Thermal safety margins and warming

ecology tutorial

thermal ecology

climate change

biogeography

Thermal safety margins in R: why tropical ectotherms, living close to their optimum, lose performance under warming while temperate species gain.

Author

Tidy Ecology

Published

2026-07-29

The thermal performance curve has a steep upper side, and that asymmetry has a geography. It predicts, against first intuition, that the ectotherms most threatened by a warming climate are not the ones with the lowest heat tolerance. They are the ones already living closest to their own optimum. That is the argument of Deutsch et al. 2008 Proceedings of the National Academy of Sciences 105(18):6668-6672, and it is a few lines of R.

Two margins

Fix two quantities for a species. Its habitat temperature (call it the mean body temperature it experiences) and its performance curve. From them come two margins. The thermal safety margin is the gap between the optimum and the habitat temperature, how much warming the species can take before it passes its peak. The warming tolerance is the larger gap up to CTmax, where performance hits zero. A species can have a high CTmax and yet almost no safety margin, if it already sits near its optimum.

Tropical ectotherms are the case in point. Tropical climates are warm and thermally stable, and tropical species have adapted to them by specialising: high optima, narrow curves, and habitat temperatures pressed right up against the peak. Temperate ectotherms live in cooler, more variable climates and typically sit well below their optima for much of the year.

briere <- function(T, a, T0, Tm) { p <- a*T*(T-T0)*sqrt(pmax(Tm-T, 0)); p[T < T0 | T > Tm] <- 0; p }
tropical  <- function(dT = 0) briere(34.0 + dT, 8e-4, 35.0 - 26, 41.0)   # Thab 34, Topt 35
temperate <- function(dT = 0) briere(20.4 + dT, 8e-4, 30.2 - 26, 36.2)   # Thab 20, Topt 30
c(tropical_now = tropical(0),  tropical_plus3 = tropical(3),
  temperate_now = temperate(0), temperate_plus3 = temperate(3))

   tropical_now  tropical_plus3   temperate_now temperate_plus3 
       1.799111        1.657600        1.050906        1.285914

The tropical species starts near its peak, so three degrees of warming pushes it over the top and onto the steep descending limb: performance falls by 7.9 per cent. The temperate species starts far down its rising limb, so the same warming pushes it toward the peak: performance rises by 22.4 per cent. Same climate change, opposite sign, and the difference is entirely the safety margin.

Tg <- seq(2, 44, 0.05)
cur <- rbind(data.frame(T = Tg, P = briere(Tg, 8e-4, 9, 41),   sp = "tropical (small margin)"),
             data.frame(T = Tg, P = briere(Tg, 8e-4, 4.2, 36.2), sp = "temperate (large margin)"))
pts <- data.frame(sp = c("tropical (small margin)","temperate (large margin)"),
                  T0 = c(34, 20.4), T1 = c(37, 23.4),
                  P0 = c(tropical(0), temperate(0)), P1 = c(tropical(3), temperate(3)))
ggplot(cur, aes(T, P, colour = sp)) +
  geom_line(linewidth = 1) +
  geom_segment(data = pts, aes(x = T0, xend = T1, y = P0, yend = P1, colour = sp),
               arrow = arrow(length = unit(0.18, "cm")), linewidth = 0.8, show.legend = FALSE) +
  geom_point(data = pts, aes(T0, P0, colour = sp), size = 2.6, show.legend = FALSE) +
  scale_colour_manual(values = c("tropical (small margin)" = col_trop, "temperate (large margin)" = col_temp)) +
  labs(x = "Temperature (C)", y = "Performance", colour = NULL,
       title = "Warming helps the temperate species and hurts the tropical one")

Two thermal performance curves; on the ochre tropical curve the habitat marker sits near the peak and an arrow points down the steep side, on the green temperate curve the marker sits low and an arrow points up toward the peak. — Figure 1: Two performance curves. The tropical species (ochre) sits near its optimum, so warming (arrow) drops it down the steep side. The temperate species (green) sits low, so the same warming climbs it toward the peak.

