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Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

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  • Hu, Guang-Ping
  • Li, Xiao-Ling

Abstract

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.

Suggested Citation

  • Hu, Guang-Ping & Li, Xiao-Ling, 2012. "Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey," Chaos, Solitons & Fractals, Elsevier, vol. 45(3), pages 229-237.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:3:p:229-237
    DOI: 10.1016/j.chaos.2011.11.011
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    References listed on IDEAS

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    1. Sun, Chengjun & Lin, Yiping & Han, Maoan, 2006. "Stability and Hopf bifurcation for an epidemic disease model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 204-216.
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    Cited by:

    1. Zhang, Jia-Fang & Chen, Heshan, 2014. "Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays," Chaos, Solitons & Fractals, Elsevier, vol. 61(C), pages 69-75.
    2. Wei Yang, 2021. "Modeling COVID-19 Pandemic with Hierarchical Quarantine and Time Delay," Dynamic Games and Applications, Springer, vol. 11(4), pages 892-914, December.
    3. 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.
    4. Misra, A.K. & Mishra, S.N. & Pathak, A.L. & Srivastava, P.K. & Chandra, Peeyush, 2013. "A mathematical model for the control of carrier-dependent infectious diseases with direct transmission and time delay," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 41-53.
    5. Sahoo, Banshidhar & Poria, Swarup, 2015. "Effects of allochthonous inputs in the control of infectious disease of prey," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 1-19.
    6. Ma, Zhan-Ping & Yue, Jia-Long, 2023. "Cross diffusion induced spatially inhomogeneous Hopf bifurcation for a three species Lotka–Volterra food web model with cycle," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    7. Lu, Yang & Li, Dan & Liu, Shengqiang, 2016. "Modeling of hunting strategies of the predators in susceptible and infected prey," Applied Mathematics and Computation, Elsevier, vol. 284(C), pages 268-285.
    8. 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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