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Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge

Author

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  • Jana, Soovoojeet
  • Chakraborty, Milon
  • Chakraborty, Kunal
  • Kar, T.K.

Abstract

This paper describes a prey–predator model with Holling type II functional response incorporating prey refuge. The equilibria of the proposed system are determined and the behavior of the system is investigated around equilibria. Density-dependent mortality rate for the predator is considered as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibrium point. Discrete-type gestational delay of predators is also incorporated on the system. The dynamics of the delay induced prey–predator system is analyzed. Delay preserving stability and direction of the system is studied. Global stability of the delay preserving system is shown. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

Suggested Citation

  • Jana, Soovoojeet & Chakraborty, Milon & Chakraborty, Kunal & Kar, T.K., 2012. "Global stability and bifurcation of time delayed prey–predator system incorporating prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 85(C), pages 57-77.
    • Handle: RePEc:eee:matcom:v:85:y:2012:i:c:p:57-77
    DOI: 10.1016/j.matcom.2012.10.003
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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.
    2. Tian, Yuan & Sun, Kaibiao & Chen, Lansun, 2011. "Modelling and qualitative analysis of a predator–prey system with state-dependent impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 318-331.
    3. Yongzhen, Pei & Changguo, Li & Lansun, Chen, 2009. "Continuous and impulsive harvesting strategies in a stage-structured predator–prey model with time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(10), pages 2994-3008.
    4. Zhao, Min & Wang, Xitao & Yu, Hengguo & Zhu, Jun, 2012. "Dynamics of an ecological model with impulsive control strategy and distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(8), pages 1432-1444.
    5. Yu, Hengguo & Zhong, Shouming & Ye, Mao, 2009. "Dynamic analysis of an ecological model with impulsive control strategy and distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(3), pages 619-632.
    6. Cressman, Ross & Garay, József, 2009. "A predator–prey refuge system: Evolutionary stability in ecological systems," Theoretical Population Biology, Elsevier, vol. 76(4), pages 248-257.
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    Cited by:

    1. Roy, Banani & Roy, Sankar Kumar & Gurung, Dil Bahadur, 2017. "Holling–Tanner model with Beddington–DeAngelis functional response and time delay introducing harvesting," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 142(C), pages 1-14.
    2. Pati, N.C. & Ghosh, Bapan, 2022. "Delayed carrying capacity induced subcritical and supercritical Hopf bifurcations in a predator–prey system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 171-196.
    3. Jana, Debaldev & Agrawal, Rashmi & Upadhyay, Ranjit Kumar, 2015. "Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1072-1094.
    4. Wang, Yan & Liu, Xianning, 2017. "Stability and Hopf bifurcation of a within-host chikungunya virus infection model with two delays," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 138(C), pages 31-48.
    5. Bürger, Raimund & Ruiz-Baier, Ricardo & Tian, Canrong, 2017. "Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 28-52.
    6. Pandey, Soumik & Ghosh, Uttam & Das, Debashis & Chakraborty, Sarbani & Sarkar, Abhijit, 2024. "Rich dynamics of a delay-induced stage-structure prey–predator model with cooperative behaviour in both species and the impact of prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 49-76.
    7. Jana, Soovoojeet & Kar, T.K., 2013. "Modeling and analysis of a prey–predator system with disease in the prey," Chaos, Solitons & Fractals, Elsevier, vol. 47(C), pages 42-53.
    8. Anjana Das & M. Pal, 2019. "Theoretical Analysis of an Imprecise Prey-Predator Model with Harvesting and Optimal Control," Journal of Optimization, Hindawi, vol. 2019, pages 1-12, January.
    9. Thirthar, Ashraf Adnan & Majeed, Salam J. & Alqudah, Manar A. & Panja, Prabir & Abdeljawad, Thabet, 2022. "Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).

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