两物种分支过程中的粒子分布

发布时间：2018-07-04 09:35

本文选题：分支过程 + 分支树　；参考：《南京大学》2017年硕士论文

【摘要】：本文对分支过程的基本理论进行了介绍,主要介绍分支过程的定义、概率母函数,生成后代数量的均值或均值矩阵与灭绝概率之间的关系,判断分支过程是否可约的方法等。在两物种分支过程中,我们详细介绍了分支树的构造,分支树与随机游走间的联系,以及与两物种分支树相对应的由变异节点组成的分支树,并说明了两者之间的关联。本文探究在最终必然灭绝的临界或是下临界分支过程中,各种类的粒子总数的分布。在两物种情形下,对可约的分支过程,各种类的粒子总数的概率分布已有相应的研究成果。结合分支树的相关理论性质,本文给出在不可约条件下各种类的粒子总数的概率分布。然后类比单物种分支过程中树叶的分布,本文给出在两物种过程中类似的分布。最后,同样类比单物种分支过程中后代数不多于k个的粒子个数的分布性质,本文给出在两物种过程中后代数不多于(k1,k2)的粒子个数的分布性质。
[Abstract]:In this paper, the basic theory of branching process is introduced, including the definition of branching process, the probability generating function, the relationship between the mean value or mean matrix of generating the number of offspring and the extinction probability, and the method of judging whether the branching process is reducible or not. In the process of two species branching, we introduce in detail the structure of branch tree, the relation between branch tree and random walk, and the branch tree which is composed of mutation nodes corresponding to branch tree of two species, and explain the relation between them. This paper explores the distribution of the total number of particles in the critical or lower critical branching processes of eventual extinction. In the case of two species, the probability distribution of the total number of particles in the reducible branching process has been studied. In this paper, the probability distribution of the total number of particles of various classes under irreducible conditions is given in combination with the related theoretical properties of branching trees. Then the distribution of leaves in the process of single species branching is compared, and the similar distribution in the process of two species is given in this paper. Finally, similar to the distribution properties of the number of particles with no more than k offspring in the process of single species branching, this paper gives the distribution properties of the number of particles not more than (k1k2) in the process of two species.
【学位授予单位】：南京大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.65

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