Threshold dynamics of Wolbachia invasion with proportion-dependent Allee effects and encounter-mediated suppression

Releasing Wolbachia-infected mosquitoes is an important biological strategy for reducing mosquito-borne disease transmission, but successful establishment depends on threshold effects and interactions with resident wild populations. Motivated by the boosted incompatible insect technique (Boosted-IIT), we develop a planar continuous-time model for Wolbachia spread, in which cytoplasmic incompatibility is coupled with two additional mechanisms: a proportion-dependent Allee threshold and encounter-mediated pyriproxyfen suppression. We obtain explicit threshold conditions for positive interior equilibria, classify their local stability, and prove convergence to equilibria and the absence of periodic orbits throughout the feasible region. We characterize the common basin boundary away from the critical thresholds. Numerical simulations suggest that stronger suppression increases the basin area for Wolbachia dominance, whereas greater Allee sensitivity decreases it. The model provides analytically tractable criteria for equilibrium regimes and basin-threshold structure, offering a framework for assessing when Wolbachia-infected mosquitoes can invade and replace the wild mosquito population.