Analysing noisy time-series: describing regional variation in the cyclic dynamics of red grouse.

We develop a method for describing the periodicity of noisy 'quasi-cyclic' time-series based on integrals of their power spectra corresponding to different frequency intervals that we use to classify time-series as 'strongly cyclic', 'weakly cyclic' or 'non-cyclic'. We apply this analysis to over 300 time-series of shooting records of red grouse from 289 moors located in 20 regions of the UK. Time-series from 63 of these populations were not distinguishable from white noise, but significant evidence of cyclic behaviour in the 2-15 year range was detected in time-series from 183 other populations. Time-series from the remaining 43 populations, though distinguishable from white noise, did not exhibit consistently recognizable cyclic behaviour in the same period range. Cyclic populations exhibit an average periodicity of 8.3 years, but only 20% of these populations cycle with a period of four to six years. Geographically, grouse populations are remarkable more for their dynamic heterogeneity than for any observable regularity. The relationship between the location of populations and their dynamical behaviour is weak. The prevalence of cyclic time-series within a region did not significantly differ from the overall average value. Moor region explained 22% of the variation in periodicity, differing from the overall mean in three regions. Average periodicity increases significantly from 6.8 to 8.9 years from the most southerly to most northerly populations. However, latitude explains only 5.3% of the variation in periodicity of the cycles.