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

When testing hypotheses about rates and divergence times, it is important to estimate the probability of any given branch duration and its implicit stratigraphic gap on a phylogeny. The two components to estimating these probabilities are the probability of sampling predecessors (ancestors) and the probability of sampling other relatives (i.e., a sister taxon). The former reflects only sampling whereas the latter reflects sampling, origination and extinction. A concern is that sampling, origination and extinction all vary substantially over time. In this paper, I present a modified version of birth death-sampling models allowing interval to interval variation in rates. I explore the effects of a small number of idealized macroevolutionary scenarios on the probabilities of branch durations. Simulations indicate that this approach does a very good job at estimating the probability of sampling a clade of any size at any time after its divergence. I show that the effects of radiations and extinctions on phylogenies precede the actual events, particularly true when sampling rates are low. For radiations, this reflects unsampled ancestors having a higher probability of numerous progeny that are ultimately sampled than have unsampled ancestors from intervals with the same origination rates but no subsequent radiation. In the case of extinctions, it is the opposite: unsampled ancestors are much less apt to leave progeny and thus have much lower chances of having any progeny sampled. Taking advantage of paleobiological estimates of diversification and sampling over time should improve tree based paleobiological studies in the future.