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Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies

Author

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  • Wenjie Qin

    (Department of Mathematics, Yunnan Minzu University, Kunming 650500, China)

  • Zhengjun Dong

    (Department of Mathematics, Yunnan Minzu University, Kunming 650500, China)

  • Lidong Huang

    (Department of Mathematics, Yunnan Minzu University, Kunming 650500, China)

Abstract

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order- k ( k ≥ 2 ) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations.

Suggested Citation

  • Wenjie Qin & Zhengjun Dong & Lidong Huang, 2024. "Impulsive Effects and Complexity Dynamics in the Anti-Predator Model with IPM Strategies," Mathematics, MDPI, vol. 12(7), pages 1-25, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1043-:d:1367641
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    References listed on IDEAS

    as
    1. Liu, Jingna & Qi, Qi & Liu, Bing & Gao, Shujing, 2023. "Pest control switching models with instantaneous and non-instantaneous impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 926-938.
    2. Wenjie Qin & Sanyi Tang & Robert A. Cheke, 2014. "The Effects of Resource Limitation on a Predator-Prey Model with Control Measures as Nonlinear Pulses," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, March.
    3. Qin, Wenjie & Tan, Xuewen & Tosato, Marco & Liu, Xinzhi, 2019. "Threshold control strategy for a non-smooth Filippov ecosystem with group defense," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.
    4. Tang, Guangyao & Qin, Wenjie & Tang, Sanyi, 2014. "Complex dynamics and switching transients in periodically forced Filippov prey–predator system," Chaos, Solitons & Fractals, Elsevier, vol. 61(C), pages 13-23.
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