具有空间扩散和尺度结构的非线性害鼠模型的最优不育控制

Optimal contraception control for a nonlinear vermin model with spatial diffusion and size-structure

中文关键词  空间扩散  尺度结构  不育控制  可分离死亡率  有限差分法

英文关键词  spatial diffusion  size-structure  contraception control  separable mortality  finite difference method

中文摘要

      本文讨论了一类具有尺度结构的非线性时变害鼠扩散模型的适定性及最优不育控制问题. 状态系统由二阶偏微分–积分方程描述, 此系统有一种重要的特殊情形, 即死亡率分为自然死亡率和额外死亡率, 系统的解关于尺度和空间位置可分离, 从而将系统分为两个子系统, 利用比较原则和不动点定理证明了变量分离型解的存在唯一性和非负有界性. 本文运用Mazur定理证明了最优策略的存在性, 导出共轭系统并借助凸集的切锥–法锥技巧给出了最优策略的必要性条件, 为模型的实际应用奠定了理论基础. 最后, 采用向后差分格式和追赶法分别对子系统的解进行了数值模拟.

英文摘要

      The paper investigates the well-posedness and optimal contraception control problem for a class of sizestructured nonlinear time-varying vermin diffusion model. The state system is described by a second-order partial integrodifferential equation. This system has an important particular situation, that is, the mortality rate is divided into atural mortality rate and additional mortality rate. The solution of the system is separable with respect to size and spatial position, thus dividing the system into two subsystems. The comparison principle and fixed point theorem are used to prove the existence, uniqueness, non-negativity and boundedness of the separable form of the solution. The existence of the optimal strategy is proved by Mazur’s theorem. The adjoint system is derived and the necessary conditions for the optimal strategy are given by means of tangent-normal cones technique of the convex set. The results lay a theoretical foundation for the practical applications of the model. Finally, the backward difference scheme and chasing method are used to simulate the solutions of the subsystems.