  We show how highly-diverse ecological communities may display persistent
abundance fluctuations, when interacting through resource competition and
subjected to migration from a species pool. This turns out to be closely
related to the ratio of realized species diversity to the number of resources.
This ratio is set by competition, through the balance between species being
pushed out and invading. When this ratio is smaller than one, dynamics will
reach stable equilibria. When this ratio is larger than one, fixed-points are
either unstable or marginally stable, as expected by the competitive exclusion
principle. If they are unstable, the system is repelled from fixed points, and
abundances forever fluctuate. While marginally-stable fixed points are in
principle allowed and predicted by some models, they become structurally
unstable at high diversity. This means that even small changes to the model,
such as non-linearities in how resources combine to generate species' growth,
will result in persistent abundance fluctuations.
