Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

We consider a branching particle model in which particles move inside a Euclidean domain according to the following rules. The particles move as independent Brownian motions until one of them hits the boundary. This particle is killed but another randomly chosen particle branches into two particles, to keep the population size constant. We prove that the particle population does not approach the boundary simultaneously in a finite time in some Lipschitz domains. This is used to prove a limit theorem for the empirical distribution of the particle family.

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“…Our aim is to construct a Brownian motion X t in such a way that X t is adapted to (W t , Z t ) and there exists a constant c 2 …”

Section: Remark 53mentioning

confidence: 99%

“…We will show: Theorem 8. 2 If D is a polyhedral domain and X t = (X 1 t , X 2 t ) is a Fleming-Viot process with jump times τ i then τ i → ∞ as i → ∞ almost surely.…”

mentioning

confidence: 99%

Non-extinction of a Fleming-Viot particle model

Bieniek

Burdzy

Finch

2011

Probab. Theory Relat. Fields

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“…Let D([0,1], M 1 (S)) be the space of all functions from [0, 1] to M 1 (S) that are right continuous and have left limits (left continuous at 1) furnished with the Skorokhod topology. Let T t be the semigroup associated with generator A.…”

Section: Central Limit Theoremmentioning

confidence: 99%

“…In [1], [2], and [13], finite particle systems are studied where each particle moves independently of the others until one particle hits a certain boundary or a catalyst or, more generally, is killed and a new birth occurs at a location uniformly chosen among the remaining particles. This sampling mechanism clearly resembles that of the Moran particle system or the Fleming-Viot process.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic Bahavior of the Moran Particle System

Feng¹,

Xiong²

2013

Adv. Appl. Probab.

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The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches ∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

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Toward a “modern coexistence theory” for the discrete and spatial

Ellner

Snyder

Adler

et al. 2022

Ecological Monographs

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The usual theoretical condition for coexistence is that each species in a community can increase when it is rare (mutual invasibility). Traditional coexistence theory implicitly assumes that the invading species is common enough that we can ignore demographic stochasticity but rare enough that it does not compete with itself, even after it has reached a stationary spatial distribution. However, short‐distance dispersal of discrete individuals leads to locally dense population clusters, and existing theory breaks down. We have an intuition that when we account for invader–invader competition, shorter‐range dispersal should reduce the invader's ability to escape competition, but exactly how does this translate into lower population growth? And how will invader discreteness affect outcomes? We need a way of partitioning the contributions to coexistence, but current modern coexistence theory (MCT) does not apply under these conditions. Here we present a computationally based partitioning method to quantify the contributions to coexistence from different mechanisms, as in MCT. We also build up an intuition for how invader clumping and discreteness will affect these contributions by analyzing a case study, a lattice‐based spatial lottery model. We first consider fluctuation‐dependent coexistence, partitioning the contributions of variable environment, variable competition, demographic stochasticity, and their correlations and interactions. Our second example examines fluctuation‐independent coexistence maintained by a fecundity–survival trade‐off, and partitions the contributions to coexistence from interspecific differences in fecundity, in mortality, and in dispersal. We find that demographic stochasticity harms an invader, but only slightly. Localized invader dispersal, on the other hand, can have a strong effect. When invaders are more clumped, they compete with each other more intensely when rare, so they too become limited by environment‐competition covariance. More invader clumping also means that variation in competition changes from helping the invader to harming it. More broadly, invader clumping is likely to weaken any coexistence mechanism that relies on the invader escaping competition from the resident, because invader clumping means that the resident is no longer the only source of competition.

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Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Cited by 25 publications

References 7 publications

Non-extinction of a Fleming-Viot particle model

Non-extinction of a Fleming-Viot particle model

Asymptotic Bahavior of the Moran Particle System

Toward a “modern coexistence theory” for the discrete and spatial

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