The pattern across a gradient

Scale this up to many species along a latitudinal gradient, letting the safety margin shrink toward the equator, and warm each by the same three degrees.

lat    <- seq(0, 50, 5)
Thab   <- 34 - 0.34 * lat            # habitat temperature: hot in the tropics
margin <- 1  + 0.22 * lat            # safety margin: small in the tropics, large at high latitude
Topt   <- Thab + margin; CTmax <- Topt + 6
perf   <- function(i, dT = 0) briere(Thab[i] + dT, 8e-4, Topt[i] - 26, CTmax[i])
dperf  <- 100 * (sapply(seq_along(lat), perf, 3) / sapply(seq_along(lat), perf, 0) - 1)
round(setNames(dperf, paste0("margin", round(margin, 1)))[c(1, 6, 11)], 1)

  margin1 margin6.5  margin12 
     -7.9      12.4      30.4

Every species with a small safety margin loses; every species with a large one gains. Here 2 of the 11 species, the most tropical, drop in performance under warming, while the temperate end gains up to 30 per cent. The threatened species are not the least heat-tolerant in absolute terms; they are the ones with the least room between where they live and where they peak.

ggplot(grad, aes(margin, dperf)) +
  geom_hline(yintercept = 0, colour = col_ref, linetype = "dashed") +
  geom_line(colour = col_temp, linewidth = 1) +
  geom_point(aes(colour = dperf < 0), size = 3) +
  scale_colour_manual(values = c("FALSE" = col_temp, "TRUE" = col_trop), guide = "none") +
  labs(x = "Thermal safety margin (optimum minus habitat, C)",
       y = "Change in performance under +3 C (%)",
       title = "Small safety margin, big warming risk")

Points and a line showing per cent change in performance rising from negative at small thermal safety margins to strongly positive at large margins, crossing zero at a small margin. — Figure 2: Change in performance under three degrees of warming against the thermal safety margin. Species with small margins (tropical) lose; species with large margins (temperate) gain. The dashed line is no change.

The honest limits

The pattern holds in outline and is fragile in detail, and it is worth saying why. The curve is measured for one trait at one life stage; whole-organism fitness is many traits, and they need not share an optimum. The habitat temperature is a single number standing in for a whole distribution, and the previous post showed that the distribution, not its mean, is what performance integrates over. Above all, the calculation freezes the curve in place: it allows no acclimation, no evolution, and no behaviour. A lizard can move into shade; a population can shift its curve over generations (Sinclair et al. 2016 Ecology Letters 19(11):1372-1385). Thermal safety margins are a way to see which species have the least slack, not a forecast of who survives. That distinction is what the checking post is for.

References

Deutsch CA et al. 2008. Proceedings of the National Academy of Sciences 105(18):6668-6672 (10.1073/pnas.0709472105).
Kingsolver JG 2009. The American Naturalist 174(6):755-768 (10.1086/648310).
Sinclair BJ et al. 2016. Ecology Letters 19(11):1372-1385 (10.1111/ele.12686).

