The Annals of Probability

Genealogical constructions of population models

Alison M. Etheridge and Thomas G. Kurtz

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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.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 1827-1910.

Received: July 2016
Revised: March 2018
First available in Project Euclid: 4 July 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60J25: Continuous-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92} 92D15: Problems related to evolution 92D25: Population dynamics (general) 92D40: Ecology
Secondary: 60F05: Central limit and other weak theorems 60G09: Exchangeability 60G55: Point processes 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 60J68: Superprocesses

Population model Moran model lookdown construction genealogies voter model generators stochastic equations Lambda Fleming–Viot process stepping stone model


Etheridge, Alison M.; Kurtz, Thomas G. Genealogical constructions of population models. Ann. Probab. 47 (2019), no. 4, 1827--1910. doi:10.1214/18-AOP1266.

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