Quasi-stationary distributions and population processes

TL;DR

This survey explores quasi-stationary distributions in ecological and population models, analyzing their long-term behavior, spectral properties, and numerical methods across various stochastic processes.

Contribution

It provides a comprehensive overview of quasi-stationarity in population processes, including theoretical results, detailed examples, and an algorithmic approach using Fleming-Viot particle systems.

Findings

Quasi-stationary distributions relate to spectral properties of killed processes.

Population models exhibit long-lasting fluctuations before extinction.

Numerical algorithms effectively approximate quasi-stationary distributions.

Abstract

This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.