Height and the total mass of the forest of genealogical trees of a large population with general competition

TL;DR

This paper analyzes branching processes with competition in continuous time, exploring conditions under which the genealogical trees of large populations have finite height and total mass, highlighting the impact of interaction strength.

Contribution

It provides a theoretical framework for understanding how competition influences the finiteness of genealogical tree height and mass in large populations.

Findings

Strong competition can lead to finite extinction times.

The total mass of genealogical trees can remain finite under certain conditions.

The model applies to both integer-valued and continuous population sizes.

Abstract

We consider branching processes with interaction in continuous time, both with values in the integers and in the reals (in the second case we restrict ourselves to continuous processes), which model the evolution of the size of a population. We assume that for large population size the interaction is of the type of a competition, which limits the size of the population. We discuss in which cases the interaction is strong enough so that the extinction time (or equivalently the height of the forest of genealogical trees) remains finite, as the number of ancestors tends to infinity, or even such that the length of the forest of genealogical trees (which in the case of continuous state is rather called its total mass) remains finite, as the ancestral population size tends to infinity.