A coral reef cross-section driven by grazing: see why a healthy reef can flip to algae and not flip back.
| Total grazing g (fish + urchins) | - |
| Coral-mortality d (with stress) | - |
| Bistable window of g (analytic) | - |
| Regime at the current g | - |
| Coral-only equilibrium C* = 1 − d/r | - |
Between the two dotted lines the reef has two stable states: the same grazing can mean a coral reef or an algal flat, depending on its history. Climate stress shifts both lines.
| Lower tipping g (coral → algae) | - |
| Upper tipping g (algae → coral) | - |
| Coral-only state | - |
| Cover sum conserved | - |
Model. Cover fractions of coral C, macroalgae M and algal turf T (M + C + T = 1) followdM/dt = a·M·C − g·M/(M+T) + γ·M·TdC/dt = r·T·C − d·C − a·M·C, T = 1 − M − C.
Macroalgae overgrow coral (a), spread over turf (γ), and are removed by grazers at rate g (per unit of grazeable algal area M+T). Coral recruits and grows into turf (r) and dies (d).
Grazing. g = gfish + 0.5·U/(U+1), a saturating urchin term (U in individuals/m²). Urchins follow logistic growth to the carrying capacity K (rate 0.6/yr) and collapse after a disease event, as Diadema did in the Caribbean in 1983. The urchin food-web feedback is not modelled.
Tipping points. Two closed forms: with no macroalgae the coral state has C* = 1 − d/r; it loses stability to macroalgal invasion when g < (1−C*)(a·C* + γ(1−C*)). The algal state M* = 1 − g/γ lets coral invade when g > γ(d + a)/(r + a). Between them the reef is bistable (hysteresis). The page checks both thresholds against full RK4 simulations.
Honest limits. A mean-field (well-mixed) model: no space, waves or depth. The parameters are illustrative values in the spirit of the published Mumby–Hastings–Edwards (2007) model, not fitted data. The 3D cells are a display of the fractions (each cell is coral/algae/turf according to the fractions), not a separate spatial simulation; each urchin marker stands for several animals.