几类离散尺度结构种群模型的定量分析与调控问题

发布时间：2018-05-12 14:24

本文选题：尺度结构 + 离散模型　；参考：《杭州电子科技大学》2015年硕士论文

【摘要】：生物数学顾名思义是生物学与数学的交叉学科,它用数学方法来研究并且解决生物学问题。数学生态学是生态学的一个重要分支。种群是物种在自然界存在的基本单位,是生态学研究的基本对象。它是由同一物种个体组成,并且占有一定的空间、同时具有潜在的繁殖能力、独立特征、结构和功能。为了保护生态环境、保护生物多样性、维护生态平衡以及合理利用可再生生物资源,就需要深入分析生物种群的演变规律。为此,国内外学者们就生物种群建立了大量的数学模型,可分为连续模型和离散模型两大类。离散结构种群模型有其优势,适于描述规模不大、世代不重叠的种群演化。自1945年Leslie提出一类非常重要的离散种群模型以来,学者们对离散生物种群模型的关注日益增加。研究离散结构的种群模型一方面可以分析种群的长期演化行为,另一方面可以根据种群的变化规律,制定科学的资源开发管理办法,比如怎样捕捞(砍伐)才既不会毁灭资源,又能够得到最佳持续的经济利益。本文主要讨论具有个体尺度结构的离散种群模型,分为线性、非线性两种。对于线性模型重点研究的是种群处于平衡态时的最优收获模式,以及种群系统的能控性和反馈控制;对非线性模型则主要研究的是模型解的有界性、持续生存、平衡态的存在性和稳定性,以及最优收获问题等。主要用到的工具有矩阵理论、离散系统控制理论及数值分析等。第2章在考虑幼体投放、成体捕捞、跨组生长和延迟生长等现象的基础上,建立一类线性尺度结构模型。研究一类种群平衡条件下的最优收获问题,应用拉格朗日乘数法求出了最优解。之后我们在只考虑跨组生长的情况下,分析了种群状态的能控性、镇定性,并给出了反馈控制方法。第3章主要讨论的是繁殖率受密度制约的一类非线性模型,在只考虑跨组生长的情形下,运用点耗散定理、本原矩阵等知识获得了种群模型解的非负性、有界性、持续生存条件;之后用圆盘定理等工具讨论了各类平衡态的存在性和稳定性。第4章的非线性模型是以第2章的线性模型为基础,进一步考虑密度制约建立起来的。该章研究具有2个尺度小组的种群模型的最优收获问题,证明了使得种群达到平衡的最优收获策略是存在的,并进一步分析了最优收获的影响因子以及他们之间的相关性。最后用MATLAB模拟软件对相关研究成果进行了数值模拟,验证了理论结果。
[Abstract]:Biological mathematics, as its name implies, is an interdiscipline between biology and mathematics, which studies and solves biological problems by mathematical methods. Mathematical ecology is an important branch of ecology. Population is the basic unit of species in nature and the basic object of ecological research. It is composed of individuals of the same species and occupies a certain space. It also has potential reproductive ability, independent characteristics, structure and function. In order to protect ecological environment, protect biodiversity, maintain ecological balance and make rational use of renewable biological resources, it is necessary to analyze the evolution law of biological population. For this reason, scholars at home and abroad have established a large number of mathematical models on biological population, which can be divided into two categories: continuous model and discrete model. The discrete structure population model has its advantages and is suitable to describe the population evolution with small scale and no overlapping generations. Since Leslie proposed a class of very important discrete population models in 1945, scholars have paid more and more attention to discrete biological population models. On the one hand, the study of discrete structure population model can analyze the long-term evolution behavior of the population, on the other hand, we can formulate scientific methods of resource development and management according to the law of population change, such as how to catch (cut down) to not destroy the resources. And get the best sustained economic benefits. In this paper, the discrete population model with individual scale structure is discussed, which can be divided into linear model and nonlinear model. The linear model focuses on the optimal harvesting model when the population is in equilibrium, as well as the controllability and feedback control of the population system, while the nonlinear model mainly studies the boundedness of the model solution and its persistence. The existence and stability of equilibrium state and the problem of optimal harvesting. The main tools are matrix theory, discrete system control theory and numerical analysis. In chapter 2, a kind of linear scale structure model is established on the basis of considering the phenomena of larval feeding, adult fishing, cross-group growth and delayed growth. The optimal harvesting problem under population equilibrium condition is studied and the optimal solution is obtained by using Lagrange multiplier method. Then we analyze the controllability and stabilization of the population state under the condition of only considering the cross-group growth, and give a feedback control method. In chapter 3, we mainly discuss a class of nonlinear model whose reproduction rate is restricted by density. In the case of only considering cross-group growth, we obtain the nonnegative and boundedness of the solution of the population model by using the point dissipation theorem, primitive matrix and other knowledge. The existence and stability of all kinds of equilibrium states are discussed by means of disk theorem. The nonlinear model in Chapter 4 is based on the linear model in Chapter 2 and further considers the density constraints. In this chapter, the optimal harvesting problem of a population model with two scale groups is studied. It is proved that the optimal harvesting strategy exists for the population to achieve equilibrium, and the influence factors of optimal harvest and their correlation are analyzed. Finally, the related research results are numerically simulated with MATLAB software, and the theoretical results are verified.
【学位授予单位】：杭州电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175;O231

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