How far from the edge need a population be to survive? A probability model

TL;DR

This work analyzes a branching random walk on a finite interval $I_N={-N,...,N}$ with death rate $1$ and birth rate $\lambda$ to neighboring sites, focusing on how edge proximity affects survival. By coupling the finite-interval process to the full line and applying Liggett's eigenvalue criterion for finite trees via the generator $A_N$, the authors establish sharp survival thresholds: if $\lambda>\sqrt{2}/2$ the population survives for all $N$, while for $1/2<\lambda\le\sqrt{2}/2$ there exists a minimal $N_c$ such that survival occurs for $N\ge N_c$ and extinction for $N<N_c$. They also show that the finite-tree critical values satisfy $\lambda_c(N)\to 1/2$ as $N\to\infty$, with explicit small-$N$ values $\lambda_c(1)=\sqrt{2}/2$ and $\lambda_c(2)=\sqrt{3}/3$, highlighting how boundary effects, not changes in local rates, govern persistence. The results offer insight into edge effects in habitat fragmentation, illustrating that proximity to a boundary can drive extinction even when central birth rates are high.

Abstract

Let $N$ be a natural number. We consider a population which lives on $I_N=\{-N,-N+1,\dots,N-1,N\}$. Each individual gives birth at rate $λ$ 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. The population on the whole line (i.e. $N=+\infty$) survives with positive probability if and only if $λ>1/2$. On the other hand for any $1/2< λ\leq \sqrt 2/2$ 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.