北京上海人口容纳量的数学建模分析

1、密级,编号分类号,UDC3中仰鸵大岑硕士学位论文北京上海人D夜纳量的教学建模分析学位申请人姓名： Merfat A.Q. Basha申请学位学生类别：全自制硕士申请学位学科专业：应用欢学指导教师姓名：张兴安教授硕士学位论文北京上海人口容纳的数学建模分析论文作者：Merfat A. Q. Basha指云弗：张兴安她利专业：应用数学研究方向,生物数学华中师范大学数学与统计学学院2019年5月Mathematical Modeling of theCarrying Capacity of thePopulation in Beijing andShanghaiA ThesisSubmitted in

2、 Partial Fulfillment of the Requirementfor the M.S. Degree in MathematicsByMERFAT AQ BASHAPostgraduate ProgramSchool of Mathematics and StatisticsSignatureMay, 2019Central China Normal UniversitySupervisor: Xingan Zhang Academic Title: Associate Professor华中师濡火学券佳论丈原创性声明和使用授权说明原创性声明本人郑重声明：所呈交的学位论文，是本

3、人在导师指导下，独立进行研究工作 所取得的研究成果。除文中己经标明引用的内容外，本论文不包含任何其他个人或 集体己经发表或撰写过的研究成果。对本文的研究做出贡献的个人和集体，均已在 文中以明确方式标明。本声明的法律结果由本人承担。日期：2必年北月2日学枚论文版权使用授权书学位论文作者完全了解华中师范大学有关保留、使用学位论文的规定，即：研 究生在校攻读学位期间论文工作的知识产权单位属华中师范大学。学校有权保留并 向国家有关部门或机构送交论文的复印件和电子版，允许学位论文被查阅和借阅； 学校可以公布学位论文的全部或部分内容，可以允许采用影印、缩印或其它复制手 段保存、汇编学位论文。（保密的学位论

4、文在解密后遵守此规定） 保密论文注释：本学位论文属于保密，在年解密后适用本授权书。非保密论文注释：本学位论文不属于保密范围，适用本授权书导师签名:日期:K作者签名b伊日期：热？年以月2日本人已经认真阅读“CALIS高校学位论文全文数据库发布章程”，同意将本人的 学位论文提交“CALIS高校学位论文全文数据库”中全文发布，并可按“章程”中的 规定享受相关权益。同意论文提交后滞后：口半年：口一年：口二年发布，作者签名:崩行日期：2汹年佻月忆日导师签名:女岩给日期物用年/月方日AbstractCarrying capacity is one of the important tools on org

5、anizing development of population, resources, and environment. It can be defined as the maximum plenty of the population that an environment can maintain under limited resources and it is also deemed the limiting factor of population or an ecosystem.In this thesis, we propose a mathematical model th

6、at describes the population and use it to simulate the real data of Beijing and Shanghai cities, P.R China from 1980-2016. The logistic equation was used in this study because it is more useful and widely used for examining population growth. The carrying capacity of Beijing and Shanghai population

7、is estimated using simulation method.Our results show that the maximum carrying capacity of population size in Beijing and Shanghai are 2 756 million and 2 729 million, respectively. However, the population sizes in both cities are approaching to the maximum carrying capacity. The optimal control st

8、rategy is the best method fbr decreases the population size of both cities by establishing sub-cities fbr example Xiongan Xinqu of Beijing.This thesis contains four chapters.The first chapter consists of five sections. In section one, we presented an introduction about the fundamental concepts and d

9、efinitions of canying capacity and the importance of mathematical models in analyzing and solving real problems. The literature review of carrying capacity is introduced in section two. Section three and section four contains the background of Malthusian growth model and Logistic growth model respec

10、tively. Overview of the Chi-Square test and its types are introduced in section five.In second chapter, we used our model to simulate the real data of Beijing population from 1980 to 2016 using Matlab software to find the optimal values of parameters: the carrying capacity of the population (K) and

11、the intrinsic growth rate (r) and calculate the confidence interval of carrying capacity. We used the ode45 to solve the model in this numerical simulation.In third chapter, we used our model to simulate the real data of Shanghai population from 1980 to 2016 using Matlab software to find the optimal

12、 values of K and r and calculate the confidence interval of carrying capacity.The ode45 was used to solve the model in this numerical simulation. The brief discussion and conclusion are presented in chapter four.Key Words: Carrying capacity, logistic equation, confidence interval, numerical simulati

