The time-dependent reconstructed evolutionary process with a key-role for mass-extinction events
Sebastian Höhna
(Submitted on 9 Dec 2013)

The homogeneous reconstructed evolutionary process is a birth-death process without observed extinct lineages. Each species evolves independently with the same diversification rates (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 reconstructed 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 time-dependent birth-death process. I elaborate on how to adapt this probability density if conditioned on survival of one or two initial lineages, or having sampled n species and show how to transform between the probability density of a reconstructed 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 enrich this compendium by providing and discussing several special cases, including: the pure birth process, the pure death process, the birth-death process and the critical branching process. Thus, I provide here most of the commonly used birth-death models in a unified framework (e.g., same condition and same data) with common notation.