Population models at stochastic times

TL;DR

This paper explores stochastic population models using time-changed processes with subordinators, analyzing distributions, intertimes, explosion conditions, extinction probabilities, and sojourn times for birth, death, and birth-death processes.

Contribution

It provides a comprehensive analysis of population models under stochastic time changes, including new insights into distributions, extinction, and sojourn times for various processes.

Findings

Distribution forms for time-changed processes identified

Conditions for explosion in pure birth processes derived

Extinction probabilities for death processes analyzed

Abstract

In this article, we consider time-changed models of population evolution $\mathcal{X}^f(t)=\mathcal{X}(H^f(t))$, where $\mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $\mathcal{X}$ is a pure birth process, we study the form of the distribution, the intertimes between successive jumps and the condition of explosion (also in the case of killed subordinators). We also investigate the case where $\mathcal{X}$ represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size $n_0$. Finally, the subordinated linear birth-death process is considered. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.