具有Size结构的生物种群动力系统的行为分析和最优控制
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摘要
在生物学和人口统计学中,建立基于年龄结构的种群模型是进行数学建模及控制的一种传统方法。深入研究生物种群的演变规律,对保护生物多样性、管理可再生资源、控制病虫害及预防流行病等具有重要的意义。近年来,考虑个体Size结构的种群模型引起了很多关注,并得到了一些研究成果。这里的Size表示与种群个体有关的某个连续指标,例如重量、长度、直径、体积、成熟度,或者显示种群个体生理或统计特征的其它数量指标。因此年龄结构是Size结构的一种特殊情形。一般来说,Size指标比年龄指标更直观、更容易测量,并且Size结构模型能更好地描述许多种群的演化过程,特别是对海洋无脊椎生物(如藤壶、珊瑚)和许多变温动物(如鱼、蛇)。另一方面,Size结构模型对开发利用可再生资源具有重要作用。
本文考虑几类具有Size结构的种群动力系统,研究它们的动力学性态(如解的存在性,唯一性,非负性,有界性,稳定性,解对模型参数的连续依赖性等)和控制问题(如最优收获控制,最优出生率控制)。综合应用(线性和非线性)泛函分析(如算子半群理论,Mazur定理,Ekeland变分原理等),微分方程,积分方程,以及现代控制论等工具,得到一些理论成果,为模型的实际应用提供了必需的科学理论依据。本文的主要工作如下:众所周知,种群个体的生存离不开资源(如食物)。基于如此考虑,在第二章建立并分析了一个带有两种资源制约的Size结构种群模型,还考虑了有新生个体从外界环境迁入的情形。在第一节,以偏微分方程、微分-积分方程组建立了一个非线性模型,给出了正平衡解的求解方法。接着在第二节,对非线性系统进行线性化,并利用算子半群相关理论得到了线性化系统的一些性质。然后在第三节,推导出特征方程,并借此给出了平衡解的稳定和不稳定条件。最后在第四节,给出了两个例子及相应的数值模拟,用以说明理论结果的有效性,并将该模型推广到多种资源制约的情形。
第三章研究具有Size结构种群模型的最优控制策略。第一节主要处理最优收获问题:首先给出基本模型及模型参数的相关假设;接着利用Banach不动点定理和Bellman不等式,证明了状态系统解的适定性、有界性,以及共轭系统解的存在唯一性;然后借助Mazur定理,证明了最优控制的存在性;最后应用切锥法锥技巧,给出了最优性条件,并证明了最优控制的唯一性。第二节则以出生率为控制变量讨论了一个最优控制问题:先给出基本模型及相关假设,证明了状态系统及其共轭系统解的存在唯一性,并给出了几个关于它们解的估计式,这些估计式将在证明最优控制的唯一性时被用到。然后借助切锥法锥技巧,给出了最优性条件,并利用Ekeland变分原理证明了最优控制的存在唯一性。最后给出了例子及相应的数值结果。我们在这章中得到的结论推广了年龄结构种群模型的相应成果。虽然我们的基本模型并不复杂,但是研究结果显示,种群个体的价格因素在处理带Size结构的种群模型的控制问题时起着关键作用。
Age-structured population models are a traditional tool for mathematical modeling and control in biology and demography. The study of population systems are of great significance for species conservation, management of renewable resources, containment of pests and parasites, and epidemic intervention. Recently, size-structured models have attracted much attention and a number of research results appeared. By size we mean a set of continuous indexes related to target individuals, such as mass, length, diameter, volume, maturity, or any other quantity displaying their physiologic or statistical characteristics. Consequently, age structure is only a special kind of size structure. Generally speaking, size is more intuitive and easier to measure than age, and size-structured models better describe the dynamics of populations than age-structured ones, especially for marine invertebrates (e.g., Barnacles and Corals) and many poikilothermal animal species (e.g., fishes and snakes). On the other hand, size-structured models are expected to play more important role in the exploitation of renewable resources.
