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Dynamics and optimal harvesting of a tri-trophic food chain model with strong Allee effect

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Listed:
  • Shan, Yun
  • Liu, Wenjun
  • Yue, Jiaxin
  • Moussaoui, Ali
  • Auger, Pierre

Abstract

In recent years, many species have been overexploited, pushing them to the brink of extinction. In response to the growing call for sustainable development, there is increasing awareness of the importance of developing suitable mathematical models to study how fishing activities can help maintain ecological balance. This topic has attracted significant attention from researchers in recent times. In this paper, we examine the impact of top predator harvesting on a modified Rosenzweig–MacArthur model incorporating an Allee effect on the prey population. We analyze the stability properties of all possible equilibrium points in this deterministic system. Furthermore, by exploring bifurcations associated with the Allee threshold parameter A and the fishing effort parameter E near the coexistence equilibrium, we observe that the combined presence of a strong Allee effect and fishing effort leads to various interesting bifurcation phenomena, which play a critical role in altering the system’s stability. As a natural next step, we conduct a deeper investigation into the influence of fishing effort under the Allee effect by formulating and solving an optimal control problem using Pontryagin’s Maximum Principle. Finally, numerical simulations are presented to visually and intuitively support the theoretical findings. The results reveal that the system is susceptible to instability and potential collapse. Therefore, in ecosystems where a strong Allee effect is present, it is essential to carefully monitor and strictly regulate fishing limits to reduce the risk of species extinction.

Suggested Citation

  • Shan, Yun & Liu, Wenjun & Yue, Jiaxin & Moussaoui, Ali & Auger, Pierre, 2025. "Dynamics and optimal harvesting of a tri-trophic food chain model with strong Allee effect," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p1:s0960077925011075
    DOI: 10.1016/j.chaos.2025.117094
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    References listed on IDEAS

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    1. Panja, Prabir & Mondal, Shyamal Kumar & Jana, Dipak Kumar, 2017. "Effects of toxicants on Phytoplankton-Zooplankton-Fish dynamics and harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 389-399.
    2. Karim, Siti Nurnabihah & Ang, Tau Keong, 2024. "Co-dimension 2 bifurcation analysis of a tri-trophic food chain model with strong Allee effect and Crowley–Martin functional response," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    3. 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.
    4. Sarif, Nawaj & Kumar, Arjun & Anshu, & Sarwardi, Sahabuddin & Dubey, Balram, 2024. "Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
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