# Approximations of Cumulants of the Stochastic Power Law Logistic Model

## TL;DR

This paper derives explicit asymptotic approximations for the first three cumulants of the quasi-stationary distribution in a stochastic power law logistic model, improving upon previous methods by avoiding system closure.

## Contribution

It introduces a novel approach that approximates cumulants without closing the system of equations, providing explicit formulas and error bounds.

## Key findings

- Explicit asymptotic formulas for cumulants derived
- Conditions for approximation validity established
- Errors and spurious solutions identified and addressed

## Abstract

Asymptotic approximations of the first three cumulants of the quasi-stationary distribution of the stochastic power law logistic model are derived. The results are based on a system of ODEs for the first three cumulants. We deviate from the classical moment closure approach by determining approximations without closing the system of equations. The approximations are explicit in the model's parameters, conditions for validity of the approximations are given, magnitudes of approximation errors are known, and spurious solutions are easily detected and eliminated. In these ways, we provide improvements on previous results for this model.