13、on, Chi-square Test.摘要容纳是衡人口，资源和环境发展是否和谐的重要工具之一具体的说，容纳是指 环境在有限的资源中可以维持的人口数的最大值.它也是人口或生态系统的限制因素。 在本文中，我们提出了一个描述人口的数学模型并用其来模拟中国北京和上海两个城 市在1980-2016年间的真实数据。本研究使用了有效且广泛用于检验人口增长的对数 方程,采用模拟方法对北京和上海的人口容纳进行了估算。结果表明，北京和上海 的人口最大容纳分别为275.6万和272.9万。但是两个城市的人口规模远大于最大 容纳量，所以对这两个城市而言最优控制策略是通过建立卫星城市的方法来减少人口 规模，例如北京市

14、的雄安新区。本论文共分四章。第一章有五节在第一节中，我们介绍了容纳量的基本概念及定义， 并对数学模型在分析和解决实际问题中的宣要性进行介绍。第二节是关于容纳量的文 献综述。第三节和第四节分别介绍马尔萨斯增长模型和Logistic增长模型的背景。卡 方检验及其类型的概述在第五节中介绍。第二章中利用Matlab软件编程模拟1980. 2016年北京市人口实际数据，找出如下两个参数的最佳值：容纳量人口( K )和内在 增长率(r),并计算了容纳的信区间，其中我们用ode45算法来求解这个数值模 拟问题。第三章我们用同样的方法对1980年至2016年之间上海人口的实际数据进行 计算及分析。一些简要讨论

15、和结论见第四章0关催词：容纳最，对数方程，置信区间，数值模拟，卡方检验。Contents TOC o 1-5 h z HYPERLINK l bookmark4 o Current Document AbstractI HYPERLINK l bookmark7 o Current Document 摘要 III HYPERLINK l bookmark35 o Current Document Chapter 1 Background 1 HYPERLINK l bookmark38 o Current Document Introduction. 1 HYPERLINK l bookmar

16、k42 o Current Document Literature review5 HYPERLINK l bookmark46 o Current Document Malthusian Growth Model7 HYPERLINK l bookmark50 o Current Document The Malthusian Equation8 HYPERLINK l bookmark54 o Current Document The Solution of Malthusian Equation8 HYPERLINK l bookmark58 o Current Document Ass

17、umptions of the Malthusian Model. 10 HYPERLINK l bookmark61 o Current Document The Logistic Growth Model11 HYPERLINK l bookmark65 o Current Document The Logistic Equation12 HYPERLINK l bookmark69 o Current Document Interpretation of parameters of the logistic Equation13 HYPERLINK l bookmark73 o Curr

18、ent Document The Solution of Logistic Equation13 HYPERLINK l bookmark76 o Current Document The Equilibrium Points of Logistic Equation15 HYPERLINK l bookmark79 o Current Document Assumptions of the Logistic Model17 HYPERLINK l bookmark83 o Current Document Chi-Square Test17 HYPERLINK l bookmark87 o

19、Current Document Overview of The Chi-Square Test 17Types of Chi-Square Test18The Chi-Square Goodness-of-Fit Test18 HYPERLINK l bookmark100 o Current Document The Chi-Square Test of Homogeneity19 HYPERLINK l bookmark106 o Current Document The Chi-Square Test of Independence20 HYPERLINK l bookmark110

20、o Current Document Chapter 2 Mathematical Model in Beijing22 HYPERLINK l bookmark113 o Current Document study Area and Data22 HYPERLINK l bookmark116 o Current Document Model and Fitting23 HYPERLINK l bookmark119 o Current Document Results23 HYPERLINK l bookmark122 o Current Document Chapter 3 Mathe

21、matical Modelin Shanghai25 HYPERLINK l bookmark125 o Current Document 3.1. Study Area and Data 25 HYPERLINK l bookmark128 o Current Document 3 2 Model and Fitting26 HYPERLINK l bookmark131 o Current Document Results26 HYPERLINK l bookmark134 o Current Document Chapter 4 Discussion andConclusion*. 28

22、 HYPERLINK l bookmark137 o Current Document References30 HYPERLINK l bookmark187 o Current Document Acknowledgements35Chapter 1 BackgroundIntroductionThe Carrying Capacity Conception is an evolving instrument for observation sustainable development.The concept produces from the ecology of animal, ba

23、sed on the logistic growth bend theory by Verhulst in 1838 1-4. According to the logistic growth model, the growth of animal population can be restricted to a higher asymptote, i.e., the capability of carrying, due to the population raise leads to negative impacts through interaction amongst the mem

