Diversity–Stability: May's Ecological Network Simulator

Community Network

Species richness S 16

Connectance C 0.30

Interaction strength σ 0.35

Self-regulation d 1.00

Stability Diagnostics

Stabilizing…

Stability radius R = σ√(SC)

0.00

Growth rate λ (measured)

0.00

Mean links / species

0.0

R − d (theory)

0.00

How it works

Each of the S spheres is a species; its population deviates from equilibrium by amount x_i(t). Near equilibrium, deviations evolve under the linearized community model (Robert May, 1972):

dx/dt = A x
A_ii = −d                       (self-regulation / limited resources)
A_ij = σ·N(0,1) with prob. C    (random interaction, i ≠ j)
A_ij = 0 otherwise               (no direct link)

Every animation frame numerically integrates this system (Euler steps, clamped dt, renormalized each step to prevent overflow) and accumulates ln‖x(t)‖ over elapsed simulated time. That running average converges to the real part of the dominant eigenvalue of A — the same quantity power iteration extracts — shown live as growth rate λ. λ < 0 means every perturbation decays back to equilibrium (stable); λ > 0 means it grows without bound (unstable).

May's celebrated result: for a large random network, the off-diagonal eigenvalues fill a disk of radius R = σ√(S·C) in the complex plane (the circular law), so the system is almost certainly stable only when R < d. This is the diversity–stability paradox — piling on more species (S), more connections (C), or stronger interactions (σ) makes a randomly wired ecosystem statistically less likely to be resilient, the opposite of the old "diversity begets stability" intuition. Real food webs dodge the trap through non-random structure: modularity, a skew toward many weak links and few strong ones, and feedback loops shaped by natural selection rather than chance.

S / C / σ / d sliders — rebuild the random community matrix live; watch the verdict flip as R crosses d.
New random network — redraws a fresh random matrix at the current settings (same statistics, different realization).
Perturb populations — kicks the current populations away from equilibrium without changing the network, so you can watch recovery (or collapse) in isolation.

Facilitation (A_ij > 0) Antagonism (A_ij < 0) Growth-rate trend λ(t)

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