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Bifurcation Analysis in a Harvested Modified Leslie–Gower Model Incorporated with the Fear Factor and Prey Refuge

Author

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  • Seralan Vinoth

    (Centre for Nonlinear Systems, Chennai Institute of Technology, Chennai 600069, India)

  • R. Vadivel

    (Department of Mathematics, Faculty of Science and Technology, Phuket Rajabhat University, Phuket 83000, Thailand)

  • Nien-Tsu Hu

    (Graduate Institute of Automation Technology, National Taipei University of Technology, Taipei 10608, Taiwan)

  • Chin-Sheng Chen

    (Graduate Institute of Automation Technology, National Taipei University of Technology, Taipei 10608, Taiwan)

  • Nallappan Gunasekaran

    (Eastern Michigan Joint College of Engineering, Beibu Gulf University, Qinzhou 535011, China)

Abstract

Fear and prey refuges are two significant topics in the ecological community because they are closely associated with the connectivity of natural resources. The effect of fear on prey populations and prey refuges (proportional to both the prey and predator) is investigated in the nonlinear-type predator-harvested Leslie–Gower model. This type of prey refuge is much more sensible and realistic than the constant prey refuge model. Because there is less research on the dynamics of this type of prey refuge, the current study has been considered to strengthen the existing literature. The number and stability properties of all positive equilibria are examined. Since the calculations for the determinant and trace of the Jacobian matrix are quite complicated at these equilibria, the stability of certain positive equilibria is evaluated using a numerical simulation process. Sotomayor’s theorem is used to derive a precise mathematical confirmation of the appearance of saddle-node bifurcation and transcritical bifurcation. Furthermore, numerical simulations are provided to visually demonstrate the dynamics of the system and the stability of the limit cycle is discussed with the help of the first Lyapunov number. We perform some sensitivity investigations on our model solutions in relation to three key model parameters: the fear impact, prey refuges, and harvesting. Our findings could facilitate some biological understanding of the interactions between predators and prey.

Suggested Citation

  • Seralan Vinoth & R. Vadivel & Nien-Tsu Hu & Chin-Sheng Chen & Nallappan Gunasekaran, 2023. "Bifurcation Analysis in a Harvested Modified Leslie–Gower Model Incorporated with the Fear Factor and Prey Refuge," Mathematics, MDPI, vol. 11(14), pages 1-25, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3118-:d:1194377
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    References listed on IDEAS

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    1. Evan L Preisser & Daniel I Bolnick, 2008. "The Many Faces of Fear: Comparing the Pathways and Impacts of Nonconsumptive Predator Effects on Prey Populations," PLOS ONE, Public Library of Science, vol. 3(6), pages 1-8, June.
    2. Jianglin Zhao & Min Zhao & Hengguo Yu, 2013. "Complex Dynamical Behavior of a Predator-Prey System with Group Defense," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, July.
    3. Panday, Pijush & Samanta, Sudip & Pal, Nikhil & Chattopadhyay, Joydev, 2020. "Delay induced multiple stability switch and chaos in a predator–prey model with fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 134-158.
    4. Li, Yajing & He, Mengxin & Li, Zhong, 2022. "Dynamics of a ratio-dependent Leslie–Gower predator–prey model with Allee effect and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 417-439.
    5. Binhao Hong & Chunrui Zhang, 2023. "Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    6. 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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    1. Haoyu Wang & Xiaoyuan Wan & Junyao Hou & Jing Lian & Yuzhao Wang, 2024. "A Study on Effects of Species with the Adaptive Sex-Ratio on Bio-Community Based on Mechanism Analysis and ODE," Mathematics, MDPI, vol. 12(14), pages 1-22, July.

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