周期环境中基于个体尺度的种群模型的最优收获策略

Optimal Harvesting of a Size-structured Population Model in a Periodic Environment

本文研究一类周期环境中具有尺度结构的种群模型的适定性及最优收获问题.首先应用积分方程及算子理论证明了系统非负解的存在唯一性,然后由Mazur定理确立了最优策略的存在性,再借助切锥法锥的特征结构导出了最大值原理,给出最优控制为Bang-Bang型的判别条件.最后陈述了数值方法与计算实例.

It is readily observed that the habitat of many biological species often under- goes some periodic changes because of the such effects as seasoning variations. On the other side, lots of ecological studies show that the vital parameters of an individual are closely connected with its body size, such as mass, length, surface area, volume, etc. Motivated by these considerations, we in this paper investigate an exploitation problem of renewable biological resources incorporating the individual＇s size-structure and periodical changes into the population model. Firstly we propose an integro-partial system to describe the pop- ulation dynamics, in which the mortality, fertility, growth rate and harvesting effort are time-periodic functions and the boundary condition （i.e. renewal equation） is of global feed- back form. Then we treat the well-posedness problem of the state system. By means of characteristics an integral equation is established for the population fertility, which is put into an abstract framework in a suitable space of functions. Roughly speaking, the model will be well posed if the reproducing number is less than one. Secondly we prove the exis- tence of optimal policies via a maximizing sequence and a use of Mazur＇s theorem in convex analysis. Following that is a careful derivation of necessary optimality conditions, which is finished by tangent-normal cones and adjoint system techniques, and provide an exact description for the optimal strategies. Excluding the singular cases enable us to assert that optimal controllers are unique and take the form of bang-bang, but we cannot expect an explicit formula for them due to complexities. Finally, we present an algorithm to compute the optimal group and test it with an example.