  Highly-diverse ecosystems exhibit a broad distribution of population sizes
and species turnover, where species at high and low abundances are exchanged
over time. We show that these two features generically emerge in the
fluctuating phase of many-variable model ecosystems with disordered species
interactions, when species are supported by migration from outside the system
at a small rate. We show that these and other phenomena can be understood
through the existence of a scaling regime in the limit of small migration, in
which large fluctuations and long timescales emerge. We construct an exact
analytical theory for this asymptotic regime, that provides scaling predictions
on timescales and abundance distributions that are verified exactly in
simulations. In this regime, a clear separation emerges between rare and
abundant species at any given time, despite species moving back and forth
between the rare and abundant subsets. The number of abundant species is found
to lie strictly below a well-known stability bound, maintaining the system away
from marginality. At the same time, other measures of diversity, which also
include some of the rare species, go above this bound.
  In the asymptotic limit where the migration rate goes to zero, trajectories
of individual species abundances are described by non-Markovian jump-diffusion
processes, which proceeds as follows: A rare species remains so for some time,
then experiences a jump in population sizes after which it becomes abundant (a
species turnover event) and later sees its population size gradually decreasing
again until rare, due to the competition with other species. The asymmetry of
abundance trajectories under time-reversal is maintained at small but finite
migration rate. These features may serve as fingerprints of endogenous
fluctuations in highly-diverse ecosystems.
