Stability and bifurcation of difference equations from stochastic logistic models

This paper investigates the stochastic logistic difference equation, $ X_{n+1} = r X_n (1 - X_n)\varepsilon_n $, where $ X_n $ is a random variable of population size, and $ \{\varepsilon_n\} $ represents independent random perturbations with $ E[\varepsilon_n] = 1 $ and $ E[\varepsilon_n^2] = v > 1 $. Under the Gaussian moment-closure approximation, we derived a closed system of difference equations for the mean and variance of $ X_n $. The analysis of the system of difference equations identified two classes of equilibria: a trivial equilibrium $ (0, 0) $ representing extinction, and nontrivial equilibria corresponding to positive steady population levels. Explicit conditions for the existence and local stability of these equilibria were obtained, showing that the extinction state is stable when $ r^2v < 1 $, whereas nontrivial equilibria arise for $ r > 1 $ with stability dependent on the stochastic intensity $ v $. The saddle-node (fold) bifurcation induced by variations in the stochastic intensity $ v $ was explicitly formulated. Monte Carlo simulations confirmed the analytical analysis.