716 (A) 0.05
k n pest
0.005 0.01 0.02
0.002 (B) 0.02 0.01 0.005 Word Capi
Reference l ine: 1 n Wu
Areal cells ~200,000 km2
MICHAEL FOOTE
[μlog, σlog, pmax] from the corresponding multivariate normal distribution; and calculating the value of p for each parameter trio, thereby effecting a parametric bootstrap of p.) Because some of the bootstrap distributions
Ch In Ol Anis Ladi Areal cells ~50,000 km2 0.002
are skewed, the results are presented as the median value of p and the interquartile range. It appears that occupancy is maximal in the latest Permian (Changhsingian) rather than the Induan; that the peak in the Induan may partly be an artifact of a low value of n; and that occupancy through the Early and Middle Triassic may be more constant than suggested by the raw data. However, the uncertainty in the Induan occupancy estimate is large, so these inferences must be regarded with caution. The increase in occupancy from the Guadalupian to the Lopingian, in both the raw data and in the bias-corrected estimate, agrees with raw, area-based geographic-range data reported by Clapham et al. (2009: Fig. 10). In any study of occupancy, it is worth
Word 265 Capi 260 Wu 255 Ch In 250 Geologic time (Ma)
FIGURE 9. Raw mean occupancy and estimated mean occupancy probability for the middle Permian through Middle Triassic. Timescale in this and the following figure is from Gradstein et al. (2012). A, Larger equal-area cells, corresponding to those used in the foregoing analyses. B, Smaller cells. Estimates of p are medians of 100 bootstrap replicates; error bars are interquartile ranges. Raw occupancy data suggest a steady rise and fall with an Induan peak, but this pattern parallels a fall and rise in the number of sites. Bias correction suggests peak occupancy in the Changhsingian but still relatively high values in the Induan and Olenekian if we consider a finer partitioning of the globe.
I applied the fitting procedure to 100 bootstrap samples of the data in each stage, resampling the values of k with replacement. Thus, the observed number of genera (S) and sites (n) remain fixed; it is simply the distribution of k that varies among bootstrap samples. (I obtain similar results [not shown] using the asymptotic estimate of the covariance matrix of the parameters, equal to the inverse of the observed information matrix [i.e., minus the matrix of second-order partial derivatives of the log-likelihood function, evaluated at the maximum-likelihood parameter estimates] [Edwards 1991]; drawing random trios of
Ol Anis 245 Ladi 240
exploring sensitivity of results to the choice of what constitutes a site. In the Permian exam- ple, we get a slightly different view if we use smaller equal-area cells (Fig. 9B). The bias- corrected occupancy history suggests that, even though the peak is still in the Changh- singian, occupancy was also higher in the Early Triassic than during most of the focal interval, a pattern that is not quite so clear in the ana- lysis using coarser spatial resolution. Age and Area.—In a pioneering study,
Miller (1997) analyzed occurrence data on Ordovician marine animal genera and found that, as a whole, they tended to occur on more paleocontinents later in time, and that, during the Late Ordovician, the more widespread genera tended to be older. These results supported Willis’s (1926) age-and-area hypothesis. Figure 10 presents a complementary approach to this question, tracking occupancy of Early Ordovician (Tremadoc and Floian) cohorts as they age. Each point in this figure represents the occupancy of the genera that originated in one of these two stages and that were still extant and sampled during the stage in question, whether their stage of first appearance or a subsequent stage. As in Figure 9, distributions are fitted to each subset of data, and corresponding occupancy
Estimated occupancy probability
Estimated occupancy probability
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