单种群生态模型的解析解.docVIP

PAGE 24 PAGE 5 分类号 O15 陕西师范大学学士学位论文 作 者 单 位 数学与信息科学学院 指 导 老 师 唐 三 一 作 者 姓 名 孙 小 丹 专 业、班 级 数学与应用数学专业05级3班 提 交 时 间 2009年5月  单种群生态模型的解析解 孙小丹 (数学与信息科学学院2005级3班) 指导教师 唐三一教授 摘要：单种群模型的解析求解在生物数学中具有非常重要的作用。本文总结了各类可求解的自治和非自治连续、离散单种群模型， 分别给出了自治单种群模型与非自治单种群模型的解析解和稳定性分析． 自治单种群模型中具有解析解的连续模型有：Logistic模型，Gompertz模型、Rosenzweig模型、反向Rosenzweig模型、Von Bertalanffy增长方程和具有Allee效应的Logistic模型；离散模型有Bverton-Holt模型．同时将以上模型的求解过程详细给出．针对以上自治模型分别给出了其对应的非自治模型，以及其解析解的求解过程，最后运用定理分析证明了各个模型周期解的存在性与稳定性． 关键词：单种群模型；解析解；自治模型；非自治模型；周期解 The analytical solution of single species models SUN xiao-dan (Class 3, Grade 2005, College of Advisor: Professor TANG san-yi 重写下面的话 Abstract: In this paper,we give the definition and classification of the single species models at first.Based on whether the parameter is the function of the time ,the models can be classified as autonomous single species model and nonautonomous single species model;based on whether the different generations can be exist at the same time and the number of the population,the models can be classified as continuous model and difference model.Then the analytical solutions of the autonomous single species models and nonautonomous single species modes are discussed respectively.The autonomous continuous single species models which have analytical solutions including:Logistic model,Gompertz model,Rosenzweig model,Rosenzweig model on contrary,Von Bertalanffy’s growth equation and Logistic model with Allee effect.The autonomous different single species models including ver ton-Holt model.also,how the analytical solutions of the models relationed above are obtained are given in detail.Corresponding to the autonomous models the nonautonomous models are also given respectively,at the same time how the analytical solutions of the models are obtained are given in detail,too.At last,the existence and stability of each model’s periodic solution is proved and analyzed use the th