Viot Process Research Articles

The spine of the Fleming–Viot process driven by Brownian motion

The spine of the Fleming–Viot process driven by Brownian motion

Demographic inference for spatially heterogeneous populations using long shared haplotypes

We introduce a modified spatial Λ-Fleming–Viot process to model the ancestry of individuals in a population occupying a continuous spatial habitat divided into two areas by a sharp discontinuity of the dispersal rate and effective population density. We derive an analytical formula for the expected number of shared haplotype segments between two individuals depending on their sampling locations. This formula involves the transition density of a skew diffusion which appears as a scaling limit of the ancestral lineages of individuals in this model. We then show that this formula can be used to infer the dispersal parameters and the effective population density of both regions, using a composite likelihood approach, and we demonstrate the efficiency of this method on a range of simulated data sets.

Measure-valued growth processes in continuous space and growth properties starting from an infinite interface

The k-parent and infinite-parent spatial Lambda-Fleming Viot processes (or SLFV), introduced in Louvet (2023), form a family of stochastic models for spatially expanding populations. These processes are akin to a continuous-space version of the classical Eden growth model (but with local backtracking of the occupied area allowed when k is finite), while being associated with a dual process encoding ancestry.In this article, we focus on the growth properties of the area occupied by individuals of type 1 (type 0 encoding units of empty space). To do so, we first define the quantities that we shall use to quantify the speed of growth of the occupied area. Using the associated dual process and a comparison with a first-passage percolation problem, we show that the growth of the occupied region in the infinite-parent SLFV is linear in time. Because of the possibility of local backtracking of the occupied area, the result we obtain for the k-parent SLFV is slightly weaker. It gives an upper bound on the probability that a given location is occupied at time t, which also shows that growth in the k-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed for the infinite-parent SLFV, and we observe that the actual speed may be higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics.

Existence, uniqueness and ergodicity for the centered Fleming–Viot process

Motivated by questions of ergodicity for shift invariant Fleming–Viot process, we consider the centered Fleming–Viot process Ztt⩾0 defined by Zt≔τ−id,Yt♯Yt, where Ytt⩾0 is the original Fleming–Viot process. Our goal is to characterize the centered Fleming–Viot process with a martingale problem. To establish the existence of a solution to this martingale problem, we exploit the original Fleming–Viot martingale problem and asymptotic expansions. The proof of uniqueness is based on a weakened version of the duality method, allowing us to prove uniqueness for initial conditions admitting finite moments. We also provide counter examples showing that our approach based on the duality method cannot be expected to give uniqueness for more general initial conditions. Finally, we establish ergodicity properties with exponential convergence in total variation for the centered Fleming–Viot process and characterize the invariant measure.

The spatial Λ-Fleming–Viot process in a random environment

We study the large scale behaviour of a population consisting of two types which evolve in dimension d=1,2 according to a spatial Lambda-Fleming–Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher–KPP equation in a random potential. The proofs build on two-scale Schauder estimates and semidiscrete approximations of the Anderson Hamiltonian.

Stochastic measure-valued models for populations expanding in a continuum

We model spatially expanding populations by means of two spatial Λ-Fleming Viot processes (or SLFVs) with selection: the k-parent SLFV and the ∞-parent SLFV. In order to do so, we fill empty areas with type 0 “ghost” individuals with a strong selective disadvantage against “real” type 1 individuals, quantified by a parameter k. The reproduction of ghost individuals is interpreted as local extinction events due to stochasticity in reproduction. When k → +∞, the limiting process, corresponding to the ∞-parent SLFV, is reminiscent of stochastic growth models from percolation theory, but is associated to tools making it possible to investigate the genetic diversity in a population sample. In this article, we provide a rigorous construction of the ∞-parent SLFV, and show that it corresponds to the limit of the k-parent SLFV when k → +∞. In order to do so, we introduce an alternative construction of the k-parent SLFV which allows us to couple SLFVs with different selection strengths and is of interest in its own right. We exhibit three different characterizations of the ∞-parent SLFV, which are valid in different settings and link together population genetics models and stochastic growth models.

Instantaneous support propagation for [formula omitted]-Fleming–Viot processes

For a probability-measure-valued neutral Fleming–Viot process (Zt:t≥0) with Lévy mutation and resampling mechanism associated to a general Λ-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time t>0, with probability one the closed support S(Zt) of the Fleming–Viot process satisfies S(ν∗Zt)⊆S(Zt), where ν is the Lévy measure of the mutation process. To show this result, we apply Donnelly–Kurtz’s lookdown particle representation for Fleming–Viot processes.

