Extinction time of the logistic process

TL;DR

This paper provides a comprehensive classification of the extinction time behavior in the logistic birth-death process, analyzing how it scales with system size and reproductive parameters.

Contribution

It completes the existing understanding by classifying all parameter sequences for which the scaled extinction time converges in distribution.

Findings

Complete classification of extinction time scaling limits.

Identification of convergence conditions for scaled extinction times.

Unified framework for various reproductive rate and initial population scenarios.

Abstract

The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction, and a phase transition, and a lot can be learned about the process by studying its extinction time, $\tau_n$, as a function of system size $n$. A number of existing results describe the scaling of $\tau_n$ as $n\to\infty$, for various choices of reproductive rate $r_n$ and initial population $X_n(0)$ as a function of $n$. We collect and complete this picture, obtaining a complete classification of all sequences $(r_n)$ and $(X_n(0))$ for which there exist rescaling parameters $(s_n)$ and $(t_n)$ such that $(\tau_n-t_n)/s_n$ converges in distribution as $n\to\infty$, and identifying the limits in each case.