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Decline in extinction rates and scale invariance in the fossil record

We show that the decline in the extinction rate during the Phanerozoic can be accurately parameterized by a logarithmic fit to the cumulative total extinction. This implies that extinction intensity is falling off approximately as the reciprocal of time. We demonstrate that this observation alone is sufficient to explain the existence of the proposed power-law forms in the distribution of the sizes of extinction events and in the power spectrum of Phanerozoic extinction, results which previously have been explained by appealing to self-organized critical theories of evolutionary dynamics.


Introduction
It has been widely accepted for some time now that the mean rate of extinction, either of genera or of families, in the known fossil record, appears to decline throughout the Phanerozoic (Raup and Sepkoski 1982, Sepkoski 1984, 1996, Van Valen 1984, Flessa and Jablonski 1985, Gilinsky and Bambach 1987, Raup 1988, Gilinsky 1994. This decline is illustrated in Figure 1, where we show the number of marine families becoming extinct per unit time in each stage since the beginning of the Cambrian, using data drawn from the compilation by J. J. Sepkoski, Jr. (Sepkoski 1992). The decline is only an average trend. Certainly there are deviations from it; two of the largest extinctions in the plot, for example, occur in the second half of the data set. Overall, however, the drop in extinction is clear in the figure.
The same decline is visible in other extinction metrics as well. For example, it has become increasingly common in recent years to quote extinction figures in terms of the percentage of taxa becoming extinct in a given interval, i.e., the ratio of the number of taxa becoming extinct to the number in existence. This metric shows an even more dramatic decline than the simple extinction rate shown in Figure 1, because of the steady increase in diversity throughout the Phanerozoic (Sepkoski 1993, 1996, Benton 1995. The origin of the decline in extinction rate has been much debated. Some authors have argued that it is a real biological phenomenon arising from a slow average increase in the fitness of species (Raup and Sepkoski 1982), from taxon sorting (Sepkoski 1984, Gilinsky 1994, from changes in taxonomic structure (Flessa and Jablonski 1985), or from taxon aging effects (Boyajian 1986). Others, by contrast, have suggested that it may be an artifact stemming from sampling biases (Pease 1992). In this paper however, we are concerned not with the causes but with the implications of extinction decline.
In recent years there has been a considerable amount of interest in the socalled self-organized critical theories of evolution and extinction. Theories of this type contend that patterns of connectivity in ecosystems have a tendency to evolve to a "critical" state in which small perturbations can trigger disturbances of arbitrary magnitude. Starting with Kauffman and Johnsen (1991), a number of authors have suggested that the fossil record of extinction shows evidence of processes of this kind (Kauffman 1993, Sneppen et al. 1995, Solé and Bascompte 1996, Bak and Paczuski 1996, principally because of the existence of apparently scale-free distributions in the fossil data, particularly in the sizes of extinction events and in the power spectrum of extinction. Kauffman (1993) for instance has suggested that a histogram of the number of families becoming extinction per stage has a power-law form. Distributions following power laws are often taken as indicative of critical behavior, although there are also many other ways in which power laws can be generated. Solé et al. (1997) have suggested that the power spectrum of fossil extinction intensity-the square of the Fourier transform-also appears to follow a power law.
In this paper we wish to make a number of observations. First, we note that the decline in extinction rate during the Phanerozoic can be quite accurately parameterized if we consider the cumulative total extinction, i.e., the total number of species which become extinct over a period of time, as the length of that period is increased. We show that this quantity can be well fit by a logarithmic function. We then use this parameterization to calculate the expected distribution of the sizes of extinction events in the fossil record, and show that the results are in precise agreement with the power-law form found by other authors. Thus there is no need to invoke self-organized critical theories to explain this form. Finally, we calculate the power spectrum of our declining extinction rate and show that it too takes a power-law form similar to that found by Solé et al. (1997).

Parameterization of extinction decline
Sepkoski's database of marine families gives us an estimate of the total number of families becoming extinct in each of 77 stages dating from the Vendian-Cambrian boundary at about 544 Ma. (The time scale used in this paper is that of Harland et al. (1990), updated with corrections kindly supplied by J. J. Sepkoski, Jr. and D. H. Erwin.) Let us denote by x(t) the number of families becoming extinct in the stage beginning at time t. Then the cumulative total extinction c(t) at time t is defined to be the total number of families which became extinct during or prior to this stage. Mathematically we can write this as where time is measured from an origin at the start of the data set (i.e., at approximately 544 Ma). This cumulative extinction metric was first proposed and studied by Sibani et al. (1995Sibani et al. ( , 1998. It is useful for two reasons. First, it is much less susceptible to noise in the data than raw extinction metrics. As we will see, the plot of c(t) as a function of t is quite smooth, making it much easier to interpret than most other metrics. (This immunity to noise is a general characteristic of integrated metrics, and the use of such metrics is common in many branches of science and engineering where noise levels are high enough to be problematic.) Second, the cumulative extinction does not depend on how time is partitioned in the data set. In the present study our data are divided into stratigraphic stages, but the numerical value of c(t) would be the same if we were to choose any other partitioning. To see this, we need only notice that the number of taxa becoming extinct before a certain time is not dependent on how we divide those taxa up.
In Figure 2 we show the cumulative total extinction of marine families, calculated from Sepkoski's data, as a function of time. Note how we have plotted the data. The horizontal axis in the figure is logarithmic and the vertical one is linear. Plotted in this way, the cumulative extinction appears as a straight line. This implies that c(t) increases logarithmically in time. In mathematical terms, where the best fit has an R 2 of 0.996 and is obtained by choosing values for the constants of A = −17700, B = 3130, and t 0 = −262 My. The last of these is a curious result. It is interesting to speculate (although we will not do so here) why the best fit should be obtained by measuring time from a point 260 My before the start of the Cambrian. Sibani et al. (1995Sibani et al. ( , 1998 proposed that the curve of cumulative extinction increases as a power law in time, meaning that it should appear straight on dual logarithmic scales. We have replotted the data in this way in the inset to on the idea of increasing mean species fitness. As it turns out, the predictions of their model are precisely in agreement with the form, Equation (2), found here, rather than with their power law form.
Combining Equations (1) and (2) and differentiating, we now deduce the following form for the actual (non-cumulative) total extinction rate: In other words, our analysis reveals that the decline in extinction during the Phanerozoic is well parameterized by a function which falls off as the reciprocal of time. This functional form is shown as the dotted line in Figure 1. Similar results apply for other extinction metrics as well. The cumulated per-taxon extinction, for instance, also fits a logarithmic growth law, with an R 2 of 0.991.
In the next two sections we study the consequences of the decline in extinction rate for the distributions of the sizes of extinction events and for the power spectrum of extinction.

