2015第八届网络建模第一阶段c题特等奖荒漠区动植物关系的研究

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2015年第八届“认证杯”数学中国

数学建模网络挑战赛第一阶段论文

题目荒漠区动植物关系的研究

关键词动植物关系生物量残差分析灰色模糊模型

摘要：

本文主要针对荒漠区动植物的关系探究不同干扰对这种关系的影响问题，运用层次分析法、灰色模糊评价法、因子分析法、t检验法、残差分析法等一系列方法，建立了层次分析模型、灰色关联模型、灰色测度模型、结构方程模型（SEM）等数学模型，综合分析了荒漠区的啮齿动物与植物在过牧和轮牧，夏季与秋季等不同干扰下生物量的变化趋势，并通过研究的数据揭示了啮齿动物生物量与植物生物量的变化关系。最终，通过对啮齿动物群落稳定性进行评估，从评估得到的数据再进一步研究不同干扰环境下啮齿动物群落的影响机制。

针对问题一，利用附件一提供的数据，对应不同问题，建立相应数学模型，从而获得不同干扰对植物生长量与啮齿动物生长量的变化趋势，并揭示不同干扰下的变化关系。首先我们将问题一划分成两个小问题。在研究第一小问时，我们对附件一中的数据按不同层次进行处理，运用层次分析模型，通过EXCEL建立折线图，进行对比，分析出不同干扰下，植物地上生物量与动物生物量的变化趋势。而在研究第二小问时，建立灰色关联模型，灰色关联分析方法对受多种因素影响的事物和现象从整体观念出发进行综合评价，它是一个被广为接受的方法，所以我们通过建立灰色关联模型，在计算出相应指标权重后，得到植物生物量与动物生物量之间的关联度，从而通过MATLAB进行绘图，以获得啮齿动物生物量与植物生物量的变换关系。

针对问题二，要求对啮齿动物群落稳定性进行研究，并揭示干扰对于啮齿动物群落的影响机制。首先，就群落稳定性而言，由于群落内部食物网络关系及天敌的“跟随”现象,当种类数增加时，个体数相应增加。由灰色系统理论知，累加生成可使离乱的原始数据中蕴含的规律充分显露出来，因此我们此处选用灰色测度模型和灰色聚类法来评估群落稳定性，将反映群落稳定性的指标因子纳入该模型，使得群落稳定性的评价更具有科学性和实用性。其次，求解数据和图像数据来总结不同的干扰下不同的因素指标对啮齿动物群落的影响机制。对于这个问题我们利用结构方程模型（SEM）来同时处理多个因变量。再利用SPSS软件，建立SEM模型的概念模型，运用其中的回归分析，因子分析，

t检验和方差分析来估计整个模型的拟合程度。最后运用模型方法检验不同干扰对啮齿动物优势种的作用途径，即可求得不同干扰对啮齿动物的影响关系产生影响机制。

同时，本文在最后进行了模型的优缺点评价，以及对各模型进行了改进和推广，分析了某些模型在其他领域的广阔应用前景。

参赛队号: 4612

所选题目: C 题

Abstract

In this paper,the study focuses on the relationship between plantsand animals in the desert to explore t he impact of different interferences of this ing Analytic Hierarchy Process,Grey fuzzy evalu ation method,the method of factor analysis,Method of test for t,the residual analysis method and a series of methods.Analytc hierarchy model,the grey correlation model,Grey evaluation model,Structural equation m odel and some other models have been established,in order to analyse the change biomass of the rodent and plants in the desert under the different interferences of over grazing or rational grazing between summer an d autumn.The relationship between rodent biomass and plant biomass were studied by data.Finally,through the assessment of the stability of rodent community ,we could get data.With the data we can do fuher resear ch toget the effect to rodent communities of different environmental interference.

For the first problem, using the data which provided by the first attachment to establish different mathematical models for different kinds of questions, after which can not only help us get the changing trends on the plant growth and the rodent growth in different interferences, but also reveal the change relationship in the same interferences. Above all, we divided the first problem into two small problems. After that, we began to deal with the data in the first attachment in different administrative levels. During this part of work, we use the Analytical Hierarchy Model and the Line chart created by Excel to compare data, so that we can analyze the change trend of the above-ground plant biomass and the animal biomass in different interferences. When we deal with the second problem, we established the Grey Relational Model. We can see that the Grey Relational Analysis can make a comprehensive evaluation for things that affect by a variety of factors from the overall concepts. Because it is a widely accepted method, we established the Grey Relational Model to obtain the associate degree between the plant biomass and the animal biomass after calculating the corresponding index weights. And then, we painted the pictures on the MATLAB to obtain the Transform relationship between the plant growth and the rodent growth.

For the second problem,we were required to study on the stability of the rodent community,and to reva l the effects of interference in rodent community.First of all,in the case of community’s stbility,for the food network of the community and “enemy follow” phenomenon,when the number of species increses,the num ber of individuals have a corresponding increase.But the trend is grey,and can not expose competely.By the theory of grey system,the rule contained by original data can be revealed by accumulated generating.So we choose the grey evaluation model and the grey clustering method to evaluate the stability of community.Th e factor which can reflect the stability of community would be token into the model,which makes the evalu ation of stability of the community more scientific and practical.Secondly ,solving data and the data for ima ge summarized the degree of influence to the rodent community with the different inference.For this proble m ,we use the Structural equation model(SEM) to handle numbers of dependent variables simultaneously. We use SEM to establish conceptual model,using the regressuion analysis,factor analysis,Model of test for t and variance analysis to estimate the fitting degree of the model.Finally,using the model test method to che ck the diffenent inference on pathway of rodent species,which cn obtain the different effect on the impact m echanism between rodent.

Meanwhile, this article carried out the advantages and disadvantages of all the models and the analysis of the broad application prospects in other areas for some models while each model is improved and promoted in the final evaluation.