Theoretical size distribution of fossil taxa: analysis of a null model

New species arise through speciations occurring independently and at random at a fixed probability rate, while extinctions either occur independently and at random (background extinctions) or cataclysmically. In addition new genera are assumed to arise through speciations of a very radical nature, again assumed to occur independently and at random at a fixed probability rate.The size distributions of the pioneering genus (following a cataclysm) and of derived genera are determined. Also the distribution of the number of genera is considered along with a comparison of the probability of a monospecific genus with that of a monogeneric family.Mathematical modelling of the evolution of lineages goes back at least to Yule[1] who developed the eponymous Yule process (homogeneous pure birth process) in which speciations occur independently and at random. Yule's model did not include extinctions per se, because he believed that they resulted only from cataclysmic events. This issue was discussed at greater length by Raup[2], who distinguished between background and episodic extinctions. Raup started from a homomogeneous birth-and-death process model (in which background extinctions occur, like speciations, independently and at random) for which he presented mathematical results, and described more complex models of extinction including episodic extinctions and a mixture of episodic and background extinctions. However he gave no mathematical results for these models. Stoyan[3] considered a time in-homogeneous birth-and death process, in which speciation and background extinction rates varied with time, based on the idea that younger paraclades have higher speciation rates, while older ones have higher background extinction rates.There has been considerable discussion (e.g. Raup[2]; Patzkowsky[4]; Przeworski and Wall[5]) about the suitability of the null birth-and-death process model (with constant birth and death rates) as a macroevolutionary model of species diversification