Dynamical Analysis of a Wolbachia Spread Model with Male Fitness Cost and Seasonal Variation

Abstract. Mathematical models are crucial for predicting the spread of the dengue-blocking bacterium Wolbachia in mosquito populations. Existing discrete-generation models often overlook two critical factors, the fitness cost to infected males and the seasonal variation of key parameters. To address this, we develop and analyze a nonautonomous frequency model with [Formula: see text]-periodic parameters that capture temperature-dependent fluctuations. For the autonomous case, we provide a complete analytical characterization of the bistable region in the parameter space, deriving two explicit thresholds that separate successful invasion, critical, and extinction regimes. For the periodic model, the analysis is complicated by the need to study [Formula: see text]-compositive map. We overcome this by employing tools for Allee maps, establishing sufficient conditions for the existence of positive [Formula: see text]-periodic solutions and an upper bound on their number, thereby confirming the persistence of threshold dynamics under seasonal forcing. Numerical bifurcation analysis reveals the critical boundary that defines the invasion boundary in parameter space. Our results provide actionable guidance on strain selection and release timing, advocating adaptive strategies aligned with favorable seasonal conditions. A key open question remains the analytical characterization of the bifurcation manifold for the periodic system.