May's Stability Limit — Random Food Web Simulator

Random Food Web

Species count S 14

Connectance C 0.30

Interaction strength σ 0.25

Self-regulation d 1.00

Stability Diagnosis

—

σ√(SC)

0.00

vs. d

1.00

Surviving species

14 / 14

Mean population

1.00

Sim time

0.0

Links drawn

How it works

Each species' abundance N_i follows a generalized (multi-species) Lotka-Volterra equation, integrated live with RK4:

dNi/dt = Ni · ( ri − d·Ni + Σj Aij·Nj )

ri  = intrinsic growth rate (random, per species)
d   = self-regulation / intraspecific crowding
Aij = random interaction matrix entry, nonzero
      with probability C (connectance), drawn
      from a Normal(0, σ²) distribution

This is the random-matrix food web studied by Robert May (Nature, 1972): instead of hand-picking who eats whom, interactions are wired randomly with connectance C and typical strength σ. May's classic result, using Girko's circular law for the eigenvalues of a large random matrix, is that the equilibrium is expected to be linearly stable when

σ · √(S · C)  <  d

i.e. the rightmost eigenvalue of the community (Jacobian) matrix has real part ≈ σ√(SC) − d. Push species count, connectance or interaction strength up (or self-regulation down) and this margin crosses zero — May's famous, counter-intuitive "complexity begets instability" result, in contrast to the older ecological intuition that more species and more links make a web more robust. (Real food webs are not fully random — predator-prey sign structure tends to be more stabilizing than random signs, a refinement due to Allesina & Tang, 2012 — but the connectance/strength trade-off shown here is the same one driving both results.)

Species count / Connectance / Interaction strength — set S, C and σ, then Reseed draws a fresh random community matrix and restarts the populations near their non-interacting equilibrium N_i* = r_i/d.
Self-regulation d — how strongly each species suppresses its own growth (crowding); raising it stabilizes the web without changing who interacts with whom.
Reset populations — restarts the ODE from near-equilibrium with the current matrix, without redrawing it.
Node height/size = current population; green→red = healthy→collapsing. Lines = nonzero interactions, green tinted for a net positive effect on the pair, red for net negative, brighter/thicker for larger |A_ij|.

Healthy population Collapsing / extinct Net-positive link Net-negative link

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