The time-dependent reconstructed evolutionary process with a key-role for mass-extinction events

The homogeneous reconstructed evolutionary process is a birth-death process without observed extinct lineages. Each species evolves independently with the same diversification rate-speciation rate, λ(t), and extinction rate, μ(t)-that may change over time. The process is commonly applied to model species diversification where the data are reconstructed phylogenies, e.g. trees estimated from present-day molecular data, and used to infer diversification rates. In the present paper I develop the general probability density of a reconstructed tree under any homogeneous, time-dependent birth-death process. I demonstrate how to adapt this probability density when conditioning on the survival of one or two initial lineages, or on the process realizing n species, and also how to transform between the probability density of a reconstructed tree and the probability density of the speciation times. I demonstrate the use of the general time-dependent probability density functions by deriving the probability density of a reconstructed tree under a birth-death-shift model with explicit mass-extinction events. I extend these functions to several special cases, including the pure-birth process, the pure-death process, the birth-death process, and the critical-branching process. Thus, I specify equations for the most commonly used birth-death models in a unified framework (e.g. same condition and same data) using a common notation.