Asymptotic regimes for the partition into colonies of a branching process with emigration

TL;DR

This paper studies the asymptotic behavior of how a spatial branching process with emigration partitions into colonies, revealing convergence to a random measure under large population and rare migration conditions.

Contribution

It introduces a new asymptotic analysis framework for the partition into colonies in spatial branching processes with emigration, linking it to Poisson point measures.

Findings

Weak convergence of the rescaled partition to a Poisson-based random measure

Derivation of a simple integral equation for the cumulant of the limiting measure

Asymptotic regimes characterized for large populations with rare migrations

Abstract

We consider a spatial branching process with emigration in which children either remain at the same site as their parents or migrate to new locations and then found their own colonies. We are interested in asymptotics of the partition of the total population into colonies for large populations with rare migrations. Under appropriate regimes, we establish weak convergence of the rescaled partition to some random measure that is constructed from the restriction of a Poisson point measure to a certain random region, and whose cumulant solves a simple integral equation.