具有尺度结构和时滞的种群系统遍历性与最优控制

Ergodicity and Optimal Control of a Size-Structured Population Model with Delay

研究带有时滞的尺度结构种群模型，它是一类包含全局反馈的偏泛函积分微分方程．利用特征线方法确立了状态系统的适定性，借助积分方程和积分变换证明了系统的强遍历性：种群各尺度类别的个体数量占总量之比渐近于常数，与初始分布无关．运用非线性分析中的切锥法锥理论描述了最优策略的结构，证实了最优策略的存在唯一性．

We investigate a class of size-dependent population model, which is a partial functional integro-differential equation with global feedbacks and a time delay in incubation period. In addition, a distributive harvesting is incorporated in the state equation. Firstly, the existence and uniqueness of long time solutions are established by characteristic curves method, which is bounded in any finite interval. Then, a strong ergodicity of the population is found via an integral equation, integral transformations and residue theory of functions in complex variables, which shows that the population profiles will be asymptotically constant. Next, we regard the population as a renewable resource and consider the optimal harvesting problem： How to choose a harvest function that maximizes economic profits？ By means of extremum sequences and Mazur＇s theorem, we prove that there is at least one optimal policy. To describe the structure of optimal strategies, we use a normal cone and construct an adjoint system. The conclusion shows that any optimal policy should take minimal or maximal harvest efforts in most of situations. Finally, we claim that optimal strategies are unique by excluding the singular cases.