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Conceived and designed the experiments: BL. Analyzed the data: AM. Wrote the paper: BL AM.

The authors have declared that no competing interests exist.

We re-examine the evidence for a 62 million year (Myr) periodicity in biodiversity throughout the Phanerozoic history of animal life reported by

One of the most controversial, yet provocative, paleobiological topics is the evidence for cyclicity in patterns of extinction and diversity in the fossil record. The history of discussion on this topic is extensive.

More recently, ^{87}Sr/^{86}Sr isotope ratios.

The results of

The possible existence of a cycle operating on such long time scales of course begs the question of what causal factors might produce such a cycle. Thus far, two distinct mechanisms have been proposed that may operate with roughly 62 Myr periodicity: one involves geologic and tectonic processes intrinsic to the Earth ^{87}Sr/^{86}Sr ratios. These ratios are known to be sensitive to weathering rates of continental rock, which may be common to both mechanisms through climate change. We stress, however, that our present work is an examination of the statistical patterns in the data, not causal mechanisms.

The data are derived from

One thing that is essential and standard practice in any study that looks to uncover evidence for cyclicity is that the data be detrended. This removes any overlying trend in the data that may obscure periodic fluctuations, e.g., ^{2} = 1- σ_{r}/σ_{m}, where σ_{r }is the sum of squared residuals about the fit, and _{m} is the sum of squares about the mean. With r^{2} = 1 as a perfect fit, the value for the cubic fit is 0.95. Visual inspection suggests a cubic is the best simple fitting function, and the r^{2} statistic bears this out. The best competing fit is an exponential, with r^{2} = 0.88 (also see

Analysis used AutoSignal with significance levels computed assuming Gaussian fluctuations and lines denote 0.1, 0.05, and 0.01 levels of significance. Frequencies are given in per Myr; there is a peak at a frequency of approximately 0.0162/Myr which is equivalent to 61.9±3.4 Myr; that peak is significant at the .01 level. All other significant peaks occur at less than 15 Myr and thus are near or below the Nyquist frequency.

There are two primary approaches to examining the time dependence of data: time series analysis and spectral analysis. They have different strengths and weaknesses. An impulsive event, such as the sound of an object striking a surface, is most easily recognized in the time domain, even though it may be decomposed into a sum of sounds of many frequencies. On the other hand, the existence of a chord would be most easily deduced from spectral analysis in the frequency domain, in which the sounds of various frequencies that would be hard to recognize in the time domain can be separated

In addition, we examined patterns of total origination, fraction of origination, extinction, and fraction of extinction (all in genera) to see if cycles were manifest in these time series as well.

There has been a significant and extensive debate about whether or not extinction and origination throughout the Phanerozoic largely occur in continuous or discrete time and we refer the interested reader to discussions in

We also conducted analyses that considered patterns in short-lived and long-lived genera (those living less than or more than 45 Myr), following

Finally, we analyzed the geological time scale and the temporal boundaries used herein to see if these contained any evidence of periodicity in their boundaries. No statistical evidence for periodicity was found with the temporal boundaries employed at 62 Myr, 32 Myr, 26 Myr, or indeed at any other period above the Nyquist frequency. On the other hand, we did test for and find a feature in the

Note that since our study involves repetition of tests for

A harmonic component f of a time-dependent quantity can be described by the equation f(t) = A sin(νt+ϕ), where ^{2}, but this is not important for us at the moment), and ϕ is the phase angle

Re-analysis of the total fossil biodiversity data used by

Lines denote 0.1, 0.05, 0.01, and 0.001 levels of significance. Frequencies are given per Myr.

Each of these analyses (and all subsequent ones) shows some other fluctuations that are statistically significant or nearly so. These fluctuations, however, occur near or above the Nyquist frequency, whose inverse (the period) is ∼10–15 Myr: about two times the duration of the average stratigraphic interval. Peaks at such frequencies often represent spurious artifacts

The fractional biodiversity spectrum has a peak at 31.9±1.0 Myr at the 0.1 confidence level. Due to the confidence level, we make no strong claims, but this may merit further investigation. Note that this peak appears in our analysis of

Frequencies are given per Myr.

Purely as an additional test of level of robustness of the 62 Myr cycle we also removed key extinction episodes from

After partition of the data set into two sections, the analysis of fluctuations in fractional diversity from 150–519 Ma revealed a peak at 61.0±3.2 Myr significant at the .001 level; there is also another peak at 32.2±1.1 Myr significant at the .05 level (

Frequencies are given in per Myr.

There are two spectral peaks in fractional origination intensity significant at the

Frequencies are given in per Myr.

