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Abstract

The sterile insect technique has emerged recently as a biologically secure and effective tool for suppressing wild mosquito pests. To improve the performance of this strategy, understanding the interaction between wild and sterile mosquitoes is critical. Although the common models for this biological problem are scalar equations, they are remarkably resistant to the mathematical analysis. In a series of papers, Due\~nas, Nu\~nez, and Obaya have developed a powerful approach to describe the dynamical behavior of scalar equations with d-concave nonlinearities, a property typically related to the sign of the third derivative. In this paper, we show that, for periodic equations coming from population dynamics, this condition is typically associated with the positive sign of the third derivative of the inverse of the Poincar\'e map. This remark allows us to simplify some arguments in the periodic case and obtain a deep geometrical understanding of the global bifurcation patterns. Consequently, the dynamical behavior of the models is analyzed in terms of simple and testable conditions. Our methodology allows us to describe precisely the dynamical behavior of the common mosquito population suppression models, even incorporating seasonality. This paper generalizes and improves many recent results in the literature.