Dynamic analysis of a phytoplankton-fish model with the impulsive feedback control depending on the fish density and its changing rate

Huidong Cheng; Hui Xu; Jingli Fu; Huidong Cheng; Hui Xu; Jingli Fu

doi:10.3934/mbe.2023352

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8103-8123. doi: 10.3934/mbe.2023352

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Dynamic analysis of a phytoplankton-fish model with the impulsive feedback control depending on the fish density and its changing rate

1.
College of Information and Control Engineering, Shandong Foreign Affairs Vocational University, Weihai 264504, China
2.
College of Mathematics and System Sciences, Shandong University of Science and Technology, Qingdao 266590, China
3.
Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Academic Editor: Rana Parshad

Received: 12 November 2022 Revised: 18 December 2022 Accepted: 01 February 2023 Published: 27 February 2023

This paper proposes and studies a comprehensive control model that considers fish population density and its current growth rate, providing new ideas for fishing strategies. First, we established a phytoplankton-fish model with state-impulse feedback control based on fish density and rate of change. Secondly, the complex phase sets and impulse sets of the model are divided into three cases, then the Poincar$ \acute{\mbox{e}} $ map of the model is defined and its complex dynamic properties are deeply studied. Furthermore, some necessary and sufficient conditions for the global stability of the fixed point (order-$ 1 $ limit cycle) have been provided even for the Poincar$ \acute{\mbox{e}} $ map. The existence conditions for periodic solutions of order-$ k $($ k \ge 2 $) are discussed, and the influence of dynamic thresholds on system dynamics is shown. Dynamic thresholds depend on fish density and rate of change, i.e., the form of control employed is more in line with the evolution of biological populations than in earlier studies. The analytical method presented in this paper also plays an important role in analyzing impulse models with complex phase sets or impulse sets.
- Poincar$ \acute{\mbox{e}} $ map,
- current growth rate,
- population density of fish,
- impulse feedback control,
- periodic solution
Citation: Huidong Cheng, Hui Xu, Jingli Fu. Dynamic analysis of a phytoplankton-fish model with the impulsive feedback control depending on the fish density and its changing rate[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8103-8123. doi: 10.3934/mbe.2023352

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Abstract

This paper proposes and studies a comprehensive control model that considers fish population density and its current growth rate, providing new ideas for fishing strategies. First, we established a phytoplankton-fish model with state-impulse feedback control based on fish density and rate of change. Secondly, the complex phase sets and impulse sets of the model are divided into three cases, then the Poincar$ \acute{\mbox{e}} $ map of the model is defined and its complex dynamic properties are deeply studied. Furthermore, some necessary and sufficient conditions for the global stability of the fixed point (order-$ 1 $ limit cycle) have been provided even for the Poincar$ \acute{\mbox{e}} $ map. The existence conditions for periodic solutions of order-$ k $($ k \ge 2 $) are discussed, and the influence of dynamic thresholds on system dynamics is shown. Dynamic thresholds depend on fish density and rate of change, i.e., the form of control employed is more in line with the evolution of biological populations than in earlier studies. The analytical method presented in this paper also plays an important role in analyzing impulse models with complex phase sets or impulse sets.

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