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

TL;DR

This paper models a population on a finite line segment with birth and death rates, analyzing how the distance from the edge affects survival probability, revealing a critical size for survival depending on the birth rate.

Contribution

It introduces a probability model for population survival on finite intervals, identifying a critical size for survival based on birth rate and edge proximity.

Findings

Population survives on the infinite line if λ > 1/2.

Existence of a critical size N_c for finite intervals when 1/2 < λ ≤ √2/2.

Population cannot grow at the edges if too close, despite high central birth rates.

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 $\lambda$ 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. 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.