How Far from the Edge Does a Population Need to Be to Survive? A Probability Model

Let \(N\) be a natural number. We consider a population that lives on \(I_{N} = {\{{- N},{{- N} + 1},\ldots,{N - 1},N\}}\). Each individual gives birth at rate \(\lambda\) on each of its neighboring sites and dies at rate 1. No births are allowed from the inside of \(I_{N}\) to the outside or vice versa. There is no limit on the number of individuals per site and therefore on the total population. The population on the whole line (i.e., \(N = {+ \infty}\)) survives with positive probability if and only if \(\lambda > {1/2}\). On the other hand, for any \({1/2} < \lambda \leq {\sqrt{2}/2}\), there exists a natural number \(N_{c}\) such that the population survives on \(I_{N}\) for \(N \geq N_{c}\) but dies out for \(N < N_{c}\). There is no limit on the number of individuals per site, so the population could grow at the center where the birth rates are maximum. Our result shows that it does not if the edge is too close.