arXiv subjects

Sarah Penington

Publications and source records attributed to Sarah Penington.

At least 19 recordsLinked to original sources

arXiv2026

A functional law of large numbers for a spatial model of Muller's ratchet

The spatial Muller's ratchet is a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population. The model is an interacting particle system consisting of particles performing symmetric random walks that reproduce and die with rates that depend on the local number of particles. For each particle, we keep track of the number of deleterious mutations that it carries, and after each birth event, with some positive probability, the offspring particle can acquire an additional mutation that gives it a lower reproduction rate than its parent. We show that, under an appropriate scaling, the process converges weakly to the solution of an infinite system of partial differential equations (PDEs), confirming non-rigorous computations of Foutel-Rodier and Etheridge. Combining the weak convergence with analytical results for the limiting PDE system, we derive quantitative lower and upper bounds on the proportion of particles with mutations that hold with high probability for the particle system. A key obstacle is the absence of uniform bounds on the number of particles per site, together with the presence of infinitely many types of particle and the nonlinear interactions. To address this, we establish a new tightness criterion for interacting particle systems in general Lp spaces based only on local properties of the dynamics.

math.PR↗

arXiv2026

Can deleterious mutations surf deterministic population waves?

In spatially structured populations, rare neutral mutations can spread through large regions during a range expansion, a phenomenon known as gene surfing. Whether deleterious mutations can also surf remains poorly understood. To address this question, we study a deterministic version of the spatial Muller's ratchet, given by an infinite system of reaction-diffusion equations describing an asexual population subject to mutation, migration, and density-dependent reproduction and death. After establishing that the system of PDEs is well-posed, we analyse the distribution of deleterious mutations within the population. In the monostable regime, we derive quantitative bounds on the ratio between the density of individuals carrying a given number of mutations and the density of mutation-free individuals. Under a Fisher-KPP condition, we further determine the spreading speed of the population into an empty habitat, confirming non-rigorous computations of Foutel-Rodier and Etheridge. Finally, using a tracer dynamics approach, we show that deleterious mutations cannot surf deterministic waves: although they are present at the expansion front, they only arise as recent descendants of the wild type.

math.AP↗

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

Existence, uniqueness and moment bounds for a spatial model of Muller's ratchet

In this article, we consider a generalisation of the spatial Muller's ratchet introduced by Foutel-Rodier and Etheridge. This particle system is a spatial model of an asexual population, with birth and death rates that depend on the local population density. Particles live in discrete demes and migrate to neighbouring demes. Each particle carries some number of mutations (its `type'), and additional mutations can occur during birth events. Mutations are assumed to be deleterious, i.e.~carrying a higher number of mutations results in a lower birth rate. Our main result shows that this interacting particle system can be constructed even when the total initial number of particles is infinite. We also prove moment bounds on the local density of particles; these bounds are a crucial ingredient of the proof of a law of large numbers result for the particle system in the companion article. The construction of the particle system uses a sequence of approximating processes. Proving weak convergence of this sequence of processes is non-trivial because the particle system is non-monotone and interactions are non-local in type space. The uniqueness of the limit relies on a delicate coupling argument.

math.PR↗

arXiv2025

Convergence and front position for an FKPP-type free boundary problem

The free boundary problem\[ \begin{cases} \partial_tu=\frac{1}{2}Δu+u,\quad &t>0, \, x>L_t,\\ u(t,x)=0,\quad &t>0,\, x\le L_t,\\ \int_{L_t}^{\infty}u(t,y)dy=1,\quad &t> 0,\\ u(t,x)dx \to u_0(dx)&\text{weakly as }t\to 0, \end{cases}\] has long been conjectured to be in the universality class of the so-called FKPP reaction-diffusion equation. It appears naturally as the hydrodynamic limit of a branching-selection particle system, the $N$-BBM. In the present work, we show that for any initial condition $u_0(dx)$ that decays fast enough as $x\to\infty$, the solution of the free boundary problem converges to the minimal travelling wave solution. We further show how the decay of the initial condition precisely determines the position of the free boundary $L_t$ at large times $t$, mirroring the celebrated results of Bramson \cite{Bramson1983} in the context of the FKPP equation. Our conditions for convergence to the minimal travelling wave, and for $L_t$ to have the Bramson asymptotics \[ L_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t+c+o(1)\quad\text{as }t\to\infty,\] are necessary and sufficient. We also apply our results to a more general free boundary problem that depends on a parameter $β$, where we see a transition from \emph{pulled} to \emph{pushed} behaviour (with \emph{pushmi-pullyu} behaviour at the critical value of $β$). We obtain analogous sharp conditions for convergence to the minimal travelling wave, along with precise asymptotics for the front position, in each of these regimes. To our knowledge, such necessary and sufficient conditions had not previously been established in the pushmi-pullyu or pushed regimes, even for classical monostable reaction-diffusion equations. Our results prove and extend non-rigorous predictions in the physics literature of the first author, Brunet and Derrida.

