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Life Cycles

Are there periodic booms and busts in the diversity of life on Earth? Hear a tale of fossils and Fourier transforms

Brian Hayes

How to Date a Fossil

Even after dates have been assigned to the stratigraphic layers, the lifespans of the fossil organisms are still not quite pinned down. Consider again the genus Tellinimera. Under the GTS2004 calibration, its first appearance in the lower Campanian could have been at any time between 83.5 and 77.05 million years ago (mya), and its last gasp in the Danian was somewhere between 65.5 and 60.2 mya. Depending on how the dates of origination and extinction are chosen within those intervals, Tellinimera could have lasted for anywhere from 11 to 23 million years.

For genera whose dates are known with the greatest precision—to the substage level of detail—Muller and Rohde adopt a simple convention: If a genus first appears within a substage, they set its date of origination to the beginning of that substage. By this rule Tellinimera is assumed to arise at 83.5 mya. Likewise a last appearance within a substage is assigned to the end of that substage. Where the data specify only a stage rather than a substage, Muller and Rohde follow a more complicated policy, allocating fractions of a genus to each possible subdivision. Thus the extinction of Tellinimera is shared equally between the two substages of the Danian stage; half of the genus dies out at the end of the lower Danian (62.85 mya) and half at the end of the upper Danian (60.2 mya). For genera dated only at the epoch or period level, an even more elaborate algorithm comes into play.

The net effect of this procedure is to divide geologic time into a series of nonoverlapping units, with an average duration of roughly three million years. Although the fractional allocations spread some events over several of these units, it is still the case that all originations and extinctions occur at the boundaries between units. Nothing ever happens during a substage.

The decision to locate all changes at stratum boundaries has a plausible argument in its favor. The boundaries were defined in the first place because they mark distinctive shifts in fossil biota, and so originations and extinctions ought to be clustered there. Still, it can't be true that all taxa began and ended their existence at those selected transition moments. So I decided to try distributing the events more evenly, a decision made in the spirit of idle experimentation, to see whether it would have any effect on the outcome.

For each genus I assigned a date of origination by selecting a moment at random from within the whole interval in which the earliest fossil was reported. For Tellinimera all dates in the lower Campanian, between 83.5 and 77.05 mya, would be equally likely. Extinction dates are chosen in the same way, by picking a number at random within the interval of last appearance. (Special care is needed when a genus begins and ends in the same time unit: It must not die before it is born.)

Under this plan, the average longevity of a genus is halfway between the minimum and the maximum possible. Moreover, the scheme has the attractive property that greater uncertainty in the dating of a fossil automatically translates into greater variance in the randomly assigned dates. If all we know about a genus is that it arose sometime in the Permian, then the randomizing procedure can assign it any date in the 48-million-year span of that period.

The major drawback of a randomized date assignment is that it makes the analysis nondeterministic. Every run of the program gives a slightly different result. But the law of large numbers protects us. Although any particular genus may be assigned quite different dates in successive runs, the outcome averaged over all 36,000 genera is highly predictable.

The final step in converting Sepkoski's database into a chronicle of biological diversity is to construct a histogram giving the number of extant genera as a function of time. My histograms have bins 1 million years wide, so 542 bins span the interval since the start of the Cambrian. Once the bins are set up, a program scans through the list of genera, placing each of the 36,000 origination and extinction events in its proper bin. Then a pass through the bins from earliest to latest increments the number of extant genera for each origination and decrements it for each extinction. The result is the graph at the top of this column.

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