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Investigating Long-Term Ecological Variability Using the Global Population Dynamics Database
Pablo Inchausti and John Halley
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Supplementary Material
Methods Used
Variance Growth Exponent
The variance growth exponent,
, was used to estimate the rate of increase in variance present in a time series
{
x1,
x2,...,
xn}, which throughout our calculations is the log-transformed time series of ecological population size over time. Since several of the series contain values of zero counts, which are problematic when taking logarithms, we consistently used ln (1 +
Nt) instead of ln (
Nt) for natural logarithm to avoid infinities. Although this solution is unsatisfactory in some ways (
1), the distortion is not so great because the actual number of zeros (or very low values) is not a large fraction of the observations for most of the time series investigated. For each time series, we first calculate the average variance
for all subsequences or 'windows' of size
k (3
k n) as a function of
k (
2,
3). The exponent is the regression slope in a doubly logarithmic scale of the average variance
as a function of
k (
4). This estimation of
involves a set of partially overlapping windows but no claim is made on the statistical significance of the regression slope calculated using these non-independent data points. Since
is estimated as a slope calculated in doubly logarithmic scale, one is effectively estimating the exponent in the equation (
4):
V (t - t0) ~ |t - t0| ,
for the sequence beginning at
t0. Values of
=0 correspond to strictly stationary time series [one whose average, variance and auto-covariances are invariant over time (
5)]. For any finite series,
0 , but for a stationary process,
will converge to zero as the series length increases (
6). A value of
= 1 corresponds to variance which increases linearly with time, such as a random walk. Values of
between 0 and 1 mean that variance increases with the length of the time series but at a decelerating rate (
3,
4,
6).
Spectral Redness Exponent
The spectral exponent was used to measure of the degree of redness of the ecological time series
{x1, x2,..., xn}. The discrete Fourier transform is applied to this series, using an appropriate windowing function (7), in order to find the spectrum {y1, y2,..., yn}. In general the y-values are complex numbers. The power spectral density is a plot of the square amplitudes of these { |y1|2 , |y2|2 ,..., |yn|2 } as a function of the spectral frequencies {1,2..., n/2}(7). The spectral redness exponent is the slope of the least squares regression line on this plot for a log-log scale (2-4). A reddened spectrum is associated with a line of negative slope, since a reddened spectrum is one in which the low frequency events (those happening on long time-scales) explain more of the total variability than those occurring with high frequency (8). In a white noise process, each frequency interval explains the same amount of total variation (i.e. it has a similar spectral power), and thus the slope of the regression line, the spectral exponent, is zero. Blue spectra are those for which the slope of the line is positive. In this procedure one is effectively estimating minus the exponent, v, of the closest 1/f v-noise process.
References
1. B. MacArdle, K. Gaston, J. Lawton, J. Anim. Ecol. 59, 439 (1990).
2. P. Allegrini, M. Barbi, P. Grigolini, B. West. Phys. Rev. E 52, 5281 (1995).
3. P. Inchausti, J. Halley. The long-term temporal variability and the spectral color of ecological time series (unpublished).
4. J. Feder, Fractals (Plenum Press, New York, 1988).
5. R. May, Stability and Complexity in Model Ecosystems (Princeton University Press, Princeton, NJ, 1973)
6. J. Halley, W. Kunin, Theor. Pop. Biol. 56, 215 (1999).
7. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C (Cambridge Univ. Press, ed. 2, Cambridge, 1992).
8. S. Pimm, A. Redfearn, Nature 334, 613 (1988).
