  When can complex ecological interactions drive an entire ecosystem into a
persistent non-equilibrium state, where species abundances keep fluctuating
without going to extinction? We show that high-diversity spatially-extended
systems, in which conditions vary somewhat between spatial locations, can
exhibit chaotic dynamics which persist for extremely long times. We develop a
theoretical framework, based on dynamical mean-field theory, to quantify the
conditions under which these fluctuating states exist, and predict their
properties. We uncover parallels with the persistence of externally-perturbed
ecosystems, such as the role of perturbation strength, synchrony and
correlation time. But uniquely to endogenous fluctuations, these properties
arise from the species dynamics themselves, creating feedback loops between
perturbation and response. A key result is that the fluctuation amplitude and
species diversity are tightly linked, in particular fluctuations enable
dramatically more species to coexist than at equilibrium in the very same
system. Our findings highlight crucial differences between well-mixed and
spatially-extended systems, with implications for experiments and their ability
to reproduce natural dynamics. They shed light on the maintenance of
biodiversity, and the strength and synchrony of fluctuations observed in
natural systems.
