Temperature & Ecosystems

1.1 Thermal Performance Curves

Every ectothermic organism has a thermal performance curve(TPC) — a relationship between body temperature and a fitness-relevant trait (growth rate, metabolic rate, locomotion speed). TPCs are typically left-skewed: performance rises gradually with temperature, peaks at an optimum \(T_{\text{opt}}\), then drops precipitously at the critical thermal maximum \(CT_{\max}\).

The Sharpe-Schoolfield Model

The most widely used mechanistic model for thermal performance is the Sharpe-Schoolfield equation, which derives from enzyme kinetics. The rate of a biological process at temperature \(T\) (in Kelvin) is:

where:

• • \(r_{\text{ref}}\) is the rate at reference temperature \(T_{\text{ref}}\)
• • \(E_a\) is the activation energy of the rate-limiting enzyme (\(\sim 0.6\;\text{eV}\))
• • \(E_h\) is the deactivation energy for high-temperature enzyme inactivation (\(\sim 2{-}5\;\text{eV}\))
• • \(T_{1/2}\) is the temperature at which 50% of enzymes are denatured
• • \(k = 8.617 \times 10^{-5}\;\text{eV/K}\) is Boltzmann's constant

The numerator captures the Boltzmann-Arrhenius rise in reaction rate with temperature. The denominator models the sharp decline caused by protein denaturation above \(T_{1/2}\).

Thermal Safety Margin

The thermal safety margin (TSM) quantifies how close a species currently lives to its thermal limit:

A critical finding from Deutsch et al. (2008) and subsequent work: tropical ectotherms have the narrowest thermal safety margins despite experiencing less warming than polar species. This is because tropical species have evolved in a thermally stable environment and their \(CT_{\max}\) is only slightly above their habitat temperature.

With warming of \(\Delta T \sim 2{-}4\;^\circ\text{C}\) projected this century, many tropical species will exceed their \(CT_{\max}\) during extreme heat events, even if mean warming appears modest.

1.2 Metabolic Theory of Ecology

The Metabolic Theory of Ecology (MTE) unifies the effects of body size and temperature on metabolic rate, and from metabolic rate derives predictions for nearly every ecological rate and pattern.

The fundamental equation for individual metabolic rate:

where:

• • \(B\) is whole-organism metabolic rate (W)
• • \(B_0\) is a normalisation constant
• • \(M\) is body mass (g)
• • \(M^{3/4}\) is Kleiber's allometric scaling (from fractal vascular networks)
• • \(E_a \approx 0.65\;\text{eV}\) is the average activation energy of respiration
• • \(k\) is Boltzmann's constant, \(T\) is absolute temperature

Ecological Consequences of Warming

Taking the temperature derivative of \(\ln B\):

A 1 °C warming increases metabolic rate by approximately 8.8%. This has cascading effects:

The MTE predicts that warming will restructure communities: favouring small-bodied, fast-reproducing species over large, slow-lived ones. This is consistent with observed trends in marine fish communities showing a shift toward smaller species.

1.3 The Velocity of Climate Change

As climate zones shift, species must migrate to track their preferred conditions. The velocity of climate change measures how fast isotherms are moving across the landscape:

The numerator is the rate of local warming (\(^\circ\text{C/yr}\)) and the denominator is the spatial temperature gradient (\(^\circ\text{C/km}\)). For typical values:

This velocity varies dramatically by landscape:

Species Dispersal Rates

Whether a species can track its climate envelope depends on whether its dispersal rate exceeds the climate velocity. Typical maximum dispersal rates from the literature:

• • Trees: 0.1–1.0 km/yr (seed dispersal limited). Most tree species cannot keep up.
• • Herbaceous plants: 1–10 km/yr (wind/animal dispersal). Marginal.
• • Mammals: 1–10 km/yr. Habitat fragmentation is the main barrier.
• • Birds: 10–100 km/yr. Most can track climate, but depend on habitat availability.
• • Marine fish: 10–70 km/decade observed poleward shift. Many are tracking, but with ecosystem disruption.

The mismatch between climate velocity and dispersal rate creates an extinction debt — species that are already committed to extinction but have not yet disappeared because the full consequences of habitat loss take generations to manifest.

1.4 Treeline Advance

The alpine and Arctic treeline — the boundary above or beyond which trees cannot grow — is advancing upward and poleward as temperatures rise. The treeline is primarily controlled by the growing degree day (GDD) threshold.

Growing degree days accumulate thermal units above a base temperature:

For most boreal and alpine tree species, the treeline corresponds to \(\text{GDD}_5 \approx 300{-}500\;^\circ\text{C}\cdot\text{days}\)(base temperature 5 °C). This GDD threshold maps to a mean growing-season temperature of approximately 6.7 °C.

