Abstract：
Ecological balance and stable economic development are crucial for the fishery. This study proposes a predator-prey system for marine communities, where the growth of predators follows the Allee effect and takes into account the rapid fluctuations in resource prices caused by supply and demand. The system predicts the existence of catastrophic equilibrium, which may lead to the extinction of prey, consequently leading to the extinction of predators, but fishing efforts remain high. Marine protected areas are established near fishing areas to avoid such situations. Fish migrate rapidly between these two areas and are only harvested in the nonprotected areas. A three-dimensional simplified model is derived by applying variable aggregation to describe the variation of global variables on a slow time scale. To seek conditions to avoid species extinction and maintain sustainable fishing activities, the existence of positive equilibrium points and their local stability are explored based on the simplified model. Moreover, the long-term impact of establishing marine protected areas and levying taxes based on unit catch on fishery dynamics is studied, and the optimal tax policy is obtained by applying Pontryagin\'s maximum principle. The theoretical analysis and numerical examples of this study demonstrate the comprehensive effectiveness of increasing the proportion of marine protected areas and controlling taxes on the sustainable development of fishery.
摘要：
生态平衡和经济稳定发展对渔业至关重要。本研究提出了一种适用于海洋群落的捕食者-食饵系统，其中，捕食者的增长遵循Allee效应，并考虑了供需引起的资源价格的快速波动。该系统预测了灾难性均衡的存在，这可能会导致猎物的灭绝，从而导致捕食者的灭绝,但是捕鱼努力仍然很高。为了避免这种情况，在捕鱼区附近建立了海洋保护区。鱼类在这两个地区之间迅速迁移，仅在非保护区收获。通过应用变量聚合来描述全局变量在慢时间尺度上的变化，得出了三维简化模型。寻求条件以避免物种灭绝并维持可持续的捕鱼活动，基于简化模型，探讨了正平衡点的存在性及其局部稳定性。此外，研究了建立海洋保护区和根据单位渔获量征税对渔业动态的长期影响，并运用Pontryagin的最大值原理得到最优税收政策。本研究的理论分析和数值算例证明了提高海洋保护区比例和控制税收对渔业可持续发展的综合有效性。
