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

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.

💡 Research Summary

The paper studies a spatial branching random walk (BRW) on a finite one‑dimensional lattice segment
(I_N={-N,-N+1,\dots ,N}). Each individual lives at a site, gives birth to a new individual on each of its two neighboring sites at rate (\lambda), and dies at rate 1. There is no limit on the number of individuals per site, and births are prohibited across the boundary of (I_N); the population must survive entirely within the finite set.

Infinite‑line benchmark.
On the whole integer line (\mathbb Z) the total birth rate per individual is (2\lambda). Classical martingale arguments for branching processes imply that the process survives with positive probability if and only if the net reproduction exceeds one, i.e. (2\lambda>1) or (\lambda>1/2). When (\lambda\le 1/2) extinction occurs almost surely. This result provides a reference point for the finite‑segment model.

Construction on a common probability space.
The authors first construct the BRW on (\mathbb Z) using independent Poisson processes (rate (\lambda)) for left‑ and right‑hand births and independent exponential(1) lifetimes for deaths. The same collection of random objects is then used to define the process on each finite segment (I_N) by simply suppressing any birth that would cross the boundary. This coupling yields two monotonicity properties:

1. Monotonicity in (\lambda). If (\lambda_1<\lambda_2), the (\lambda_2) process can be thinned (by independent coin flips with probability (\lambda_1/\lambda_2)) to obtain the (\lambda_1) process. Consequently, at any fixed time and site the (\lambda_2) process contains at least as many individuals as the (\lambda_1) process.

2. Monotonicity in (N). If (N_1<N_2), the construction guarantees that every birth allowed in (I_{N_1}) is also allowed in (I_{N_2}). Hence the process on the larger segment dominates the one on the smaller segment pointwise.

These monotonicities are crucial for comparing survival probabilities across different parameters.

Liggett’s finite‑tree framework.
The core of the analysis adapts Tom Liggett’s 1999 work on branching random walks on finite homogeneous trees. In the one‑dimensional case the tree (T_{1,N}) coincides with the segment (I_N) and the root is the central site 0. The authors classify individuals by their distance from the root, giving rise to (N+1) “types”. The mean offspring matrix (A_N) (size ((N+1)\times(N+1))) has entries

\