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Stability and Hopf bifurcation analysis of a ratio-dependent predator–prey model with two time delays and Holling type III functional response

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  • Wang, Xuedi
  • Peng, Miao
  • Liu, Xiuyu

Abstract

In this paper, a delayed ratio-dependent predator–prey model with Holling type III functional response and stage structure for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcations at the coexistence equilibrium is established. By utilizing normal form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations supporting the theoretical analysis are given.

Suggested Citation

  • Wang, Xuedi & Peng, Miao & Liu, Xiuyu, 2015. "Stability and Hopf bifurcation analysis of a ratio-dependent predator–prey model with two time delays and Holling type III functional response," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 496-508.
    • Handle: RePEc:eee:apmaco:v:268:y:2015:i:c:p:496-508
    DOI: 10.1016/j.amc.2015.06.108
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    References listed on IDEAS

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    1. Xu, Rui & Ma, Zhien, 2008. "Stability and Hopf bifurcation in a ratio-dependent predator–prey system with stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 38(3), pages 669-684.
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    Cited by:

    1. Peng, Miao & Zhang, Zhengdi & Qu, Zifang & Bi, Qinsheng, 2020. "Qualitative analysis in a delayed Van der Pol oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 544(C).
    2. Du, Wentong & Xiao, Min & Ding, Jie & Yao, Yi & Wang, Zhengxin & Yang, Xinsong, 2023. "Fractional-order PD control at Hopf bifurcation in a delayed predator–prey system with trans-species infectious diseases," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 414-438.
    3. Zhou, Weigang & Huang, Chengdai & Xiao, Min & Cao, Jinde, 2019. "Hybrid tactics for bifurcation control in a fractional-order delayed predator–prey model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 183-191.
    4. Chen, Xiaoxiao & Wang, Xuedi, 2019. "Qualitative analysis and control for predator-prey delays system," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 361-372.

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