Each of the S species follows an independent mean-reverting stochastic process (an Ornstein–Uhlenbeck walk) around its own baseline abundance μᵢ:
dxᵢ = -θ·xᵢ·dt + σ·dWᵢ
Pᵢ(t) = μᵢ·(1 + xᵢ(t)) — population of species i
B(t) = Σ Pᵢ(t) — total community biomass
The driving noise dWᵢ is a blend of a shared shock (drought, disease, temperature swing) and an independent shock, mixed by the synchrony slider ρ:
dWᵢ = ρ·dW_common + √(1-ρ²)·dW_i,independent
When species respond independently (low ρ), their ups and downs average out — this is the portfolio effect (also called the insurance hypothesis): biodiversity buffers the aggregate ecosystem function even though every individual species is just as volatile as before. For S equally-volatile species with pairwise correlation ρ, the coefficient of variation of the total is:
CV_total / CV_species = √[ (1 + (S-1)·ρ) / S ]
- S (richness) — more independent species pack more statistical averaging into the total; the theoretical ratio shrinks as 1/√S when ρ = 0.
- ρ (synchrony) — when every species reacts to the same driver (ρ → 1) the portfolio effect vanishes and CV_total → CV_species, no matter how many species there are.
- σ (volatility) — scales every species' swings equally; it changes the absolute size of the fluctuations but not the stabilization ratio.
Real-world relevance: this is the mechanism behind why species-rich grasslands, reefs and fisheries show steadier total productivity year to year than any single dominant species would on its own (Tilman 1999; Doak et al. 1998) — and why correlated stressors like climate change, which synchronize species' responses, erode that buffering even without a single extinction.