Abstract
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 c\`adl\`ag 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 with competition between individuals through their past trajectories.
Highlights
The evolution of genealogies in population dynamics is a major problem, which motivated an abundant literature and has applications to evolution and population genetics
We are interested in keeping track of the genealogies of individuals with small weights, in large populations with allometric demographies
We introduce a parameter n which scales the size of the population
Summary
The evolution of genealogies in population dynamics is a major problem, which motivated an abundant literature and has applications to evolution and population genetics. Let us remark that our model allows the description of both genealogies and population densities in the forward physical time, with variable population size including extinction phenomena This can be a first step to reconstruct the past biodiversity from the phylogenies of living species with ecological interactions. Barton, Etheridge and Veber [4] study the genealogies of a spatial version of Λ-Fleming-Viot processes and of their various limits All these models allow the incorporation of selection and mutation This example suggests that the historical processes may allow one to understand evolution without the assumption of rare mutations and time scale separation. These spaces are embedded with the topology of weak convergence
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