一类可再生资源系统的最优动态平衡收获

研究一类可再生资源系统的最优利用问题．首先，引进一个新的效用函数， 它依赖于收获努力度和资源量，由此导出最优控制问题．其次证明该控制问题最优解的存在性．然后，利用无穷区间上控制问题的最大值原理，得到一个非线性的四维最优系统．通过对上述系统正平衡解的详细分析，借助 Hopf 分支定理证明了极限环的存在性．之后考虑中心流形上的简化系统， 分析极限环的稳定性．最后，解释所得结果的生物经济学意义．

Optimal Dynamical Balance Harvesting for a Class of Renewable Resources System

An optimal utilization problem for a class of renewable resources system is investigated. Firstly, a control problem was proposed by introducing a new utility function which depends on the harvesting effort and the stock of resources.Secondly, the existence of optimal solution for the problem was discussed. Then, using a maximum principle for infinite horizon problem, a nonlinear four-dimensional differential equations system was attained. After a detailed analysis of the unique positive equilibrium solution, the existence of limit cycles for the system is demonstrated. Next a reduced system on the central manifold is carefully derived, which assures the stability of limit cycles. Finally significance of the results in bioeconomics is explained.