Related papers: Duality, Ancestral and Diffusion Processes in Mode…
Mathematical models of genetic evolution often come in pairs, connected by a so-called duality relation. The most seminal example are the Wright-Fisher diffusion and the Kingman coalescent, where the former describes the stochastic…
Consider the diffusion process defined by the forward equation $u_t(t, x) = \tfrac{1}{2}\{x u(t, x)\}_{xx} - \alpha \{x u(t, x)\}_{x}$ for $t, x \ge 0$ and $-\infty < \alpha < \infty$, with an initial condition $u(0, x) = \delta(x - x_0)$.…
The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [25]. The first step, which is easily accomplished using results of…
In population genetics, diffusions on the unit interval are often used to model the frequency path of an allele. In this setting we derive approximations for fixation probabilities, expected hitting times and the expected…
Consider a genetic locus carrying a strongly beneficial allele which has recently fixed in a large population. As strongly beneficial alleles fix quickly, sequence diversity at partially linked neutral loci is reduced. This phenomenon is…
In this article, we focus on Bienaym\'e-Galton-Watson processes with linear-fractional offspring distributions. At a fixed generation, we consider a sample of the individuals alive, drawn in two different ways: either through Bernoulli…
The Ancestral Selection Graph (ASG) is an important genealogical process which extends the well-known Kingman coalescent to incorporate natural selection. We show that the number of lineages of the ASG with and without mutation is…
We consider the mutation--selection differential equation with pairwise interaction (or, equivalently, the diploid mutation--selection equation) and establish the corresponding ancestral process, which is a random tree and a variant of the…
We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. http://www.ams.org/mathscinet-getitem?mr=MR2123250) and for the self-duality of Feller's branching…
A proper understanding of the links between varying gene expression levels and complex trait adaptation is still lacking, despite recent advances in sequencing techniques leading to new insights on their importance in some evolutionary…
We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation and drift. The number of beneficial alleles in a multi-locus system can be considered a trait under exponential selection.…
Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed…
We introduce a general diploid population model with self-fertilization and possible overlapping generations, and study the genealogy of a sample of $n$ genes as the population size $N$ tends to infinity. Unlike traditional approach in…
Coalescent processes, including mutation, are derived from Moran type population models admitting large offspring numbers. Including mutation in the coalescent process allows for quantifying the turnover of alleles by computing the…
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…
The sample frequency spectrum of a segregating site is the probability distribution of a sample of alleles from a genetic locus, conditional on observing the sample to have more than one clearly different phenotypes. We present a model for…
$\Lambda$-Wright--Fisher processes provide a robust framework to describe the type-frequency evolution of an infinite neutral population. We add a polynomial drift to the corresponding stochastic differential equation to incorporate…
We consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves. For one particular model, that we shall call the exponential model, the…
Solving the recombination equation has been a long-standing challenge of \emph{deterministic} population genetics. We review recent progress obtained by introducing ancestral processes, as traditionally used in the context of…
Compared to a neutral model, purifying selection distorts the structure of genealogies and hence alters the patterns of sampled genetic variation. Although these distortions may be common in nature, our understanding of how we expect…