# Population Extinction under Bursty Reproduction in a Time Modulated Environment

## TL;DR

This paper investigates how combined non-demographic variability, specifically bursty reproduction and time-varying environmental factors, influences the extinction risk of isolated populations, using advanced analytical and numerical methods.

## Contribution

It introduces a combined analysis of bursty reproduction and environmental variability on population extinction using a momentum-space WKB approach.

## Key findings

- Derived time-dependent Hamilton equations for the system.
- Numerically computed instantons for various time-dependent environments.
- Validated theoretical results with Monte Carlo simulations.

## Abstract

In recent years non-demographic variability has been shown to greatly affect dynamics of stochastic populations. For example, non-demographic noise in the form of a bursty reproduction process with an a-priori unknown burst size, or environmental variability in the form of time-varying reaction rates, have been separately found to dramatically impact the extinction risk of isolated populations. In this work we investigate the extinction risk of an isolated population under the combined influence of these two types of non-demographic variation. Using the so-called momentum-space WKB approach we arrive at a set of time-dependent Hamilton equations. In order to account for the explicit time dependence, we find the instanton of the time-perturbed Hamiltonian numerically, where analytical expressions are presented in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically-varying rates corresponding to seasonal effects, and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.