Rare mutations in the spatial Lambda-Fleming-Viot model in a fluctuating environment and SuperBrownian Motion

Chetwynd-Diggle, Jonathan A.; Klimek, Aleksander

doi:10.48550/arxiv.1901.04374

Cited by 5 publications

(8 citation statements)

References 33 publications

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“…Using the bounds given by Eq. ( 8), ( 9), (10) and Lemma 33, we can apply the dominated convergence theorem and obtain the desired result.…”

Section: Extended Martingale Problem For the ∞-Parent Ancestral Processmentioning

confidence: 86%

“…This lemma, along with Eqs. (8,9,10,11), will allow us to use the dominated convergence theorem in the proof of Lemma 32.…”

Section: Lemma 32mentioning

confidence: 99%

“…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%

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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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“…Using the bounds given by Eq. ( 8), ( 9), (10) and Lemma 33, we can apply the dominated convergence theorem and obtain the desired result.…”

Section: Extended Martingale Problem For the ∞-Parent Ancestral Processmentioning

confidence: 86%

“…This lemma, along with Eqs. (8,9,10,11), will allow us to use the dominated convergence theorem in the proof of Lemma 32.…”

Section: Lemma 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stochastic measure-valued models for populations expanding in a continuum

Louvet¹

2021

Preprint

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“…) t∈[0,∞) for every finite D ∈ N, in the limit as D → ∞ a finite number of fixed components become independent since they only "depend" on each other through the deterministic process (E[δ Mt (1) ]) t∈[0,∞) . Theorem 4.1 in Gärtner [8] implies (3) and (4) under more general assumptions including strict positivity of σ and Proposition 4.29 in Hutzenthaler [11] implies (3) and (4) for certain cases where σ is locally Hölder- 1 2 -continuous in the first argument and does not depend on the second argument. For further results on propagation of chaos see, e.g., McKean [20], Sznitman [27], Oelschläger [23], Méléard & Roelly-Coppoletta [21], Lasry & Lions [18], and Buckdahn et al [1].…”

Section: Introductionmentioning

confidence: 94%

“…For example, it is a classical result that the number of alleles of one type in a Wright-Fisher model (Moran model) converges to a branching process in discrete (continuous) time as the population size converges to infinity if the initial numbers of alleles of this type are bounded. For results with SuperBrownian motion appearing in suitable rescalings see, e.g., Durrett & Perkins [6], Cox & Perkins [4], Chetwynd-Diggle & Etheridge [2], and Chetwynd-Diggle & Klimek [3].…”

Section: Introductionmentioning

confidence: 99%

Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime

Hutzenthaler,

Pieper

2018

Preprint

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Propagation of chaos is a well-studied phenomenon and shows that weakly interacting diffusions may become independent as the system size converges to infinity. Most of the literature focuses on the case of exchangeable systems where all involved diffusions have the same distribution and are "of the same size". In this paper, we analyze the case where only a few diffusions start outside of an accessible trap. Our main result shows that in this "sparse regime" the system of weakly interacting diffusions converges in distribution to a forest of excursions from the trap. In particular, initial independence propagates in the limit and results in a forest of independent trees.

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Spatial Population Models

Drechsler¹

2020

Ecological-Economic Modelling for Biodiversity Conservation

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These notes are intended to provide a very rapid introduction to some of the mathematical models of theoretical population genetics. We shall be particularly concerned with models that are designed to capture the interplay between spatial structure and various forces of evolution. However, we begin with a brief review of some classical non-spatial models (the Wright-Fisher and Moran models and their diffusion approximations) and the corresponding (backwards in time) Kingman coalescent which describes the genealogical relationships between individuals sampled at random from the population.Our first spatial model is Kimura's stepping stone model of subdivided populations, which we use to capture information about the way in which the correlation between genetic types of individuals sampled from the population decays with spatial separation.The stepping stone model is not a particularly natural model for populations evolving in a spatial continuum and we next turn our attention to the challenges of modelling evolution in a spatial continuum, describing, in particular, Felsenstein's 'pain in the torus'. The spatial Lambda-Fleming-Viot model is then presented as one resolution of the pain in the torus, and the rest of the notes are devoted to developing that model and illustrating how it captures some of the features seen in data, before using it to investigate the interplay between spatial structure and natural selection.

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Rare mutations in the spatial Lambda-Fleming-Viot model in a fluctuating environment and SuperBrownian Motion

Cited by 5 publications

References 33 publications

Stochastic measure-valued models for populations expanding in a continuum

Stochastic measure-valued models for populations expanding in a continuum

Propagation of chaos and the many-demes limit for weakly interacting diffusions in the sparse regime

Spatial Population Models

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