【作者】 张贝贝；

【导师】 吕广迎；

【作者基本信息】 河南大学 ， 应用数学， 2019， 硕士

【摘要】 本文分别讨论了共栖带切换模型与谣言传播模型的随机动力学行为.首先,讨论了随机切换共栖模型中物种的生存与灭绝问题;其次,研究了随机谣言传播模型解渐近稳定性.具体分为以下两个部分:一、随机切换共栖模型.通过考虑环境切换与噪声干扰等因素,在共栖模型的基础上建立了与生态环境更为相符的随机切换共栖模型.首先,利用马尔可夫链与遍历性等基本知识,得到了随机切换过程存在唯一平稳分布.其次,通过构造适当的辅助过程,并根据其常返性与暂留性来判断随机切换过程的常返性与暂留性.最后,利用It?o’s公式,给出了物种生存与灭绝的充分条件.值得一提的是,本文比较了随机切换共栖模型与随机共栖模型的区别,并给出了随机切换共栖模型的优势.二、随机谣言传播模型.首先,运用可积的马尔可夫半群的Foguel二择一性质,并通过构造Khasminskiˇ?函数,讨论了随机谣言传播模型解的渐近稳定性.其次,通过构造一个合适的辅助随机微分方程并运用遍历性等知识,得到了解的分布收敛到一个测度.更多还原

【Abstract】 In this thesis,we discuss the stochastic dynamical behavior of the mutualism model and the rumor propagation model,respectively.Firstly,we discuss the permanence and extinction of species to the stochastic regime-switching mutualism model.Secondly,we study the asymptotic stability of the solution to the stochastic rumor propagation model.The main content of the thesis is divided into the following two aspects:The first part is the stochastic regime-switching mutualism model.According to considering the factors of the environment switching and the noise perturbation,we establish a stochastic regime-switching mutualism model which is more consistent with the ecological environment than the one without regime-switching.Firstly,utilizing the basic knowledge of Markov chain and ergodicity,the unique stationary distribution of stochastic switching process is obtained.Secondly,we construct two appropriate auxiliary processes and we can judge their recurrence and persistence.Therefore,the recurrence and persistence of stochastic switching process are decided.Finally,the sufficient conditions of the permanence and extinction of species are derived by using It?o’s formula.Moreover,compared with the system without regime-switching,the advantages of the stochastic regime-switching mutualism model are given.The second part is the stochastic rumor propagation model.Firstly,the asymptotic stability of the solution to the stochastic rumor propagation model is discussed by using the integrable Markov semigroup and by constructing a Khasminskiˇ? function.Then,constructing an appropriate auxiliary stochastic differential equation and applying the ergodicity,we obtain that the distribution of the solution converges to a measure.更多还原