arXiv subjects

Alison Etheridge

Publications and source records attributed to Alison Etheridge.

17 recordsLinked to original sources

arXiv2026

Fluctuations for fully pushed stochastic fronts

We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.

math.PR↗

arXiv2026

How seed banks evolve in plants: a stochastic dynamical system subject to a strong drift

We study how changes in population size and fluctuating environmental conditions influence the establishment of seed banks in plants. Our model is a modification of the Wright-Fisher model with seed bank, introduced by Kaj, Krone and Lascoux. We distinguish between wild type individuals, producing only nondormant seeds, and mutants, producing seeds with dormancy. To understand how changing population size shapes the establishment of seed banks, we analyse the process under a diffusive scaling. The results support the biological insight that seed banks are favoured in a declining population, and disfavoured if population size is constant or increasing. The surprise is that this is true even when population sizes are changing very slowly -- over evolutionary timescales. We also investigate the influence of short-term fluctuations, such as annual variations in rainfall or temperature. Mathematically, our analysis reduces to a stochastic dynamical system forced onto a manifold by a large drift, which converges under scaling to a diffusion on the manifold. Inspired by the Lyapunov--Schmidt reduction, we derive an explicit formula for the limiting diffusion coefficients by projecting the system onto its linear counterpart. This provides a general framework for deriving diffusion approximations in models with strong drift and nonlinear constraints.

math.PR↗

arXiv2025

Multi-type $\Xi$-coalescents from structured population models with bottlenecks

We introduce an individual-based model for structured populations undergoing demographic bottlenecks, i.e. drastic reductions in population size that last many generations and can have arbitrary shapes. We first show that the (non-Markovian) allele-frequency process converges to a Markovian diffusion process with jumps in a suitable relaxation of the Skorokhod J1 topology. Backward in time we find that genealogies of samples of individuals are described by multi-type $\Xi$-coalescents presenting multiple simultaneous mergers with simultaneous migrations. These coalescents are also moment-duals of the limiting jump diffusions. We then show through a numerical study that our model is flexible and can predict various shapes for the site frequency spectrum, consistent with real data, using a small number of interpretable parameters.

math.PR↗

arXiv2023

Branching stable processes and motion by mean curvature flow

We prove a new result relating solutions of the scaled fractional Allen--Cahn equation to motion by mean curvature flow, motivated by the motion of hybrid zones in populations that exhibit long range dispersal. Our proof is purely probabilistic and takes inspiration from Etheridge et al. to describe solutions of the fractional Allen--Cahn equation in terms of ternary branching $α$-stable motions. To overcome technical difficulties arising from the heavy-tailed nature of the stable distribution, we couple ternary branching stable motions to ternary branching Brownian motions subordinated by truncated stable subordinators.

math.PR↗

arXiv2020

Genealogies in bistable waves

We study a model of selection acting on a diploid population (one in which each individual carries two copies of each gene) living in one spatial dimension. We suppose a particular gene appears in two forms (alleles) $A$ and $a$, and that individuals carrying $AA$ have a higher fitness than $aa$ individuals, while $Aa$ individuals have a lower fitness than both $AA$ and $aa$ individuals. The proportion of advantageous $A$ alleles expands through the population approximately according to a travelling wave. We prove that on a suitable timescale, the genealogy of a sample of $A$ alleles taken from near the wavefront converges to a Kingman coalescent as the population density goes to infinity. This contrasts with the case of directional selection in which the corresponding limit is thought to be the Bolthausen-Sznitman coalescent. The proof uses 'tracer dynamics'.

