang, G.D., Zhu, L.L., Chen, B.S.: Hopf bifurcation and stability for a differential-algebraic biological economic system. Appl. Math. Comput. 217, 330–338 (2010)
Zhang, G.D., Zhu, L.L., Chen, B.S.: Hopf bifurcation in a delayed differential-algebraic biological economic system. Nonlinear Anal., Real World Appl. 12, 1708–1719 (2011)
Zhang, X., Zhang, Q.L., Zhang, Y.: Bifurcations of a class of singular biological economic models. Chaos Solitons Fractals 40, 1309–1318 (2009)
Chen, B.S., Liao, X.X., Liu, Y.Q.: Normal forms and bifurcations for the difference-algebraic systems. Acta Math. Appl. Sin. 23, 429–433 (2000) (in Chinese)
Faria, T., Magalhães, L.T.: Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995)
Faria, T., Magalhães, L.T.: Normal form for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differ. Equ. 122, 201–224 (1995)
Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
Boothby, W.M.: An Introduction to Differential Manifolds and Riemannian Geometry, 2nd edn. Academic Press, New York (1986)
Cooke, K.L., Grossman, Z.: Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl. 86, 592–627 (1982)
Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10, 863–874 (2003)
Hale, J.K., Lunel, S.: Introduction to Functional Differential Equations. Springer, New York (1993)
Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)
Acknowledgements
The authors gratefully acknowledge two anonymous referees’ comments and patient work. This work is supported by the National Science Foundation of China under Grant No. 11271146, the Key Program of National Natural Science Foundation of China under Grant No. 61134012, the Science and Technology Program of Wuhan under Grant No. 2013010501010117 and the Development Program of the Department of Science and Technology of Henan Province under Grant No. 122300410276.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, G., Shen, Y. & Chen, B. Hopf bifurcation of a predator–prey system with predator harvesting and two delays. Nonlinear Dyn 73, 2119–2131 (2013). https://doi.org/10.1007/s11071-013-0928-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-013-0928-2