【作者】 徐文科；

【导师】 蔡体久；

【作者基本信息】 东北林业大学 ， 生态学， 2009， 博士

【摘要】 本文在系统分析国内外生物统计与生态数学理论发展的基础上,总结了几类具有代表性的生态数学模型,评述了生物统计与生态数学各自的研究领域,以及它们的交叉与融合的缺失,并详细地论述了Logistic模型和GM(1,1)模型的统计建模方法与原理；论述了运用似乎不相关线性模型的基本原理与方法于具有相关关系的Logistic模型组及两种群Lotka-Volterra模型。对于Logistic模型、GM(1,1)模型、具有相关关系的Logistic模型组及两种群Lotka-Volterra模型利用双向差分广义加权最小二乘法对其参数进行了估计,并应用于具体的实际问题中。生物数学中的数学生态学包括生物统计及生态种群数学模型,而生态种群模型多为微分方程形式。生物统计与生态种群数学模型(微分方程)在各自的研究领域的发展历史久远,体系完整、内容丰富、理论完善,但如何把这两个方向的研究交叉、融合在一起,从而推进生物统计与生物种群数学模型向前发展,进而解决更实际的具体问题,这是本文的焦点。生态种群数学模型一般只对种群做定性描述与分析。本文利用生物统计方法与原理对生态种群数学模型(微分方程)的参数进行了估计,并在Logistic模型与GM(1,1)模型中多方位、多角度的对初始预测值及参数估计进行了优化,从而使得生态种群微分方程模型有了进一步的拓展和具体应用。以下是本文的一些主要观点和结论：(1)以多元统计分析原理与方法为工具,对Logistic方程进行建模,提出了双向差分广义加权最小二乘估计方法,同时对初始预测值进行了加权修正,进一步完善了Logistic模型的拟合曲线,指出了对于初始预测值原来认识上的不足。而改进后的初始预测值会优化系统预测。(2)灰色系统GM(1,1)模型为微分方程形式,其参数估计是GM(1,1)模型的主体,本文利用统计学原理与方法对参数估计进行了精细加工,优化了参数的估计。在双向差分中对系统预测模型调整其加权值及初始预测值,使其参数估计得到优化,从而达到减小预测误差的目的。(3)对于多个具有相互关系的种群,且每个种群都适于Logistic方程曲线,如果单独考虑每个种群的Logistic方程的参数估计,不免失去了具有相互关系多个种群的系统性,从而失去了系统的功能。本文利用数理统计学中的似乎不相关线性模型的原理与方法,为多个具有相互关系的Logistic模型搭建了相互联系的建模平台。从而使其参数估计具有了统计估计的原理依据,进而达到了优化参数估计的目的。最后,利用双向差分广义加权最小二乘估计方法估计了Logistic方程组的参数,从具体应用的实例结果说明,考虑到Logistic方程组相关性、系统性的结果要优于各自独立建模的结果。(4)关于两种群Lotka—Volterra模型,包括捕食与被捕食模型、相互竞争模型、互惠共存模型的定性描述与分析的研究历史久远、理论完善。而生物统计学的研究亦是如此。但是两种群Lotka—Volterra模型的参数估计等统计问题的研究缺失。本文应用比较成熟的生物统计学原理与方法对两种群Lotka—Volterra模型进行了统计分析,如参数估计、假设检验等。这为解决具体的应用问题提供了一个可供参考的方法。本文运用数理统计学中似乎不相关线性模型的方法与原理对两种群Lotka—Volterra模型中的参数进行了估计,并应用于生态学中的典型例子猞猁与雪兔两种群的关系中,从定量分析的角度说明了这两个种群的捕食与被捕食关系,得到了这两个种群的环境容纳量及稳定平衡点的估计值。(5)在根据样本资料建立线性统计模型时,如何利用先后不同批次的样本资料来建立广义线性模型与似乎不相关线性模型,从而提出了广义线性模型与似乎不相关线性模型的广义最小二乘估计吐故纳新问题。给出了广义最小二乘估计纳新问题、吐故问题及吐故纳新问题的递推算法公式,从而简化了建立模型时求解广义最小二乘估计的计算方法,提高了运算速度与效率,节省了计算机的存储空间。从而有效地解决了广义线性模型与似乎不相关线性模型未知参数广义最小二乘估计的吐故纳新递推算法问题。更多还原