math.AP↗

arXiv2025

Analysis of a household-scale model for the invasion of Wolbachia into a resident mosquito population

In areas infested with Aedes aegypti mosquitoes it may be possible to control dengue, and some other vector-borne diseases, by introducing Wolbachia-infected mosquitoes into the wildtype population. Thus far, empirical and theoretical studies of Wolbachia release have tended to focus on the dynamics at the community scale. However, Ae. aegypti mosquitoes typically dwell in and around the same houses as the people they bite and it can be insightful to explore what happens at the household scale where small population sizes lead to inherently stochastic dynamics. Here we use a continuous-time Markov framework to develop a stochastic household model for small populations of wildtype and Wolbachia-infected mosquitoes. We investigate the transient and long term dynamics of the system, in particular examining the impact of stochasticity on the Wolbachia invasion threshold and bistability between the wildtype-only and Wolbachia-only steady states previously observed in deterministic models. We focus on the influence of key parameters which determine the fitness cost of Wolbachia infection and the probability of Wolbachia vertical transmission. Using Markov and matrix population theory, we derive salient characteristics of the system including the probability of successful Wolbachia invasion, the expected time until invasion and the probability that a Wolbachia-infected population reverts to a wildtype population. These attributes can inform strategies for the release of Wolbachia-infected mosquitoes. In addition, we find that releasing the minimum number of Wolbachia-infected mosquitoes required to displace a resident wildtype population according to the deterministic model, only results in that outcome about 20% of the time in the stochastic model; a significantly larger release is required to reach a steady state composed entirely of Wolbachia-infected mosquitoes 90% of the time.

q-bio.PE↗

arXiv2024

Epidemiological dynamics in populations structured by neighbourhoods and households

Epidemiological dynamics are affected by the spatial and demographic structure of the host population. Households and neighbourhoods are known to be important groupings but little is known about the epidemiological interplay between them. In order to explore the implications for infectious disease epidemiology of households with similar demographic structures clustered in space we develop a multi-scale epidemic model consisting of neighbourhoods of households. In our analysis we focus on key parameters which control household size, the importance of transmission within households relative to outside of them, and the degree to which the non-household transmission is localised within neighbourhoods. We construct the household reproduction number $R_*$ over all neighbourhoods and derive the analytic probability of an outbreak occurring from a single infected individual in a specific neighbourhood. We find that reduced localisation of transmission within neighbourhoods reduces $R_*$ when household size differs between neighbourhoods. This effect is amplified by larger differences between household sizes and larger divergence between transmission rates within households and outside of them. However, the impact of neighbourhoods with larger household sizes on an individual's risk of infection is mainly limited to the individuals that reside in those neighbourhoods. We consider various surveillance scenarios and show that household size information from the initial infectious cases is often more important than neighbourhood information while household size and neighbourhood localisation influences the sequence of neighbourhoods in which an outbreak is observed.

q-bio.PE↗

arXiv2024

Branching random walk with non-local competition

We study the Bolker-Pacala-Dieckmann-Law (BPDL) model of population dynamics in the regime of large population density. The BPDL model is a particle system in which particles reproduce, move randomly in space, and compete with each other locally. We rigorously prove global survival as well as a shape theorem describing the asymptotic spread of the population, when the population density is sufficiently large. In contrast to most previous studies, we allow the competition kernel to have an arbitrary, even infinite range, whence the term non-local competition. This makes the particle system non-monotone and of infinite-range dependence, meaning that the usual comparison arguments break down and have to be replaced by a more hands-on approach. Some ideas in the proof are inspired by works on the non-local Fisher-KPP equation, but the stochasticity of the model creates new difficulties.

math.PR↗

arXiv2021

A branching process with deletions and mergers that matches the threshold for hypercube percolation

We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. Individuals have Poisson offspring distribution with mean $1+p$ and certain deletions and mergers occur with probability $q$; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice $\mathbb{Z}^d$ and the hypercube $Q_d$. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold. The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of $\mathcal{G}(p,q)$ is monotone in $p$ or $q$; we discuss this and some other open problems regarding the new graph process.

math.PR↗

arXiv2021

Genealogy and spatial distribution of the $N$-particle branching random walk with polynomial tails

The $N$-particle branching random walk is a discrete time branching particle system with selection. We have $N$ particles located on the real line at all times. At every time step each particle is replaced by two offspring, and each offspring particle makes a jump of non-negative size from its parent's location, independently from the other jumps, according to a given jump distribution. Then only the $N$ rightmost particles survive; the other particles are removed from the system to keep the population size constant. Inspired by work of J. Bérard and P. Maillard, we examine the long term behaviour of this particle system in the case where the jump distribution has regularly varying tails and the number of particles is large. We prove that at a typical large time the genealogy of the population is given by a star-shaped coalescent, and that almost the whole population is near the leftmost particle on the relevant space scale.

