Releasing Wolbachia-infected mosquitoes to replace the wild mosquito population represents an innovative biocontrol strategy currently implemented in over 15 countries to combat mosquito-borne diseases. This study investigates the population dynamics of Wolbachia invasion under periodic release strategies where only infected males are additionally introduced to accelerate population replacement, a strategy that yields a challenging non-autonomous difference equation model. The analytical complexity stems from the infinite composition of distinct rational maps, which renders conventional methods ineffective. To address this challenge, we develop a novel framework based on Poincaré map theory and geometric analysis. Our approach identifies a critical release threshold α* that fully determines system behavior. The main result reveals a sharp dichotomy. When release ratios surpass α*, the Wolbachia-fixed equilibrium achieves global stability, guaranteeing successful population replacement. Below this threshold, the system maintains bistability. This theoretical advancement establishes a quantitative criterion for optimizing intervention strategies, resolving computational obstacles inherent in non-autonomous systems while providing practical guidance for designing effective Wolbachia-based control programs.
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