【Abstract】 The paper summed up several kinds of typical biomathematical models, by all-around analyzing the home and abroad developments of biostatistical and ecological mathematical theories. It outlined studies of the two fields and the deletions of their integrations. Logistic model and GM(1,1) model are definitely discussed on basis of statistical modeling methods and principles; the basic theories and principles of the seemingly unrelated linear model are applied in Logistic model groups with correlativity and the two-group Lotka-VIterra model. For the Logistic model,the GM(1,1) model, Logistic model groups with correlativity and the two-group Lotka-Vlterra model, the parameters are estimated with the two direction difference generalized weighted least squares estimate method, and the approach is used in practical problems. Mathematical ecologic in biomathematics including biostatistics and ecological species mathematical models, most of ecological species mathematical models are differential equations. Though the studies of biostatistics and ecological species mathematical models (differential equation) both have long history, complete system, abundant contents and perfect theory, the main point in the paper is how we can fuse and cross the two direction studies to advance the developments of biostatistics and ecological species mathematical model so that more practical problems can be solved. Ecological species mathematical models usually made qualitative representations and analyses only for species. Biostatistical methods and principles are used to estimate the parameters of ecological species mathematical models(differential equation), and optimize the parameters of the Logistic model and the GM(1,1)model from multi-direction and multi-angle, thereby the more practical application of ecological species differential equation model is obtained, and new development in estimating parameters came into being. The following are the main points and conclusions of this article:(1) By using multivariate statistical analytical methods and principles as a tool, Logistic function model is constructed, and the two direction difference generalized weighted least squares estimate method is put forward, and the initial predictive value is modified through weighed, and the fitting curve of the Logistic model is improved. The paper pointed out the inadequacy of the original understanding for the initial predictive value and the initial predictive value which is improved can optimize system prediction. (2) The GM(1,1) model of grey system whose parameter estimation is the main part of the GM (1,1) model is form of differential equations. The parameter estimation is processed well and optimized through modifying the weighted value and the initial predictive value in two direction difference so as to achieve the purpose of reducing the prediction error in this paper.(3) For some species, which are fitted for Logistic equation curve with correlativity, if the parameter estimation of the Logistic equation of every specie is studied respectively, inevitably lost the systematic of some species with correlativity, and thus lost the function of the system. In this paper, by using principles and methods of the seemingly unrelated linear model, interrelated modeling platform for Logistic models with correlativity is built. The parameter estimation has principles and basis of statistical estimation, then parameter estimation is optimized. In addition, the approach of issue-analyzing and problem-solving is holistic and systematic, rather than apart the system, analyze and study respectively, otherwise inevitably lost the function of the system. At last, the parameters of the Logistic equation system were estimated with the two direction difference generalized weighted least squares estimate method, examples of practical applications indicated that the results taking into account the relevance and systemic of Logistic quation system are superior to them without doing it.(4) The qualitative representation and analyses of the two species Lotka-Vlterra model, besides predator-prey model competitive model and cooperative model, have long history, complete system, abundant contents and perfect theory, so is the studies of biostatistics. On the other hand, statistical studies of the two-group Lotka-Vlterra model are missing. In this paper, the better ripe principles and methods of biostatistics are used to make statistical studies, such as parameter estimation and hypothesis test, and so on. a reference method for specific applications is obtained. Principles and methods of the seemingly unrelated linear model are used to estimate parameters of the two species Lotka-Vlterra model, and the approach is applied in two species relation of lynx and snowshoe hare. Quantitative analysis indicated predator-prey relationship of the two species, and got the environmental capital of the two species and the estimating value of stable equilibrium.(5) Based on using the sample data to build a linear statistical model and How to use different batches of samples and information to establish generalized linear model and seemingly unrelated linear models, the paper put forward to the air refreshing problem on the generalized least squares estimation of generalized linear models and seemingly unrelated linear models, present the problem taken in the fresh of the generalized least squares estimation, the problem got rid of the stale and the recursive algorithm formula of the air refreshing problem. Thus simplifying the calculation method of the generalized least squares estimation, increasing computing speed and efficiency and saving storage space of the computer. Finally the paper effectively solved the air refreshing problem of the recursive algorithm of the generalized least squares estimation of unknown parameters of generalized linear models and seemingly unrelated linear models.更多还原