【作者】 张勇；

【导师】 田宝单；

【作者基本信息】 西南科技大学 ， 控制科学与工程， 2021， 硕士

【摘要】 近年来,在生物数学研究领域,种群生态数学模型的研究一直是热门。随着世界经济持续发展,人类对自然环境的破坏日益加剧,环境灾害变得越来越频繁,自然生态也随之发生变化。因此,在生物种群模型中考虑环境噪声干扰更具现实意义。本文正是基于上述背景,建立几类具有随机扰动的生态传染病模型,并讨论相应模型的动力学行为。论文主要内容安排如下:第一章介绍研究背景及意义,以及将会使用到的基本概念及定理等。第二章建立一类带有改进的Leslie-Gower和Holling Ⅱ功能反应的随机传染病捕食系统,利用伊藤公式、随机微分比较定理、微分不等式等技巧得到系统正解的存在唯一性以及种群在时间均值意义下持续生存及灭绝的条件,最后进行数值模拟来验证所得理论结果。第三章考虑季节变化、昼夜交替以及环境噪声的干扰等因素,建立一类带有改进的LG-Holling Ⅱ功能反应的非自治型随机传染病捕食系统,利用伊藤公式、微分不等式等分析技巧证明了该系统存在唯一正解。此外,通过比较定理、强大数定律、切比雪夫不等式等方法得到了种群灭绝、均值弱持久生存、均值强持久生存以及系统随机持久生存的充分条件,最后利用数值模拟仿真进一步验证了理论结果的正确性。第四章建立一类带有改进的LG-Holling Ⅱ功能反应和Ornstein–Uhlenbeck过程的随机传染病捕食系统,通过伊藤积分公式、随机微分比较定理等方法得到系统全局正解的存在唯一性以及种群均值持续生存及灭绝的条件,并利用数值模拟实验进行理论成果验证,讨论不同回归速率和不同波动强度对系统动力学行为的影响。第五章对论文进行总结,并对今后工作提出展望。更多还原

【Abstract】 In recent years,research on population ecological mathematical models has become a hot topic in the field of biomathematics research.With the continuous development of the world economy,the destruction of the natural environment by human beings is increasing,environmental disasters are becoming more and more frequent and the natural ecology also changes.Therefore,considering the perturbation of the environment noise in the natural ecological model will be more practical.Based on the above background,this dissertation establishes several kinds of ecological epidemic models with stochastic perturbation,and discusses the dynamic behavior of the corresponding models.The main contents of this dissertation are arranged as follows:In chapter 1,we introduce the research background and significance,as well as the basic concepts and theorems that will be used.In chapter 2,we establish a stochastic diseased predator system with modified LG-Holling type Ⅱ functional response.By using It?’s formula,stochastic differential comparison theorem,differential inequality and other analytical techniques,we obtain the existence and uniqueness of positive solution of the system and the conditions for being stable in time average and extinction of the population.Finally,numerical simulation experiments are carried out to verify the theoretical results.In chapter 3,considering the seasonal variation,the alternation of day and night,and the interference of environmental noise,we establish a non-autonomous stochastic diseased predator system with modified LG-Holling type Ⅱ functional response,By using It?’s formula,differential inequality and other analytical techniques,it is proved that the system has a unique positive solution.In addition,the sufficient conditions for extinction,weak and strong persistence in the mean,as well as stochastic permanence of the system are deduced by using stochastic differential comparison theorem,strong law of large numbers,Chebyshev inequality and other mathematical skills.In chapter 4,we establish a stochastic diseased predator system with modified LG-Holling Ⅱ functional response and Ornstein–Uhlenbeck process.By using mathematical methods such as It?’s formula,stochastic differential comparison theorem,we obtain the existence and uniqueness of the global positive solution of the system and the conditions for being stable in time average and extinction of the population.The theoretical results are verified by numerical simulation experiments,and the effects of different regression rates and fluctuation intensities on the dynamic behavior of the system are discussed.In chapter 5,we summarize the full dissertation and put forward the outlook for the future work.更多还原