Joint Probability of A Clade Surviving Infinitely or Being Sampled Once
Given the rates of branching, extinction and sampling, calculates the joint probability of a random clade (of unknown size, from 1 to infinite) either (a) never going extinct on an infinite time-scale or (b) being sampled at least once, if it does ever go extinct. As we often assume perfect or close to perfect sampling at the modern (and thus we can blanket assume that living groups are sampled), we refer to this value as the Probability of Being Sampled, or simply P(s). This quantity is useful for calculating the probability distributions of waiting times that depend on a clade being sampled (or not).
Instantaneous rate of speciation (lambda). If the underlying model assumed is anagenetic (e.g. taxonomic change within a single lineage, 'phyletic evolution') with no branching of lineages, then p will be used as the rate of anagenetic differentiation.
Instantaneous rate of extinction (mu)
Instantaneous rate of sampling
If TRUE, an exact solution developed by Emily King is used; if FALSE, an iterative, inexact solution is used, which is somewhat slower (in addition to being inexact...).
Note that the use of the word 'clade' here can mean a monophyletic group of any size, including a single 'species' (i.e. a single phylogenetic branch) that goes extinct before producing any descendants. Many scientists I have met reserve the word 'clade' for only groups that contain at least one branching event, and thus contain two 'species'. I personally prefer to use the generic term 'lineage' to refer to monophyletic groups of one to infinity members, but others reserve this term for a set of morphospecies that reflect an unbroken anagenetic chain.
Obviously the equation used makes assumptions about prior knowledge of the time-scales associated with clades being extant or not: if we're talking about clades that originated a short time before the recent, the clades that will go extinct on an infinite time-scale probably haven't had enough time to actually go extinct. On reasonably long time-scales, however, this infinite assumption should be reasonable approximation, as clades that survive 'forever' in a homogenous birth-death scenario are those that get very large immediately (similarly, most clades that go extinct also go extinct very shortly after originating... yes, life is tough).
Both an exact and inexact (iterative) solution is offered; the exact solution was derived in an entirely different fashion but seems to faithfully reproduce the results of the inexact solution and is much faster. Thus, the exact solution is the default. As it would be very simple for any user to look this up in the code anyway, here's the unpublished equation for the exact solution:
Ps = 1-(((p+q+r)-(sqrt(((p+q+r)^2)-(4*p*q))))/(2*p))
Returns a single numerical value, representing the joint probability of a clade generated under these rates either never going extinct or being sampled before it goes extinct.
This function is entirely the product of a joint effort between the package author (David W. Bapst), Emily A. King and Matthew W. Pennell. In particular, Emily King solved a nasty bit of calculus to get the inexact solution and later re-derived the function with a quadratic methodology to get the exact solution. Some elements of the underlying random walk model were provided by S. Nalayanan (a user on the website stackexchange.com) who assisted with a handy bit of math involving Catalan numbers.
Bapst, D. W., E. A. King and M. W. Pennell. In prep. Probability models for branch lengths of paleontological phylogenies.
Bapst, D. W. 2013. A stochastic rate-calibrated method for time-scaling phylogenies of fossil taxa. Methods in Ecology and Evolution. 4(8):724-733.
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