Community
Drag a knob and the eigenvalues move. Push self-limitation to 0 and
watch the rightmost eigenvalue cross into the unstable half-plane — a Hopf bifurcation,
the predator–prey loop breaking into oscillation.
Eigenvalue spectrum (per day)
Each dot is an eigenvalue of the community matrix on the complex
plane. Left of the dashed line is stable (perturbations decay); the rightmost
dot is the verdict. Height = oscillation frequency (predator–prey ringing).
Community matrix — ∂(growthᵢ)/∂(biomassⱼ)
green = + (j helps i),
red = − (j harms i). The diagonal is
self-regulation; the predator row/column shows top-down control.
Keystones — who moves the web when pressed (normalised)
From −J⁻¹: a sustained nudge to a species shifts the whole
equilibrium by its column. Long bar ⇒ pressing it ripples far (Bender press perturbation).