Diversity–Resilience theorem

This page states two results connecting diversity to resilience under explicit assumptions.

I. Stochastic insurance (variance reduction)

Setup. For i=1..n let x_i follow Ornstein–Uhlenbeck dynamics around means μ_i: $\mathrm {d} x_{i}=-a(x_{i}-\mu _{i})\,\mathrm {d} t+\sigma \,\mathrm {d} W_{i}$, with a>0, σ>0, and $\mathrm {Corr} (\mathrm {d} W_{i},\mathrm {d} W_{j})=\rho _{ij}$. Define output $Y=\sum _{i}w_{i}x_{i}$. At stationarity $\mathrm {Var} [Y]={\tfrac {\sigma ^{2}}{2a}}\left(\sum _{i}w_{i}^{2}+\sum _{i\neq j}w_{i}w_{j}\rho _{ij}\right)$.

Theorem (insurance). If increasing diversity lowers the weighted average correlation ${\bar {\rho }}={\frac {\sum _{i\neq j}w_{i}w_{j}\rho _{ij}}{\sum _{i\neq j}w_{i}w_{j}}}$, then $\mathrm {Var} [Y]$ decreases monotonically, so any variance-based resilience metric increases.

II. Deterministic niche-spread (stability bound)

Setup. Linearize near equilibrium: ${\dot {\boldsymbol {x}}}=J{\boldsymbol {x}}$ with $J=-aI+A$, a>0, $A_{ii}=0$, $A_{ij}\leq 0$, and $|A_{ij}|\leq \phi (d_{ij})$ where d_{ij} is trait/functional distance and φ is nonincreasing.

Let $R_{\max }=\max _{i}\sum _{j\neq i}|A_{ij}|$. By Gershgorin, $\mathrm {Re} \,\lambda (J)\leq -a+R_{\max }$ for all eigenvalues.

Theorem (niche-spread). If functional diversity increases (pairwise distances do not shrink), then each $|A_{ij}|$ weakens and $R_{\max }$ decreases, so the spectral abscissa $s(J)=\max \mathrm {Re} \,\lambda (J)$ decreases (faster return to equilibrium ⇒ higher resilience).

Notes

• “Diversity” here means lower response correlation and/or larger trait distances that weaken competitive overlap.
• Results are sufficient, not necessary; with strong mixed-sign random interactions, diversity can destabilize (outside these assumptions).