This dissertation is concerned with several classes of size-structured population systems, for which we investigate dynamical properties (e.g., existence, uniqueness, non-negativeness, boundedness, stability of stationary solutions, continuous dependence on model parameters, etc.) and control problems such as optimal harvesting and optimal birth control. By means of (linear and nonlinear) functional analytic approach (e.g., semigroups theory, Mazur’s theorem, Ekeland variational principle, etc.), differential equations, integral equations, and modern control theory, we obtain some theoretical results, which provide a solid ground for the practical applications of the involved models.
The principal works of this dissertation are as follows:
It is undoubted that the survival of individuals of population is dependent on resources (e.g., foods). Therefore, we formulate and analyze a size-structured population model with resources-dependence and inflow in chapter 2. In section 2.1, we propose the basic model, which is a nonlinear hybrid system of ordinary and partial differential and integral equations, and present a method to solve the positive stationary of the basic system. Then in section 2.2, we linearize the nonlinear system and derive some regularity properties for the linearized system by means of the semigroups theory, following that we deduce the characteristic equation and establish some conditions for stability and instability of the stationary solution in section 2.3. Section 2.4 consists of two examples and their computer simulations, which are used to show the effectiveness of the theoretical results. Finally, we extend this model to the multi-resources-dependent situations.
Chapter 3 is devoted to the study on the optimal control problems of size-structured population models. Section 3.1 focuses on an optimal harvesting problem. We firstly state the control problem and assumptions. Then by making use of the Banach fixed point theorem and Bellman’s inequality, we prove well-posedness and boundedness of solutions to the basic system, and derive existence of a unique solution to the adjoint system. Next we show the existence of optimal controls by means of Mazur's theorem, and establish optimality conditions of first order in the form of maximum principle via normal cone technique. Finally we demonstrate the uniqueness of optimal control. Section 3.2 handles a control problem with birth rate as control variable. Firstly, the basic model and assumptions are proposed. Then we show existence and uniqueness of solutions both to the state system and the adjoint system, and make several estimates, which will be useful in the derivation of the uniqueness of optimal control. Furthermore, we establish first order necessary conditions of optimality via tangent-normal cone techniques, and prove the existence of a unique optimal control by means of the Ekeland variational principle. Finally, we establish an example and its numerical result. Our conclusions in this chapter cover the corresponding results of age-structured population models. Although our index and the underlying model are simple, we still get an insight that the price factor of individuals plays a key role in the structure of the optimal controller.
本文考虑几类具有Size结构的种群动力系统,研究它们的动力学性态(如解的存在性,唯一性,非负性,有界性,稳定性,解对模型参数的连续依赖性等)和控制问题(如最优收获控制,最优出生率控制)。综合应用(线性和非线性)泛函分析(如算子半群理论,Mazur定理,Ekeland变分原理等),微分方程,积分方程,以及现代控制论等工具,得到一些理论成果,为模型的实际应用提供了必需的科学理论依据。本文的主要工作如下:众所周知,种群个体的生存离不开资源(如食物)。基于如此考虑,在第二章建立并分析了一个带有两种资源制约的Size结构种群模型,还考虑了有新生个体从外界环境迁入的情形。在第一节,以偏微分方程、微分-积分方程组建立了一个非线性模型,给出了正平衡解的求解方法。接着在第二节,对非线性系统进行线性化,并利用算子半群相关理论得到了线性化系统的一些性质。然后在第三节,推导出特征方程,并借此给出了平衡解的稳定和不稳定条件。最后在第四节,给出了两个例子及相应的数值模拟,用以说明理论结果的有效性,并将该模型推广到多种资源制约的情形。
第三章研究具有Size结构种群模型的最优控制策略。第一节主要处理最优收获问题:首先给出基本模型及模型参数的相关假设;接着利用Banach不动点定理和Bellman不等式,证明了状态系统解的适定性、有界性,以及共轭系统解的存在唯一性;然后借助Mazur定理,证明了最优控制的存在性;最后应用切锥法锥技巧,给出了最优性条件,并证明了最优控制的唯一性。第二节则以出生率为控制变量讨论了一个最优控制问题:先给出基本模型及相关假设,证明了状态系统及其共轭系统解的存在唯一性,并给出了几个关于它们解的估计式,这些估计式将在证明最优控制的唯一性时被用到。然后借助切锥法锥技巧,给出了最优性条件,并利用Ekeland变分原理证明了最优控制的存在唯一性。最后给出了例子及相应的数值结果。我们在这章中得到的结论推广了年龄结构种群模型的相应成果。虽然我们的基本模型并不复杂,但是研究结果显示,种群个体的价格因素在处理带Size结构的种群模型的控制问题时起着关键作用。
