arXiv subjects

Jason Schweinsberg

Publications and source records attributed to Jason Schweinsberg.

At least 19 recordsLinked to original sources

arXiv2026

Branch lengths for geodesics in the directed landscape and mutation patterns in growing spatially structured populations

Consider a population that is expanding in two-dimensional space. Suppose we collect data from a sample of individuals taken at random either from the entire population, or from near the outer boundary of the population. A quantity of interest in population genetics is the site frequency spectrum, which is the number of mutations that appear on $k$ of the $n$ sampled individuals, for $k = 1, \dots, n-1$. As long as the mutation rate is constant, this number will be roughly proportional to the total length of all branches in the genealogical tree that are on the ancestral line of $k$ sampled individuals. While the rigorous literature has primarily focused on models without any spatial structure, in many natural settings, such as tumors or bacteria colonies, growth is dictated by spatial constraints. Many such two dimensional growth models are expected to fall in the KPZ universality class. In this article we adopt the perspective that for population models in the KPZ universality class, the genealogical tree can be approximated by the tree formed by the infinite upward geodesics in the directed landscape, a universal scaling limit constructed in \cite{dov22}, starting from $n$ randomly chosen points. Relying on geodesic coalescence, we prove new asymptotic results for the lengths of the portions of these geodesics that are ancestral to $k$ of the $n$ sampled points and consequently obtain exponents driving the site frequency spectrum as predicted in \cite{fgkah16}. An important ingredient in the proof is a new tight estimate of the probability that three infinite upward geodesics stay disjoint up to time $t$, i.e., a sharp quantitative version of the well studied N3G problem, which is of independent interest.

math.PR↗

arXiv2026

Estimating the growth rate of a birth and death process using data from a small sample

The problem of estimating the growth rate of a birth and death processes based on the coalescence times of a sample of $n$ individuals has been considered by several authors (\cite{stadler2009incomplete, williams2022life, mitchell2022clonal, Johnson2023}). This problem has applications, for example, to cancer research, when one is interested in determining the growth rate of a clone. Recently, \cite{Johnson2023} proposed an analytical method for estimating the growth rate using the theory of coalescent point processes, which has comparable accuracy to more computationally intensive methods when the sample size $n$ is large. We use a similar approach to obtain an estimate of the growth rate that is not based on the assumption that $n$ is large. We demonstrate, through simulations using the R package \texttt{cloneRate}, that our estimator of the growth rate performs well in comparison with previous approaches when $n$ is small.

stat.ME↗

arXiv2025

Two waves of adaptation: speciation induced by dormancy in a model with changing environment

We consider a population model in which the season alternates between winter and summer, and individuals can acquire mutations either that are advantageous in the summer and disadvantageous in the winter, or vice versa. Also, we assume that individuals in the population can either be active or dormant, and that individuals can move between these two states. Dormant individuals do not reproduce but do not experience selective pressures. We show that, under certain conditions, over time we see two waves of adaptation. Some individuals repeatedly acquire mutations that are beneficial in the summer, while others repeatedly acquire mutations that are beneficial in the winter. Individuals can survive the season during which they are less fit by entering a dormant state. This result demonstrates that, for populations in fluctuating environments, dormancy has the potential to induce speciation.

q-bio.PE↗

arXiv2024

The Accumulation of Beneficial Mutations and Convergence to a Poisson Process

We consider a model of a population with fixed size $N$, which is subjected to an unlimited supply of beneficial mutations at a constant rate $μ_N$. Individuals with $k$ beneficial mutations have the fitness $(1+s_N)^k$. Each individual dies at rate 1 and is replaced by a random individual chosen with probability proportional to its fitness. We show that when $μ_N \ll 1/(N \log N)$ and $N^{-η} \ll s_N \ll 1$ for some $η< 1$, the fixation times of beneficial mutations, after a time scaling, converge to the times of a Poisson process, even though for some choices of $s_N$ and $μ_N$ satisfying these conditions, there will sometimes be multiple beneficial mutations with distinct origins in the population, competing against each other.

math.PR↗

arXiv2024

The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift

Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-ρ$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $ρ\geq \sqrt{2}$. We show that in the subcritical case when $ρ> \sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t \rightarrow \infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $ρ\downarrow \sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $ε^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations.

math.PR↗

arXiv2024

Phase transition of the consistent maximal displacement of branching Brownian motion

Consider branching Brownian motion in which we begin with one particle at the origin, particles independently move according to Brownian motion, and particles split into two at rate one. It is well-known that the right-most particle at time $t$ will be near $\sqrt{2} t$. Roberts considered the so-called consistent maximal displacement and showed that with high probability, there will be a particle at time $t$ whose ancestors stayed within a distance $ct^{1/3}$ of the curve $s \mapsto \sqrt{2} s$ for all $s \in [0, t]$, where $c = (3 π^2)^{1/3}/\sqrt{2}$. We consider the question of how close the trajectory of a particle can stay to the curve $s \mapsto (\sqrt{2} + \varepsilon) s$ for all $s \in [0, t]$, where $\varepsilon> 0$ is small. We find that there is a phase transition, with the behavior changing when $t$ is of the order $\varepsilon^{-3/2}$. This result allows us to determine, for branching Brownian motion in which particles have a drift to the left of $\sqrt{2} + \varepsilon$ and are killed at the origin, the position at which a particle needs to begin at time zero for there to be a high probability that the process avoids extinction until time $t$.

math.PR↗

arXiv2023

The all-time maximum for branching Brownian motion with absorption conditioned on long-time survival

