arXiv subjects

Julie Tourniaire

Publications and source records attributed to Julie Tourniaire.

6 recordsLinked to original sources

arXiv2026

Convergence of spatial branching processes to $α$-stable CSBPs: Genealogy of semi-pushed fronts

We consider an inhomogeneous branching diffusion on an unbounded domain of $\mathbb{R}^d$ and propose a simple condition under which we expect the size process (i.e., the number of particles) and the genealogy of the system to converge to those of an $α$-stable continuous-state branching process, with $α\in(1,2)$. This condition can be seen as the spatial analogue of the classical assumption that the tail of the offspring distribution of a Galton--Watson process is regularly varying. We make a first step towards establishing this result by providing a set of sufficient conditions under which the branching diffusion, seen as a random marked metric measure space that captures both the positions and the genealogical structure of the population, converges to an $α$-stable genealogy. These conditions are based on the convergence of the moments of the process, which can be efficiently computed via recursive formulas. We apply this framework to a one-dimensional branching Brownian motion with inhomogeneous branching rate and negative drift. This model was introduced by Tourniaire as a toy model to investigate the internal dynamics of fluctuating pushed fronts. By using our general set of conditions we prove convergence of the genealogy of the process in the semipushed regime, which was conjectured to hold by Birzu, Hallatschek, and Korolev.

math.PR↗

arXiv2026

Power-law scaling of the effective population size in a branching particle system for moderate mutation-selection

We consider a one-dimensional dyadic branching Brownian motion on $\mathbb{R}$ with positive drift $β\in (0,1)$, branching rate $1/2$, reflected at $0$ and killed at a boundary $L > 0$. The killing boundary $L$ is chosen so that the total population size remains approximately constant, proportional to $N \in \mathbb{N}$. This branching process models a population accumulating deleterious mutations. In the large-$N$ limit, we prove that when the typical width of the particle cloud is of order $c \log(N)$, with $c \in (0,1)$, the demographic fluctuations follow a Yaglom law on a polynomial time scale. Moreover, the limiting genealogy of the system involves only binary mergers, concentrated near the reflecting boundary. Our model is a version of the branching Brownian motion with absorption introduced by Berestycki, Berestycki, and Schweinsberg to study the effect of beneficial mutations on genealogies. In sharp contrast with their model, whose genealogy is given by a Bolthausen--Sznitman coalescent, we show that our system falls into the universality class of Kingman's coalescent.

math.PR↗

arXiv2025

Stochastic neutral fractions and the effective population size

The dynamics of a general structured population is modelled using a general stochastic differential equation (SDE) with an infinite decomposability property. This property allows the population to be divided into an arbitrary number of allelic components, also known as stochastic neutral fractions. When demographic noise is small, a fast-slow principle provides a general formula for the effective population size in structured populations. To illustrate this approach, we revisit several examples from the literature, including expansion fronts.

math.PR↗

arXiv2024

Spreading speed of locally regulated population models in macroscopically heterogeneous environments

We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order.

math.PR↗

arXiv2024

Spectral analysis and $k$-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift $-μ$ and killed upon reaching $0$. More precisely, the particles branch at rate $r(x)=(1+W(x))/2,$ where $W$ is a compactly supported and non-negative smooth function and the drift $μ$ is chosen in such a way that the system is critical in some sense. This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semi pushed and fully pushed fronts. Here, we focus on the fully pushed regime. We establish a Yaglom law for this branching process and prove that the genealogy of the particles converges to a Brownian Coalescent Point Process using a method of moments. In practice, the genealogy of the BBM is seen as a random marked metric measure space and we use spinal decomposition to prove its convergence in the Gromov-weak topology. We also carry the spectral decomposition of a differential operator related to the BBM to determine the invariant measure of the spine as well as its mixing time.

math.PR↗

arXiv2024

A branching particle system as a model of semipushed fronts

We consider a system of particles performing a one-dimensional dyadic branching Brownian motion with space-dependent branching rate, negative drift $-μ$ and killed upon reaching $0$, starting with $N$ particles. More precisely, particles branch at rate $ρ/2$ in the interval $[0,1]$, for some $ρ>1$, and at rate $1/2$ in $(1,+\infty)$. The drift $μ(ρ)$ is chosen in such a way that, heuristically, the system is critical in some sense: the number of particles stays roughly constant before it eventually dies out. This particle system can be seen as an analytically tractable model for fluctuating fronts, describing the internal mechanisms driving the invasion of a habitat by a cooperating population. Recent studies from Birzu, Hallatschek and Korolev suggest the existence of three classes of fluctuating fronts: pulled, semipushed and pushed fronts. Here, we rigorously verify and make precise this classification and focus on the semipushed regime. More precisely, we prove the existence of two critical values $1<ρ_1<ρ_2$ such that for all $ρ\in(ρ_1,ρ_2)$, there exists $α(ρ)\in(1,2)$ such that the rescaled number of particles in the system converges to an $α$-stable continuous-state branching process on the time scale $N^{α-1}$ as $N$ goes to infinity. This complements previous results from Berestycki, Berestycki and Schweinsberg for the case $ρ=1$.

math.PR↗