On the uniqueness of quasi-stationary distributions for population models with spatial structure

TL;DR

This paper investigates the conditions under which quasi-stationary distributions (QSDs) are unique in spatially structured population models, demonstrating that adding genealogical or branching random walk information ensures uniqueness.

Contribution

It shows that incorporating genealogical data or branching random walks into spatial population models guarantees the uniqueness of QSDs, addressing a key open problem.

Findings

Branching processes with genealogy have a unique QSD.

Branching random walks with spatial structure have a unique QSD.

Adding extra information ensures QSD uniqueness in these models.

Abstract

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.