具有时滞的红松种群数学模型研究

发布时间：2019-04-26 00:26

【摘要】：红松是我国东北地区的重点保护树种.红松的物种价值不仅体现在保护生态环境方面,也体现在经济方面.因此,对于红松生态系统的研究具有重要的意义.考虑到雌鼠的乳汁为刚出生的幼鼠提供了所有身体所需的营养,这让正在哺乳期的雌鼠将加大采集、捕食和埋藏松籽的力度,继而使得松籽被埋藏于地表的概率增大,增加了松籽被遗漏的可能性,处在特别适宜松籽萌发成幼苗环境的被遗漏的松籽就萌发.因此,要想让对红松种群的研究更切合实际情况把鼠类的哺乳期考虑在内很有必要.本文考虑了鼠类的哺乳期这一实际现象,建立了四类线性和非线性的具有时滞的反映松籽、鼠类和幼苗三者之间相互关系的红松种群数学模型.对本文的四类具有时滞的红松种群模型,时滞微分方程的知识被用来对模型进行研究,数值模拟了松籽、鼠类、幼苗及三者对时滞参数鼠类的哺乳期的走势,得出时滞参数对三者呈现有规律的稳定特性的影响.本文研究成果是天然红松林系统的松籽、鼠类和幼苗三者之间在一定条件下可以形成一个随时间变化互为消长并且有规律的周期性振荡,并最终保持动态稳定.全文包括五章内容.绪论是第一章,主要阐述课题的研究背景及研究的目的和意义、红松种群模型的国内外研究现状、主要内容及本文所用到的基础知识.第二章建立了一类线性时滞红松种群的模型,这里的时滞参数是鼠类哺乳期,本章最先分析和判定了正平衡点的稳定性,给出了时滞界限,导出存在Hopf分支的条件.第三章构造了两个非线性时滞红松种群的模型,本章先分析和判定正平衡点的稳定性,接着通过运用理论知识分析得Hopf分支方向和分支周期解的稳定性的相关公式.第四章构造了人工开发线性时滞红松种群模型,本章最先分析和判定了正平衡点的稳定性,获得了发生Hopf分支条件及产生了分支周期解.第五章是本文的主要结论和展望.
[Abstract]:Pinus koraiensis (Pinus koraiensis) is a key protected tree species in northeast China. The species value of Pinus koraiensis is not only reflected in the protection of ecological environment, but also in the economic aspect. Therefore, it is of great significance for the study of Pinus koraiensis ecosystem. Considering that the milk of the female mice provides all the nutrients needed by the body for the newborn mice, which makes the lactating female rats more likely to gather, hunt and bury the pine seeds, which in turn increases the probability that the pine seeds are buried on the surface, The possibility of pine seeds being missed is increased, and the missed pine seeds germinate in a suitable environment for pine seeds to germinate into seedlings. Therefore, it is necessary to consider the lactation period of rodents in order to make the study of Pinus koraiensis population more realistic. In this paper, four kinds of linear and nonlinear mathematical models of Pinus koraiensis population, which reflect the relationship among pine seeds, rodents and seedlings, are established by taking into account the actual phenomenon of the lactation period of rodents, and four kinds of linear and nonlinear mathematical models of Korean pine population with time delay are established. The knowledge of delay differential equation is used to study the four types of Pinus koraiensis population models in this paper. The trends of lactation period of pine seeds, rodents, seedlings and three kinds of delay parameter rodents are numerically simulated. The effects of time-delay parameters on the stability of the three parameters are obtained. The results of this study are the pine seeds of the natural Korean pine forest system. Under certain conditions, the rodents and seedlings can form a periodic oscillation which fluctuates and fluctuates regularly with each other with time, and finally keeps the dynamic stability. The full text includes five chapters. The introduction is the first chapter, which mainly describes the research background, the purpose and significance of the research, the domestic and foreign research status of Pinus koraiensis population model, the main contents and the basic knowledge used in this paper. In the second chapter, the model of a class of linear delay Korean pine population is established, where the delay parameter is mouse lactation. In this chapter, the stability of positive equilibrium point is first analyzed and determined, the delay bound is given, and the conditions for the existence of Hopf bifurcation are derived. In chapter 3, two nonlinear delay Korean pine population models are constructed. In this chapter, the stability of positive equilibrium point is analyzed and determined, and then the stability formulas of Hopf bifurcation direction and bifurcation periodic solution are obtained by using theoretical knowledge. In chapter 4, we construct the artificial development linear delay Korean pine population model. In this chapter, we first analyze and determine the stability of the positive equilibrium point, obtain the Hopf bifurcation condition and produce the bifurcation periodic solution. The fifth chapter is the main conclusion and prospect of this paper.
【学位授予单位】：北京建筑大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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