Remarks on stochastic non-linear birth and death models

The first part of this paper is an attempt to formulate and motivate additional work on the important problem of obtaining global bounds applicable to the controlled truncation of the paper relates specifically to the linear birth, quadratic death model. Asymptotic results are given for the first finite difference ΔTm where Tm is the exactly known mean time to extinction starting from state m (m= 0,1,…). These results are in terms of the environmental carrying capacity n∗ taken to be large. For m near zero ΔTm∼en∗/(n∗)2; whereas, for m near n∗ΔTm ≈ (π/2)1/2/(n∗)3/2. This indicates the vastly different time scales in those two regions of state space - with considerably slower action near extinction than near n∗.