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Complex dynamic analysis of a big fish-small fish system by using the Poincaré map

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  • Cheng, Huidong
  • Li, Wei
  • Zhang, Tonghua

Abstract

In terms of fishery management, if it is not managed carefully, it will lead to the extinction of the species and other unfavorable conditions. In order to strengthen the protection, development and rational utilization of fishery resources, this paper proposes a population capture model with state feedback control based on previous studies, in which the small fish is influenced by the Allee effect, and the catch of the big fish is linearly related with the release of the small fish. Firstly, we give the definition of the Poincaré map of the catch system based on impulsive point series, and study the complex dynamic properties of the map, including single peak, multi-peak functions and multiple jumping points. Moreover, the conditions of existence and the number of jumping points are discussed. At the same time, we analyze its properties including monotonicity, differentiability and fixed point. Secondly, the conditions under which the existence and uniqueness of the order-1 periodic solution are provided, and we address the necessary and sufficient conditions for stability of the order-1 periodic solution. Furthermore, existence of the order-k(k≥2) periodic solution is obtained. Finally, correctness of our theoretical results is illustrated by numerical simulations. The results show that under human control, the density of big fish and small fish will maintain periodic benign changes, and fishery resources are sufficient for sustainable development.

Suggested Citation

  • Cheng, Huidong & Li, Wei & Zhang, Tonghua, 2024. "Complex dynamic analysis of a big fish-small fish system by using the Poincaré map," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    • Handle: RePEc:eee:apmaco:v:482:y:2024:i:c:s0096300324004338
    DOI: 10.1016/j.amc.2024.128972
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    References listed on IDEAS

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