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

Brownian bees in the infinite swarm limit

The Brownian bees model is a branching particle system with spatial selection. It is a system of $N$ particles which move as independent Brownian motions in $\mathbb{R}^d$ and independently branch at rate 1, and, crucially, at each branching event, the particle which is the furthest away from the origin is removed to keep the population size constant. In the present work we prove that as $N \to \infty$ the behaviour of the particle system is well approximated by the solution of a free boundary problem (which is the subject of a companion paper), the hydrodynamic limit of the system. We then show that for this model the so-called selection principle holds, i.e. that as $N \to \infty$ the equilibrium density of the particle system converges to the steady state solution of the free boundary problem.

math.PR↗

arXiv2020

A free boundary problem arising from branching Brownian motion with selection

We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.

math.AP↗

arXiv2018

Global existence for a free boundary problem of Fisher-KPP type

Motivated by the study of branching particle systems with selection, we establish global existence for the solution $(u,μ)$ of the free boundary problem \[ \begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and $x>μ_t$,}\\ u(x,t)=1 &\text{for $t>0$ and $x \leq μ_t$}, \\ \partial_x u(μ_t,t)=0 & \text{for $t>0$}, \\ u(x,0)=v(x) &\text{for $x\in \mathbb{R}$}, \end{cases} \] when the initial condition $v:\mathbb{R}\to[0,1]$ is non-increasing with $v(x) \to 0$ as $x\to \infty$ and $v(x)\to 1$ as $x\to -\infty$. We construct the solution as the limit of a sequence $(u_n)_{n\ge 1}$, where each $ u_n$ is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. Recent results of De Masi \textit{et al.}~\cite{DeMasi2017a} show that this global solution can be identified with the hydrodynamic limit of the so-called $N$-BBM, {\it i.e.} a branching Brownian motion in which the population size is kept constant equal to $N$ by killing the leftmost particle at each branching event.

math.AP↗

arXiv2017

Branching Brownian motion with decay of mass and the non-local Fisher-KPP equation

In this work we study a non-local version of the Fisher-KPP equation, \begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\tfrac{1}{2}Δu +u (1- ϕ\ast u), \quad t>0, \quad x\in \mathbb{R}, u(0,x)=u_0(x), \quad x\in \mathbb{R} \end{cases} \end{equation*} and its relation to $\textit{branching Brownian motion with decay of mass}$ as introduced by Addario-Berry and Penington (2017), i.e. a particle system consisting of a standard branching Brownian motion (BBM) with a competitive interaction between nearby particles. Particles in the BBM with decay of mass have a position in $\mathbb{R}$ and a mass, branch at rate 1 into two daughter particles of the same mass and position, and move independently as Brownian motions. Particles lose mass at a rate proportional to the mass in a neighbourhood around them (as measured by the function $ϕ$). We obtain two types of results. First, we study the behaviour of solutions to the partial differential equation above. We show that, under suitable conditions on $ϕ$ and $u_0$, the solutions converge to 1 behind the front and are globally bounded, improving recent results of Hamel and Ryzhik (arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBM with decay of mass is the solution of the non-local Fisher-KPP equation. We then harness this to obtain several new results concerning the behaviour of the particle system.

math.PR↗

arXiv2017

The spreading speed of solutions of the non-local Fisher-KPP equation

We consider the Fisher-KPP equation with a non-local interaction term. Hamel and Ryzhik showed that in solutions of this equation, the front location at a large time $t$ is $\sqrt 2 t +o(t)$. We study the asymptotics of the second order term in the front location. If the interaction kernel $ϕ(x)$ decays sufficiently fast as $x\rightarrow \infty$ then this term is given by $-\frac{3}{2\sqrt 2 }\log t +o(\log t)$, which is the same correction as found by Bramson for the local Fisher-KPP equation. However, if $ϕ$ has a heavier tail then the second order term is $-t^{β+o(1)}$, where $β\in (0,1)$ depends on the tail of $ϕ$. The proofs are probabilistic, using a Feynman-Kac formula. Since solutions of the non-local Fisher-KPP equation do not obey the maximum principle, the proofs differ from those in Bramson's work, although some of the ideas used are similar.

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↗