Rate of coalescence of lineage pairs in the Spatial [formula omitted]-Fleming–Viot process

We revisit the Spatial Λ-Fleming–Viot process introduced in Barton and Kelleher (2010). Particularly, we are interested in the time T0 to the most recent common ancestor for two lineages. We distinguish between the cases where the process acts on the two-dimensional plane and on a finite rectangle. Utilizing a differential equation linking T0 with the physical distance between the lineages, we arrive at computationally efficient and reasonably accurate approximation schemes for both cases. Furthermore, our analysis enables us to address the question of whether the genealogical process of the model “comes down from infinity”, which has been partly answered before in Véber and Wakolbinger (2015).

Diffusion with stochastic resetting of interacting particles emerging from a model of population genetics

In this paper, we put forward a connection between diffusion with resetting and a certain extension of Ohta–Kimura model, inspired on what was carried out in Da Silva and Fragoso (2018 J. Phys. A: Math. Theor. 51 505002). The contribution is twofold: (1) we derive a new extension of Ohta–Kimura model, dubbed here new extended version of Ohta–Kimura ladder model (NOKM) which bears a strong liaison with the so-called jump-type Fleming–Viot process. The novelty here, when we compare with the classical Ohta–Kimura model, is that we now have an operator which allows simultaneous interaction among many individuals. It has to do with a generalized branching mechanism i.e. m individual types extinguish and one individual type splits into m copies. The system of evolution equations arising from NOKM can be seen as a system of n-dimensional Kolmogorov forward equations (or Fokker–Planck equations). The analysis requires an amenable armory of concepts and mathematical technique to analyze some relevant issues such as correlation, indistinguishability of individuals and stationarity; (2) nudged by the ideas brought to bear in Da Silva and Fragoso (2018 J. Phys. A: Math. Theor. 51 505002), we advance in this agenda here by making an initial incursion on the connection between diffusion with resetting and the NOKM. The connection which relies on the similarities between the models allows, in some cases, that relevant results obtained for one model can be translated to the other model framework by taking advantage of the technique used to derive a result in one of the models. Through the development of the population genetic model and its reinterpretation in terms of diffusion with stochastic resetting, we show an invariance property of the correlation between two interacting particles that reset at a time-inhomogeneous resetting rate. Pushing forward the ideas we obtain the stationary state of a new model for an n-particle system under an anisotropic diffusion with resetting. Although the results using this approach are of recent vintage, we believe that this avenue of research seems to be very encouraging.

Stein’s method for the Poisson–Dirichlet distribution and the Ewens sampling formula, with applications to Wright–Fisher models

We provide a general theorem bounding the error in the approximation of a random measure of interest—for example, the empirical population measure of types in a Wright–Fisher model—and a Dirichlet process, which is a measure having Poisson–Dirichlet distributed atoms with i.i.d. labels from a diffuse distribution. The implicit metric of the approximation theorem captures the sizes and locations of the masses, and so also yields bounds on the approximation between the masses of the measure of interest and the Poisson–Dirichlet distribution. We apply the result to bound the error in the approximation of the stationary distribution of types in the finite Wright–Fisher model with infinite-alleles mutation structure (not necessarily parent independent) by the Poisson–Dirichlet distribution. An important consequence of our result is an explicit upper bound on the total variation distance between the random partition generated by sampling from a finite Wright–Fisher stationary distribution, and the Ewens sampling formula. The bound is small if the sample size n is much smaller than N1/6log(N)−1/2, where N is the total population size. Our analysis requires a result of separate interest, giving an explicit bound on the second moment of the number of types of a finite Wright–Fisher stationary distribution. The general approximation result follows from a new development of Stein’s method for the Dirichlet process, which follows by viewing the Dirichlet process as the stationary distribution of a Fleming–Viot process, and then applying Barbour’s generator approach.

On Λ-Fleming–Viot processes with general frequency-dependent selection

AbstractWe construct a multitype constant-size population model allowing for general selective interactions as well as extreme reproductive events. Our multidimensional model aims for the generality of adaptive dynamics and the tractability of population genetics. It generalises the idea of Krone and Neuhauser [39] and González Casanova and Spanò [29], who represented the selection by allowing individuals to sample several potential parents in the previous generation before choosing the ‘strongest’ one, by allowing individuals to use any rule to choose their parent. The type of the newborn can even not be one of the types of the potential parents, which allows modelling mutations. Via a large population limit, we obtain a generalisation of$\Lambda$-Fleming–Viot processes, with a diffusion term and a general frequency-dependent selection, which allows for non-transitive interactions between the different types present in the population. We provide some properties of these processes related to extinction and fixation events, and give conditions for them to be realised as unique strong solutions of multidimensional stochastic differential equations with jumps. Finally, we illustrate the generality of our model with applications to some classical biological interactions. This framework provides a natural bridge between two of the most prominent modelling frameworks of biological evolution: population genetics and eco-evolutionary models.

