Figure legends and Tables
Figure 1 A demonstration of the species and environmental variation properties in our model. (a) Following the concept of biological performance curves, the difference between the two species can be depicted by the per capita growth rates across all environments (i.e. temperatures). We assume that there is a high temperature-adapted species (orange line) and a low temperature-adapted species (blue line) competing against one another (see methods for more details about the thermal performance curves). The solid black line describes the time series of temperature variation, whereas the dashed black line indicates the average temperature. (b) To more easily visualize environmental fluctuation at different temporal scales, the temperature time series can be decomposed into waves of different frequencies through fast Fourier transformation. (c) For each wave, the amplitude and frequency are plotted as points on a continuous spectrum that characterizes the temperature time series. (d) If the short-term temperature variation increases, the spectrum, (e) would have a larger amplitude in the high frequency regions than in the original spectrum (c, and the grey line ine ). (f) Conversely, increasing long-term variation causes the spectrum, g , to have a larger amplitude in the low frequency regions than in the original spectrum.
Figure 2 Short- and long-term temperature variation have contrasting impacts on species coexistence. (a-d) Proportion of simulations producing species coexistence dynamics, where brighter colors indicate greater proportions of coexisting species. Each panel has constant long-term variation (\(\sigma_{\text{long}}\), labeled above each panel) but variable mean temperature (\(T_{\text{mean}}\), x-axis) and short-term variation (\(\sigma_{\text{short}}\), y-axis). Species coexistence occurs if both species sustain through 20,000 short-term variations and 286 long-term variations (4-million calculations of population change). Each combination of mean and variability is repeated for 100 times. (e-h) Proportion of coexistence with the same definition, but each panel has constant short-term variation (\(\sigma_{\text{short}}\), labeled above each panel) and variable long-term variation (\(\sigma_{\text{long}}\), y-axis).
Figure 3 Examples of the complex patterns of variability and mean temperature on species coexistence. (a-c) In the first case, we set the mean and short-term variation as constant and only change the magnitude of long-term variation. We find that coexistence occurs most commonly under intermediate magnitudes of long-term variation (b). (d-f) In the second case, we fix the size of the mean and long-term variation. By changing short-term variation alone, we see that coexistence is greatest when short-term variation is high (f). (g-i) In the third case, we set the mean at another value (compared to the first case) and only alter the magnitude of the long-term variation. We find that species coexistence is more likely to occur when long-term variation is low (g).
Figure 4 Patterns of species coexistence changes with different mean temperatures. (a-c) At a high mean temperature (Tmean=31), increasing both short- and long-term variation results in greater species coexistence. (d-f) When the mean temperature is lower (Tmean=27), higher short- and long-term temperature variation may still promote coexistence. (g-i) However, if the mean temperature decreases further (Tmean=23), coexistence occurs at both low levels (g) or high levels (i) of short- and long-term variation.
Figure 5 Different combinations of short-term temperature variation, long-term temperature variation, and mean temperature can generate diverse patterns of species coexistence. (a) Coexistence can be promoted by greater temperature variability. (b) Coexistence may also be supported when temperature variability is intermediate (Hutchinson 1961; Connell 1978) (solid line arrow), or (3) hindered by temperature variability (May & MacArthur 1972) (dashed line arrow). (c) Finally, it is also possible that coexistence is promoted when temperature variability is either high or low. The size of short- and long-term temperature variation are labeled above each panel.