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A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation

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  • Sk, Nazmul
  • Tiwari, Pankaj Kumar
  • Pal, Samares

Abstract

In this article, we study the dynamics of a three species food chain model for a predator–prey system with hunting cooperation in predators; due to fear of predation risk, prey population shows anti-predator behavior. Fear of middle-predator affects the growth rate of prey and the fear of top-predator reduces the predation of middle-predator on prey species. The prey and middle-predator are assumed to take refuge against predation. Numerical results confirm the stabilizing role of refuge whereas hunting cooperation destroys stability and induces limit cycle oscillations. We find that the fear of middle-predator on prey has tendency to stabilize the system while the fear of top-predator on middle-predator creates instability in the system. Further, time delays are considered in the processes of taking refuges and hunting cooperations. Delay in hunting cooperations creates multiple stability switches in the system. Furthermore, seasonal forcing of parameters related to refuges and hunting cooperations depicts a more realistic scenario. We investigate the behaviors of seasonally forced system in the absence as well as presence of time delays. The seasonally forced system exhibits different types of periodic solutions. Complex bursting patterns are observed due to periodicity in refuge of middle-predator (cooperation of top-predator).

Suggested Citation

  • Sk, Nazmul & Tiwari, Pankaj Kumar & Pal, Samares, 2022. "A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 136-166.
    • Handle: RePEc:eee:matcom:v:192:y:2022:i:c:p:136-166
    DOI: 10.1016/j.matcom.2021.08.018
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    References listed on IDEAS

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    1. Hossain, Mainul & Pal, Nikhil & Samanta, Sudip, 2020. "Impact of fear on an eco-epidemiological model," Chaos, Solitons & Fractals, Elsevier, vol. 134(C).
    2. Zhimin Zhang, 2013. "Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-9, August.
    3. Justin P. Suraci & Michael Clinchy & Lawrence M. Dill & Devin Roberts & Liana Y. Zanette, 2016. "Fear of large carnivores causes a trophic cascade," Nature Communications, Nature, vol. 7(1), pages 1-7, April.
    4. Pati, N.C. & Layek, G.C. & Pal, Nikhil, 2020. "Bifurcations and organized structures in a predator-prey model with hunting cooperation," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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    Cited by:

    1. Yuke Zhang & Xinzhu Meng, 2022. "Dynamics Analysis of a Predator–Prey Model with Hunting Cooperative and Nonlinear Stochastic Disturbance," Mathematics, MDPI, vol. 10(16), pages 1-18, August.
    2. Shang, Zuchong & Qiao, Yuanhua, 2024. "Complex dynamics of a four-species food web model with nonlinear top predator harvesting and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 458-484.
    3. Pal, Debjit & Kesh, Dipak & Mukherjee, Debasis, 2024. "Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 128-147.
    4. Mondal, Bapin & Thirthar, Ashraf Adnan & Sk, Nazmul & Alqudah, Manar A. & Abdeljawad, Thabet, 2024. "Complex dynamics in a two species system with Crowley–Martin response function: Role of cooperation, additional food and seasonal perturbations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 415-434.
    5. Kumbhakar, Ruma & Hossain, Mainul & Karmakar, Sarbari & Pal, Nikhil, 2024. "An investigation of the parameter space in a tri-trophic food chain model with refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 37-59.
    6. Sk, Nazmul & Mondal, Bapin & Thirthar, Ashraf Adnan & Alqudah, Manar A. & Abdeljawad, Thabet, 2023. "Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    7. 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.

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