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

TL;DR

This paper develops a Gamma moment-closure framework for the stochastic Ricker map $X_{n+1}=X_n e^{r(1-X_n)} \varepsilon_n$ with $E[\varepsilon_n]=1$ and $E[\varepsilon_n^2]=v>1$, deriving a closed two-dimensional moment map for the mean $\mu_n$ and variance $s_n$. A key contribution is the auxiliary function $F(z;r,v)$, which yields a necessary and sufficient condition $v<(2-e^{-r})^2$ for the existence of a positive feasible equilibrium $(\mu^*,s^*)$, with explicit formulas $\mu^* = \frac{z^*(r-\ln(1+z^*))}{r\ln(1+z^*)}$ and $s^* = \frac{\mu^* z^*}{r}$. The authors show that the stability region coincides with the existence region, supported by Jacobian-based analysis and Monte Carlo simulations that validate the Gamma-closure in certain regimes and reveal its limitations under strong noise or high growth. The work provides a unified theoretical framework linking environmental variability to population persistence, and suggests avenues for deeper bifurcation analysis and rigorous proofs beyond numerical evidence.

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.

Figures (11)

• Figure 1: Comparison of stationary distribution with Gamma moment--closure dynamics.
• Figure 2: Graphs of $R(r) = (2 - e^{-r})^2$
• Figure 3: Graph of $F(\cdot;r,v)$ and $(\mu^*,s^*)$
• Figure 4: Graphs of $F'(z)=r\Phi'(z)+\frac{2}{1+2z}$
• Figure 5: Existence and uniqueness region

Theorems & Definitions (15)

• Theorem 2.1: Gamma moment closure equations for the stochastic Ricker map
• proof
• Remark 2.2
• proof
• Theorem 3.1: Auxiliary function $F$
• proof
• Theorem 3.2: Existence of Feasible Gamma closure equilibrium
• Remark 3.3
• proof
• Theorem 3.4: Uniqueness of Feasible Gamma closure equilibrium