Galton-Watson processes in varying environment and accessibility percolation

TL;DR

This paper studies branching processes with changing environments, providing conditions for survival or extinction based on offspring moments, and applies these to models of biological evolution and accessibility percolation.

Contribution

It introduces new criteria for survival or extinction in varying environment branching processes and extends these to models with selection and accessibility percolation.

Findings

Derived sufficient conditions for survival/extinction based on moments

Applied results to processes with selection and accessibility percolation

Linked branching processes to biological evolution models

Abstract

This paper deals with branching processes in varying environment, namely, whose offspring distributions depend on the generations. We provide sufficient conditions for survival or extinction which rely only on the first and second moments of the offspring distributions. These results are then applied to branching processes in varying environment with selection where every particle has a real-valued label and labels can only increase along genealogical lineages; we obtain analogous conditions for survival or extinction. These last results can be interpreted in terms of accessibility percolation on Galton-Watson trees, which represents a relevant tool for modeling the evolution of biological populations.