  We study models of biological evolution and investigate a key factor to yield
self-organized criticality (SOC). The Bak-Sneppen (BS) model is the most basic
model that shows an SOC state, which is developed based on minimal and
plausible assumptions of Darwinian competition. Another class of models, which
have population dynamics and simple rules for species migrations, has also been
studied. It turns out that they do not show an SOC state although the
assumptions made in these models are similar to those in the BS model. To
clarify the origin of these differences and to identify a key ingredient of
SOC, we study models that bridge the BS model and the Dynamical Graph model,
which is a representative of the population dynamics models. From a comparative
study of the models, we find that SOC is found when the fluctuations of the
number of species $N$ are suppressed, while it shows off-critical states when
$N$ changes according to its evolutionary dynamics. This indicates that the
assumption of the fixed system size in the BS model plays a pivotal role to
drive the system into an SOC state, and casts doubt on its applicability to
actual evolutionary dynamics.
