Studies on generalized Yule models

TL;DR

This paper generalizes the Yule model for macroevolution by incorporating point processes with order statistics and nonlinear fractional birth processes, deriving explicit distributions and introducing a time-changed mixed Poisson process.

Contribution

It introduces a comprehensive generalization of the Yule model using advanced stochastic processes and derives explicit distributions for the number of species in a genus.

Findings

Derived explicit distribution of species count in generalized models

Introduced a time-changed mixed Poisson process with fractional properties

Extended Yule model to include nonlinear and fractional growth processes

Abstract

We present a generalization of the Yule model for macroevolution in which, for the appearance of genera, we consider point processes with the order statistics property, while for the growth of species we use nonlinear time-fractional pure birth processes or a critical birth-death process. Further, in specific cases we derive the explicit form of the distribution of the number of species of a genus chosen uniformly at random for each time. Besides, we introduce a time-changed mixed Poisson process with the same marginal distribution as that of the time-fractional Poisson process.