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 a phase transition, lot can be learned about by studying its extinction time, $\tau_n$ , as function of system size n . A number existing results describe scaling $n\to\infty$ for various choices reproductive rate $r_n$ initial $X_n(0)$ We collect complete this picture, obtaining classification all sequences $(r_n)$ $(X_n(0))$ which there exist rescaling parameters $(s_n)$ $(t_n)$ such $(\tau_n-t_n)/s_n$ converges in distribution identifying limits each case.