兰州大学机构库 >数学与统计学院
一类结构种群动力系统的定性分析和精确零控制
Alternative TitleAnalysis and exact null control of a class of structured population dynamics
何源
Thesis AdvisorAinseba Bedreddine ; 李万同
2013-11-29
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword结构种群动力学 积分微分方程 精确零控制 特征线法 不动点定理
Abstract目前, 欧洲葡萄蛾(EGVM)已经对欧洲, 北非,甚至亚洲一些国家的葡萄园产生了很大的危害,很有必要从生物数学的角度来研究它. 为此, 一个与之相关的, 多阶段的,具有年龄结构的种群模型被提出来. 在本文中,我们主要对这样一类结构种群动力系统作了定性分析和精确零控制. 首先研究一个多阶段的具有生理年龄结构的动力系统,在这个系统中考虑成年的蛾子是可以在葡萄园中扩散的.由于气候变化会影响到每个阶段的种群增长, 在该系统中考虑一个决定于气候变量和生理年龄变量的增长函数. 基于压缩不动点定理,我们得到了该系统的解的存在唯一性.接着应用极大区间理论将存在性推广到全局, 并且该系统的解轨道存在一个全局吸引子. 最后, 结合紧算子理论和Krasnoselskii 不动点定理,证明了该系统非零稳态解的存在性. 鉴于葡萄蛾给葡萄园带来非常严重的经济损失, 研究这个具有年龄结构和阶段结构的动力系统的精确零控制问题显得非常有意义. 假设该系统的四个阶段: 卵、幼虫、雌蛾和雄蛾处于静止状态, 我们对一个较小年龄区间里的卵进行控制.首先建立一个优化控制问题, 然后求得原系统的向后伴随系统, 并对其建立一些先验估计, 最后采用不动点定理得出精确零控制的结果: 在未来某个时刻, 除了一个年龄足够小的组群, 可以将卵的数量控制到零. 受上述结果的启发,继续考虑当成虫具有空间扩散行为的 Lobesia botrana 动力系统. 我们通过对卵和幼虫在一定的年龄区间加以控制,并且对雌蛾在一个较小的区域采取控制, 从而得到该种群动力系统的精确零控制. 具体地,先把非局部项转化成局部项,建立一个优化控制问题, 并得到原系统的局部向后伴随系统. 对这个向后伴随系统, 建立~Carleman 不等式, 并结合一些估计, 通过对一个多重值函数应用不动点定理来选择一个控制, 即可得到除了一个足够小的年龄区间,雌蛾在有限的时间和一个非空开子区间上可以控制到零.
Other AbstractEuropean grape moth (EGVM) has been the most serious wine pest in Europe, North Africa, and even some Asian countries. It has a significant impact on the vineyards. A physiologically structured multistage population model is then presented to study the dynamics of one of the most important grapevine insect pests. In this thesis, we mainly make a qualitative analysis and exact null control for a class of structured population dynamical systems. Firstly, we consider a multistage, physiologically age structured dynamics system, with adult moths diffusing around the vineyard. Growth function of the population at each stage is modeled considering the climatic variations and the grape variety, which depends on the physiological age, and allows us to model great variability of growth within a cohort. Based on the contraction fixed point principle, the existence and uniqueness of solutions for the model can be obtained. Then we also prove the existence of a global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, the theory of compact operators and the Krasnoselskii's fixed point theorem are used to prove the existence of steady states. Next, it is meaningful to think about the control problem of this Lobesia botrana model. First of all, we investigate the exact null controllability of an age-dependent life cycle dynamics with nonlocal transition processes arising as boundary conditions. Assume that the four stages of this system: egg, larva, female moth and male moth are all in static station. We obtain the null controllability for the pest by acting on eggs in a small age interval. The main method is based on the derivation of estimations for the adjoint variables related to an optimal control problem. Then we apply a fixed point theorem to draw the conclusion that the population of egg except the small enough age groups to zero at a certain moment in the future, using an age- and time-dependent control of eggs. Inspired by the above result, it is necessary to consider the control problem for the Lobesia botrana population dynamics system, while the adult moths can be diffusive. Therefore, we describe a control by a removal of egg and larva population, and also on female moths in a small region. Specifically, we transform the nonlocal term into a local one, create one optimal control problem, and get the backward adjoint system related to the original system. Then combining some estimations and the Ca...
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Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224484
Collection数学与统计学院
Recommended Citation
GB/T 7714
何源. 一类结构种群动力系统的定性分析和精确零控制[D]. 兰州. 兰州大学,2013.
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