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

Klimek, Aleksander; Rosati, Tommaso Cornelis

doi:10.48550/arxiv.2004.05931

Cited by 4 publications

(4 citation statements)

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“…For instance, in mathematical models for populations dynamics such random potentials can be used to quantify environmental fluctuations, see e.g. [17] and the references specified therein. Several PDEs where the linear part has this structure have been investigated, see e.g.…”

Section: Examplesmentioning

confidence: 99%

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

Kuehn¹,

Neamţu²,

Sonner³

2020

Preprint

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We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approach is essential for our setting since the stochastic evolution equation cannot be transformed into a family of PDEs with random coefficients via the stationary Ornstein-Uhlenbeck process.

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Section: Examplesmentioning

confidence: 99%

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

Kuehn¹,

Neamţu²,

Sonner³

2020

Preprint

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“…While most of the studies on SLFVs with selection were carried out in a weak selection regime (see, e.g., [16,17,18] for selection against one type, [20] for general forms of selection, [6,8,29] for fluctuating selection and [15] for selection against heterozygosity), the ∞-parent SLFV corresponds to a strong selection limit, motivated by the interpretation of ghost individuals as modelling empty areas. In particular, and contrary to previous works on SLFVs with selection, the results we obtain do not require to rescale time, nor space.…”

Section: Introductionmentioning

confidence: 99%

Growth properties of the infinite-parent spatial Lambda-Fleming Viot process

Louvet¹,

Véber²

2022

Preprint

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The ∞-parent spatial Lambda-Fleming Viot process, or ∞-parent SLFV, is a model for spatially expanding populations in which empty areas are filled with "ghost" individuals. The interest of this process lies in the fact that it is akin to a continuous-space version of the classical Eden growth model, while being associated to a dual process encoding ancestry and allowing one to study the evolution of the genetic diversity in such a population.In this article, we focus on the growth properties of the ∞-parent SLFV in two dimensions.To do so, we first define the quantity that we shall use to quantify the speed of growth of the area covered with the subpopulation of real individuals. 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 ∞-parent SLFV is linear in time. We use numerical simulations to approximate the growth speed, and conjecture that the actual speed is higher than the speed expected from simple first-moment calculations due to the characteristic front dynamics.We then study a toy model of two interacting growing piles of cubes in order to understand how the growth dynamics at the front edge can increase the global growth speed of the "occupied" region. We obtain an explicit formula for this speed of growth in our toy model, using the invariant distribution of a discretised version of the model. This study is of interest on its own right, and its implications are not restricted to the case of the ∞-parent SLFV.

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“…The original version of the SLFV does not account for the presence of a selectively favoured genetic type, but it can be modified in order to incorporate selection : see [22] for different forms of fixed selection mechanisms, and [7,10,31] for ways to introduce fluctuating selection. Our approach will be based on a version of the SLFV with selection introduced in [22] and rigourously constructed in [19].…”

Section: Introductionmentioning

confidence: 99%

Stochastic measure-valued models for populations expanding in a continuum

Louvet¹

2021

Preprint

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We model spatially expanding populations by means of a spatial Λ-Fleming Viot process (SLFV) with selection : the k-parent SLFV. We fill empty areas with type 0 "ghost" individuals, which have a strong selective disadvantage against "real" type 1 individuals. This model is a special case of the SLFV with selection introduced in [19,22] : natural selection acts during all reproduction events, and the fraction of individuals replaced during a reproduction event is constant equal to 1. Letting the selective advantage k of type 1 individuals over type 0 individuals grow to +∞, and without rescaling time nor space, we obtain a new model for expanding populations, the ∞-parent SLFV.This model is reminiscent of the Eden growth model [13], but with an associated dual process of potential ancestors, making it possible to investigate the genetic diversity in a population sample. In order to obtain the limit k → +∞ of the k-parent SLFV, we introduce an alternative construction of the k-parent SLFV adapted from [38], which allows us to couple SLFVs with different selection strengths.

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

Cited by 4 publications

References 0 publications

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

Growth properties of the infinite-parent spatial Lambda-Fleming Viot process

Stochastic measure-valued models for populations expanding in a continuum

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