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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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Cited by 20 (5 self)
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL
The spatial LambdaFlemingViot process: an eventbased construction and a lookdown representation
, 2013
"... We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To th ..."
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Cited by 4 (1 self)
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We construct a measurevalued equivalent to the spatial ΛFlemingViot process (SLFV) introduced in [Eth08]. In contrast with the construction carried out in [Eth08], we fix the realization of the sequence of reproduction events and obtain a quenched evolution of the local genetic diversities. To this end, we use a particle representation which highlights the role of the genealogies in the attribution of types (or alleles) to the individuals of the population. This construction also enables us to clarify the statespace of the SLFV and to derive several path properties of the measurevalued process as well as of the labeled trees describing the genealogical relations between a sample of individuals. We complement it with a lookdown construction which provides a particle system whose empirical distribution at time t, seen as a process in t, has the law of the quenched SLFV. In all these results, the facts that we work with a fixed configuration of events and that reproduction occurs only locally in space introduce serious technical issues that are overcome by controlling the number of events occurring and of particles present in a given area over macroscopic time intervals.
Genealogical constructions of population models
, 2014
"... 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, λ], whereas in the infinite i ..."
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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, λ], whereas in the infinite intensity limit, at each time t, the joint distribution of types and levels is conditionally Poisson, with mean measure Ξ(t) × l where l denotes Lebesgue measure and Ξ(t) is a measurevalued population process. Key forces of ecology and genetics can be captured within this common framework. Models covered incorporate both individual and event based births and deaths, oneforone replacement, immigration, independent ‘thinning ’ and independent or exchangeable spatial motion and mutation of individuals. Since birth and death probabilities can depend