Equilibrium analysis of discrete stochastic population models with gamma distribution

This paper analyzes the stochastic logistic and Ricker difference equations at equilibrium with the gamma distribution. We identify mathematical relationships among the intrinsic growth rate in the stochastic equations, the parameters of the gamma distribution and a small stochastic perturbation. The mathematical relations reveal that there are 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. 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.