Related papers: Graphon-valued stochastic processes from populatio…
We study a general setting of neutral evolution in which the population is of finite, constant size and can have spatial structure. Mutation leads to different genetic types ("traits"), which can be discrete or continuous. Under minimal…
A general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the…
Background: The accumulation of deleterious mutations of a population directly contributes to the fate as to how long the population would exist. Muller's ratchet provides a quantitative framework to study the effect of accumulation.…
We investigate local-density dependent Markov processes on a class of large graphs sampled from a graphon, where the transition rates of the vertices are influenced by the states of their neighbors. We show that as the average degree…
We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and…
$\Lambda$-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…
The dynamics of adaptation is difficult to predict because it is highly stochastic even in large populations. The uncertainty emerges from number fluctuations, called genetic drift, arising in the small number of particularly fit…
We consider a particle system in continuous time, discrete population, with spatial motion and nonlocal branching. The offspring's weights and their number may depend on the mother's weight. Our setting captures, for instance, the processes…
We present stochastic, finite-population formulations of the Crow-Kimura and Eigen models of quasispecies theory, for fitness functions that depend in an arbitrary way on the number of mutations from the wild type. We include back mutations…
We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size $N\to \infty$. The model is a voter model perturbation. For typical populations, we require perturbation strengths…
We develop a global and hierarchical scheme for the forward Kolmogorov (Fokker-Planck) equation of the diffusion approximation of the Wright-Fisher model of population genetics. That model describes the random genetic drift of several…
This chapter focuses on the derivation of a doubly nonlocal Fisher-KPP model, which is a macroscopic nonlocal evolution equation describing population dynamics in the large population limit. The derivation starts from a microscopic…
We consider a general interacting particle system with interactions on a random graph, and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the…
Consider a two-type Moran population of size $N$ with selection and mutation, where the selective advantage of the fit individuals is amplified at extreme environmental conditions. Assume selection and mutation are weak with respect to $N$,…
We study voter models defined on large sets. Through a perspective emphasizing the martingale property of voter density processes, we prove that in general, their convergence to the Wright-Fisher diffusion only involves certain averages of…
We study the evolution of gene frequencies in a population living in $\mathbb{R}^d$, modelled by the spatial Lambda Fleming-Viot process with natural selection (Barton, Etheridge and Veber, 2010 and Etheridge, Veber and Yu, 2014). We…
In this note, we investigate fundamental relations between exploration processes in random graphs, and branching processes. We formulate a class of models that we call {\em rank-$k$ random graphs}, and that are special in that their…
We consider time-continuous Markovian discrete-state dynamics on random networks of interacting agents and study the large population limit. The dynamics are projected onto low-dimensional collective variables given by the shares of each…
To learn about the past from a sample of genomic sequences, one needs to understand how evolutionary processes shape genetic diversity. Most population genetic inference is based on frameworks assuming adaptive evolution is rare. But if…
Our principal aim is to observe the Markov discrete-time process of population growth with long-living trajectory. First we study asymptotical decay of generating function of Galton-Watson process for all cases as the Basic Lemma.…