Night 1Phase —Clock —Out 0 / 0Cave air — °CMean fat — gr_max — mFPS 0

Drag to orbit, wheel / pinch to zoom. Flying bats are drawn 12x life size, hanging bats 4x and insects about 45x so you can see them; wing beats are slowed for display. Hanging bats glow warm when euthermic and cool blue when in torpor.

Bat Colony: Cave Heat Balance, Torpor and Sonar Foraging Budget

A cave colony leaves at dusk to hunt insects by echolocation. Does the night's catch pay for the flight and the day's roost?

Echolocation (sonar equation)

Foraging night

Roost and cave heat balance

Live readouts

Detection range vs call frequency

Cave air temperature vs colony size

Mean fat reserve per night

Independent numerical checks

Controls

Drag / one finger orbits, wheel / pinch zooms. The three view buttons jump to the cave mouth, the foraging valley or a single bat.
Time scale = simulated seconds per real second while foraging (the day roost is fast-forwarded). Fast-forward runs the rest of the night without drawing.
Sliders change the physics live; presets set several at once. Roost sliders act on the next roost period and on the charts at once.

The science

Sonar equation. Echo level at the bat: EL = SL + TS − 40 log₁₀(r/0.1 m) − 2αr (dB, spherical spreading out and back from the 0.1 m reference, atmospheric absorption α dB/m each way). The bat detects the insect while EL ≥ DT; r_max is found by bisection. With α = 0 and a large target: r_max = 0.1·10^{(SL+TS−DT)/40} m, which the page checks.

Frequency trade-off. Absorption grows about as f^1.8 (empirical, 1.3 dB/m at 40 kHz, 20 °C, 50 % RH, scaled by a simple humidity factor), so high frequencies reach less far. But a target smaller than the wavelength scatters weakly: TS = TS₀ + 10 log₁₀(ka⁴/(1+ka⁴)), k = 2π/λ, a = insect radius. Together they give a best call frequency for each prey size. Beam half-angle is taken as asin(0.61 λ/8 mm) (piston-like emitter, rough).

Encounters. A forward cone of range r and half-angle θ sweeps new volume at ρ v π r² sin²θ; the agent model should reproduce this rate for its detection events. Detection volume of the cone is (2π/3) r³ (1 − cos θ) (Monte Carlo check).

Flight cost. Pennycuick-style: induced + parasite + profile power, divided by muscle efficiency 0.23, plus basal metabolism. The profile term (8 W/kg) is a lumped illustrative constant, giving about 0.6 W for a 12 g bat, a plausible order of magnitude, not a species value.

Roost. Euthermic bat: P = max(BMR, G₁ n^−1/3 (37 − T_c)): a cluster of n bats exposes area ∝ n^2/3, so conductance per bat falls as n^−1/3. Torpor: body temperature max(T_c+1.5, 5 °C) and rate BMR·Q₁₀^{(T_b−37)/10} (Q₁₀ = 2.5), with a regulation floor in cold caves; arousal costs the heat to re-warm the body (3.4 J/g/K).

Cave air. The colony's heat plus rock exchange equals ventilation loss through the entrance: P_col(T_c) + G_w(T_rock − T_c) = ρ_ac_p V̇ (T_c − T_out), with stack flow V̇ = 0.6 A √(2 g H ΔT / T), H = 15 m. Solved by bisection; more bats warm the cave, which lowers each bat's own heating cost.

Simplifications: one prey size, no clutter, no jamming, no group effects on foraging, fixed prey energy. Numbers are typical orders of magnitude for a ~12 g insectivorous bat; check species data before quoting. No bats are harmed in the model: when a reserve runs out the page only reports a deficit. Models (bats, cave, valley, trees, insects) built with Blender and rendered with Three.js.