Linking Population-Size-Dependent and Controlled Branching Processes

TL;DR

This paper explores the relationship between population-size-dependent and controlled branching processes, establishing conditions for their equivalence and analyzing how they approximate each other as initial populations grow.

Contribution

It develops conditions for equivalence between PSDBPs and CBPs, especially DCBPs, and provides bounds on their differences as initial population size increases.

Findings

Conditions for PSDBP and CBP equivalence established

Upper bounds on total variation distance derived

As initial population grows, models become increasingly similar

Abstract

Population-size dependent branching processes (PSDBP) and controlled branching processes (CBP) are two classes of branching processes widely used to model biological populations that exhibit logistic growth. In this paper we develop connections between the two, with the ultimate goal of determining when a population is more appropriately modelled with a PSDBP or a CBP. In particular, we state conditions for the existence of equivalent PSDBPs and CBPs, we then consider the subclass of CBPs with deterministic control functions (DCBPs), stating a necessary and sufficient condition for DCBP-PSDBP equivalence. Finally, we derive an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size, and show that under certain conditions this bound tends to zero as the initial population size becomes large.