Ecological Tipping Points

Fold (saddle-node) bifurcations govern tipping points in shallow lakes, coral reefs, and grasslands. Model: dx/dt = ax²/(1+x²) − bx − r, where x is biomass, r is external stress (harvesting/pollution). Two saddle-node bifurcations bound an S-shaped region of bistability — one stable (high biomass), one unstable, one stable (collapsed). As stress r increases past r_max, the upper stable branch vanishes: catastrophic collapse. Recovery requires reducing r below r_min — a lower threshold — creating hysteresis. The black loop traces the bifurcation diagram. Early warning: variance and autocorrelation rise as the system slows near the fold (critical slowing down).