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Highly-diverse ecosystems exhibit a broad distribution of population sizesand species turnover, where species at high and low abundances are exchangedover time. We show that these two features generically emerge in thefluctuating phase of many-variable model ecosystems with disordered speciesinteractions, when species are supported by migration from outside the systemat a small rate. We show that these and other phenomena can be understoodthrough the existence of a scaling regime in the limit of small migration, inwhich large fluctuations and long timescales emerge. We construct an exactanalytical theory for this asymptotic regime, that provides scaling predictionson timescales and abundance distributions that are verified exactly insimulations. In this regime, a clear separation emerges between rare andabundant species at any given time, despite species moving back and forthbetween the rare and abundant subsets. The number of abundant species is foundto lie strictly below a well-known stability bound, maintaining the system awayfrom marginality. At the same time, other measures of diversity, which alsoinclude some of the rare species, go above this bound.  In the asymptotic limit where the migration rate goes to zero, trajectoriesof individual species abundances are described by non-Markovian jump-diffusionprocesses, which proceeds as follows: A rare species remains so for some time,then experiences a jump in population sizes after which it becomes abundant (aspecies turnover event) and later sees its population size gradually decreasingagain until rare, due to the competition with other species. The asymmetry ofabundance trajectories under time-reversal is maintained at small but finitemigration rate. These features may serve as fingerprints of endogenousfluctuations in highly-diverse ecosystems.

Many-species ecological fluctuations as a jump process from the brink of extinction