Optimal harvesting for periodic age-dependent population dynamics with logistic term

We prove an asymptotic behavior result for an age-dependent population dynamics with logistic term and periodic vital rates. We investigate next an optimal harvesting problem related to a periodic age-structured model with logistic term. Existence of an optimal control and necessary optimality conditions are established. A conceptual algorithm to approximate the optimal pair is derived and some numerical experiments are presented.     

Numerical Optimal Harvesting for an Age-Dependent Prey-Predator System

In this article, we develop a numerical study of an optimal harvesting problem for age-dependent prey-predator system. Here, the rates of growth and decay as well as the interaction effect between species are assumed to be depending on age, time and space. Existence, uniqueness, and necessary conditions for the optimal control are assured in case of a small final time T. The discrete parabolic nonlinear dynamical systems are obtained by using a finite difference semi-implicit scheme. Then a numerical algorithm is developed to approximate the optimal harvesting effort and the optimal harvest. Results of the numerical tests are given.     

具有年龄结构的线性周期种群动力系统的最优收获控制问题

研究一类具有年龄结构的线性周期种群动力系统的最优收获控制问题,即讨论了具有周期的生死率和周期变化的收获项的Lotka Mckendrick模型.利用Mazur's定理,作者证明了控制问题最优解的存在性,同时借助于法锥概念,还得到了控制问题最优解存在的必要条件。最后，在适当的假设下，得到了最优控制问题的唯一解。该文的结论推广了某些已有的结果.      

具空间扩散和年龄结构的时变种群系统的最优收获控制

本文讨论了一类具空间扩散和年龄结构的时变种群系统的最优收获控制问题,利用Banach空间的SaksMasur引理,证明了系统最优收获控制的存在性,并利用切向锥、法向锥概念,建立了收获控制为最优的必要条件.     

一类周期种群系统的适定性及最优控制

本文研究了一类非线性周期种群系统的适定性及最优控制问题 .利用压缩不动点原理讨论了该种群系统的周期解的存在唯一性 .并用变分法给出了最优控制所满足的必要条件 .     

一类具有年龄结构的非线性种群扩散系统的最优收获控制

讨论了一类与年龄相关的非线性种群扩散系统的最优收获控制问题,证明了最优收获控制的存在性,并且给出了控制为最优的必要条件及其由偏微分方程组和变分不等式组成的最优性组.这些结果可为种群扩散系统最优控制问题的实际研究提供理论基础.     

具周期系数的单种群模型及其最优捕获策略

该文用一种新的方法, 讨论了单种群生物资源的捕获优化问题.以最大的可持续单位时间捕获量为管理目标, 得到一类非自治单种群捕获模型的最优捕获策略, 所得结果包括了文献中研究过的几乎所有单种群捕获模型的相应研究结果.     

一类具年龄结构种群动力系统的最优控制

研究一类具年龄结构种群线性动力系统的最优控制问题 .由Mazur’s定理 ,我们证明了最优控制问题 (OH)最优解的存在性 ,同时借助于法锥概念 ,我们还得到了最优控制问题 (OH)最优解存在的必要条件     

具扩散与年龄结构的三种群捕食与被捕食系统的最优收获

研究了一类具有空间扩散和年龄结构的三种群捕食与被捕食系统的最优收获问题,运用Banach不动点原理讨论了系统解的存在唯一性,证明了最优收获控制的存在性,给出了最大值原理.结果可为多种群扩散系统最优控制问题的实际研究提供理论基础.     

Optimal Harvesting and Stability for a Predator-prey System with Stage Structure

The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.     

一类具有收获率的时滞阶段结构捕食系统的多重正周期解

研究了一类具有收获率的时滞阶段结构捕食系统周期解存在性问题,利用重合度理论中的延拓定理,通过一些分析技巧,获得了该系统至少存在两个正周期解的一组易于验证的充分条件.     

Optimal control of parabolic problems with state constraints: a penalization method for optimality conditions

In this paper state constrained optimal control problems governed by parabolic evolution equations are studied. Our purpose is to obtain a (first-order) decoupled optimality system (that ensures the Lagrange multipliers existence). In a first step we are led to Slater-like assumptions and we are then allowed to extend the application field of the decoupled system we obtain. With a weaker assumption the existence of Lagrange multipliers (that are measures) for nonqualified problems may be established.     

具有年龄结构和迁移的非线性种群的最优生育控制

对一类非线性种群模型，以生育率为控制变量，研究最小代价一规模问题和最小代价一偏差问题．证明了最优控制的存在唯一性，给出了一阶最优性条件，讨论了最优反馈的合成．所得结论推广了文献中的一个结果．     

Numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm

This paper deals with an inverse problem of identifying a nonlinear source term g=g(u) in the heat equation u_t-u_xx=a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M>0, with which an optimal perturbation algorithm is applied to determine the source function g(u) on u∈[0,M]. Numerical inversions are carried out for g(u) with functional forms of polynomial, trigonometric and index functions. The inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here. By the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources. The inversion algorithm seems to be very sharp if the solution’s maximum M of the forward problem is relatively small; otherwise, the deviations in the source solutions become large especially near the endpoint of u=M.     

Feedback Stabilization for a Reaction-Diffusion System with Nonlocal Reaction Term

We consider a two-component reaction-diffusion system with a nonlocal reaction term. A necessary condition and a sufficient condition for the internal stabilizability to zero of one of the two components of the solution while preserving the nonnegativity of both components have been established in [6 S. Ani?a , W.-E. Fitzgibbon , and M. Langlais ( 2009 ). Global existence and internal stabilization for a class of predator-prey systems posed on non coincident spatial domains . Discrete Cont. Dynam. Syst. B 11 : 805 – 822 .[Crossref], [Web of Science ®] , [Google Scholar]]. In case of stabilizability, a feedback stabilizing control of harvesting type has been indicated. The rate of stabilization (for the indicated feedback control) is given by the principal eigenvalue of a certain non-selfadjoint operator. A large principal eigenvalue leads to a fast stabilization. The first main goal of this article is to approximate this principal eigenvalue. This is done in two steps. First, we investigate the large-time behavior of the solution to a logistic population dynamics with migration, and next we derive as a consequence a method to approximate the principal eigenvalue. The other main goal is to derive a conceptual iterative algorithm to improve the position of the support of the control in order to get a faster stabilization. Our results apply to prey-predator systems.