2017 Volume 7 Issue 3
Article Contents

Jingnan Wang, Weihua Jiang. HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1051-1069. doi: 10.11948/2017066
Citation: Jingnan Wang, Weihua Jiang. HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1051-1069. doi: 10.11948/2017066

HOPF-ZERO BIFURCATION OF A DELAYED PREDATOR-PREY MODEL WITH DORMANCY OF PREDATORS

  • Fund Project:
  • In this paper, We investigate Hopf-zero bifurcation with codimension 2 in a delayed predator-prey model with dormancy of predators. First we prove the specific existence condition of the coexistence equilibrium. Then we take the mortality rate and time delay as two bifurcation parameters to find the occurrence condition of Hopf-zero bifurcation in this model. Furthermore, using the Faria and Magalhases normal form method and the center manifold theory, we obtain the third order degenerate normal form with two original parameters. Finally, through theoretical analysis and numerical simulations, we give a bifurcation set and a phase diagram to show the specific relations between the normal form and the original system, and explain the coexistence phenomena of several locally stable states, such as the coexistence of multiperiodic orbits, as well as the coexistence of a locally stable equilibrium and a locally stable periodic orbit.
    MSC: 34K18;37G05;37G10;92D25
  • 加载中
  • [1] E. Beninca, J. Huisman, R. Heerkloss, K. D. Johnk, P. Branco, E. H. Van Nes, M. Scheffer and S. P. Ellner, Chaos in a long-term experiment with a plankton community, Nature, 2008, 451, 822-825.

    Google Scholar

    [2] E. Beninca, K. D. Johnk, R. Heerkloss and J. Huisman, Coupled predator-prey oscillations in a chaotic food web, Ecology Letters, 2009, 12, 1367-1378.

    Google Scholar

    [3] J. Carr, Applications of Centre Manifold Theory, Springer, 1981.

    Google Scholar

    [4] Y. Ding, W. Jiang and P. Yu, Hopf-zero bifurcation in a generalized Gopalsamy neural network model, Nonlinear Dynamics, 2012, 70, 1037-1050.

    Google Scholar

    [5] Y. Ding, W. Jiang and H. Wang, Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay, Chaos Solitons & Fractals, 2012, 45, 1048-1057.

    Google Scholar

    [6] T. Faria and L. Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, Journal of Differential Equations, 1995, 122, 201-224.

    Google Scholar

    [7] T. Faria and L. Magalhaes, Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation, Journal of Differential Equations, 1995, 122, 181-200.

    Google Scholar

    [8] S. Guo and W. Jiang, Hopf-fold bifurcation and fluctuation phenomena in a delayed rotio-dependent Gause-type predator-prey model, International Journal of Bifurcation and Chaos, 2013, 23, 1350153.

    Google Scholar

    [9] J. Ge and J. Xu, An efficient method for studying fold-Hopf bifurcation in delayed neural networks, International Journal of Bifurcation and Chaos, 2011, 21, 1393-1406.

    Google Scholar

    [10] S. Guo, Y. Chen, and J. Wu, Two-parameter bifurcations in a network of two neurons with multiple delays, Journal of Differential Equations, 2008, 244, 444-86.

    Google Scholar

    [11] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer, 1983.

    Google Scholar

    [12] X. He, C. Li, T. Huang and C. Li, Codimension two bifurcation in a delayed neural network with unidirectional coupling, Nonlinear Analysis:RealWorld Applications, 2013, 14, 1191-1202.

    Google Scholar

    [13] M. Han and P. Yu, Normal Forms, Melnikov Functions, and Bifurcations of Limit Cycles, Springer-Verlag, 2012.

    Google Scholar

    [14] W. Jiang and B. Niu, On the coexistence of periodic or quasi-periodic oscillations near a Hopf-pitchfork bifurcation in NFDE, Communications in Nonlinear Science and Numerical Simulation, 2013, 18, 464-477.

    Google Scholar

    [15] W. Jiang and H. Wang, Hopf-transcritical bifurcation in retarded functional differential equations, Nonlinear Analysis, 2010, 73, 3626-3640.

    Google Scholar

    [16] M. Kuwamura, T. Nakazawa and T. Ogawa, A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment, Journal of Mathematical Biology, 2009, 58, 459-479.

    Google Scholar

    [17] M. Kuwamura and H. Chiba, Mixed-mode oscillations and chaos in a preypredator system with dormancy of predators, Chaos, 2009, 190, 043121.

    Google Scholar

    [18] S. Ruan and J. Wei, On the zero of transcendental functions with applications to stability of delay defferential equations with two delays, Dynamics of continuous, Discrete and Impulsive Systems Series A:Mathematical Analysis, 2003, 10, 863-874.

    Google Scholar

    [19] J. Wang and W. Jiang, Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dynamics, 2012, 69, 1541-1558.

    Google Scholar

    [20] Y. Wang, H. Wang and W. Jiang, Hopf-transcritical bifurcation in toxic phytoplankton-zooplankton model with delay, Journal of Mathematical Analysis and Applications, 2014, 415, 574-594.

    Google Scholar

    [21] J. Wang and W. Jiang, Hopf-pitchfork bifurcation in a two-neuron system with discrete and distributed delays, Mathemtical methods in the Applied Sciences, 2015, 38, 4967-4981.

    Google Scholar

    [22] H. Wang and W. Jiang, Hopf-pitchfork bifurcation in van der Pols oscillator with nonlinear delayed feedback, Journal of Mathematical Analysis and Applications, 2010, 368, 9-18.

    Google Scholar

    [23] B. Zhen and J. Xu, Fold-Hopf bifurcation analysis for a coupled FitzHughNagumo neural system with time delay, International Journal of Bifurcation and Chaos, 2010, 20, 3919-3934.

    Google Scholar

Article Metrics

Article views(2836) PDF downloads(1535) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint