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).