Periodicity and Extinction of a Physiologically Structured System with Pulse-Switching Cytoplasmic Incompatibility Interventions

Abstract. We study a pulse-switching planar system that arises from describing the population dynamics of wild mosquitoes under the pulse-switching releases of Wolbachia-infected males. The periodic pulse switching between two systems renders the traditional qualitative method of planar systems ineffective. We develop a geometric approach to determine the maximal invariant set and the fixed points of the associated Poincaré map. This enables us to describe scenarios of global dynamics—extinction and bistability—in the sense of an extinction equilibrium coexisting with a periodic solution with a large amplitude and to locate the periodic solutions. Examples will be given to illustrate how our theoretical results can be used to inform the optimal design of pulse-releasing strategies.