In all cases except for the British flora, *S*_{t} increased with *N*_{t−1} for the early discoveries (*a–h*). This was reflected in an exponential increase in *N*_{t} over time (*a–h*), and meant that for all groups except the British flora and birds of the world, bounded confidence limits on *N* could not be estimated when the entire dataset was included (). For all groups except New World grasses (*a*), the discovery rate declined after this initial ‘start-up’ period. The New World grasses were omitted from further analyses, as there was no indication of decline in *S*_{t}, and subsequently no point estimate of *N*_{tot} could be made. Subsets of the data for the other groups were then fitted, which omitted the early increasing *S*_{t} phase.

| **Table 1**Estimates and 95% confidence limits for total species number based on subsets of data for eight groups. (Models were fitted with constant *k*. Data range gives the lower and upper proportions of known species included in the model. Date range is the range (more ...) |

Gymnosperms, ants and mosses all showed a similar discovery rate dynamic (*b–d*). Although *S*_{t} declined after an initial increase, this was followed by another increase for the most recently discovered species. Model fits that included only the central declining *S*_{t} phase gave bounded confidence limits for *N*_{tot} (). However, inclusion of the most recently discovered species either gave very large, or unbounded, confidence limits ().

Ferns and lycopods show a slightly different pattern (

*e*,

*f*). In these groups,

*S*_{t} remained roughly constant, precluding estimation of

*N*_{tot} even when the increasing

*S*_{t} phase was omitted. Bounded confidence limits could be obtained for the latest 10–20% of discoveries, however, owing to recent declines in discovery rates (). In other words, predictions at some point in the past would have been impossible. For ferns, the upper 95% confidence limit of the best estimate that includes the most recent data was only 271 species more than the current total of 14

891 species (). For lycopods, the upper limit was 22 species more than the current total of 484 species ().

The best-behaved groups, in terms of long-term declines in *S*_{t}, were the British flora and birds of the world (*g*,*h*). For the British flora, the first data point (Linnaeus) was omitted from the model fit as it contained many more species than any other point. Fitting the remaining data gave very small confidence limits for *N*_{tot}, but omission of just the most recent 10% of discoveries lead to unbounded confidence limits (). The best estimate for British flora gave 95% confidence limits of 1459–1488 species, with a current known total of 1458 species. Implementing *k* as a linear function of *N*_{t−1} (the ‘varying *k*’ model) gave a best estimate of 1493 species with 95% confidence limits of 1459–1559 species, using the 131 most recent discoveries ().This interval is wider than the best estimate from the constant *k* model. The varying *k* model was able to give bounded, though wide, confidence limits when the most recent 10% of discoveries were omitted (). Omission of more than 10% of the most recent discoveries made prediction impossible.

| **Table 2**Estimates and 95% confidence limits for total species number with *k* as a linear function of *N*_{t−1}. (Only British flora and birds of the world are shown, as the other groups showed no decline in *S*_{t}.) |

For the birds of the world, omission of earlier data gave progressively smaller estimates of the dispersion index, and tighter confidence limits on

*N*_{tot} (). However, omission of just a few late discoveries widened the confidence limits dramatically. Once again, prediction in the absence of just a few species was impossible. The 95% confidence limits on the best estimate for birds were 9994–10

061, with a current total of 9968 species. This estimate included only the most recent 25% of species in the model. Use of the varying

*k* model did not lead to improvements in prediction over the constant

*k* model ().