Studies on generalized Yule models

Federico Polito

doi:10.15559/18-vmsta125

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.

Highlights

In 1925, Udny Yule published the paper [45] in which he described a possible model for macroevolution
Genera appear in the system at random times according to a linear birth process
As soon as the genus appears, an independent linear birth process modelling the evolution of the species belonging to it, starts

Summary

Introduction

In 1925, Udny Yule published the paper [45] in which he described a possible model for macroevolution. The finite-dimensional distributions of the degree of a vertex in the Barabási–Albert model converges to the finite-dimensional distributions of the number of individuals in a Yule process with initial population size equal to the number m of attached edges in each time step. This further entails that the asymptotic degree distribution of a vertex chosen uniformly at random in the Barabási–Albert model coincides with the asymptotic distribution of the number of species belonging to a genus chosen uniformly at random from the Yule model with an initial number m of species This result suggests that asymptotic models similar to the Yule model can be linked to different preferential attachment random graph processes in discrete time. In particular we will connect a member of the class of the suitably time-changed mixed Poisson processes with the time-fractional Poisson process, a non-Markov renewal process governed by a time-fractional difference-differential equations involving the Caputo–Džrbašjan derivative

Preliminaries

Generalized Yule model

A critical macroevolutionary model with species deletion