Extinction times for a birth-death process with weak competition

TL;DR

This paper analyzes the asymptotic behavior of extinction times in a birth-death process with weak competition, covering all reproductive regimes, and provides insights into how small competition rates affect population extinction times.

Contribution

It offers a comprehensive, self-contained analysis of the asymptotic properties of extinction times in a logistic branching process with weak competition across all regimes.

Findings

Extinction times grow large as competition weakens in the supercritical case.

Different regimes exhibit distinct asymptotic behaviors for extinction times.

The study provides explicit asymptotic formulas for various reproductive regimes.

Abstract

We consider a birth-death process with the birth rates $i\lambda$ and death rates $i\mu +i(i-1)\theta$, where $i$ is the current state of the process. A positive competition rate $\theta$ is assumed to be small. In the supercritical case when $\lambda>\mu$ this process can be viewed as a demographic model for a population with a high carrying capacity around $\lambda-\mu\over\theta$. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as $\theta\to0$. All three reproduction regimes $\lambda>\mu$, $\lambda<\mu$, and $\lambda=\mu$ are studied.