--- title: "Thermal safety margins and warming" description: "Thermal safety margins in R: why tropical ectotherms, living close to their optimum, lose performance under warming while temperate species gain." date: "2026-07-29 11:00" categories: [ecology tutorial, R, thermal ecology, climate change, biogeography] image: thumbnail.png image-alt: "Change in performance under warming plotted against thermal safety margin, negative for small margins and rising to strongly positive for large ones." --- ```{r setup} #| include: false knitr::opts_chunk$set(echo = TRUE, message = FALSE, warning = FALSE, dev = "png", dpi = 120, fig.path = "figures/", fig.width = 7, fig.height = 4.4) library(ggplot2) theme_te <- theme_minimal(base_size = 12) + theme(panel.background = element_rect(fill = "white", colour = NA), plot.background = element_rect(fill = "white", colour = NA), panel.grid.minor = element_blank(), plot.title = element_text(face = "bold", size = 12)) theme_set(theme_te) col_trop <- "#b5651d"; col_temp <- "#3a6b52"; col_ref <- "#555555" briere <- function(T, a, T0, Tm) { p <- a*T*(T-T0)*sqrt(pmax(Tm-T, 0)); p[T < T0 | T > Tm] <- 0; p } ## latitudinal gradient: safety margin shrinks toward the tropics lat <- seq(0, 50, 5) Thab <- 34 - 0.34 * lat margin <- 1 + 0.22 * lat Topt <- Thab + margin CTmax <- Topt + 6 warming <- 3 perf <- function(i, dT = 0) briere(Thab[i] + dT, 8e-4, Topt[i] - 26, CTmax[i]) now <- sapply(seq_along(lat), perf, 0) fut <- sapply(seq_along(lat), perf, warming) dperf <- 100 * (fut / now - 1) grad <- data.frame(lat, Thab, margin, Topt, CTmax, dperf) n_lose <- sum(dperf < 0) ## two representative species it <- 1; ie <- 9 # tropical (lat 0), temperate (lat 40) tp <- function(dT = 0) briere(Thab[it] + dT, 8e-4, Topt[it] - 26, CTmax[it]) te <- function(dT = 0) briere(Thab[ie] + dT, 8e-4, Topt[ie] - 26, CTmax[ie]) d_trop <- 100 * (tp(warming) / tp(0) - 1) d_temp <- 100 * (te(warming) / te(0) - 1) ``` The [thermal performance curve](../thermal-performance-curves/) has a steep upper side, and that asymmetry has a geography. It predicts, against first intuition, that the ectotherms most threatened by a warming climate are not the ones with the lowest heat tolerance. They are the ones already living closest to their own optimum. That is the argument of Deutsch et al. 2008 Proceedings of the National Academy of Sciences 105(18):6668-6672, and it is a few lines of R. ## Two margins Fix two quantities for a species. Its habitat temperature (call it the mean body temperature it experiences) and its performance curve. From them come two margins. The **thermal safety margin** is the gap between the optimum and the habitat temperature, how much warming the species can take before it passes its peak. The **warming tolerance** is the larger gap up to CTmax, where performance hits zero. A species can have a high CTmax and yet almost no safety margin, if it already sits near its optimum. Tropical ectotherms are the case in point. Tropical climates are warm and thermally stable, and tropical species have adapted to them by specialising: high optima, narrow curves, and habitat temperatures pressed right up against the peak. Temperate ectotherms live in cooler, more variable climates and typically sit well below their optima for much of the year. ```{r two-species} briere <- function(T, a, T0, Tm) { p <- a*T*(T-T0)*sqrt(pmax(Tm-T, 0)); p[T < T0 | T > Tm] <- 0; p } tropical <- function(dT = 0) briere(34.0 + dT, 8e-4, 35.0 - 26, 41.0) # Thab 34, Topt 35 temperate <- function(dT = 0) briere(20.4 + dT, 8e-4, 30.2 - 26, 36.2) # Thab 20, Topt 30 c(tropical_now = tropical(0), tropical_plus3 = tropical(3), temperate_now = temperate(0), temperate_plus3 = temperate(3)) ``` The tropical species starts near its peak, so three degrees of warming pushes it *over* the top and onto the steep descending limb: performance falls by `r sprintf("%.1f", -d_trop)` per cent. The temperate species starts far down its rising limb, so the same warming pushes it *toward* the peak: performance rises by `r sprintf("%.1f", d_temp)` per cent. Same climate change, opposite sign, and the difference is entirely the safety margin. ```{r fig-two} #| fig-cap: "Two performance curves. The tropical species (ochre) sits near its optimum, so warming (arrow) drops it down the steep side. The temperate species (green) sits low, so the same warming climbs it toward the peak." #| fig-alt: "Two thermal performance curves; on the ochre tropical curve the habitat marker sits near the peak and an arrow points down