SOME EXAMPLE DIVERSITY CHANGES

As an example of the application of such a model, we have applied it to groups of dinosaurs, mammals, and angiosperms. A detailed theoretical discussion of sources of bias in these data is beyond the purpose of this article. We know that filtering data and different classifications may affect the parameters of our model, but its overall properties still stand.

When calculating the parameters of our mathematical model for a particular group of organisms we always start from the date of the first record in that group as t=0, so that the parameters of the various groups of organisms can be compared with each other and the behaviour of the model analysed more easily.

Dinosaurs

The diversification pattern of saurichian dinosaurs from The Fossil Record 2 and its fit to our model are shown in Figure 2. The earliest record is dated at 240 million years ago. We can see from the figure that saurichian dinosaur families diversified and became extinct suddenly about 65 million years ago. The fact that the global trajectory curve from the dinosaur data deviates suddenly from the model and becomes extinct suggests influence by external factors. Without that catastrophic event, saurichian dinosaurs may have continued well into the Cenozoic. We can see from the figure that the global trajectory did not show a downward trend at the mass-extinction event. This means that the data are insufficient to calculate the parameter (which indicates how fast the extinction would have taken place under normal circumstances).

It is interesting to consider the families assigned in The Fossil Record 2 to the dinosaur group and those assigned to the birds (Aves) as a single group to be analysed since it is commonly accepted that birds arose from within the saurichian clade. Figure 3 shows that although the Saurichia + Aves group did not become extinct during the K-T mass extinction, this event had a very deep effect on its diversification pattern. Group diversity quickly recovered from the mass-extinction minima, however, and has been restored to its previous trajectory. Actually, expansion after the K-T event is seen only from the birds' diversity, indicating that the birds may continue to diversify exponentially for some time in the future and that the current diversity may still represent the early part of their bell curve.

Mammals

Figure 4 shows the diversification pattern of mammals from The Fossil Record 2 and its fit to the model. From Figure 4 we can see that the modern mammal groups show a clear downward trend. Based on data from The Fossil Record 2 and the modified logistic model the extinction of mammals can roughly as predicted to occur some 900 million years from now.

To test the validity of these results we have compared the output for mammals and angiosperms with plots from two other databases. One is of our family assignations of Tertiary mammals listed by John Alroy on his "North American Fossil Mammal Systematics Database" website. Our purpose in making this comparison is to see if these independently-compiled data show similar trends. Alroy’s website gives good interpretations of mammalian appearance events and discusses ways of correcting sampling bias. We can also see a global downward trend from the diversification pattern of North American mammals that are calculated from Alroy's data (see Figure 5). The earliest record is dated at 78 Ma. The projected intersection point with the time axis of the curve is approximately 360 million years in the future.

The North American Tertiary mammals curve shows a radiation after the K-T event and reaches a diversity peak around the time of the Eocene-Oligocene boundary. However, for North American mammals, family diversity starts to decline, from more than 50 just before the Oligocene to about 30 families by the end of the Tertiary.

Another mammalian electronic database we have compiled is from the data in the appendix of Kurten and Anderson’s (1980) monograph of Pleistocene mammal species. The diversity curve for this dataset are shown in Figure 6. The curve of data from the Pleistocene shows a dramatic fall in the number of families beginning around 250,000 years ago. Debate will continue as to whether this decline was caused by climate change or by the intrusion of humans into the North American environment. However, there is no doubt that the last data point (the Recent) coincides, in a geological sense, with the industrial revolution and dramatic increases in the use of fossil fuels. This level is matched by the Kurten and Anderson data, further confirming some stability in the recognised fossil record. A number of mammal orders show family extinctions through the Pleistocene: Pilosa, Cingulata, Rodentia, Artiodactyla, Perissodactyla and Proboscidea. The Primates are the only group to show an increase over this same interval.

Angiosperms

Figure 7 shows the diversification pattern of angiosperms from The Fossil Record 2. The angiosperm diversity curve shows that angiosperm families went through slow long-tailed initial diversification , and then a quicker exponential radiation. The rate of the diversification decreases towards the end of the curve. The intersection point of the global trajectory with the time axis is at around 3,000 million years from now. The earliest record is dated at 243.05 Ma (after The Fossil Record 2).

The angiosperm family diversity reaches a peak at the Eocene-Oligocene boundary, marking the end of the grand radiation of the group after the Cretaceous-Tertiary mass-extinction event. Although the rate of radiation falls dramatically, the level of diversity remains very high. So, unlike the mammals, angiosperm family diversity continues to increase through the Late Mesozoic and Cenozoic and there are no extinctions in the Pleistocene.

Agnatha and Cimolesta

Our final examples showing bell-shaped diversity are from two extinct groups of animals, the Agnatha and the Cimolesta. We have chosen these because the full extent of their stratigraphic range is known. It is interesting that the much older group gives a more or less symmetrical bell curve, while the younger Cimolesta shows faster radiation than extinction.

The diversification of Agnatha and their fit to our mathematical model is shown in Figure 8. The earliest record is dated at 484.55 Ma; the youngest record is dated at 372.2 Ma.

The diversification pattern of Cimolesta and its fit to the mathematical model are shown in Figure 9. The earliest record is dated at 118.25 Ma, while the youngest record is dated at 19.8 Ma.