The defining feature of a population cycle or oscillation is regularity: a peak (or trough) every x years. (Of course, x varies from case to case, and a certain degree of variation around x is inevitable; even in a '3-year cycle', the occasional interval of 2 or 4 years is to be expected.) The statistical methods applied to a time series, to determine whether the claim of 'cyclicity' can justifiably be made, usually involve the use of an autocorrelation function (Royama, 1992; Turchin & Hanski, 2001). This sets out the correlations between pairs of abundances one time interval apart, two time intervals apart, and so on (Figure 14.12a). The correlation between abundances just one time interval apart can often be high simply because one abundance has led directly to the next. Thereafter, a high positive correlation between pairs, for example, 4 years apart would indicate a regular cycle with a period of 4 years; while a further high negative correlation between pairs 2 years apart would indicate a degree of symmetry in the cycle: peaks and troughs typically 4 years apart; with peaks typically 2 years from troughs.
It must be remembered, however, that it is not just the pattern of an autocorrelation function that is important but also its statistical significance. Even a single clear rise and fall in a relatively short time series may hint at a cycle (Figure 14.12b), but this pattern would need to be repeated in a much longer series before the autocorrelations were significant, and only then could a cycle be said to have been identified (and require explanation). It is no surprise that major investments in time and effort are required to study cycles in natural populations. Even where those investments have been made, the resulting 'ecological' time series are shorter than those commonly generated in, say, physics - and shorter than those probably envisaged by the statisticians who devised methods for analyzing them. Ecologists need always to be cautious in their interpretations.
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