# Quasi-Stationary Distributions and Resilience: What to get from a sample?

## TL;DR

This paper investigates how to estimate the resilience of multi-species birth-and-death processes by linking macroscopic dynamical stability to microscopic fluctuations, especially in large but finite populations.

## Contribution

It establishes relations between resilience and process fluctuations, providing estimators to assess resilience from observed data on large stochastic systems.

## Key findings

- Relations between resilience and microscopic fluctuations are derived.
- Estimators for resilience are developed for different time scales.
- The approach connects stochastic process behavior with dynamical system stability.

## Abstract

We study a class of multi-species birth-and-death processes going almost surely to extinction and admitting a unique quasi-stationary distribution (qsd for short). When rescaled by $K$ and in the limit $K\to+\infty$, the realizations of such processes get close, in any fixed finite-time window, to the trajectories of a dynamical system whose vector field is defined by the birth and death rates. Assuming that this dynamical has a unique attracting fixed point, we analyzed in a previous work what happens for large but finite $K$, especially the different time scales showing up. In the present work, we are mainly interested in the following question: Observing a realization of the process, can we determine the so-called engineering resilience? To answer this question, we establish two relations which intermingle the resilience, which is a macroscopic quantity defined for the dynamical system, and the fluctuations of the process, which are microscopic quantities. Analogous relations are well known in nonequilibrium statistical mechanics. To exploit these relations, we need to introduce several estimators which we control for times between $\log K$ (time scale to converge to the qsd) and $\exp(K)$ (time scale of mean time to extinction).