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Fold Bifurcation (Saddle-Node): Ecological systems can exhibit catastrophic tipping points via
fold bifurcations. The normal form is ẋ = r·x(1 − x/K) − u·x²/(x² + h²) where the first term
is logistic growth and the second is an Allee-type predation pressure. As the stress parameter u
increases slowly, the system has two stable equilibria (high and low vegetation) separated by
an unstable saddle. At the tipping point, the stable and unstable branches collide and
annihilate — a fold bifurcation — and the system jumps catastrophically to the alternative state.
This is hysteresis: the reverse transition requires reducing u far below the forward tipping point.
Early warning signals (EWS) include: critical slowing down (variance ↑, autocorrelation ↑)
as the system's recovery rate → 0 near the bifurcation. The bifurcation diagram shows all
equilibria (solid = stable, dashed = unstable) as a function of stress.