The shape of red grouse cycles

1. We examined 223 time-series of red grouse (Lagopus lagopus scoticus) numbers compiled from shooting bag-records from grouse moors located across England, Scotland and Wales for evidence of non-time-reversibility, asymmetry in cycle shape and non-linearity. 2. We found 44% of time-series to be non-time-reversible. Over 80% of time-series spent more time increasing than decreasing (but only 32% did so significantly). 3. Non-linearity was detected in 44% of time-series examined, and these non-linear time-series were 1.7 times more likely to be non-time-reversible compared to the linear time-series. 4. The occurrence of non-linear and non-time-reversible series was not related to location, number of days with rain, temperature, moor area or population growth rate. The predominance of symmetrical peaks increased with easting but was not related to number of days with rain, temperature, northing or moor area. 5. There were no significant relationships between time-reversibility or non-linearity status and either tendency to cycle or cycle periodicity. However, populations that cycled with longer periods exhibited slower rates of increase and more equal lengths of time increasing and decreasing in numbers. 6. Our results provide little evidence that grouse cycles observed in different parts of the species range in the United Kingdom arise from fundamentally different processes.