一类具有植化相克的捕食模型的最优收获

Optimal Harvest of a Predation Model with Allelopathy

摘要: 通过引入捕食者-食饵模型中的非线性捕获函数，研究了一类具有植化相克的捕食模型，得到了该模型平衡点的存在性和局部稳定性的条件，通过构造 Lyapunov 函数来判断正平衡点周围的全局稳定性。 确定了非线性收获下的生态经济平衡点，使用 Pontryagin 极大值原理得到了最优收获策略， 从而揭示了在保证浮游生物种群不灭绝的情况下，对渔业资源进行利用和开发的同时，不仅给渔民提供了最大的经济利润，而且维持了海洋生态系统的平衡。

Abstract: By introducing the nonlinear capture function in the predator-prey model, a kind of predator-prey model with planting phase is studied. The conditions for the existence and local stability of the equilibrium point of the model are obtained. The global stability around the positive equilibrium point is judged by constructing Lyapunov function, the ecological and economic equilibrium point under nonlinear harvesting is determined, and the optimal harvesting strategy is obtained by using Pontryagin maximum principle, which reveals that while ensuring the non extinction of plankton population, the utilization and development of fishery resources not only provides fishermen with the maximum economic profit, but also maintains the balance of marine ecosystem.