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2012 Nonlinear historical superprocess approximations for population models with past dependence
Sylvie Méléard, Viet Chi Tran
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Electron. J. Probab. 17: 1-32 (2012). DOI: 10.1214/EJP.v17-2093


We are interested in the evolving genealogy of a birth and death process with trait structure and ecological interactions. Traits are hereditarily transmitted from a parent to its offspring unless a mutation occurs. The dynamics may depend on the trait of the ancestors and on its past and allows interactions between individuals through their lineages. We define an interacting historical particle process describing the genealogies of the living individuals; it takes values in the space of point measures on an infinite dimensional càdlàg path space. This individual-based process can be approximated by a nonlinear historical superprocess, under the assumptions of large populations, small individuals and allometric demographies. Because of the interactions, the branching property fails and we use martingale problems and fine couplings between our population and independent branching particles. Our convergence theorem is illustrated by two examples of current interest in biology. The first one relates the biodiversity history of a population and its phylogeny, while the second treats a spatial model where individuals compete through their past trajectories.


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Sylvie Méléard. Viet Chi Tran. "Nonlinear historical superprocess approximations for population models with past dependence." Electron. J. Probab. 17 1 - 32, 2012.


Accepted: 18 June 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1258.60051
MathSciNet: MR2946154
Digital Object Identifier: 10.1214/EJP.v17-2093

Primary: 60J80
Secondary: 60J68 , 60K35

Keywords: Evolution models , Genealogical interacting particle system , limit theorem , Nonlinear historical superprocess

Vol.17 • 2012
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