a–f, As in Fig. 1, but for surrogate data, obtained by time scrambling—that is, randomly drawing 360 values with replacement from the predator and prey time series in experiment C1—yielding a time series without temporal correlation (white noise) but with the same abundance distribution as the experimental data. The figure shows a typically outcome. Occasionally, for short bursts the randomized predator and prey time series oscillate with a common frequency (a), yielding areas of high coherence (c; WCO > 0.83; spurious coherent oscillation regimes) with random phase differences (e). Whereas most time intervals with coherent predator–prey cycles in the surrogate data have a short duration, in this example (by chance) there is one regime (days 187–209) with a duration of 23 days. g, h, Box plots of oscillation regime length. Boxes show the interquartile range, orange lines indicate the median, whiskers range from the lowest to the highest data point within 1.5× the interquartile range and markers indicate outliers. g, Length distribution of coherent oscillation regimes for the experimental data under free-running conditions (C1–C5) and for a large ensemble of surrogate data. This shows that time intervals with coherent oscillations are significantly longer in the experiments (n = 20; median, 16 days; mean ± s.d., 29 ±28 days) than in the surrogate data (n = 13,719; median, 8 days; mean ± s.d., 9 ± 7 days). The two outliers in the left box plot correspond to the two long coherent oscillation regimes of 111 days and 85 days in C1, which (even though they are outliers in the length distribution) effectively dominate the dynamic behaviour in C1. Consequently, we calculate the typical duration \(\tilde{T}\) of coherent oscillations as the expected regime duration for a randomly chosen time instance taken from a coherent oscillation regime (Methods) (\(\tilde{T}\) = 58 days in C1–C5; \(\tilde{T}\) = 15 days in the surrogate data). h, As in g, but for durations of non-coherent oscillation regimes, showing that time intervals of non-coherent oscillations are significantly shorter in the experiments (n = 15; median, 17 days; mean ± s.d., 16 ± 11 days; typical duration \(\tilde{T}\) = 23 days) than in the surrogate data (n = 12,719; median, 43 days, mean ± s.d., 60 ± 11 days; typical duration \(\tilde{T}\) = 113 days).