24、bers, which is overwhelmingly evinced as a height density and the linked overcrowding effects 5.The conception of carrying capacity was advanced in the domains of biology and ecology, whereby it was used to wildlife administration to make sure “the numeral of animals whose can be maintained in a par

25、ticular domicile prior unnecessary biological or ecological influences occur 6.Carrying capacity was then used to garden administration when rising visitors* numbers became anxiety. The additional human ingredient expanded the conception of carrying capacity from a simple biological one to a ganglio

26、n matter with both social and biological ingredients.Carrying capacity definition is not an issue of calculating and sternly perform a single, candid maximum value. Instead of, carrying capacity contains an element of comprehension from entertainment area users and directors, who collect the compone

27、nt of human to entertainment carrying capacity. For this cause, the carrying capacity locating is never a frugally objective. As such, capacity is predominating announced in the form of a domain of assessment as opposed to an optimum or top value. Capacity shall change from site to site in conformit

28、y with the behavior of visitor and predilection, besides goals of management 7.There is a fully-recognized style that advanced countries with richness are ordinarily correlating with loud levels of civilization for instance, the USAs 82% and Japans 91% 8, 9.It is predestined that population of urban

29、 will grow to over 70% of the worlds overall population by 205010 areas inexorably encourage abundant challenges to the destination of cities 115 12. The rapid growing exigencies on carrying capability in numerous cities have been a considerable burden reason by urbanization.Over-carrying capacity i

30、n urban evolution is sense as huge and over-fastened development behind its ingrained limits 13, and leads to different urban problems, like passage overcrowding, housing decrease, unaffordable habitation prices, congested streets, decadent ecosystem, water and air pollution, increasing request fbr

31、rubbish elimination, social tussles, gathered fortune apportionment and community cutoff 13-16. Those urban problems of over*urban carrying capacity (UCC) evolution can be summed up as four areas of unsustainability: slender urban services, environmental retrogression, natural resources scarcity and

32、 social tussles.These days, the issue of over-UCC urban expansion has produced diverse city typologies 17, given the hugely various socioeconomic and demographic status across diverse cities. It has become a large defy fbr urban planners and city directors Oo supply residents with a perfect goodness

33、 life in their towns 10. Since advancing sustainability is a fundamental liability for urban planning and evolution, the significance of UCC has drawn big anxieties at the global level. The concept of UCC supplies substantial steering fbr managers of a city and urban planners to best manage, structu

34、re and dispense urban fortunes, thus addressing the large requests from the rising urban inhabitance 15. However, given a boost of explications and debate surrounding this object, present studies still scarcity a universally known qualifier on UCC 18 and standardization for a methodical estimate 19.

35、The accomplishment of sustainable growth has been a centrally significant target fbr urban projects and managers of the city 20. Anyway, the managers and planners of city oftentimes fail to pledge timely and convenient estimations on the situations of urban sustainable growth, especially in growing

36、countries.In China, penetrating solemn policy and schedules, like Chinas 2th Five-Year Plan Report 21, have highlighted the significance of UCC refinements in the agenda of government.For advancing to realization of sustainable urban development, UCC deserves methodical and scholarly studies.Procedu

37、res of UCC situations in lighting of concerning sustainable gauges are helpful for characterizing the relativism significance and investing exigencies for capacity optimizations. Both UCC and sustainability have to be viewed as “a persisting procedure rather than a sole juvenile 22. It means that th

38、e statuses of UCC or sustainability at one moment do not include the same situation in different circumstances* Both of them must be constantly and recursive observed, measured, and enhanced to maximize the socioeconomic and ecological luxuriousness 5.Cities have played extremely paramount roles in

39、nationalist economic growth, but because the elevated concentricity of population and consumption of resources, the demolition of the urban ambiance have become further dangerous. It is mostly approved that the towns economic evolution and human actions shall largely influence the ecological environ

40、ment 23.A mathematical model is a characterization of a system utilization mathematical notions and language. The procedure of evolving mathematical model is named mathematical modeling.Mathematical models are applied in analyses of the systems logistics. A model assists to dissect a system, study t

41、he impacts of various ingredients, and make foretelling about behavior. Mathematical models may take shapes like statistical methodologies, dynamical systems, differential equations, mathematical programming, queuing theory, and others. Mathematical models can be split into two kinds: simulation and