Distribution of the sizes of extinction events
One quantity which has received a lot of attention from the proponents of self-organized critical theories of evolution is the distribution of the sizes of extinction events in the fossil record. In fact, the temporal resolution of the fossil record is insufficient to distinguish individual extinction events, and so attention has usually focussed on the number of taxa becoming extinct per stratigraphic stage. As Newman (1996) has pointed out, it is not clear that the distribution of this quantity obeys a power law, but the data are compatible with the power-law form, with an exponent of −2.0 ± 0.2. We now show that this is exactly what one should expect from an extinction profile which decays over time according to Equation (3). Sibani et al. (1995) in their original paper proposing the cumulative extinction measure demonstrated that given a declining extinction rate one can extract an estimate of the expected distribution of number of taxa becoming extinct per unit time. Here we use their methods to calculate this distribution for our data.
The number dt of unit time intervals in which the extinction rate has a value in some range from x to x + dx is proportional to the derivative where we have made use of Equation (3) twice. In other words, the probability P (x) of a number x of families becoming extinct in any randomly-chosen stage should vary as This is precisely the power-law form observed in the fossil data, with an exponent −2 which agrees exactly with the fossil record. Given the form of the decline in extinction intensity therefore, no further assumptions are needed to explain this power-law form. In particular, it is not necessary to assume a self-organized critical dynamics.
The argument above is not perfect. In particular, it neglects the variation in stage lengths, which could contribute significantly to variation in the number of taxa becoming extinct per stage. It also assumes Equation (3) to be an exact representation of the decline in extinction rates, whereas it is of course only an approximation, since the real extinction rate contains fluctuations about the average form, of which the most important are the large mass extinction events. Nonetheless, the crucial point is that the distribution of the sizes of extinction events need not have anything to do with the proposed critical processes. A declining extinction rate is all that is needed to explain the observed data.
4 The power spectrum of extinction Solé et al. (1997) have analysed the Phanerozoic extinction record using Fourier transform techniques and proposed that the power spectrum P (f ) of extinction falls off with frequency f according, once more, to a power law: with values of the exponent β in the vicinity of 1. They suggested that this too could be a sign of critical dynamics in the processes giving rise to extinction (although they also conceded that other explanations are possible). In this section we demonstrate that a mean extinction rate which declines according to Equation (3) is also sufficient to explain the observed power spectrum, without resorting to critical theories.
The power spectrum for an extinction rate declining as (t − t 0 ) −1 is given by where we measure time again from the beginning of the Phanerozoic and t 1 is the latest time for which we have data, in this case the end of the Pleistocene. This integral is unfortunately not analytically tractable, but it is trivial to perform numerically. We have done this and the results are shown in Figure 3 for the value of t 0 extracted from our fit to the fossil data. As the figure shows, the results follow a power-law form (i.e., a straight line on the logarithmic scales employed in the figure) closely over the range of frequencies of interest, with an exponent β = 1.96±0.01. Although, as we mentioned above, Solé et al. power P(f) Figure 3 The power spectrum for an extinction profile declining in accordance with Equation (3), calculated using Equation (7). The units on the vertical axis are arbitrary.
values closer to 2 (Newman and Eble 1998), so the form found here is perfectly plausible.
It is worth mentioning that this is not the only alternative explanation that has been proposed for the form of the fossil power spectrum. In another paper we discuss this issue in more detail and propose a different interpretation of the results (Newman and Eble 1998).

Conclusions
In this paper we have studied the cumulative total extinction for fossil families during the Phanerozoic and shown that it can be well approximated by a logarithmic growth law. This in turn implies that the extinction intensity itself falls off as the reciprocal of time. We have made use of this observation to demonstrate possible explanations of two previously published results: the power-law form of the distribution of the sizes of extinction events, and the power-law form of the power spectrum of the fossil extinction record. In both cases we show that the decline in extinction intensity is all that is necessary to explain the observed results; we do not need to appeal to an underlying self-organized critical dynamics as other authors have done.
The results presented in this paper are only one step in an argument. We have shown that a declining extinction rate can account for a number of observations made by other authors. The other step in the argument is to explain what gives rise to the decline. This is a question of great interest and one about which arguments will no doubt continue for many years to come.