MY SIMULATOR

Arctic Fox Year

A white coat on bare ground, a brown coat on snow — watch the mismatch grow.

INTERACTIVE 3D

Date 1 JanAir 0 °CDaylight 0 hSnow 0%Coat white 0%Mismatch 0%FPS 0

Arctic Fox Year Lab: Coat Moult, Snow Cover Mismatch and the Energy Budget

An arctic fox changes its coat on a schedule set by day length, but the snow arrives and leaves on a schedule set by temperature. Warm the climate and the two drift apart. Scrub through the year, change the climate, and read how many days the fox is the wrong colour, and what that does to its hunting and energy budget.

Scenario

Time speed

Right now matched

The fox is on tundra.

The year at a glance

Blue: fraction of ground under snow. White: coat whiteness. Red: mismatch = |coat - snow|. The dashed line is today.

Energy budget (kJ per day)

Need follows Scholander heat loss (coat conductance times body-air temperature gap, floored at the basal rate). Intake is hunts per day times success times energy per lemming. The lower plot is the running balance in MJ.

1 · Climate

2 · Snow

Degree-day model: snow accumulates when the day is below 0 °C; above 0 °C it melts by DDF x T each day.

3 · Moult

The coat relaxes toward white once days shorten below the autumn trigger, and toward brown once they lengthen past the spring trigger. The relaxation takes weeks.

4 · Hunting

Experiment (measured by running the whole model)

Run the sweep: it recomputes a full year at every warming step with your current settings.

Independent numerical checks

Also run outside the browser: node src/check.js (day length integral, degree-day closed form, moult exponential, Scholander lower critical temperature, snow mass balance, monotone sweep).

Educational model. Temperature, precipitation, conductances, hunting effort and the effect of mismatch on capture success and predation risk are illustrative [assumptions] chosen to give plausible magnitudes, not fitted to field data. The thing to take from the model is the mechanism: a fixed photoperiod cue plus a warming-driven snow calendar produces growing mismatch.

Controls

Drag / one finger: orbit; wheel / pinch: zoom; Follow, Close-up and Overview buttons.
Drag the day-of-year slider to jump to any date; Pause stops time.
Every slider recomputes the entire year instantly; the charts and the 3D scene both read the same numbers.
Low quality (button or ?q=low) uses a smaller shadow map and a pixel ratio of 1.

The science

Day length. Declination δ = 23.44° sin(2π(d-80)/365); the hours of daylight are (24/π) arccos(-tan φ tan δ), clamped to 0 h (polar night) or 24 h (midnight sun). Refraction is ignored. This is the cue the moult follows.

Temperature. An annual cosine, coldest on day 23, with a mean and amplitude that vary with latitude, plus your warming offset. This is an illustrative climatology, not a station record.

Snow (degree-day model). Snow water equivalent S grows by the snowfall rate on days below 0 °C and shrinks by DDF · T on days above 0 °C (DDF typically 2-6 mm w.e. per °C per day for tundra). Cover is min(1, S/25 mm). It ignores wind redistribution, sublimation, radiation and rain-on-snow.

Moult. Whiteness w relaxes toward a target (1 white, 0 brown) with time constant τ: w(t+1) = w + (target - w)(1 - e^-1/τ). The target flips to white when the days are shortening and below the autumn threshold, and to brown when they are lengthening and above the spring threshold. Real moult is also modulated by temperature and condition; here it is photoperiod only (or snow, if you switch the cue). Note that even a perfect snow cue lags by the moult time, which is why switching the cue does not remove the mismatch.

Mismatch. m = |w - cover|, between 0 (matched) and 1 (white fox on bare ground, or brown fox on snow). Days with m > 0.3 are counted as mismatched.

Energy. Scholander model: heat production M = max(BMR, C (T_b - T_a)), where the conductance C interpolates between a thin summer coat and a thick winter coat. The lower critical temperature is T_b - BMR/C; with the defaults it is about -42 °C in the full winter coat, in the range reported for arctic foxes (typical value, please spot-check). Daily need is the activity factor times M times 86.4 kJ per W-day.

Hunting. Intake per day is attempts · s · E_lemming with capture success s = s₀(1 - γ m). The reduction of success with mismatch (γ) and a relative predation-risk index 1 + 2m are assumptions: field studies of other species (for example snowshoe hares) find survival falls when coat and background mismatch, but the size of the effect for foxes is not given by this model.

Not modelled: wind chill, sea-ice foraging, scavenging, den use, fat as insulation, population dynamics and the lemming cycle, evolutionary adaptation, regional snow patchiness.