Abstract: Assuming that a population growth follows the Gilpin-Ayala equation, the multiple impulsive harvest optimal policies for the maximum stock level of the population at the end of a harvesting season are investigated. Firstly, the necessary condition of the optimal harvesting moments is obtained by using the maximum principle of impulsive differential equation and some analysis techniques. Moreover, for any given initial population and harvesting season length, it obtains the exact expression of the optimal harvest strategy when the stock level at the end of the season reaches its maximum. Further, it studies the maximum number of impulsive harvest with same quantity of impulsive harvest during the exploitation period. The related numerical simulations are also provided to support the theoretical results.

Key words: Gilpin-Ayala system, impulsive harvesting, maximum principle, optimal control strategy, maximum stock level

摘要： 主要研究在有限时间周期内，由Gilpin-Ayala模型描述的脉冲收获系统的优化控制问题。收获函数包括比例收获和常量收获，在收获量一定的条件下，以种群在周期末的存储量最大为目标函数，对于不同的初值条件，研究不同的收获时刻对种群的影响，并获得最优的收获策略。首先通过脉冲微分方程的极值原理和一些分析技巧，得到了最优收获时刻应满足的必要条件，讨论了在时间周期足够长的条件下具有多次脉冲收获的最优收获策略；进一步考虑了在给定时间范围内的最大收获次数及相应的最优收获策略问题；最后通过实例及数值模拟验证了所得到的主要结果。

关键词: Gilpin-Ayala系统, 脉冲收获, 极值原理, 优化控制策略, 最大存储量