Stochastic Wolbachia Dynamics: Persistence, Invariant Measures, and Probability-Based Release Strategies

We develop a stochastic discrete-time framework for Wolbachia invasion in mosquito populations under environmental variability. Starting from a deterministic frequency-dependent difference equation with an Allee-type threshold, we consider the random iteration Xn+1=FΘn(Xn), where Θn represents fluctuations in fitness cost, cytoplasmic incompatibility, and maternal transmission. We establish existence, uniqueness, positivity, forward invariance of [0,1], continuity, and pathwise order preservation, and formulate the model as a random dynamical system. Under independent environmental forcing, the process is Markovian; the associated Markov operator is Feller, and compactness of the state space implies the existence of invariant probability measures. Local behavior near extinction is characterized by the stochastic Lyapunov exponent λ=Elog(1−M)(1−Sf), with λ<0 yielding local exponential stability of the Wolbachia-free state. We further introduce finite-horizon establishment probabilities and probability-based release thresholds, whose monotonicity follows from the pathwise comparison principle. Numerical simulations show how the magnitude, source, and dependence structure of environmental variability affect invasion probabilities and release requirements. The framework combines nonlinear difference equations, random dynamical systems, Markov operators, invariant measure theory, and stochastic stability in a unified analysis of threshold-dependent Wolbachia invasion.