Models, mathematical or stochastic, which move from one functional form to another through pathway parameters, so that in between stages can be captured, are examined in this article. Models which move from generalized type-1 beta family to type-2 beta family, to generalized gamma family to generalized Mittag-Leffler family to Lévy distributions are examined here. It is known that one can likely find an approximate model for the data at hand whether the data are coming from biological, physical, engineering, social sciences or other areas. Different families of functions are connected through the pathway parameters and hence one will find a suitable member from within one of the families or in between stages of two families. Graphs are provided to show the movement of the different models showing thicker tails, thinner tails, right tail cut off etc.
The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this article, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide range of parameters. It is proposed that the probability distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this article.
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