Necessary and Sufficient Conditions for Existence of a Unique Solution to Gamma Moment Closure for the Stochastic Ricker Equation

Abstract

This paper investigates the stochastic Ricker difference equation $X_{n+1} = X_n \exp(r(1-X_n)) \varepsilon_n$, where $X_n$ is a random variable representing the population size and $\{\varepsilon_n\}$ denotes independent random perturbations with $E[\varepsilon_n] = 1$ and $E[\varepsilon_n^2] = v > 1$. We derive a closed system of difference equations for the mean and variance of $X_n$ using the Gamma moment-closure technique and numerically verify the validity of the Gamma moment-closure approximation. By constructing an auxiliary function, we establish the necessary and sufficient condition, $v < (2 - e^{-r})^2$, for the existence of the positive unique feasible equilibrium. We further verify its local stability with numerical analysis. Monte Carlo simulations confirm the validity of the Gamma moment approximation and illustrate how the interplay between the intrinsic growth rate $r$ and noise intensity $v$ determines population persistence. The results provide a unified theoretical framework for analyzing stochastic Gamma dynamics, offering new biological insights into the stabilizing and destabilizing effects of environmental variability.