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Raphaël Forien

Publications and source records attributed to Raphaël Forien.

14 recordsLinked to original sources
arXiv2026

Eco-evolutionary cycles in a matching type predator-prey interaction

We study the population dynamics of a predator-prey system with two types in each species. Within a species, predator or prey, dynamics are described by a neutral competitive Lotka-Volterra model, i.e., birth, death and competition parameters are equal for both types. Additionally, we assume that the intra- and inter-type competition parameters are equal. The predator-prey interaction is defined by a matching-types model where predators of type $i$ exclusively interact with prey of type $i$. The individual-based model is described by a birth-death process with immigration, where immigration reflects mutations between the types of the same species. We completely describe the deterministic dynamics arising as a large population limit of this birth-death process. We find that depending on the parameters, potential equilibria are the coexistence of all four types, coexistence of a non-matching or matching pair of predators and prey, or the extinction of the predator or prey species resulting in a line of two-type equilibria. When mutations are sufficiently rare, then the predator-prey dynamics are described by successive jumps between the different deterministic equilibria on this mutational time scale. These jumps describe eco-evolutionary cycles of repeated prey or predator invasions and declines. When coexistence of all the types is possible, we show that these cycles accumulate on this time scale. Lastly, to prove that after the accumulation point the system converges to the coexistence equilibrium, we consider a slightly modified model with unequal intra- and inter-type competition parameters. This modified setting allows us to conclude that after the accumulation point all four populations remain macroscopic and converge to the coexistence equilibrium.

math.PR↗
arXiv2026

Correction to "The stepping stone model in a random environment" -- limit theorems for the occupation time and intersection time of reversible random walks in random environments

This note corrects a mistake in the author's above-mentionned published work regarding the intersection local time of two independent random walks in a random environment. The random walks each behave as nearest neighbour random walks in random conductances in the same random environment. The proof of the main result in the original work relied on the derivation of the scaling limit of an additive functionnal of the environment at the locations where the two independent random walks meet, which was shown to be given by a constant times the intersection local time of two Brownian motions. This note shows how the value of the constant denoted by $ γ$ in this work should be updated to correct the statement. The derivation of this constant uses new results on the scaling limit of the occupation time of reversible random walks in a random environment.

math.PR↗
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↗
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↗
arXiv2025

Central limit theorems describing isolation by distance under various forms of power-law dispersal

In this paper, we uncover new asymptotic isolation by distance patterns occurring under long-range dispersal of offspring. We extend a recent work of the first author, in which this information was obtained from forwards-in-time dynamics using a novel stochastic partial differential equations approach for spatial $Λ$-Fleming-Viot models. The latter were introduced by Barton, Etheridge and Véber as a framework to model the evolution of the genetic composition of a spatially structured population. Reproduction takes place through extinction-recolonisation events driven by a Poisson point process. During an event, in certain ball-shaped areas, a parent is sampled and a proportion of the population is replaced. We generalize the previous approach of the first author by allowing the area from which a parent is sampled during events to differ from the area in which offspring are dispersed, and the radii of these regions follow power-law distributions. In particular, while in previous works the motion of ancestral lineages and coalescence behaviour were closely linked, we demonstrate that local and non-local coalescence is possible for ancestral lineages governed by both fractional and standard Laplacians.

math.PR↗
arXiv2025

Stochastic epidemic models with varying infectivity and waning immunity

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if $ R_0 $ is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if $ R_0 $ is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.

math.PR↗
arXiv2025

Central limit theorems describing isolation by distance under varying population size

We derive a central limit theorem for a spatial $Λ$-Fleming-Viot model with fluctuating population size. At each reproduction, a proportion of the population dies and is replaced by a not necessarily equal mass of new individuals. The mass depends on the local population size and a function thereof. Additionally, as new individuals have a single parental type, with growing population size, events become more frequent and of smaller impact, modelling the successful reproduction of a higher number of individuals. From the central limit theorem we derive a Wright-Malécot formula quantifying the asymptotic probability of identity by descent and thus isolation by distance. The formula reflects that ancestral lineages are attracted by centres of population mass and coalesce with a rate inversely proportional to the population size. Notably, we obtain this information despite the varying population size rendering the dual process intractable.