Age-structured population models are a traditional tool for mathematical modeling and control in biology and demography. The study of population systems are of great significance for species conservation, management of renewable resources, containment of pests and parasites, and epidemic intervention. Recently, size-structured models have attracted much attention and a number of research results appeared. By size we mean a set of continuous indexes related to target individuals, such as mass, length, diameter, volume, maturity, or any other quantity displaying their physiologic or statistical characteristics. Consequently, age structure is only a special kind of size structure. Generally speaking, size is more intuitive and easier to measure than age, and size-structured models better describe the dynamics of populations than age-structured ones, especially for marine invertebrates (e.g., Barnacles and Corals) and many poikilothermal animal species (e.g., fishes and snakes). On the other hand, size-structured models are expected to play more important role in the exploitation of renewable resources.
This dissertation is concerned with several classes of size-structured population systems, for which we investigate dynamical properties (e.g., existence, uniqueness, non-negativeness, boundedness, stability of stationary solutions, continuous dependence on model parameters, etc.) and control problems such as optimal harvesting and optimal birth control. By means of (linear and nonlinear) functional analytic approach (e.g., semigroups theory, Mazur’s theorem, Ekeland variational principle, etc.), differential equations, integral equations, and modern control theory, we obtain some theoretical results, which provide a solid ground for the practical applications of the involved models.
The principal works of this dissertation are as follows:
It is undoubted that the survival of individuals of population is dependent on resources (e.g., foods). Therefore, we formulate and analyze a size-structured population model with resources-dependence and inflow in chapter 2. In section 2.1, we propose the basic model, which is a nonlinear hybrid system of ordinary and partial differential and integral equations, and present a method to solve the positive stationary of the basic system. Then in section 2.2, we linearize the nonlinear system and derive some regularity properties for the linearized system by means of the semigroups theory, following that we deduce the characteristic equation and establish some conditions for stability and instability of the stationary solution in section 2.3. Section 2.4 consists of two examples and their computer simulations, which are used to show the effectiveness of the theoretical results. Finally, we extend this model to the multi-resources-dependent situations.
Chapter 3 is devoted to the study on the optimal control problems of size-structured population models. Section 3.1 focuses on an optimal harvesting problem. We firstly state the control problem and assumptions. Then by making use of the Banach fixed point theorem and Bellman’s inequality, we prove well-posedness and boundedness of solutions to the basic system, and derive existence of a unique solution to the adjoint system. Next we show the existence of optimal controls by means of Mazur's theorem, and establish optimality conditions of first order in the form of maximum principle via normal cone technique. Finally we demonstrate the uniqueness of optimal control. Section 3.2 handles a control problem with birth rate as control variable. Firstly, the basic model and assumptions are proposed. Then we show existence and uniqueness of solutions both to the state system and the adjoint system, and make several estimates, which will be useful in the derivation of the uniqueness of optimal control. Furthermore, we establish first order necessary conditions of optimality via tangent-normal cone techniques, and prove the existence of a unique optimal control by means of the Ekeland variational principle. Finally, we establish an example and its numerical result. Our conclusions in this chapter cover the corresponding results of age-structured population models. Although our index and the underlying model are simple, we still get an insight that the price factor of individuals plays a key role in the structure of the optimal controller.
