The two major correlational relationships (Fig. 11.3) are well constrained over a realistic range of diversity and turnover levels, making it reasonable to offer specific predictions about the recovery from the mass extinction that is clearly underway (Myers and Knoll, 2001; Wake, 2008).

220 / John Alroy

First, however, the relationships need to be modeled as accurately as possible, which requires transforming the data appropriately and then fitting linear regression functions.

The detrended diversity curve falls in such a narrow range that it is normally distributed on either a linear, log, or square root scale according to a Shapiro-Wilk test. However, logging diversity data is intuitive because diversification is a multiplicative process. For the detrended extinction rates, the raw data are far too skewed to be normal (P < 0.001), and taking square roots does not really help (P = 0.030). However, logging does render them normal, and the parametric correlation r between log diversity and log future extinction (0.462) is much the same as the rank-order correlation p (0.439). Like extinction rates, origination rates are too skewed to be normal (P = 0.011). Technically, logging does normalize them (P = 0.090), but the square root transform does a slightly better job (P = 0.455).

For the data starting after the earliest Ordovician, the median extinction rate is 0.380, and the diversity:extinction regression line is so strong that it implies a near-zero rate when diversity is 1 (i.e., the intrinsic rate). For example, at diversity levels 50, 90, and 99% below the median, the predicted extinction rates are 0.133, 0.014, and 0.0005, respectively. Thus, there would be hardly any extinction if not for major environmental perturbations and the ecological interactions that generate density dependence. Of course, that does not mean the rates are entirely predictable; it simply means that if abundant data were to extend all the way down to a diversity level of 1, we would find that density dependence explained a large majority of the variance. Indeed, the residual variance of the actual regression (Fig. 11.3A) is substantial and, therefore, likely to be real.

The second step is to model origination as a function of past extinction. However, origination rates cannot be predicted solely from the contemporary extinction rates produced in the first step without producing pathological results, because the initially low extinction rates would imply low, not high, origination rates. Instead, the extinction rates put into the equation need to reflect the entire loss of diversity relative to the starting point. The solution is to add the log ratio of preextinction diversity to current diversity to the predicted rate.

Put together, the two functions paint a grim picture (Fig. 11.4). A mass extinction on the scale of the Permo-Triassic event would probably leave diversity still 20% below its equilibrium level after =40 Myr, which is nearly as long as a typical geological period. Indeed, a comparable recovery from even the weakest modeled extinction is expected to take =10 Myr, and a 90% recovery would take =20 Myr. The worst-case scenario is not unthinkable for marine invertebrates: any increase in global atmospheric CO2 by >500 ppm would cause coral reef ecosystems to collapse (Hoegh-Guldberg et al., 2007), and, putting everything else aside, biotic

homogenization through species introductions could cause up to a 58% mass extinction (McKinney, 1998).

Origination rates do vary significantly from one 11.0-Myr bin to the next (Figs. 11.1A and 11.2A), so stochastically achieved high rates in a few bins could push diversity to its equilibrium much faster. Indeed, full recoveries seem to have taken place within the span of one bin after the end-Ordovician and Cretaceous-Tertiary mass extinctions (Alroy et al., 2008). Nonetheless, a minimum 10-Myr recovery time is consistent with qualitative assessments of the pace of recovery in the geological record (Erwin, 2001).

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