Measuring Patterns in Population Fluctuations

E. Ranta et al. state (1) that Canadian lynx populations exhibit time-varying synchrony analogous to spatiotemporal patterns produced by their simulation model. Similarities between their model and the lynx data, however, appear to be superficial and exaggerated by their analytical methods. First, a cross-correlation coefficient is a meaningful measure of synchrony only when the population data are stationary. Overall, the lynx data are stationary; however, there is a prominent upward trend for 1947 to 1961. Cross-correlations measured over a window as small as 15 years are sensitive to such long phases of nonstationarity. This causes a problem with the moving window analysis presented in figure 1D in the report (1): as the window is moved into and out of the nonstationary portion of the data, the cross-correlation swings high and then low. Thus, oscillatory patterns in this figure are an illusion created by the single transition from negative cross-correlations around year 28 to positive ones around year 34, and back to negative ones by year 40. The impression of oscillations is further enhanced by correlation coefficients being inherently positively autocorrelated as a result of the sharing of data points among successive windows. Also, clustering of correlograms in the 30to 40-year window of figure 1D erroneously suggests that alleged shifts in synchrony are numerous, but this clustering is the result of nonstationarity being more pronounced in central populations (Manitoba, Saskatchewan, and Alberta) than in satellite populations (British Columbia, Yukon, Northwest Territories, Ontario, and Quebec). My reanalysis of the lynx data using four other techniques—differencing original data, splicing out nonstationary sections of data, using a window larger than 15 years, and moving the window in increments larger than 1 year—reveals that synchrony has not changed over time. What requires explanation are regional differences in nonstationarity of the pelt harvest reported for 1947 to 1961—and these could be a result of regional differences in trapping practices associated with changing post-war socioeconomic conditions. Second, a cross-correlation coefficient seems an inaccurate measure of synchrony for populations that cycle in phase and then suddenly snap out of phase, as does the pair of populations shown in figure 1G of the report (1). In these data, the smooth transition from a high correlation around generation 600 to no correlation around generation 695 to a negative correlation around generation 730 is not indicative of the abruptness of the transition at generation 695. It is, in part, the artificial smoothness of oscillations in figure 1F that leads to the superficial resemblance with correlograms of lynx data in figure 1D. The resemblance disappears when cross-correlations are calculated on differenced data or over a series of nonzero temporal lags.