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Bifurcation and Stability Analysis of a New Fractional-Order Prey–Predator Model with Fear Effects in Toxic Injections

Author

Listed:
  • Cuimin Liu

    (Yellow River Middle School, Dongying 257068, China
    These authors contributed equally to this work.)

  • Yonggang Chen

    (College of Science, China University of Petroleum, Qingdao 266580, China
    These authors contributed equally to this work.)

  • Yingbin Yu

    (Qingdao No. 66 High School of Shandong Province, Qingdao 266031, China)

  • Zhen Wang

    (College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China)

Abstract

This paper proposes a prey–predator model affected by fear effects and toxic substances. We used the Lipschitz condition to prove the uniqueness of the model solution and Laplace transform to prove the boundedness of the model solution. We used the fractional-order stability theorem to provide sufficient conditions for the local stability of equilibrium points, and selected fractional-order derivatives as parameters to perform Hopf bifurcation analysis on the system. Finally, the theoretical results are verified via numerical simulation. The results show that a value of α will affect the stability of the system and that the population size and the effect of toxic substances have a huge impact on the stability of the system.

Suggested Citation

  • Cuimin Liu & Yonggang Chen & Yingbin Yu & Zhen Wang, 2023. "Bifurcation and Stability Analysis of a New Fractional-Order Prey–Predator Model with Fear Effects in Toxic Injections," Mathematics, MDPI, vol. 11(20), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4367-:d:1264091
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    References listed on IDEAS

    as
    1. Barman, Dipesh & Roy, Jyotirmoy & Alrabaiah, Hussam & Panja, Prabir & Mondal, Sankar Prasad & Alam, Shariful, 2021. "Impact of predator incited fear and prey refuge in a fractional order prey predator model," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    2. Das, Meghadri & Samanta, G.P., 2020. "A delayed fractional order food chain model with fear effect and prey refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 218-245.
    3. Wang, Zhen & Xie, Yingkang & Lu, Junwei & Li, Yuxia, 2019. "Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 360-369.
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