42、 analytical models. Analytical models are applied, when the frame of the system is comparatively not complex and there occur analytical shape in what frame of systems may be written. Most of statuses simulation models are applied while the systems cannot be written in an analytical method. Simulatio

43、n is applied to study cases described by incertitude.Mathematical modeling gets further and further significant for analysis of logisticssystems as systems get complicated and there is a requirement for an instrument that assists to understand the systems and to grant the desirable responses to ques

44、tions in time as stumpy as conceivable 24.For modeling growth of biological systems, abundant models were introduced. These differently address dynamics of population, either modelled separately or, for big populations, mostly constantly. Others model factual physical evolution of some possessions o

45、f benefit for a creature or creatures.The natural exponential growth model can supply a sufficient approach to such growth fbr the first time. However, for inhabitancy no ferocity or intraspecific rivalry is included. The population would thus keep rising unimpeded (or imperatively decrease to zero

46、if an incipient growth decrease was present). Even in the status where ferocity was mainly trivial, the model does not conform to lowering because of intraspecific rivalry fbr resources of environmental like eating and habitat. For the status of growth per se? unrestrained growth is also not practic

47、al. For example, as plants access ripeness, the physical advantages of attention will arrive a limiting remoteness 25.In this thesis, we propose a model fbr the populations of Beijing and Shanghai. The aim is to use mathematical modeling to control and reduce the problems that arise from increasing

48、of the total populations fbr example traffic, the population of children in schools, a spread of infection diseases, etc. The model described by an ordinary differential equation, it is called the logistic equation. This equation is independent of the constant parameters r and K. Its nature is such

49、that the characteristics of the solutions, fbr all positive initial conditions, approach the constant value of the carrying capacity K as time t tends to infinity. We study the basic properties of the model, including the solution and the equilibrixim points. Then, to validate the model, we use it t

50、o simulate the data of the populations of Beijing and Shanghai cities-(PRC) from 1980 to 2016. We use MATLAB software to estimate the values of K and r which allow us to make good decisions and solve the problems. Numerical simulations correspond to the data rationally well.Literature reviewThe conn

51、otation of urban carrying capacity is undergoing to hot argument but lacks scholarly consideration.Presently, farthest pertinent literature is devoted to human or environmental carrying capacity. Studies of urban carrying capacity, focusing on urban framing, is a hugely various research object due t

52、o the various concentration, concepts, rules, and impacts. For example, UCC examines the system of urban, consideration not solely the naturalism elements but else the human-made properties, like cultural, economic, infrastructural parts, etc.There is as yet a rarity of studies focusing on the signi

53、ficant subject of UCC, and current studies extremely raise questions of fractionated and insufficient comprehension, a shortage of feasible estimation modes, and bounded implementations 5. 26 developed a collection of UCC indictors fundamentally concentrated on water equipping, sewerage, and fiscal

54、ability of social establishments in India. According to these valuation indicators, a conceptually framing base was suggested. However, this conceptually framing has not been studied in an actual state study, leaving its implementing doubtful Also 14 presented sundry significant contributions. He re

55、ferenced key standards for the option of urban indexes and debated how urban ecological indexes able to serve urban administration. With a sustainable view, he proposed some economic and ecological urban indexes for town management, and a public framework is offered to support managers of a city in

56、transaction with urban affairs. Based on the carrying capacity conception and valuation of carbon revival 27 evolved an arithmetical model to estimate the sustainability of a constructing venture. The mode entails estimation of the carbon revival of the total life cycle of the structure from constru

57、ction to structure process. Using flat modeling as the prime research procedure, 28 appreciated Chinas food frugality capacities. The research feedbacks characterize the preposterous agrarian frame as the main reason for the sitting food scarcity. Numerous propositions were presented to promote Chin

58、as carrying capacity by a further equiponderant cultivation product frame. After Contrasting the efficacity of different estimate materials, 29 and 30 fbund that the estimate materials advanced for international, patriotic, rustic levels might not be efficient for provincial usage. Versus this backg

59、round, 30 evolved a modem analytic instrument for estimating sustainability at the territorial level. The named sustaining human carrying capacity appreciated the prospective level of human activities by promoting the efforts decreed on the regional ecosystem. The efficiency is studied in South East

60、 Queensland as the situation study. 13 evolved a framework for estimating UCC which enable assessment the optimize population intensity based on the present infrastructures and earth usage. The restriction is that because the hardness of gaining data, this study fails to amalgamate institutional and