Temporal scales of environmental variation
We introduce stochasticity into the model by allowing environmental conditions (i.e. temperature) to change over time. We design two types of variation with different temporal scales that differ in resampling frequency to capture this idea. Specifically, temperatures are determined by the mean and variation of the short-term distribution,
\(T\sim Norm(T_{\text{smean}},\sigma_{\text{short}}^{2})\), (3a)
where the mean of the short-term distribution, \(T_{\text{smean}}\), is sampled from a long-term distribution,
\(T_{\text{smean}}\sim Norm\left(T_{\text{mean}},\sigma_{\text{long}}^{2}\right).\)(3b)
Under this scenario, environmental fluctuation is controlled by three factors: (1) the mean temperature of the long-term distribution (\(T_{\text{mean}}\)); (2) long-term variation (\(\sigma_{\text{long}}^{2}\)); and (3) short-term variation (\(\sigma_{\text{short}}^{2}\)). Thus, we model a nonequilibrium system in the sense that environmental conditions vary stochastically and, therefore, the carrying capacities of competing species change dynamically, which prevents populations from reaching equilibrium states.
The characteristics of short- and long-term variation can be visualized using fast Fourier transformation (FFT) to decompose the time series of temperature into frequency and amplitude domains (Fig. 1a-c) (Dillonet al. 2016). Since frequencies with the greatest amplitudes contribute more to the pattern of variation, if we change one type of variation at a time, we can see that increasing short-term variation increases the amplitude of higher frequencies more than of lower frequencies (Fig. 1d-e). Similarly, increasing long-term variation expands the amplitude of lower frequencies more than of higher frequencies (Fig. 1e-f). These settings capture the characteristics of stochastic fluctuation at different temporal scales. For instance, body temperature may vary over the course of few minutes due to a difference in wind speed, whereas variation in weather (e.g. a tropical storms or heat waves) may last for weeks. Further, global events like El Nino may occur at considerably irregular intervals at the temporal scale of years. These examples illustrate how stochastic events can occur at both short- and long-term temporal scales. Importantly, long- and short-term variation should be viewed as two points along a continuum of temporal environmental variation. Although we do not directly focus on the number of variation types or the relative temporal scale of updating the environmental conditions (i.e. \(\delta\)), it is possible to extend our model to incorporate more complex and realistic patterns of environmental variation (see Table 1 for summary of all parameters).