Optimal Dynamical Balance Harvesting for a Class of Renewable Resources System
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摘要: 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.Abstract: 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.
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Key words:
- renewable resource /
- optimal harvesting /
- stability /
- bifurcation /
- limit cycle
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