Rich and complex dynamics of a time-switched differential equation model for wild mosquito population suppression with Ricker-type density-dependent survival probability

Abstract

Dengue presents over 390 million cases worldwide yearly. Releasing Wolbachia-infected male mosquitoes to suppress wild mosquitoes via cytoplasmic incompatibility has proven to be a promising method for combating the disease. As cytoplasmic incompatibility causes early developmental arrest of the embryo during the larval stage, we introduce the Ricker-type survival probability to assess the resulting effects. For periodic and impulsive release strategies, our model switches between two ordinary differential equations. Owing to a Poincaré map and rigorous dynamical analyses, we give thresholds $ T^*, c^* $ and $ c^{**} (>c^*) $ for the release period $ T $ and the release amount $ c $. Then, we assume $ c > c^* $ and prove that our model admits a globally asymptotically stable periodic solution, provided $ T > T^* $, and it admits at most two periodic solutions when $ T < T^* $. Moreover, for the latter case, we assert that the origin is globally asymptotically stable if $ c\ge c^{**} $, and there exist two positive numbers such that whenever there is a periodic solution, it must initiate in an interval composed of the aforementioned two numbers, once $ c^* < c < c^{**} $. We also offer numerical examples to support the results. Finally, a brief discussion is given to evoke deeper insights into the Ricker-type model and to present our next research directions.