  How diversity is maintained in natural ecosystems is a long-standing question
in Theoretical Ecology. By studying a system that combines ecological dynamics,
heterogeneous interactions and spatial structure, we uncover a new mechanism
for the survival of diversity-rich ecosystems in the presence of demographic
fluctuations. For a single species, one finds a continuous phase transition
between an extinction and a survival state, that falls into the universality
class of Directed Percolation. Here we show that the case of many species with
heterogeneous interactions is different and richer. By merging theory and
simulations, we demonstrate that with sufficiently strong demographic noise,
the system exhibits behavior akin to the single-species case, undergoing a
continuous transition. Conversely, at low demographic noise, we observe unique
features indicative of the ecosystem's complexity. The combined effects of the
heterogeneity in the interaction network and migration enable the community to
thrive, even in situations where demographic noise would lead to the extinction
of isolated species. The emergence of mutualism induces the development of
global bistability, accompanied by sudden tipping points. We present a way to
predict the catastrophic shift from high diversity to extinction by probing
responses to perturbations as an early warning signal.