the steep side, on the green temperate curve the marker sits low and an arrow points up toward the peak." Tg <- seq(2, 44, 0.05) cur <- rbind(data.frame(T = Tg, P = briere(Tg, 8e-4, 9, 41), sp = "tropical (small margin)"), data.frame(T = Tg, P = briere(Tg, 8e-4, 4.2, 36.2), sp = "temperate (large margin)")) pts <- data.frame(sp = c("tropical (small margin)","temperate (large margin)"), T0 = c(34, 20.4), T1 = c(37, 23.4), P0 = c(tropical(0), temperate(0)), P1 = c(tropical(3), temperate(3))) ggplot(cur, aes(T, P, colour = sp)) + geom_line(linewidth = 1) + geom_segment(data = pts, aes(x = T0, xend = T1, y = P0, yend = P1, colour = sp), arrow = arrow(length = unit(0.18, "cm")), linewidth = 0.8, show.legend = FALSE) + geom_point(data = pts, aes(T0, P0, colour = sp), size = 2.6, show.legend = FALSE) + scale_colour_manual(values = c("tropical (small margin)" = col_trop, "temperate (large margin)" = col_temp)) + labs(x = "Temperature (C)", y = "Performance", colour = NULL, title = "Warming helps the temperate species and hurts the tropical one") ``` ## The pattern across a gradient Scale this up to many species along a latitudinal gradient, letting the safety margin shrink toward the equator, and warm each by the same three degrees. ```{r gradient} lat <- seq(0, 50, 5) Thab <- 34 - 0.34 * lat # habitat temperature: hot in the tropics margin <- 1 + 0.22 * lat # safety margin: small in the tropics, large at high latitude Topt <- Thab + margin; CTmax <- Topt + 6 perf <- function(i, dT = 0) briere(Thab[i] + dT, 8e-4, Topt[i] - 26, CTmax[i]) dperf <- 100 * (sapply(seq_along(lat), perf, 3) / sapply(seq_along(lat), perf, 0) - 1) round(setNames(dperf, paste0("margin", round(margin, 1)))[c(1, 6, 11)], 1) ``` Every species with a small safety margin loses; every species with a large one gains. Here `r n_lose` of the `r length(lat)` species, the most tropical, drop in performance under warming, while the temperate end gains up to `r sprintf("%.0f", max(dperf))` per cent. The threatened species are not the least heat-tolerant in absolute terms; they are the ones with the least room between where they live and where they peak. ```{r fig-gradient} #| fig-cap: "Change in performance under three degrees of warming against the thermal safety margin. Species with small margins (tropical) lose; species with large margins (temperate) gain. The dashed line is no change." #| fig-alt: "Points and a line showing per cent change in performance rising from negative at small thermal safety margins to strongly positive at large margins, crossing zero at a small margin." ggplot(grad, aes(margin, dperf)) + geom_hline(yintercept = 0, colour = col_ref, linetype = "dashed") + geom_line(colour = col_temp, linewidth = 1) + geom_point(aes(colour = dperf < 0), size = 3) + scale_colour_manual(values = c("FALSE" = col_temp, "TRUE" = col_trop), guide = "none") + labs(x = "Thermal safety margin (optimum minus habitat, C)", y = "Change in performance under +3 C (%)", title = "Small safety margin, big warming risk") ``` ## The honest limits The pattern holds in outline and is fragile in detail, and it is worth saying why. The curve is measured for one trait at one life stage; whole-organism fitness is many traits, and they need not share an optimum. The habitat temperature is a single number standing in for a whole distribution, and the [previous post](../jensens-inequality-and-thermal-variability/) showed that the distribution, not its mean, is what performance integrates over. Above all, the calculation freezes the curve in place: it allows no acclimation, no evolution, and no behaviour. A lizard can move into shade; a population can shift its curve over generations (Sinclair et al. 2016 Ecology Letters 19(11):1372-1385). Thermal safety margins are a way to see which species have the least slack, not a forecast of who survives. That distinction is what the checking post is for. ## Related tutorials - [Thermal performance curves in R](../thermal-performance-curves/) - [Jensen's inequality and thermal variability](../jensens-inequality-and-thermal-variability/) - [Checking a thermal performance analysis](../checking-a-thermal-performance-analysis/) - [Species distribution models with GLM](../species-distribution-model-glm/) ## References - Deutsch CA et al. 2008. Proceedings of the National Academy of Sciences 105(18):6668-6672 (10.1073/pnas.0709472105). - Kingsolver JG 2009. The American Naturalist 174(6):755-768 (10.1086/648310). - Sinclair BJ et al. 2016. Ecology Letters 19(11):1372-1385 (10.1111/ele.12686).