Saber Elaydi

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.

1. Discrete-time epidemic models are often the natural mathematical setting when surveillance data are collected in daily, weekly, or other discrete reporting intervals. They are also appropriate w...

We study the global dynamics and bifurcation structure of an evolutionary Beverton–Holt model with Allee effects, a framework that couples ecological constraints with adaptive trait evolution. The model accounts for density dependence, mate limitation, and predator saturation, while traits evolve according to selection gradients that influence reproduction and competition. From an ecological perspective, we show that weak Allee effects create bistability between extinction and survival, while strong Allee effects generate a critical threshold below which populations collapse and above which they persist at carrying capacity. Evolutionary feedback further reshapes these outcomes by shifting thresholds, modifying stability regions, and producing multiple long-term attractors. Biologically, this reveals how demographic pressures such as scarce mates or high predation interact with trait evolution to determine persistence or extinction, and how adaptive responses may rescue populations facing critical density barriers. Our rigorous analysis and simulations demonstrate that eco–evolutionary processes not only alter classical Beverton–Holt outcomes but also provide insight into mechanisms underlying species persistence, extinction risk, and invasion success under Allee constraints.