On the probabilities of branch durations and stratigraphic gaps in phylogenies of fossil taxa when rates of diversification and sampling vary over time

The time separating the first appearances of species from their divergences from related taxa affects assessments of macroevolutionary hypotheses about rates of anatomical or ecological change. Branch durations necessarily posit stratigraphic gaps in sampling within a clade over which we have failed to sample predecessors (ancestors) and over which there are no divergences leading to sampled relatives (sister taxa). The former reflects only sampling rates, whereas the latter reflects sampling, origination, and extinction rates. Because all three rates vary over time, the probability of a branch duration of any particular length will differ depending on when in the Phanerozoic that branch duration spans. Here, I present a birth–death-sampling model allowing interval-to-interval variation in diversification and sampling rates. Increasing either origination or sampling rates increases the probability of finding sister taxa that diverge both during and before intervals of high sampling/origination. Conversely, elevated extinction reduces the probability of divergences from sampled sister taxa before and during intervals of elevated extinction. In the case of total extinction, a Signor-Lipps will reduce expected sister taxa leading up to the extinction, with the possible effect stretching back many millions of years when sampling is low. Simulations indicate that this approach provides reasonable estimates of branch duration probabilities under a variety of circumstances. Because current probability models for describing morphological evolution are less advanced than methods for inferring diversification and sampling rates, branch duration priors allowing for time-varying diversification could be a potent tool for phylogenetic inference with fossil data.