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Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart

Author

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  • Sk, Nazmul
  • Mondal, Bapin
  • Thirthar, Ashraf Adnan
  • Alqudah, Manar A.
  • Abdeljawad, Thabet

Abstract

The paper focuses on studying the global dynamics of a simple prey–predator model. The model incorporates cooperative hunting behavior among predators and takes into account the effect of harvesting on the predator population. The interaction between the prey and predator follows the Crowley–Martin functional response. The study involves analyzing the equilibrium points of the system and investigating their stability properties through mathematical analysis. Various types of bifurcations, including Hopf bifurcation, saddle–node bifurcation, and transcritical bifurcation, are numerically demonstrated in the figures, highlighting the dynamic behavior of the model. One of the intriguing findings of the study is the occurrence of bistability and tristability in the model. To incorporate stochasticity into the model, white noise is added to the deterministic system. This allows for the examination of transitions between different steady states in the stochastic system. We have conducted an analysis of species persistence and extinction in relation to the presence of noise. The paper presents the stochastic sensitivity function (SSF) technique and the use of confidence ellipses to assess the likelihood of such transitions occurring in the system. Overall, the study provides insights into the complex dynamics of prey–predator interactions, considering factors such as cooperation, harvesting, and stochasticity. The results contribute to our understanding of population dynamics and the potential effects of environmental and human-induced perturbations on ecosystem stability.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:chsofr:v:176:y:2023:i:c:s0960077923009748
    DOI: 10.1016/j.chaos.2023.114073
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    References listed on IDEAS

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    1. Bashkirtseva, Irina & Perevalova, Tatyana & Ryashko, Lev, 2020. "Noise-induced shifts in dynamics of multi-rhythmic population SIP-model," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    2. Bashkirtseva, Irina & Perevalova, Tatyana & Ryashko, Lev, 2022. "Regular and chaotic variability caused by random disturbances in a predator–prey system with disease in predator," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    3. Yang, Anji & Wang, Hao & Yuan, Sanling, 2023. "Tipping time in a stochastic Leslie predator–prey model," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).
    4. 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.
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