The science
Light. Beer-Lambert attenuation: I(z) = I0 e−Kd z. Clear oceanic water has Kd ≈ 0.05–0.1 /m, coastal bays 0.2–0.5, murky estuaries > 1. The 1 % light depth is ln(100)/Kd ≈ 4.6/Kd.
Photosynthesis. A saturating P–I curve (Jassby–Platt): P = Pmax tanh(I / Ik), with Ik the saturation onset. Over a day with sinusoidal sunlight of length D, N(z) = ∫0D Pmax tanh(I(z,t)/Ik) dt − 24 R. The compensation depth zc is where N = 0 (production only pays for respiration); below it a meadow cannot persist. For a flat-topped day the root is exact: zc = ln( I0 / (Ik atanh(24R/(Pmax D))) ) / Kd. The page checks that closed form against numerical bisection, and the sinusoidal day against Simpson and midpoint quadrature.
Standing crop is assumed proportional to N(z)/N(0), reduced near the shore by an exponential depletion term (prior grazing) — an illustrative simplification, not a measured relation.
Diving. Resting metabolism follows Kleiber, 3.4 M0.75 W, scaled by 0.4 here because sirenians have unusually low metabolic rates; diving costs a factor f. The aerobic dive limit is ADL = (usable O2 store) / (O2 consumption) with 20.1 J per mL O2. A real dugong usually dives for 1–3 minutes, far below the ADL, because foraging economics, not oxygen, sets the dive.
Foraging (Charnov marginal value theorem). Energy gained at depth z saturates with bottom time tb: G = Gmax(z) (1 − e−tb/τ); a cycle lasts T = 2z/v + tb + tsurface with tsurface = t0 + ρ(dive time). The best tb maximises G/T, and the best depth balances shorter commute against sparser or depleted grass. Gmax, τ, ρ and the energy per bite are illustrative round numbers.
Simplifications: 1-D light attenuation, no tides, waves or shading by blades, no seagrass species differences, no nutrients; a single vertical dive per cycle; the dugong's body, swim speed and breath recovery are schematic. Real dugongs also graze rhizomes and leave visible feeding trails; here grazed tufts shrink then regrow only for illustration.
Independent educational model; the dugong and seagrass are stylised illustrations.