Air temperature falls with altitude at the environmental lapse rate L ≈ 6.5 °C/km. Each species is adapted to its own starting elevation h₀ — its optimal temperature is set once, at t=0, to whatever the lapse rate predicts at h₀ — and tolerates ±τ around it:
T(h,t) = T0(t) − L·h, T0(t) = T0(0) + warmingRate·(t/10 yr)
Topt_i = T0(0) − L·h0_i (fixed per species at t=0)
Comfort band_i(t) = [ (T0(t) − Topt_i − τ)/L , (T0(t) − Topt_i + τ)/L ]
Substituting Topt_i shows every species' band shifts upslope by the same Δ(t) = warmingRate·(t/10)/L metres, regardless of where it started:
bandLo_i(t) = h0_i + Δ(t) − τ/L
A species goes extinct once its band's lower edge is pushed past the summit (no cooler ground remains at any elevation). Because Δ(t) is identical for every species, the one that started closest to the summit has the least headroom and always runs out first — the "escalator to extinction" measured across the whole community, not just one species:
extinctionYear_i ≈ 10·L·(Hpeak − h0_i)/warmingRate + 10·τ/warmingRate
Population per species follows logistic growth toward the carrying capacity implied by the conical mountain's shrinking habitat area at its current band, and declines once no elevation on the mountain is left inside that band.
- Warming rate — sets Δ(t); doubling it roughly halves every species' remaining time.
- Thermal tolerance — a wider τ gives every species a thicker band and buys extra years before the same Δ(t) exhausts it.
- The live chart plots each extinction as it happens: starting elevation on the x-axis, extinction year on the y-axis — the downward trend is the escalator effect made visible.