Macroscopic and microscopic structures of the family tree for decomposable critical branching processes

TL;DR

This paper analyzes the detailed structure of family trees in a decomposable, strongly critical multitype Galton-Watson process, modeling geographically structured populations with migration, focusing on the distribution of ancestors over time.

Contribution

It introduces a novel analysis of the family tree structure and ancestor distributions in a decomposable critical branching process with migration.

Findings

Characterization of the family tree structure

Distribution of the most recent common ancestor

Insights into population genealogy over time

Abstract

A decomposable strongly critical Galton-Watson branching process with $N$ types of particles labelled $1,2,...,N$ is considered in which a type~$i$ parent may produce individuals of types $j\geq i$ only. This model may be viewed as a stochastic model for the sizes of a geographically structured population occupying $N$ islands, the location of a particle being considered as its type. The newborn particles of island $i\leq N-1$ either stay at the same island or migrate, just after their birth to the islands $% i+1,i+2,...,N$. Particles of island $N$ do not migrate. We investigate the structure of the family tree for this process, the distributions of the birth moment and the type of the most recent common ancestor of the individuals existing in the population at a distant moment $n.$