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White dashed: prediction of a plain lapse-rate shift, DeltaT / Gamma. The measured range rises less because mountains run out of land at the top.
The science
Temperature on a mountain. Mean annual temperature falls with height at the lapse rate Γ (about 6.5 K/km for moist air, up to 9.8 K/km for dry air): T(x,y) = T0 - Γ h(x,y)/1000 + ΔT + δpool + δaspect. Warming by ΔT therefore moves every isotherm up by ΔT/Γ metres (3 K at 6.5 K/km is 462 m). A species with a thermal niche [Tlo, Thi] is climatically suitable where T lies in that interval.
Why the range area shrinks. A mountain narrows toward the top. For a cone of base radius R and height H the area above elevation z is πR²(1 - z/H)², so moving a band upward trades a wide lower band for a smaller upper one; if the isotherm rises above the summit the habitat disappears (the "summit squeeze"). The page checks its area counting against this formula.
Refugia. Macro-refugia are places that stay suitable (the overlap of the starting and the warmed range). Microrefugia are small sites that stay suitable only because local effects keep them cooler than the lapse rate predicts. Two are modelled: cold-air pooling, where dense cold air drains into concave terrain (valleys, basins) and the nights are colder, δpool = -P·clamp(-TPI/250 m, 0, 1) with TPI the elevation minus the mean elevation within 1.5 km; and aspect, δaspect = -A·(northness)·clamp(slope/0.4, 0, 1) for north-facing slopes in the northern hemisphere. Microrefugia are drawn in violet: cells that are occupied and suitable now but would be unsuitable on the plain lapse-rate temperature.
Dispersal and extinction lag. Each year an unoccupied suitable cell is colonised if an occupied cell lies within the dispersal distance (Chebyshev neighbourhood, cell = 0.21 km), and an occupied cell that has been unsuitable for more than the tolerance (years) is lost. Slow dispersers therefore trail the climate (amber cells: suitable but empty); tolerant species linger (the occupied area exceeds the suitable area).
Species richness. The species-area relationship S/S0 = (A/A0)z with z = 0.25 turns a lost fraction of range into a rough fraction of species committed to extinction. The exponent is a typical literature value, not a property of this mountain.
Honest simplifications. One synthetic massif with made-up heights (not a real mountain); annual mean temperature only (no precipitation, snow cover, soil, competition, biotic interactions or evolution); a species is a pure thermal niche, deterministic and with a hard edge; warming is linear over 100 years; lapse rate is constant with height and season; cold-air pooling and aspect are simple empirical offsets, not an energy-balance model. The pikas are decorative samples of the occupied area; the trees are scenery and do not respond to climate; the vertical exaggeration of the view is shown on screen. Results illustrate mechanisms; they are not forecasts for any real species.