math.PR↗

arXiv2020

Rescaling limits of the spatial Lambda-Fleming-Viot process with selection

We consider the spatial Lambda-Fleming-Viot process model for frequencies of genetic types in a population living in R^d, with two types of individuals (0 and 1) and natural selection favouring individuals of type 1. We first prove that the model is well-defined and provide a measure-valued dual process encoding the locations of the `potential ancestors' of a sample taken from such a population. We then consider two cases, one in which the dynamics of the process are driven by events of bounded radii and one incorporating large-scale events whose radii have a polynomial tail distribution. In both cases, we consider a sequence of spatial Lambda-Fleming-Viot processes indexed by n, and we assume that the fraction of individuals replaced during a reproduction event and the relative frequency of events during which natural selection acts tend to 0 as n tends to infinity. We choose the decay of these parameters in such a way that when reproduction is only local, the measure-valued process describing the local frequencies of the less favoured type converges in distribution to a (measure-valued) solution to the stochastic Fisher-KPP equation in one dimension, and to a (measure-valued) solution to the deterministic Fisher-KPP equation in more than one dimension. When large-scale extinction-recolonisation events occur, the sequence of processes converges instead to the solution to the analogous equation in which the Laplacian is replaced by a fractional Laplacian. We also consider the process of `potential ancestors' of a sample of individuals taken from these populations, which we see as a system of branching and coalescing symmetric jump processes. We show their convergence in distribution towards a system of Brownian or stable motions which branch at some finite rate. In one dimension, in the limit, pairs of particles also coalesce at a rate proportional to their collision local time.

math.PR↗

arXiv2018

The spatial Lambda-Fleming-Viot process with fluctuating selection

We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population. We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space. We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from Hallatschek and Nelson (2008) and Durrett and Fan (2016), in numerical experiments with a closely related model based on the classical Moran model.

math.PR↗

arXiv2016

The Brownian Net and Selection in the Spatial Lambda-Fleming-Viot Process

We obtain the Brownian net of Sun and Swart (2008) as the scaling limit of the paths traced out by a system of continuous (one-dimensional) space and time branching and coalescing random walks. This demonstrates a certain universality of the net, which we have not seen explored elsewhere. The walks themselves arise in a natural way as the ancestral lineages relating individuals in a sample from a biological population evolving according to the spatial Lambda-Fleming-Viot process. Our scaling reveals the effect, in dimension one, of spatial structure on the spread of a selectively advantageous gene through such a population.

math.PR↗

arXiv2016

Branching Brownian motion and Selection in the Spatial Lambda-Fleming-Viot Process

We ask the question "when will natural selection on a gene in a spatially structured population cause a detectable trace in the patterns of genetic variation observed in the contemporary population?". We focus on the situation in which 'neighbourhood size', that is the effective local population density, is small. The genealogy relating individuals in a sample from the population is embedded in a spatial version of the ancestral selection graph and through applying a diffusive scaling to this object we show that whereas in dimensions at least three, selection is barely impeded by the spatial structure, in the most relevant dimension, $d=2$, selection must be stronger (by a factor of $\log(1/μ)$ where $μ$ is the neutral mutation rate) if we are to have a chance of detecting it. The case $d=1$ was handled in Etheridge et al. (2015). The mathematical interest is that although the system of branching and coalescing lineages that forms the ancestral selection graph converges to a branching Brownian motion, this reflects a delicate balance of a branching rate that grows to infinity and the instant annullation of almost all branches through coalescence caused by the strong local competition in the population.

math.PR↗

arXiv2016

Branching Brownian Motion, mean curvature flow and the motion of hybrid zones

We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $Λ$-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as hybrid zones. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.

math.PR↗

arXiv2014

Scaling limits of weakly asymmetric interfaces

We consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with $N$ particles on a finite lattice of $2N$ sites. Our Model 1 defines an evolving bridge on $[0,1]$, our Model 1-w an evolving excursion on $[0,1]$ while our Model 2 consists of an evolving pair of non-crossing bridges on $[0,1]$. Based on the observation that the invariant measures of the dynamics depend on the area under (or between) the interface(s), we characterise the scaling limits of the invariant measures when the asymmetry of the exclusion process scales like $N^{-\frac{3}{2}}$. Then, we show that the scaling limits of the dynamics themselves are expressed in terms of variants of the stochastic heat equation. In particular, in Model 1-w we obtain the well-studied reflected stochastic heat equation introduced by Nualart and Pardoux.

