  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.