Rate of Treeline Advance

Using the environmental lapse rate (\(\Gamma \approx 6.5\;^\circ\text{C/km}\)), the elevation shift per unit warming is:

Observed treeline advance rates are typically 0.5–1.0 m/yr upward in elevation (approximately 5–11 m per decade), lagging behind the thermal equilibrium prediction due to:

• • Recruitment limitation: Seed dispersal, germination, and seedling establishment take decades
• • Soil development: Recently deglaciated terrain lacks sufficient soil
• • Competition: Existing alpine meadow vegetation resists tree invasion
• • Wind and snow: Mechanical damage at exposed sites inhibits growth even if thermal conditions are suitable

The consequence for alpine biodiversity is severe: species adapted to conditions above the treeline face a summit trap. As the treeline advances upward, alpine habitat area shrinks (mountains taper), and species with nowhere higher to go face extinction. Area loss follows approximately \(A(z) \propto (z_{\max} - z)^2\) for conical peaks.

1.5 Coral Reef Decline and Bleaching

Coral bleaching occurs when thermal stress causes corals to expel their symbiotic zooxanthellae (dinoflagellate algae of genus Symbiodinium). The threshold is typically just 1 °C above the local maximum monthly mean (MMM) sea surface temperature sustained for 4 or more weeks.

Degree Heating Weeks (DHW)

NOAA's Coral Reef Watch uses Degree Heating Weeks(DHW) as the primary metric for bleaching risk:

The sum accumulates thermal stress over the past 12 weeks (84 days), counting only days where SST exceeds MMM + 1 °C. Thresholds:

Under SSP5-8.5, annual severe bleaching is projected for over 99% of reef locations by 2050. Even under SSP1-2.6, 70–90% of tropical reefs are projected to decline at 1.5 °C warming. The relationship between warming and reef survival is:

Corals support approximately 25% of all marine species despite covering less than 0.1% of ocean floor area. Their loss would trigger cascading biodiversity collapse across tropical marine ecosystems.

Recovery and Adaptation Potential

Recovery from bleaching requires 10–15 years of thermal stability. With bleaching events now recurring every 6 years on average (compared to every 25–30 years in the 1980s), many reefs can no longer fully recover between episodes. However, some mechanisms offer hope:

• • Symbiont shuffling: Corals can associate with more heat-tolerant Symbiodinium clades (e.g., clade D), gaining 1–1.5 °C additional tolerance
• • Genetic adaptation: Standing genetic variation in thermal tolerance exists within coral populations. Selection may produce more heat-tolerant genotypes, but adaptation rates (\(\sim 0.1{-}0.5\;^\circ\text{C}\) per generation) are slow relative to warming rates
• • Assisted evolution: Selective breeding and assisted gene flow are being explored as conservation strategies (van Oppen et al. 2015)

Even with these mechanisms, the window for coral reef survival narrows rapidly above 1.5 °C global warming. The difference between 1.5 °C and 2.0 °C warming represents a near-total loss of tropical coral reef ecosystems.

Biome Shifts and Thermal Safety Margins

Global patterns of biome redistribution and thermal vulnerability under climate change:

Biome redistribution patterns (left) and thermal safety margin comparison across latitudes (right). Data synthesized from Deutsch et al. (2008), Loarie et al. (2009), and IPCC AR6 WG2.

Computational Simulations

References

• • Deutsch, C.A. et al. (2008). Impacts of climate warming on terrestrial ectotherms across latitude. Proceedings of the National Academy of Sciences, 105(18), 6668–6672.
• • Brown, J.H. et al. (2004). Toward a metabolic theory of ecology. Ecology, 85(7), 1771–1789.
• • Loarie, S.R. et al. (2009). The velocity of climate change. Nature, 462, 1052–1055.
• • Schoolfield, R.M., Sharpe, P.J.H. & Magnuson, C.E. (1981). Non-linear regression of biological temperature-dependent rate models based on absolute reaction-rate theory. Journal of Theoretical Biology, 88(4), 719–731.
• • Hughes, T.P. et al. (2018). Spatial and temporal patterns of mass bleaching of corals in the Anthropocene. Science, 359(6371), 80–83.
• • Harsch, M.A. et al. (2009). Are treelines advancing? A global meta-analysis of treeline response to climate warming. Ecology Letters, 12(10), 1040–1049.
• • Sunday, J.M. et al. (2014). Thermal-safety margins and the necessity of thermoregulatory behavior across latitude and elevation. Proceedings of the National Academy of Sciences, 111(15), 5610–5615.
• • IPCC, 2022: Climate Change 2022: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Sixth Assessment Report, Cambridge University Press.