May-Wigner Stability Criterion (1972)
Stability iff: σ√(SC) < |d|
M = random matrix, M_ij ~ N(0, σ²) w/ prob C
M_ii = d (self-regulation)
Spectral radius ρ(M) → σ√(SC) as S→∞
(Wigner semicircle law)
Stable ⟺ all Re(λ) < 0
⟺ ρ(M-dI) + d < 0
Robert May (1972) showed that complexity destabilizes ecosystems — larger, more connected communities are harder to maintain. This contradicts intuition from simple food webs, and remains debated (structured interaction matrices can stabilize large systems).