Long-time limits and occupation times for stable Fleming–Viot processes with decaying sampling rates

A class of Fleming-Viot processes with decaying sampling rates and $\alpha$-stable motions that correspond to distributions with growing populations are introduced and analyzed. Almost sure long-time scaling limits for these processes are developed, addressing the question of long-time population distribution for growing populations. Asymptotics in higher orders are investigated. Convergence of particle location occupation and inhabitation time processes are also addressed and related by way of the historical process. The basic results and techniques allow general Feller motion/mutation and may apply to other measure-valued Markov processes.

Modifiers of mutation rate in selectively fluctuating environments

We study a mutation–selection model with a fluctuating environment. More precisely, individuals in a large population are assumed to have a modifier locus determining the mutation rate u∈[0,ϑ] at a second locus with types v∈[0,1]. In addition, the environment fluctuates, meaning that individual types change their fitness at some high rate. Fitness only depends on the type of the second locus. We obtain general limit results for the evolution of the allele frequency distribution for rapidly fluctuating environments. As an application, we make use of the resulting Fleming–Viot process and compute the fixation probabilities for higher mutation rates in the special case of two bi-allelic loci in the limit of small fitness differences at the second locus.

Genealogical constructions of population models

Representations of population models in terms of countable systems of particles are constructed, in which each particle has a “type,” typically recording both spatial position and genetic type, and a level. For finite intensity models, the levels are distributed on $[0,\lambda]$, whereas in the infinite intensity limit $\lambda\rightarrow\infty$, at each time $t$, the joint distribution of types and levels is conditionally Poisson, with mean measure $\Xi(t)\times\ell$ where $\ell$ denotes Lebesgue measure and $\Xi(t)$ is a measure-valued population process. The time-evolution of the levels captures the genealogies of the particles in the population. Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, one-for-one replacement, immigration, independent “thinning” and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend on type, they also include natural selection. The primary goal of the paper is to present particle-with-level or lookdown constructions for each of these elements of a population model. Then the elements can be combined to specify the desired model. In particular, a nontrivial extension of the spatial $\Lambda$-Fleming–Viot process is constructed.

Probability measure-valued polynomial diffusions

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

The sequential loss of allelic diversity

Abstract In this paper we give a new flavour to what Peter Jagers and his co-authors call `the path to extinction'. In a neutral population of constant size N, assume that each individual at time 0 carries a distinct type, or allele. Consider the joint dynamics of these N alleles, for example the dynamics of their respective frequencies and more plainly the nonincreasing process counting the number of alleles remaining by time t. Call this process the extinction process. We show that in the Moran model, the extinction process is distributed as the process counting (in backward time) the number of common ancestors to the whole population, also known as the block counting process of the N-Kingman coalescent. Stimulated by this result, we investigate whether it extends (i) to an identity between the frequencies of blocks in the Kingman coalescent and the frequencies of alleles in the extinction process, both evaluated at jump times, and (ii) to the general case of Λ-Fleming‒Viot processes.

The interplay between population genetics and diffusion with stochastic resetting

In this work we analyze the connection between the theory of diffusion with stochastic resetting and the theory of the Ohta–Kimura continuous ladder model. A first hint of this connection is readily given by the equivalence between the models’ partial differential equations for their distribution probabilities. In addition, it turns out that Fleming–Viot processes play a central role in connecting these theories. As an unfolding thereof, it is expected that relevant results obtained for one model can be translated to the other model framework. This is illustrated in this article by obtaining the mean first-passage time for the Ohta–Kimura continuous ladder model and by establishing the correlation between particles in the stochastic resetting model. We also study the non-equilibrium stationary distribution for two interacting resetting particles. In addition we make a brief excursion into another particle interaction model that does not have stationary distribution.

Weak atomic convergence of finite voter models toward Fleming–Viot processes

We consider the empirical measures of multi-type voter models with mutation on large finite sets, and prove their weak atomic convergence in the sense of Ethier and Kurtz (1994) toward a Fleming–Viot process. Convergence in the weak atomic topology is strong enough to answer a line of inquiry raised by Aldous (2013) concerning the distributions of the corresponding entropy processes and diversity processes for types.

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 $\log(1/\mu)$ where $\mu$ 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.

P.2.f.018 - SAGE-217 in subjects with major depressive disorder: efficacy and safety results from open-label Part A of a phase 2A study

We consider the empirical measures of multi-type voter models with mutation on large finite sets, and prove their weak atomic convergence in the sense of Ethier and Kurtz (1994) toward a Fleming–Viot process. Convergence in the weak atomic topology is strong enough to answer a line of inquiry raised by Aldous (2013) concerning the distributions of the corresponding entropy processes and diversity processes for types.