Stability and periodicity in a mosquito population suppression model composed of two sub-models

In this paper, we propose a mosquito population suppression model which is composed of two sub-models switching each other. We assume that the releases of sterile mosquitoes are periodic and impulsive, only sexually active sterile mosquitoes play a role in the mosquito population suppression process, and the survival probability is density-dependent. For the release waiting period T and the release amount c, we find three thresholds denoted by $$T^*$$ T ∗ , $$g^*$$ g ∗ , and $$c^*$$ c ∗ with $$c^*>g^*$$ c ∗ > g ∗ . We show that the origin is a globally or locally asymptotically stable equilibrium when $$c\ge c^*$$ c ≥ c ∗ and $$T\le T^*$$ T ≤ T ∗ , or $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T<T^*$$ T < T ∗ . We prove that the model generates a unique globally asymptotically stable T-periodic solution when either $$c\in (g^*, c^*)$$ c ∈ ( g ∗ , c ∗ ) and $$T=T^*$$ T = T ∗ , or $$c>g^*$$ c > g ∗ and $$T>T^*$$ T > T ∗ . Two numerical examples are provided to illustrate our theoretical results.