math.PR↗
arXiv2025

Lambda-Fleming-Viot processes arising in logistic Bienaymé-Galton-Watson processes with a large carrying capacity

We consider a continuous-time Bienaymé-Galton-Watson process with logistic competition in a regime of weak competition, or equivalently of a large carrying capacity. Individuals reproduce at random times independently of each other but die at a rate which increases with the population size. When individuals reproduce, they produce a random number of offspring, drawn according to some probability distribution on the natural integers. We keep track of the number of descendants of the initial individuals by adding neutral markers to the individuals, which are inherited by one's offspring. We then consider several scaling limits of the measure-valued process describing the distribution of neutral markers in the population, as well as the population size, when the competition parameter tends to zero. Three regimes emerge, depending on the tail of the offspring distribution. When the offspring distribution admits a second moment (actually a $ 2+δ$ moment for some positive $ δ$), the fluctuations of the population size around its carrying capacity are small and the neutral types asymptotically follow a Fleming-Viot process. When the offspring distribution has a power-law decay with exponent $ α\in (1,2) $, the population size remains most of the time close to its carrying capacity with some (short-lived) fluctuations, and the neutral types evolve in the limit according to a generalised $ Λ$-Fleming-Viot process. When the exponent $ α$ is equal to 1, the time scale of the fluctuations changes drastically, as well as the order of magnitude of the population size. In that case the limiting dynamics of the neutral markers is given by the dual of the Bolthausen-Sznitman coalescent.

math.PR↗
arXiv2022

Stochastic partial differential equations describing neutral genetic diversity under short range and long range dispersal

In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial $ Λ$-Fleming-Viot process introduced by Barton, Etheridge and Véber, which describes the state of the population at any time by a measure on $ \R^d \times [0,1] $, where $ \R^d $ is the geographical space and $ [0,1] $ is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Malécot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal we obtain a new formula which could open the way for a better appraisal of long range dispersal in inference methods.

math.PR↗
arXiv2022

Multi-patch multi-group epidemic model with varying infectivity

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.

math.PR↗
arXiv2021

Ancestral lineages in mutation-selection equilibria with moving optimum

We investigate the evolutionary dynamics of a population structured in phenotype, subjected to trait dependent selection with a linearly moving optimum and an asexual mode of reproduction. Our model consists of a non-local and non-linear parabolic PDE. Our main goal is to measure the history of traits when the population stays around an equilibrium. We define an ancestral process based on the idea of neutral fractions. It allows us to derive quantitative information upon the evolution of diversity in the population along time. First, we study the long-time asymptotics of the ancestral process. We show that the very few fittest individuals drive adaptation. We then tackle the adaptive dynamics regime, where the effect of mutations is asymptotically small. In this limit, we provide an interpretation for the minimizer of some related optimization problem, an Hamilton Jacobi equation, as the typical ancestral lineage. We check the theoretical results against individual based simulations.

math.AP↗
arXiv2019

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

We study a one-dimensional spatial population model where the population sizes at each site are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.

math.PR↗
arXiv2016

A central limit theorem for the spatial Lambda Fleming-Viot process with selection

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 suppose that the population is divided into two genetic types, $a$ and $A$, and consider the proportion of the population which is of type $a$ at each spatial location. If we let both the selection intensity and the fraction of individuals replaced during reproduction events tend to zero, the process can be rescaled so as to converge to the solution to a reaction-diffusion equation (typically the Fisher-KPP equation, as in Etheridge, Veber and Yu, 2014). We show that the rescaled fluctuations converge in distribution to the solution to a linear stochastic partial differential equation. Depending on whether offspring dispersal is only local or if large scale extinction-recolonization events are allowed to take place, the limiting equation is either the stochastic heat equation with a linear drift term driven by space-time white noise or the corresponding fractional heat equation driven by a coloured noise which is white in time. If individuals are diploid (i.e. either $AA$, $Aa$ or $aa$) and if natural selection favours heterozygous ($Aa$) individuals, a stable intermediate gene frequency is maintained in the population. We give estimates for the asymptotic effect of random fluctuations around the equilibrium frequency on the local average fitness in the population. In particular, we find that the size of this effect - known as the drift load - depends crucially on the dimension $d$ of the space in which the population evolves, and is reduced relative to the case without spatial structure.

math.PR↗