The generalized time variable reconstructed birth–death process

Much recent research has investigated the effect that different time variable birth–death processes have on the distribution of branching times in phylogenies of extant taxa. Previous work has shown how to calculate the distributions of number of lineages and branching times for a reconstructed constant rate birth–death process that started with one or two reconstructed lineages at some time in the past or ended with some number of lineages in the present. Here I expand that work to include any time variable birth–death process that starts with any number of reconstructed lineages and/or ends with any number of reconstructed lineages at any time, and I calculate a number of distributions under that process. I also explore the discrete time birth–death process which operates as an efficient and accurate numerical solution to any time-variable birth–death process and allows for the analytical incorporation of sampling and mass extinctions. I describe how these distributions can be used to compare different time variable models using maximum likelihood analysis, and I show how to simulate random trees under any of these models. I also introduce two visual methods for evaluating different time variable birth–death processes; these methods illustrate the shape of distributions for the number of lineages and waiting times by plotting them over time.

► Generalize all results for any reconstructed time variable birth–death process (TVBD).
► Results conditioned on any set of assumptions about the numbers of reconstructed lineages at any times.
► Add new efficient and flexible numerical solution and include random sampling and mass extinctions analytically.
► Distributions of lineage counts at any time, waiting times between lineage splits and all divergence times in a tree.
► Sample from the distribution of divergence times within a tree.