math.PR↗

arXiv2013

Conditioning the logistic branching process on non-extinction

We consider a birth and death process in which death is due to both `natural death' and to competition between individuals, modelled as a quadratic function of population size. The resulting `logistic branching process' has been proposed as a model for numbers of individuals in populations competing for some resource, or for numbers of species. However, because of the quadratic death rate, even if the intrinsic growth rate is positive, the population will, with probability one, die out in finite time. There is considerable interest in understanding the process conditioned on non-extinction. In this paper, we exploit a connection with the ancestral selection graph of population genetics to find expressions for the transition rates in the logistic branching process conditioned on survival until some fixed time $T$, in terms of the distribution of a certain one-dimensional diffusion process at time $T$. We also find the probability generating function of the Yaglom distribution of the process and rather explicit expressions for the transition rates for the so-called Q-process, that is the logistic branching process conditioned to stay alive into the indefinite future. For this process, one can write down the joint generator of the (time-reversed) total population size and what in population genetics would be called the `genealogy' and in phylogenetics would be called the `reconstructed tree' of a sample from the population. We explore some ramifications of these calculations numerically.

math.PR↗

arXiv2010

On the moments and the interface of the symbiotic branching model

In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127--160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Amer. Math. Soc. 87 (1958) 187--197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [Ann. Probab. 26 (1998) 1088--1138] for the mutually catalytic branching model. As an application, we show how a bound on the 35th moment improves the result of Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127--160] on the speed of the propagation of the interface of the symbiotic branching model.

math.PR↗

arXiv2010

Asymptotic behavior of the rate of adaptation

We consider the accumulation of beneficial and deleterious mutations in large asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations and population size $N$. We show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least $\mathcal{O}(\log^{1-δ}N)$ where $δ$ can be any small positive number, if the population size is sufficiently large. This shows that if the genome is modeled as continuous, there is no limit to natural selection, that is, the rate of adaptation grows in $N$ without bound.

math.PR↗

arXiv2008

Fixation Probability for Competing Selective Sweeps

We consider a biological population in which a beneficial mutation is undergoing a selective sweep when a second beneficial mutation arises at a linked locus and we investigate the probability that both mutations will eventually fix in the population. Previous work has dealt with the case where the second mutation to arise confers a smaller benefit than the first. In that case population size plays almost no role. Here we consider the opposite case and observe that, by contrast, the probability of both mutations fixing can be heavily dependent on population size. Indeed the key parameter is $ρN$, the product of the population size and the recombination rate between the two selected loci. If $ρN$ is small, the probability that both mutations fix can be reduced through interference to almost zero while for large $ρN$ the mutations barely influence one another. The main rigorous result is a method for calculating the fixation probability of a double mutant in the large population limit.

math.PR↗

arXiv2007

Coexistence in locally regulated competing populations and survival of branching annihilating random walk

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which do not, at present, incorporate all of the competitive strategies that a population might adopt. The second is a simplification of the first, in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching annihilating random walk. For each model, using a comparison with oriented percolation, we show that for certain parameter values, both populations will coexist for all time with positive probability. As a corollary, we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present a number of conjectures relating to the rôle of space in the survival probabilities for the two populations.

math.PR↗

arXiv2006

An approximate sampling formula under genetic hitchhiking

For a genetic locus carrying a strongly beneficial allele which has just fixed in a large population, we study the ancestry at a linked neutral locus. During this ``selective sweep'' the linkage between the two loci is broken up by recombination and the ancestry at the neutral locus is modeled by a structured coalescent in a random background. For large selection coefficients $α$ and under an appropriate scaling of the recombination rate, we derive a sampling formula with an order of accuracy of $\mathcal{O}((\log α)^{-2})$ in probability. In particular we see that, with this order of accuracy, in a sample of fixed size there are at most two nonsingleton families of individuals which are identical by descent at the neutral locus from the beginning of the sweep. This refines a formula going back to the work of Maynard Smith and Haigh, and complements recent work of Schweinsberg and Durrett on selective sweeps in the Moran model.

math.PR↗