一类具有年龄等级结构的n种群竞争模型的最优收获

The Optimal Harvest of an n-Population Competition Model with Hierarchy Structure

文章讨论一类具有年龄等级结构的n种群竞争模型的最优收获问题,首先,利用压缩映射的不动点定理和一些重要的比较定理,研究系统解的存在唯一性和非负有界性,即解的适定性;其次,运用极值化序列和相应的紧性定理以及Mazur定理来证明控制问题最优解的存在性;最后,构造适当的共轭方程组和利用法锥的概念刻画以及运用相应的连续性定理,得出最优收获问题最优解的一阶必要条件.而这些定理及其证明均能为研究多种群系统模型提供一定的理论依据.

In this paper,the optimal harvest problem of an-population competition model with age hierarchy structure is discussed.First and foremost,the first thing is that by using the fixed point theorem of contraction mapping and some important comparison theorems,the existence,uniqueness,and non-negative boundedness of the system solution,which is the well-posedness of the solution,are studied.Subsequently,the second thing is that the existence of the optimal solution of the control problem is proved by using some constructed extremum sequences,the relevant compactness theorem and Mazur theorem.Third but not least,the last thing is that by constructing an appropriate system of conjugate equations,utilizing the concept of the cone and applying the corresponding continuity theorem,the first-order necessary conditions of the optimal harvest problem are obtained.And these proofs of these lemmas,theorems,and conditions can provide a certain degree of theoretical basis for the study of many multiple population system models.