Beaver Ecosystem 2D: dam section, flow balance and trout-eagle phase plane

Dam cross-section: water level, hydrostatic pressure, weir and seepage

Schematic side view through the pond centre (not to scale horizontally). Arrows on the dam face show pressure p = rho g depth; blue arrows above the crest are weir flow, dashed arrows are seepage.

Hydrograph (last 10 days)

Rating curve: outflow vs pond level

Where outflow meets the inflow line the pond is in equilibrium.

Trout phase plane: growth rate dN/dt vs trout number N

Positive rate: trout grow; negative: eagle predation wins. Circles mark equilibria (filled = stable).

Trout, carrying capacity and meadow (last 40 days)

Controls

Dam height (courses) Mud seal Rain runoff Eagle hunting pressure Plant growth Time speed

Why: raising the dam lifts the crest so the pond holds more water and the waterfall shrinks; the mud seal cuts seepage; rain adds inflow; eagles take trout with a saturating (Holling II) appetite.

The science

This page runs the same model code as the 3D lab (model.js, eco.js copied unchanged); only the pond basin is a simple bowl instead of the baked valley, so numbers match in character but not exactly.

Pond balance: dV/dt = Q_in - Q_out(h) - E A(h), solved by an implicit (backward-Euler) step so it is stable at any time step. Q_in = seasonal base flow + snow-melt pulse + storm pulses + rain runoff.
Weir: broad-crested weir Q = C_w L H^3/2 with C_w about 1.6 (typical 1.5-1.7); the notch is 0.12 m lower than the crest.
Seepage: Dupuit-Darcy flow through the stick-and-mud body, Q_s = K (1 - seal) W (H² - h_T²) / (2T).
Hydrostatic pressure: p = rho g d on the dam face; the resultant per metre of wall is rho g H²/2.
Trout: dN/dt = r N (1 - N/K) - E a N / (1 + a h N): logistic growth plus a type-II predator. K scales with pond volume, dissolved oxygen (Elmore-Hayes saturation polynomial) and water plants. The 3D version has the eagle hunting only in clear daylight; here the slider is the time-averaged pressure.
Meadow: logistic regrowth dg/dt = r g (1 - g) with r reduced by cold and shade.
Simplifications: constant evaporation of 4 mm/day, bowl-shaped basin, single trout stock, no beaver build/cut events here. Constants marked [assumption] in the code are illustrative.

Independent checks (button above): the pond ODE against the closed-form weir level, trout integration against the equilibrium root, and the volume mass balance.