the ends) of taxon ranges. At the origin of a clade, it is standard dogma that there can be
only a small founder population, and that as the history of the clade unfolds, it will
diversify (although what this word means here requires some attention, as is discussed
below). Clearly, the chance of finding the first member, or even the first few members of
a clade, is vanishingly small. On the other hand, at the acme of the clade, when diversity
is highest, the chances of finding a fossil must be greatly enhanced. If a clade is abruptly
terminated by an extinction caused by something like a meteorite impact, a similar
argument will not apply to the end of the range of the clade. Simply looking at
preservation rates from the period of time when a clade has already diversified and then
extrapolating them back in time is thus likely to underestimate considerably the true time
of origin.
If the above point is accepted, then in order to model the potential likelihood or
probability of finding possible fossils below their lowest known occurrence, some
modelling of the earliest history of the clade is necessary (Foote et al. 1999; Tavaré et al.
2002). Although this approach has been applied to taxa such as mammals and primates, it
is potentially limited by two problems. The first is that it is difficult to decide what sort of
model is appropriate for the (unknown) diversity increase. Although on a priori grounds a
diversity-dependent model is usually chosen, such choice seems to beg the question. For
what one is trying to estimate in the first place is the time lag of diversification that took
place before the first fossil is found: yet it is this very variable that must be (tacitly)
assumed by one’s choice of diversification model. Tavaré et al. (2002) attempt to
circumvent this problem by modelling their diversification curve from the known data and
then extrapolating backwards, assuming at the very least that the curve represents the
monotonous continuation of the initial diversification event. True diversification events,
on the other hand, may be much more complex involving various types of positive
feedback, lags, and so on (Erwin 2001).
Another potentially much more serious problem is the use of diversity as a proxy for
‘rate of fossilization’. Consider the example of the planktonic graptolite Rhabdinopora.
This graptolite probably represents a single invasion of the planktonic realm by the
benthic dendroid graptoloids. It seems, as far as can be ascertained, to appear virtually
simultaneously in huge numbers all over the North Atlantic Province (Cooper et al.
1998). However, its diversity, a complex of subspecies, is tiny. If one was to model its
chance of fossilization on ‘number of lineages’ or diversity, then one would be forced to
predict that it would not have much luck. Yet, in fact, the preservation rate is clearly not
linked to number of species but to number of individuals. Determining numbers of
individuals of a fossil, or indeed extant, organism is fraught with difficulties, and is rarely
even attempted. Willis (1922) showed that there was a striking hollow-curve distribution
of species abundance, which he attempted to explain by a punctuated model of speciation.
If one were to sample such a distribution, then one’s chances of selecting a high-
abundance species would have some relationship to the number of species sampled: high-
abundance species are rare. However, during a radiation, the total abundance—related to
carrying capacity—would be the determinant of chances of fossilization, not necessarily
the number of species. As a result, estimating how total abundance of a clade might
change through the early stages of a diversification must be of more importance than
estimating diversity. Given that diversity is usually the only variable known, this would
GRAHAM E.BUDD AND SÖREN JENSEN 175