Moran model with simultaneous strong and weak selections: convergence towards a $Λ$-Wright-Fisher SDE

具有随机重采样速率的 Moran 模型

The Moran model with random resampling rates

本文研究个体重采样速率随机且独立同分布的两类型 Moran 模型，证明在可数支撑、尾部与矩条件下，类型1速率经验分布在 Meyer–Zheng 拓扑中收敛为经 Fisher–Wright 扩散缩放的分布。

$Λ$-不对称Moran模型的祖先选择图

The ancestral selection graph for a $Λ$-asymmetric Moran model

本文构建了$Λ$-不对称Moran模型的祖先选择图，通过耦合两种繁殖测度实现前向与后向过程的路径对偶，并推导了频率过程的尺度极限及优势较小类型的固定概率公式。

A Mean-Reverting SDE on Correlation matrices

We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright-Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a useful connection with Wishart processes that makes understand how we get the full SDE. Then, we focus on the simulation of this diffusion and present discretization schemes that achieve a second-order weak convergence. Last, we explain how these correlation processes could be used …

带突变的字素值Wright-Fisher扩散

The grapheme-valued Wright-Fisher diffusion with mutation

本文研究群体遗传学中的字素值Wright-Fisher扩散，构造带重采样和突变的有限图Moran型马尔可夫链，编码为字素后证明其顶点数趋于无穷时收敛于与Poisson-Dirichlet分布相关的马尔可夫扩散，为相关一般理论提供示例。

关于具有突变和选择的种群模型中区块计数过程平稳分布的研究

On the stationary distribution of the block counting process for population models with mutation and selection

本文研究Moran与$\Lambda$-Wright-Fisher模型中区块计数过程的平稳分布，刻画几何平稳分布条件，并通过线性方程组及积分微分方程求解典型模型。

Structural properties of the seed bank and the two-island diffusion

We investigate various aspects of the (biallelic) Wright-Fisher diffusion with seed bank in conjunction with and contrast to the two-island model analysed e.g. in Kermany, Zhou and Hickey, 2008, and Nath and Griffiths, 1993, including moments, stationary distribution and reversibility, for which our main tool is duality. Further, we show that the Wright-Fisher diffusion with seed bank can be reformulated as a one-dimensional stochastic delay differential equation, providing an elegant interpretation of the age structure in the seed bank also forward in time in the spirit of Kaj, Krone and Lasc…

Critical population and error threshold on the sharp peak landscape for the Wright-Fisher model

We pursue the task of developing a finite population counterpart to Eigen's model. We consider the classical Wright-Fisher model describing the evolution of a population of size $m$ of chromosomes of length $\ell$ over an alphabet of cardinality $κ$. The mutation probability per locus is $q$. The replication rate is $σ>1$ for the master sequence and $1$ for the other sequences. We study the equilibrium distribution of the process in the regime where $\ell\to+\infty$, $m\to+\infty$, $q\to0$, $\ell q\to a\in\,]0,+\infty[$, $\frac{m}{\ell}\toα\in [0,+\infty]$. We obtain an equation $αψ(a)=\lnκ$ i…

Limit Theorems for Competitive Density Dependent Population Processes

Near the beginning of the century, Wright and Fisher devised an elegant, mathematically tractable model of gene reproduction and replacement that laid the foundation for contemporary population genetics. The Wright-Fisher model and its extensions have given biologists powerful tools of statistical inference that enabled the quantification of genetic drift and selection. Given the utility of these tools, we often forget that their model - for mathematical, and not biological reasons - makes assumptions that are violated in most real-world populations. In this paper, I consider an alternative fr…

一个双尺寸Wright--Fisher模型：通过均匀更新理论的渐近分析

A two-size Wright--Fisher model: asymptotic analysis via uniform renewal theory

该研究提出双尺寸Wright-Fisher模型，利用均匀更新理论证明在进化时间尺度上频率过程收敛于Wright-Fisher型SDE，并揭示资源消费策略对漂移和扩散项的影响。

Multi-type logistic branching processes with selection: frequency process and genealogy for large carrying capacities

We present a model for growth in a multi-species population. We consider two types evolving as a logistic branching process with mutation, where one of the types has a selective advantage, and are interested in the regime in which the carrying capacity of the system goes to $\infty$. We first study the genealogy of the population up until it almost reaches carrying capacity through a coupling with an independent branching process. We then focus on the phase in which the population has reached carrying capacity. After recovering a Gillespie--Wright--Fisher SDE in the infinite carrying capacity …

一般选择模型：Bernstein对偶与最小祖先结构

General selection models: Bernstein duality and minimal ancestral structures

本文在$\Lambda$-Wright–Fisher过程中引入频率依赖选择，借助Moran模型逼近构造推广的祖先选择图，提出Bernstein对偶，并刻画最小祖先结构。

Finite populations with frequency-dependent selection: a genealogical approach

Evolutionary models for populations of constant size are frequently studied using the Moran model, the Wright-Fisher model, or their diffusion limits. When evolution is neutral, a random genealogy given through Kingman's coalescent is used in order to understand basic properties of such models. Here, we address the use of a genealogical perspective for models with weak frequency-dependent selection, i.e. N s =: α is small, and s is the fitness advantage of a fit individual and N is the population size. When computing fixation probabilities, this leads either to the approach proposed by Rousset…