引文
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[2]马知恩.种群生态学的数学建模与研究[M].合肥:安徽教育出版社, 1996.
[3] Sharpe F. R., Lotka A.. A problem in age distribution [J]. Phi. Mag., 1911, 21(6): 435-438.
[4] McKendrick A. G.. Applications of mathematics to medical problems [J]. Proceedings Edinburgh Mathematics Society, 1926, 44: 98-130.
[5] Gurtin M. E., MacCamy R. C.. Nonlinear age-dependent population dynamics [J]. Arch. Rat. Mech. Anal., 1974, 54(3): 281-300.
[6] Rorres C., Fair W.. Optimal harvesting policy for an age-specific population [J]. Math. Biosci., 1975, 24: 31-47.
[7] Webb G.. Nonlinear semigroups and age-dependent population models [J]. Ann. Mat. Pura. Appl., 1981, 4: 43-55.
[8] Gyllenberg M.. Stability of a nonlinear age-dependent population model containing a control variable [J]. SIAM J. Appl. Math., 1983, 43: 1418-1438.
[9] Brokate M.. Pontryagin's principle for control problems in age-dependent population dynamics [J]. J. Math. Biol., 1985, 23: 75-101.
[10] Chan W.-L., Guo B.-Z.. Global behavior of age-dependent logistic population models [J]. J. Math. Biol., 1990, 28: 225-235.
[11] Murphy L. F., Smith S. J.. Optimal harvesting of an age-structured population [J]. J. Math. Biol., 1990, 29: 77-90.
[12] Murphy L. F., Smith S. J.. Maximum sustainable yield of a nonlinear population model with continuous age structure [J]. Math. Biosci., 1991, 104: 259-270.
[13] Medhin N. G.. Optimal harvesting in age-structured populations [J]. Journal of Optimization Theory and Application, 1992, 74(3): 413-423
[14] Iannelli M.. Mathematical theory of age-structured population dynamics [M]. Giardini editory e stampatori Pisa., 1995.
[15] Webb G. F.. Theory of nonlinear age-dependent population dynamics [M]. New York: Marcel Dekker, 1985.
[16] Brauer F., Castillo-Chavez C.. Mathematical model in population biology and epidemiology [M]. New York: Springer-Verlag, 2001.
[17] Anita S., Iannelli M., Kim M. Y., Park E. J.. Optimal harvesting for periodic age-dependent population dynamics [J]. SIAM J. Appl. Math., 1998, 58(5): 1646-1666.
[18] Anita S.. Optimal harvesting for a nonlinear age-dependent population dynamics [J]. J. Math. Anal. Appl.,1988, 226: 6-22.
[19] Anita S.. Analysis and Control of Age-Dependent Population Dynamics [M]. Dordrecht: Kluwer Academic Publishers, 2000.
[20] Barbu V., Iannelli M.. Optimal control of population dynamics [J]. J. Optim. Theo. Appl., 1999, 102: 1-14.
[21] Barbu V., Iannelli M., Martcheva M.. On the controllability of the Lotka-Mckendrick model of population dynamics [J]. J. Math. Anal. Appl., 2001, 253: 142-165.
[22] Iannelli M., Kim M. Y., Park E. J., Pugliese–NoDEA A.. Global boundedness of the solutions to a Gurtin-MacCamy systems [J]. Nonl. Differ. Equ. Appl., 2002, 9: 197-216.
[23]宋健,于景元.人口控制理论[M].北京:科学出版社, 1985.
[24]申建中,徐宗本.时变人口系统的适定性及关于生育率的最优控制[J].系统科学与数学, 2001, 21(3): 274-282.
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