Our results here, however, do not mean there are no interesting periodicities around 27 Myr involving extinction and origination. This is because if most extinction or origination events were discrete, impulsive events, grouped predominantly once per stratigraphic interval, as some have argued, e.g.,

We therefore examined cumulative origination and extinction using the procedure outlined in the section on methods of analysis. Of course, the difference of cumulative origination and cumulative extinction is the cumulative change in biodiversity. The sum of these changes as a function of time beginning from the Cambrian was detrended and analyzed (

Since biodiversity, origination, and extinction show the same period(s) when analyzed in various ways, it may be worthwhile to explore their phase relationship which can be described by the equation f(t) = A sin(νt+ϕ) given above. Thus far we have been reporting the period T = 1/ν and this number is identical (within the errors) for biodiversity and origination; thus, it is interesting to look at ϕ, which will tell us whether the waves overlap, or how much they are offset if they are. We found considerable uncertainty in the value of ϕ, depending on some details of the fit. Because these are our primary focus in the present study, we allowed only waves 62 Myr or longer into their least-squares fit, and while

It is also worth considering the phase relationship of the three quantities that show somewhat significant spectral peaks in the vicinity of 27 Myr: fractional biodiversity (ϕ∼4); extinction intensity (ϕ∼5); and cumulative origination ϕ∼2). The results show peaks in cumulative origination following extinction intensity by perhaps 13 Myr out of the 27 Myr cycle, followed finally by a new biodiversity peak.

Both series show the same time ordering, with the extinction peak following the origination peak, and the time lag between the two differs by less than the factor of approximately two by which the overall periods differ. The reversal between intensity and rate in which processes show which periods is puzzling. Note that both periods are present in both series, but we only report those peaks that rise above the “red noise” general level of fluctuations. The “intensity” measures in

We further examined this possibility by sectioning the cumulative origination and extinction series, as we did with biodiversity, into 0–150 Ma and 150–519 Ma periods. In origination, the 62 Myr peak appeared at a lower significance level in the older series, but was absent for the period 0–150 Ma. In the cumulative extinction series, peaks around 62 and 30 Myr appeared in the 150–519 Ma partition, but not in the newer one. Taken together, this implies that the interaction of origination and extinction rates are needed to produce the full signal in biodiversity, and of course longer time series facilitate higher possible levels of significance. Our stated significance levels take into account the length of the series.

The 62±3 Myr periodicity appears in detrended fossil biodiversity whether FFT or LS methods are used; when fractional changes in biodiversity are examined instead, its significance increases.

Eliminating the downturns due to any one of three major mass extinctions does not eliminate the peak or reduce its significance. However, examining the spectra of either long-lived genera (>45 Myr) or only fossil biodiversity in the last 150 Myr does eliminate this peak.

A peak consistent with 62 Myr also appears at a significant level in origination intensity, with a time lead of about 20 Myr from the biodiversity component, and a peak in cumulative extinction appears to lag it by the same amount. Also, when fractional rather than absolute biodiversity changes are examined, a second peak at about 32 Myr emerges at the 0.1 confidence level. This marginally significant peak appears to survive in slightly modified form through most data cuts.

Significant peaks in the region around 24–27 Myr appear in origination and extinction, whether analyzed as intensity or cumulative change. Their relationship with the 32 Myr feature in biodiversity is not clear.

There is a significant spectral feature at 27 Myr in the distribution of stratigraphic interval lengths, implying some pattern that repeats on that timescale, and must be carefully taken in to account in any fluctuation analysis. Nevertheless, our analyses of cumulatives using LS which should be insensitive to this problem also provide some evidence of real changes with periods close to this. There is no such artifact near 62 Myr.

On the whole our results appear to provide statistical support to the notion that there is evidence for long term cycles in the fossil record of apparent diversity. In particular, there appears to be statistical support for the 62 Myr periodicity, the main result of

Strong statistical support emerges for a cycle in fluctuating biodiversity (and fractional biodiversity and origination) operating at roughly 62 Myr. Note further that the results of our analysis (and

As we mentioned before, spectral analysis is widely used because it separates various frequency components in data. However, some additional insight may be gained by considering autocorrelation. Formally there is no new information, since the power spectrum and the autocorrelation function are the Fourier Transform of one another. Still it may assist with visualization.

The function has peaks and minima at intervals of approximately 62 Myr; a peak at approximately 140 Myr is also visible.

At first pass it may seem unusual that evidence for 62 Myr cyclicity is essentially absent when extinction intensity is considered alone: the cyclicity is much stronger when analyzing origination intensity. However, when examining the cumulatives of origination and extinction, which difference to biodiversity, the 62 Myr signal appears stronger in extinction. We thus have a complicated relationship, which deserves further study beyond the scope of this report, between changes of large amplitude and those that happen rapidly. Mathematically though, this result found here and in

The results from our analyses when either the Ordovician/Silurian, Permo/Triassic, or Cretaceous/Tertiary mass extinctions were removed also bear on

The removal of the Cretaceous/Tertiary event should have and does have the least effect on the peak at 62 Myr periodicity because it is a somewhat off-cycle biodiversity fluctuation (

The results from analysis of the two partitions of the diversity and the fractional origination time series' serve to largely support aspects of 62 Myr periodicity. In particular, a statistically significant 62 Myr peak is present in the time series that includes the Paleozoic and roughly half of the Mesozoic, although another significant peak emerges at roughly 30 Myr: the latter peak again is quite close in duration to the peak identified by

Ironically, this objection may also call into question a separate objection to

We found the 140 Myr periodicity noted by

It appears that strong, though not unequivocal, support emerges for the results of

We thank M. Medvedev, R. Rohde, M. Patzkowsky, J. Hendricks, and P. Davies for valuable discussions and R. Bambach, PLoS ONE editor J. Hawks, and an anonymous reviewer for very helpful comments on an earlier version of this paper.