Nonlinear historical superprocess approximations for population models with past dependence

TL;DR

This paper develops a nonlinear historical superprocess framework to approximate complex population models with trait inheritance, past dependence, and ecological interactions, providing insights into biodiversity and spatial competition.

Contribution

It introduces a novel nonlinear superprocess approximation for genealogies with past dependence, extending classical models to include interactions and trait history.

Findings

Convergence of the particle process to the superprocess under large population assumptions

Application to biodiversity history and phylogeny modeling

Application to spatial competition models with past trajectory interactions

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.