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Dynamics of generalist predator in a stochastic environment: Effect of delayed growth and prey refuge

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  • Jana, Debaldev
  • Agrawal, Rashmi
  • Upadhyay, Ranjit Kumar

Abstract

In this paper, an attempt has been made to understand the dynamics of a prey–predator system with multiple time delays where the predator population is regarded as a generalist type. In this regard, we consider a modified Holling–Tanner prey–predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predator’s interference in prey–predator relationship provides better descriptions of predator’s feeding over a range of prey–predator abundances, so the predator’s functional response is considered to be Type II ratio-dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population. In accordance with previous studies, it is observed that delay destabilizes the system, in general and stability loss occurs via Hopf-bifurcation. In particular, we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. Also, environmental stochasticity in the form of Gaussian white-noise plays a significant role to describe the system and its values. Numerical computation is also performed to validate and visualize different theoretical results presented. The analysis and results in this work are interesting both in mathematical and biological point of views.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:268:y:2015:i:c:p:1072-1094
    DOI: 10.1016/j.amc.2015.06.098
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    References listed on IDEAS

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    1. Karaoglu, Esra & Merdan, Huseyin, 2014. "Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 159-168.
    2. 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.
    3. Blume Lawrence E., 1993. "The Statistical Mechanics of Strategic Interaction," Games and Economic Behavior, Elsevier, vol. 5(3), pages 387-424, July.
    4. 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.
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    Cited by:

    1. Jana, Debaldev & Pathak, Rachana & Agarwal, Manju, 2016. "On the stability and Hopf bifurcation of a prey-generalist predator system with independent age-selective harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 252-273.
    2. Rensheng He & Zuoliang Xiong & Desheng Hong, 2016. "Dynamical Behavior of a Stochastic Delayed One‐Predator and Two‐Mutualistic‐Prey Model with Markovian Switching and Different Functional Responses," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    3. Jana, Debaldev & Banerjee, Aniket & Samanta, G.P., 2017. "Degree of prey refuges: Control the competition among prey and foraging ability of predator," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 350-362.
    4. Aybar, I. Kusbeyzi & Aybar, O.O. & Dukarić, M. & Ferčec, B., 2018. "Dynamical analysis of a two prey-one predator system with quadratic self interaction," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 118-132.
    5. Cortés García, Christian, 2023. "Impact of prey refuge in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and linear functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 147-165.
    6. Bhargava, Masoom & Dubey, Balram, 2024. "Trade-off and chaotic dynamics in a two-prey one-predator model with refuge, environmental noise and seasonal effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 226(C), pages 218-245.

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