Equilibrium Analysis of Discrete Stochastic Population Models with Gamma Distribution

TL;DR

This paper investigates the stationary distributions of stochastic population models using gamma distributions, revealing how growth rates and stochastic perturbations influence population stability and dual stable states.

Contribution

It introduces a mathematical framework linking growth rate parameters with gamma distribution parameters in stochastic population models, highlighting dual stable states.

Findings

Identification of relationships between growth rate and gamma distribution parameters

Discovery of two stable states corresponding to different growth rates

Insights into population stability under stochastic perturbations

Abstract

This paper analyzes the stationary distributions of populations governed by the discrete stochastic logistic and Ricker difference equations at equilibrium examines with the gamma distribution. We identify mathematical relationships between the intrinsic growth rate in the stochastic equations and the parameters of the gamma distribution with a small stochastic perturbation. We present the biological significance of these relationships, emphasizing how the stochastic perturbation and shape parameter of the gamma distribution influence population dynamics at equilibrium. Furthermore, we identify two branches of the intrinsic growth rate, representing alternative stable states corresponding to higher and lower growth rates. This duality provides deeper insights into population stability and resilience under stochastic conditions.