We consider branching Brownian motion in which initially there is one particle at $x$, particles produce a random number of offspring with mean $m+1$ at the time of branching events, and each particle branches at rate $\beta = 1/2m$. Particles independently move according to Brownian motion with drift $-1$ and are killed at the origin. It is well-known that this process eventually dies out with positive probability. We condition this process to survive for an unusually large time $t$ and study the behavior of the process at small times $s \ll t$ using a spine decomposition. We show, in particular, that the time when a particle gets furthest from the origin is of the order $t^{5/6}$.

math.PR↗

arXiv2023

Asymptotics for the site frequency spectrum associated with the genealogy of a birth and death process

Consider a birth and death process started from one individual in which each individual gives birth at rate $λ$ and dies at rate $μ$, so that the population size grows at rate $r = λ- μ$. Lambert and Harris, Johnston, and Roberts came up with methods for constructing the exact genealogy of a sample of size $n$ taken from this population at time $T$. We use the construction of Lambert, which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with this sample. In the supercritical case $r > 0$, our results extend results of Durrett for exponentially growing populations. In the critical case $r = 0$, our results parallel those that Dahmer and Kersting obtained for Kingman's coalescent.

math.PR↗

arXiv2022

Particle configurations for branching Brownian motion with an inhomogeneous branching rate

Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distribution. We show that after a sufficiently long time, the fitness distribution from the lowest to the highest fitness levels approximately evolves as a traveling wave with a profile which is asymptotically related the the Airy function. Our work complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.

math.PR↗

arXiv2021

A Spatial Mutation Model with Increasing Mutation Rates

We consider a spatial model of cancer in which cells are points on the $d$-dimensional torus $\mathcal{T}=[0,L]^d$, and each cell with $k-1$ mutations acquires a $k$th mutation at rate $μ_k$. We will assume that the mutation rates $μ_k$ are increasing, and we find the asymptotic waiting time for the first cell to acquire $k$ mutations as the torus volume tends to infinity. This paper generalizes results on waiting for $k\geq 3$ mutations by Foo, Leder, and Schweinsberg, who considered the case in which all of the mutation rates $μ_k$ were the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates.

math.PR↗

arXiv2021

Yaglom-type limit theorems for branching Brownian motion with absorption

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.

math.PR↗

arXiv2021

A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.

math.PR↗

arXiv2020

$Λ$-coalescents arising in populations with dormancy

Consider a population evolving from year to year through three seasons: spring, summer and winter. Every spring starts with $N$ dormant individuals waking up independently of each other according to a given distribution. Once an individual is awake, it starts reproducing at a constant rate. By the end of spring, all individuals are awake and continue reproducing independently as Yule processes during the whole summer. In the winter, $N$ individuals chosen uniformly at random go to sleep until the next spring, and the other individuals die. We show that because an individual that wakes up unusually early can have a large number of surviving descendants, for some choices of model parameters the genealogy of the population will be described by a $Λ$-coalescent. In particular, the beta coalescent can describe the genealogy when the rate at which individuals wake up increases exponentially over time. We also characterize the set of all $Λ$-coalescents that can arise in this framework.

math.PR↗

arXiv2020

Mutation timing in a spatial model of evolution

Motivated by models of cancer formation in which cells need to acquire $k$ mutations to become cancerous, we consider a spatial population model in which the population is represented by the $d$-dimensional torus of side length $L$. Initially, no sites have mutations, but sites with $i-1$ mutations acquire an $i$th mutation at rate $μ_i$ per unit area. Mutations spread to neighboring sites at rate $α$, so that $t$ time units after a mutation, the region of individuals that have acquired the mutation will be a ball of radius $αt$. We calculate, for some ranges of the parameter values, the asymptotic distribution of the time required for some individual to acquire $k$ mutations. Our results, which build on previous work of Durrett, Foo, and Leder, are essentially complete when $k = 2$ and when $μ_i = μ$ for all $i$.

math.PR↗

arXiv2018

The Nested Kingman Coalescent: Speed of Coming Down from Infinity

The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from infinity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time $t$ decays as $ 2γ/ct^2$, where $c$ is the ratio of the coalescence rates at the individual and species levels, and the constant $γ\approx 3.45$ is derived from a recursive distributional equation for the number of lineages contained within a species at a typical time.

math.PR↗

arXiv2017

The size of the last merger and time reversal in $Λ$-coalescents

We consider the number of blocks involved in the last merger of a $Λ$-coalescent started with $n$ blocks. We give conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $Λ$-coalescents that have a dust component. For general $Λ$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.

math.PR↗

arXiv2017

A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process

An important property of Kingman's coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity'. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman's coalescent is the `fastest' to come down from infinity. In this article we study what happens when we counteract this `fastest' coalescent with the action of an extreme form of fragmentation. We augment Kingman's coalescent, where any two blocks merge at rate $c>0$, with a fragmentation mechanism where each block fragments at constant rate, $λ>0$, into it's constituent elements. We prove that there exists a phase transition at $λ=c/2$, between regimes where the resulting `fast' fragmentation-coalescence process is able to come down from infinity or not. In the case that $λ<c/2$ we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.

math.PR↗

arXiv2015

Rigorous results for a population model with selection I: evolution of the fitness distribution

We consider a model of a population of fixed size $N$ undergoing selection. Each individual acquires beneficial mutations at rate $μ_N$, and each beneficial mutation increases the individual's fitness by $s_N$. Each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. Under certain conditions on the parameters $μ_N$ and $s_N$, we obtain rigorous results for the rate at which mutations accumulate in the population and the distribution of the fitnesses of individuals in the population at a given time. Our results confirm predictions of Desai and Fisher (2007).

math.PR↗