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July 2019 Genealogical constructions of population models
Alison M. Etheridge, Thomas G. Kurtz
Ann. Probab. 47(4): 1827-1910 (July 2019). DOI: 10.1214/18-AOP1266


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.


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Alison M. Etheridge. Thomas G. Kurtz. "Genealogical constructions of population models." Ann. Probab. 47 (4) 1827 - 1910, July 2019.


Received: 1 July 2016; Revised: 1 March 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114706
MathSciNet: MR3980910
Digital Object Identifier: 10.1214/18-AOP1266

Primary: 60J25 , 92D10 , 92D15 , 92D25 , 92D40
Secondary: 60F05 , 60G09 , 60G55 , 60G57 , 60H15 , 60J68

Keywords: genealogies , generators , Lambda Fleming–Viot process , Lookdown construction , Moran model , population model , Stepping stone model , stochastic equations , voter model

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 4 • July 2019
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