# Quasistationarity and extinction for population processes under asymptotic reversibility conditions

## TL;DR

This paper studies population processes that eventually go extinct, focusing on their quasistationary behavior and extinction time using mathematical approximations.

## Contribution

The paper introduces analytical conditions and solutions for approximating quasistationary distributions and extinction times using Hamilton-Jacobi equations.

## Key findings

- Conditions for solving the Hamilton-Jacobi equation analytically are provided.
- An improved approximation for quasistationary distributions is derived using a transport equation.
- Applications demonstrate the effectiveness of the approximations in multitype birth and death processes.

## Abstract

We consider stochastic population processes that are almost surely absorbed at the origin within finite time. Our interest is in the quasistationary distribution, \documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}τ, from quasistationarity to extinction, both of which we study via WKB approximation. This approach involves solving a Hamilton-Jacobi partial differential equation specific to the model. We provide conditions under which analytical solution of the Hamilton-Jacobi equation is possible, and give the solution. This provides a first approximation to both \documentclass[12pt]{minimal}
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\begin{document}$${\varvec{u}}$$\end{document}u and \documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}τ. We provide further conditions under which a corresponding ‘transport equation’ may be solved, leading to an improved approximation of \documentclass[12pt]{minimal}
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\begin{document}$${\varvec{u}}$$\end{document}u. For multitype birth and death processes, we then consider an alternative approximation for \documentclass[12pt]{minimal}
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\begin{document}$${\varvec{u}}$$\end{document}u that is valid close to the origin, provide conditions under which the elements of this alternative approximation may be found explicitly, and hence derive an improved approximation for \documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}τ. We illustrate our results in a number of applications.