Extinction time of the logistic process

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.

49 pages, no figures

References in corpus (1)

• On the emergence of random initial conditions in fluid limits