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- 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.

中文翻译：

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利用庞加莱图进行大鱼-小鱼系统复杂动力学分析


在渔业管理方面，如果管理不周，将会导致物种灭绝等不利情况。为了加强渔业资源的保护、开发和合理利用，本文在前人研究的基础上，提出了一种状态反馈控制的种群捕获模型，其中小鱼受Allee效应影响，大鱼的捕获受到影响。与小鱼的释放呈线性关系。首先，给出了基于脉冲点序列的捕获系统庞加莱图的定义，并研究了该图的复杂动力学特性，包括单峰、多峰函数和多跳跃点。此外，还讨论了存在条件和跳跃点的数量。同时分析了其单调性、可微性和不动点等性质。其次，给出了1阶周期解存在唯一性的条件，并给出了1阶周期解稳定性的充要条件。进一步得到了阶周期解的存在性。最后，通过数值模拟验证了理论结果的正确性。结果表明，在人类控制下，大鱼和小鱼的密度将保持周期性良性变化，渔业资源充足，可持续发展。