Nonlinear averaging of thermal experience predicts population growth rates in a thermally variable environment

Abstract

As thermal regimes change worldwide, projections of future population and species persistence often require estimates of how population growth rates depend on temperature. These projections rarely account for how temporal variation in temperature can systematically modify growth rates relative to projections based on constant temperatures. Here, we tested the hypothesis that time-averaged population growth rates in fluctuating thermal environments differ from growth rates in constant conditions as a consequence of Jensen's inequality, and that the thermal performance curves (TPCs) describing population growth in fluctuating environments can be predicted quantitatively based on TPCs generated in constant laboratory conditions. With experimental populations of the green alga Tetraselmis tetrahele, we show that nonlinear averaging techniques accurately predicted increased as well as decreased population growth rates in fluctuating thermal regimes relative to constant thermal regimes. We extrapolate from these results to project critical temperatures for population growth and persistence of 89 phytoplankton species in naturally variable thermal environments. These results advance our ability to predict population dynamics in the context of global change.

Keywords: Jensen's inequality; phytoplankton; population growth; scale transition theory; thermal variability.

Figure 1.

(a) A TPC and its critical temperatures (T_min, T_max, T_opt; dots) and thermal breadth, w. The critical temperatures are not fitted parameters of the TPC, but are estimated by numerical optimization. This negatively skewed curve shows an exponential increase typical of processes following an Arrhenius function, with an accelerating region to the left of the inflection point (grey vertical line), followed by a decelerating region to the right of the inflection point. Notice that the accelerating region corresponds to the region with a positive second derivative (b). Figure adapted from [5], but parametrized with T. tetrahele data from this study. (Online version in colour.)

Figure 2.

TPCs for T. tetrahele populations growing under constant and variable temperature conditions. (a) Exponential growth rates under constant-temperature conditions; green line is the fitted TPC and green shading corresponds to 95% CI generated from non-parametric bootstrapping. The dashed curve represents predicted mean growth rate and 95% CI under thermally variable conditions (±5°C) based on nonlinear averaging (equation (2.1)). Points and error bars are observed mean growth rates generated by estimating exponential growth rates at each mean temperature separately (‘indirect’ approach described in the electronic supplementary material, appendix A; error bars represent 95% CI). (b) Exponential growth rates under thermally fluctuating conditions (±5°C); dashed curve is predicted based on panel (a) and equation (2.1), orange line is the fitted TPC under fluctuating conditions, orange shading corresponds to 95% CI generated from non-parametric bootstrapping. Orange points and error bars are observed mean growth rates in the fluctuating temperature regime (estimated via the ‘indirect’ approach, electronic supplementary material, appendix A; error bars represent 95% CI). (c) Predicted and observed and were statistically indistinguishable (predicted: black triangles and 95% CI error bars, observed: orange triangles and 95% CI error bars), and lower than observed T_max and T_opt in constant conditions (green triangles and 95% CI error bars).

Figure 3.

Curve skew and environmental variability predict differences between performance under constant and variable conditions. (a) Diagram showing predicted differences in key features of a TPC under constant (black line) and variable (grey line) environmental conditions. The distances labelled ‘C’, ‘D’ and ‘E’ illustrate the distances between curve features, plotted in the panels (c–e). (b) Map of all phytoplankton isolation locations used in the analysis. The colour of the ocean and the points corresponds to standard deviation of daily SST over the time period 1981–2011 [34]. (c) The difference between predicted and T_opt generated under constant lab conditions. (d,f) Predicted differences in phytoplankton growth rates that do not incorporate in situ temperature variation (r) versus predicted growth rates based on equation (2.1) (e) The difference between and T_max. Colour coding in panels (c, e and f as in b).