  In this work we study the stability of the equilibria reached by ecosystems
formed by a large number of species. The model we focus on are Lotka-Volterra
equations with symmetric random interactions. Our theoretical analysis,
confirmed by our numerical studies, shows that for strong and heterogeneous
interactions the system displays multiple equilibria which are all marginally
stable. This property allows us to obtain general identities between diversity
and single species responses, which generalize and saturate May's bound. By
connecting the model to systems studied in condensed matter physics, we show
that the multiple equilibria regime is analogous to a critical spin-glass
phase. This relation provides a new perspective as to why many systems in
several different fields appear to be poised at the edge of stability and also
suggests new experimental ways to probe marginal stability.
