How diverse natural communities stay stable despite May's random-matrix bound
In plain words
In 1972 Robert May showed that a community with many randomly interacting species should almost always be unstable, so that small disturbances grow. Real ecosystems are very diverse and persist, so real interactions must violate some assumption of his argument.
Precise statement
For a community matrix $J$ with $S$ species, connectance $C$, interaction standard deviation $\sigma$ and self-regulation d, random-matrix theory gives stability only if $\sigma \sqrt{S C} < d$, with refinements for correlated pairs (predator-prey or competitive). Given empirical interaction strengths and abundances ($J_{ij} = -N_i^{*} (\delta_{ij} + \alpha_{ij})$ at a feasible equilibrium $N^{*}$), determine which structural property (feasibility filtering during assembly, interaction-strength distribution, pair correlations, trophic coherence, consumer-resource structure) controls the leading eigenvalue of real communities. An answer is an identified property with a quantitative test on measured networks showing it accounts for observed stability at observed $S$ and $\sigma$.
What would settle it
Measurement of full interaction matrices in several replicated communities, compared with random-matrix and assembly-model predictions for the spectral edge with and without each candidate structure.