Closing the gap between palaeontological and neontological speciation and extinction rate estimates

Measuring the pace at which speciation and extinction occur is fundamental to understanding the origin and evolution of biodiversity. Both the fossil record and molecular phylogenies of living species can provide independent estimates of speciation and extinction rates, but often produce strikingly divergent results. Despite its implications, the theoretical reasons for this discrepancy remain unknown. Here, we reveal a conceptual and methodological basis able to reconcile palaeontological and molecular evidence: discrepancies are driven by different implicit assumptions about the processes of speciation and species evolution in palaeontological and neontological analyses. We present the “birth-death chronospecies” model that clarifies the definition of speciation and extinction processes allowing for a coherent joint analysis of fossil and phylogenetic data. Using simulations and empirical analyses we demonstrate not only that this model explains much of the apparent incongruence between fossils and phylogenies, but that differences in rate estimates are actually informative about the prevalence of different speciation modes.

Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding, anagenetic speciation, and extinction without replacement. The proportion of bifurcation events is kept constant (-= 0.5) to illustrate the e ect of variable rates of anagenesis (⁄ a ). These plots show that even though ⁄ a andcannot be estimated directly, ⁄ ú ≠ 2⁄ tends to be around 0 when anagenetic speciation and cladogenesis via budding are equally frequent (top row), whereas ⁄ ú ≠2⁄ tends to be positive when anagenetic speciation is more frequent than budding (bottom row).
Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate   Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding or bifurcation, anagenetic speciation, and extinction without replacement, when 90% of ranges are randomly removed (x). Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Phylogenetic estimates of the speciation and extinction rates (⁄, µ) are shown in red with black triangles representing the true values. The speciation and extinction rates (⁄ ú , µ ú ) estimated from fossil ranges are shown in blue with black stars representing the true values.  Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding or bifurcation, anagenetic speciation, and extinction without replacement, with Poisson fossil sampling (Â = 0.5).
Speciation rate Extinction rate Speciation rate Extinction rate . Phylogenetic estimates of the speciation and extinction rates (⁄, µ) are shown in red with black triangles representing the true values. The speciation and extinction rates (⁄ ú , µ ú ) estimated from fossil ranges are shown in blue with black stars representing the true values. Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding or bifurcation, anagenetic speciation, and extinction without replacement, when 90% of extinct ranges are removed (x). Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding or bifurcation, anagenetic speciation, and extinction without replacement, when 90% of extant ranges are removed (x). Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate Estimates are shown for simulations of fossil and phylogenetic data under the birth-death chronospecies model with di erent proportions of cladogenesis via budding or bifurcation, anagenetic speciation, and extinction without replacement, when diversification rate varies over time. During the period 10-20 Myr there is an elevated rate of diversification. Speciation rate Extinction rate Speciation rate Extinction rate Speciation rate Extinction rate  Figure 16: Posterior samples of the sum between bifurcating and anagenetic rates of speciation in four mammal clades (Bovidae, Cetacea, Feliformia, Cervidae), as inferred under the birth-death chronospecies model. Although the individual rates cannot be teased apart, their sum is obtained using the properties of the BDC model (⁄ a + ⁄-= ⁄ ú ≠ ⁄) based on the posterior samples of speciation rates from fossil and phylogenetic data (⁄ ú , and ⁄, respectively).
Supplementary Figure 17: Results from a joint Bayesian analysis of fossil and phylogenetic data for ferns under the skyline model. Posterior samples of origination rates (in blue) and extinction rates (in red) jointly inferred from the two data types are plotted against one another; posterior samples of the two terms of equation (5) are shown in black. Six rate shifts were used to account for rate heterogeneity. Under this model, the fern phylogenetic and fossil data conform to the BDC model, which was instead rejected under the assumption of constant rates (Fig. 3).
Supplementary Figure 18: Results from a joint Bayesian analysis of fossil and phylogenetic data for corals under the skyline model. Posterior samples of speciation rates (in blue) and extinction rates (in red) jointly inferred from the two data types are plotted against one another; posterior samples of the two terms of equation (5) are shown in black. Six rate shifts were used to account for rate heterogeneity. Under this model, the coral phylogenetic and fossil data rejects to the birth-death chronospecies model, as in the case of constant rates. There is little phylogenetic information in all estimates prior to 25 Ma (as shown by the large spread of posterior values) and in the phylogenetic rates significantly exceed fossil rates between 25 and 0 Ma, thus contradicting the expectations of the BDC model.

Supplementary tables
Supplementary Supplementary Table 2: Speciation and extinction rates estimated from simulated data under the equal, compatible and incompatible rates models. The proportion of bifurcation events is kept constant (-= 0.5) to illustrate the impact of variable rates of anagenesis (⁄a). Estimated parameters are averaged over 100 simulations under each parameter setting. Supplementary Table 8: Speciation and extinction rates estimated from simulated data under the equal and compatible and incompatible rates models when diversification rates vary. Results are shown for (1) a scenario in which an interval of high diversification is follow by an interval of equal speciation and extinction ⁄ = µ and (2) a scenario in which diversification decreases over time and ⁄ < µ during the final interval.  (0), given the level of significance (0.95 or 0.99). Likelihood values are shown for the equal (L Eq. ), compatible (L Co. ) and incompatible (L In. ) rates models. The corresponding likelihood ratio values are shown in Table 12. ı and ù indicate the best model at the 1% and 5% levels, respectively. Table 11: Parameter estimates under the BDC model applied to seven empirical clades, assuming constant diversification rates over time. The BDC model assumed equal rates for fossil and phylogenetic data for Canidae, Sphenisciformes, and Ursidae, and compatible rates for the other clades (see also Fig. 3). The analyses were carried out using the Bayesian implementation of the method and the parameter values reported here represent the mean of the posterior samples with 95% credible intervals given in parentheses. The range of possible ⁄ a andvalues are also shown, which can be obtained based on the properties of the BDC model (see main text). Under the equal rates model all speciation occurs via budding. Under the compatible rates model, anagenesis exceeds budding when ⁄ ú ≠ 2⁄ > 0, whereas budding exceeds anagenesis ⁄ ú ≠ 2⁄ < 0. The contribution of both processes will be equal when ⁄ ú ≠ 2⁄ ¥ 0 (Fig. 4).  Table 12: Results of model testing between three birth-death models in empirical clades. The equal rates model (Eq.) assumes identical rates between phylogenetic and stratigraphic data; the compatible rates model (Co.) indicates that rates are significantly different, but compatible with di erent modes of speciation; the incompatible rates model (In.) assumes independent rate parameters. Results of model testing are given for both Bayesian and maximum likelihood implementations under thresholds of 0.05 and 0.01. The column LR shows the log likelihood ratio between the compatible and independent rates models. The fern dataset, re-analysed under the skyline model (with 6 rate shifts; see main text), provided support for the BDC model (compatible rates) in all 7 time bins at a 0.01 significance threshold (results reported in parentheses). In contrast, the Scleractinia dataset supported incompatible rates even under a skyline model. Model testing under maximum likelihood using simulated datasets based on the speciation and extinction rates estimated from the empirical phylogenies (⁄ and µ) and reflecting the number of lineages observed in the fossil record and phylogenetic trees. The results are based on 100 simulations for each clade. The equal rates model (Eq.) assumes identical rates between phylogenetic and stratigraphic data; the compatible rates model (Co.) indicates that rates are significantly di erent, but compatible with di erent modes of speciation; the incompatible rates model (In.) assumes independent rate parameters. Results are shown for the 95 and 99% confidence levels.