具有中性Poissonian突变的分裂树I：小家族
Splitting trees with neutral Poissonian mutations I: Small families
摘要：
我们考虑一个生物多样性〈动〉性<动>，其中个体独立生活和繁殖。他们有身份证。寿命持续时间（不一定是指数分布的），并以恒定的速率分娩（单个）。这样的系谱树通常被称为分裂树，人口计数过程（N_t;t \ge 0）是一个齐次的二元Crump-Mode-Jagers过程。我们假设个体在其一生中以恒定的速率θ独立经历突变，在无限等位基因假设下：每个突变瞬间赋予其携带者一个全新的类型，称为等位基因。我们感兴趣的是时间t时的等位基因频率谱，即，在时间t时群体中代表的不同等位基因的数量A（t），更具体地，在时间t时k个个体代表的等位基因的数量A（k，t），k= 1，2，.，N_t。我们主要使用两类工具：合并点过程和随机特征计数的分支过程。本文给出了一个以种群大小为条件的合并点过程中A（k，t）的期望的显式表达式，它适用于分裂树的特殊情况。我们分别推导出a. s。由于随机特性，A（k，t）/N_t和A（t）/N_t的极限。最后，我们分别计算预期的纯合性，通过应用一种方法表征树分布的动态，树的起源时间在时间上向后移动，在向后Kolmogorov方程的精神。
英文摘要：
We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\ge 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate θduring their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.
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