Why do ibex climb in summer and come down in winter? Slide the elevation and the day of the year: air cools with height, snow arrives, fresh grass climbs the slope, and the ibex pays or earns energy every day.
-Air temperature (daily mean)
-Snow depth / hoof sinks
-Forage quality (0 to 1)
-Heat production / loss (day)
-Net energy per day
-Best elevation today
-Climb cost to best elevation
-Water to dump heat
The science
What the model asks. An alpine ibex can choose its elevation each day. Higher is cooler (less heat stress, fresh plants arrive later) but colder, windier and snowier in the cold half of the year. The page computes the animal's daily energy balance at any elevation and day, then finds the elevation where it is best. The best elevation follows the season: that is a migration track.
1. Climate. Air temperature: T(z, d) = Tm + A cos(2π(d-200)/365) - Γ (z - 1500 m)/1000, with annual mean 5 °C and amplitude 10 K at 1500 m, and lapse rate Γ (the slider; the standard atmosphere uses 6.5 K/km). Snow depth is a stylised function of the 20-day-lagged temperature: h = max(0, -0.1 Tlag (1 + (z-1500)/2000)) up to 2.5 m. Wind rises with elevation: u = u0(1 + 0.6 (z-1500)/1000).
2. Heat budget of the body. Basal metabolic rate follows Kleiber: B = 3.39 M0.75 W (70 M0.75 kcal/day), scaled by a seasonal factor 0.70 to 1.0 because ibex are reported to lower winter metabolism (spot-check the size of this effect). Surface area A = 0.10 M2/3 m² (Meeh). Heat flows from skin (35 °C) through the coat and the outer air layer in series:
q = (Tskin - Tair) / (l/keff + 1/(hc + hr)), keff = 0.04 (1 + 0.08 u)
The coat thickness l goes from 1.2 cm in summer to 4.5 cm in winter (or set it by hand). Wind pushes through the coat (the 8 % per m/s factor is an assumption). The outside coefficient hc comes from the Churchill-Bernstein correlation for a cylinder (trunk, 0.45 m wide) in crossflow, Nu = 0.3 + 0.62 Re1/2 Pr1/3 [1 + (Re/282000)5/8]4/5 / [1 + (0.4/Pr)2/3]1/4; hr is a linearised radiation term. A share of absorbed sunshine reaches the skin (absorptivity 0.7, 800 W/m² clear noon, 0.3 A projected, 40 % through the coat). Day air is 4 K warmer and night air 4 K colder than the daily mean. If heat loss exceeds heat production the animal burns extra energy to stay warm; if production exceeds what the body can dump (loss with the coat opened up by half), it must rest in the shade and loses grazing time.
3. Forage. Quality Q is the larger of dry standing grass (0.22) and a green-up term exp(-((Tlag - 9)/5)²): the best young plants are where the lagged temperature is near 9 °C, a front that climbs the mountain in spring ("green wave"). Snow hides plants (access = exp(-h/0.35)), except on wind-scoured patches (up to 30 % of the ground, more with wind). Intake = Q × 2.2 B0 × 86400 s (a maximum of 2.2 times basal rate on perfect forage).
4. Moving and snow. Cost of transport (Taylor et al. 1982): C = 10.7 M-0.316 J per kg per metre, times a walking distance of 3 km/day divided by √Q (poor forage means longer walks). Hoof pressure p = Mg/(4 Ahoof) is compared with snow strength s: the hoof sinks by min(h (1 - s/p), 0.6 leg) and the walking cost is multiplied by 1 + 6 d/leg (an assumed linear law in the spirit of snow-wading studies on deer, to be spot-checked). Climbing a height Δz costs M g Δz / η with muscle efficiency η = 0.25, a few MJ for 1000 m, small against the daily budget: elevation changes are decided by the food and weather, not by the climb.
5. Net energy and the optimum. Net = intake - (B + walking + extra heat production). The page scans 1200 to 3400 m every 10 m and refines the maximum by a ternary search. The annual experiment (a stylised sum, not a prediction of real weight change) adds the daily net over a year for an animal staying at one elevation and for one that follows the best elevation each day (and pays the climbing cost); the fat equivalent uses 39 MJ per kg of fat at 80 % efficiency.
Honest simplifications.
- One average slope aspect, no cliffs for escaping predators or hiding places, no rutting season, no pregnancy, no lactation, no disturbance by people. Males and females do separate things in reality.
- The climate, snowpack, green-up law, intake maximum, heat-dump factors and all the percentage factors are teaching-grade round numbers, not measurements from one valley. Literature values (Kleiber, Meeh, Taylor, Churchill-Bernstein, Zukauskas) are typical; check them before quoting.
- Daily means with a fixed day/night swing; storms and cold snaps are not modelled. Real ibex lose weight in winter and live on fat; negative winter values are expected.
- The 3D goat is a rigid model with swinging legs; the winter coat is a texture swap and a slight fattening, not real hair. The mountain is an invented massif in true metres; no real peak is meant.