arXiv math/05035122005-03-24math.PR

The stepping stone model. II: Genealogies and the infinite sites model

Iljana Zahle, J. Theodore Cox, Richard Durrett

备注：: Published at http://dx.doi.org/10.1214/105051604000000701 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Applied Probability 2005, Vol. 15, No. 1B, 671-699
DOI：: 10.1214/105051604000000701

英文摘要：

This paper extends earlier work by Cox and Durrett, who studied the coalescence times for two lineages in the stepping stone model on the two-dimensional torus. We show that the genealogy of a sample of size n is given by a time change of Kingman's coalescent. With DNA sequence data in mind, we investigate mutation patterns under the infinite sites model, which assumes that each mutation occurs at a new site. Our results suggest that the spatial structure of the human population contributes to the haplotype structure and a slower than expected decay of genetic correlation with distance revealed by recent studies of the human genome.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Continuum Space Limit of the Genealogies of Interacting Fleming-Viot Processes on $\Z$

We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on $\Z$. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on $\R$, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as…

2017-01-06math.PR

Generalized stepping stone model with $Ξ$-resampling mechanism

A generalized stepping stone model with $Ξ$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $Ξ$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $Ξ$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that i…

2021-02-03math.PR

Stepping-stone model with circular Brownian migration

In this paper we consider a stepping-stone model on a circle with circular Brownian migration. We first point out a connection between Arratia flow and the marginal distribution of this model. We then give a new representation for the stepping-stone model using Arratia flow and circular coalescing Brownian motion. Such a representation enables us to carry out some explicit computation. In particular, we find the Laplace transform for the time when there is only a single type left across the circle.

2005-09-16math.PR

Continuum-sites stepping-stone models, coalescing exhcangeable partitions, and random trees

Analogues of stepping--stone models are considered where the site--space is continuous, the migration process is a general Markov process, and the type--space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum--sites stepping--stone process has continuou…

1998-11-10math.PR

One-dimensional stepping stone models, sardine genetics and Brownian local time

Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $ν$, or a voter model on $\mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $Mν/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a o…

2008-01-22math.PR

Intermediate range migration in the two-dimensional stepping stone model

We consider the stepping stone model on the torus of side $L$ in $\mathbb{Z}^2$ in the limit $L\to\infty$, and study the time it takes two lineages tracing backward in time to coalesce. Our work fills a gap between the finite range migration case of [Ann. Appl. Probab. 15 (2005) 671--699] and the long range case of [Genetics 172 (2006) 701--708], where the migration range is a positive fraction of $L$. We obtain limit theorems for the intermediate case, and verify a conjecture in [Probability Models for DNA Sequence Evolution (2008) Springer] that the model is homogeneously mixing if and only …

2010-10-12math.PR

更多相关论文收起更多论文

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

We study a one-dimensional spatial population model where the population sizes at each site are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the eff…

2019-06-13math.PR

对“随机环境中的踏脚石模型”的修正——随机环境中可逆随机游走的占据时间和相交时间的极限定理

Correction to "The stepping stone model in a random environment" -- limit theorems for the occupation time and intersection time of reversible random walks in random environments

INRAE（法国国家农业、食品与环境研究院）

修正了原工作中关于两个独立随机游走在随机环境中相交局部时间的常数错误，利用可逆随机游走占据时间的新标度极限更新了常数γ。

2026-06-23math.PR

二维逻辑分支随机游走祖先谱系的配对合并时间

Pair coalescence times of ancestral lineages of two-dimensional logistic branching random walks

本文研究二维逻辑分支随机游走平稳状态下两条祖先谱系在大初始距离下的合并时间渐近性，证明其与踏脚石模型及Malécot连续空间近似一致，从而为用固定有效种群规模替代局部波动规模提供理论依据。

2024-05-03math.PR