Uniqueness and stability of periodic solutions for an interactive wild and Wolbachia-infected male mosquito model

We investigate a mosquito population suppression model, which includes the release of Wolbachia-infected males causing incomplete cytoplasmic incompatibility (CI). The model consists of two sub-equations by considering the density-dependent birth rate of wild mosquitoes. By assuming the release waiting period T is larger than the sexual lifespan T¯ of Wolbachia-infected males, we derive four thresholds: the CI intensity threshold sh∗, the release amount thresholds g∗ and c∗, and the waiting period threshold T∗. From a biological view, we assume sh > sh∗ throughout the paper. When g∗ < c < c∗, we prove the origin E0 is locally asymptotically stable iff T < T∗, and the model admits a unique T-periodic solution iff T ≥ T∗, which is globally asymptotically stable. When c ≥ c∗, we show the origin E0 is globally asymptotically stable iff T ≤ T∗, and the model has a unique T-periodic solution iff T > T∗, which is globally asymptotically stable. Our theoretical results are confirmed by numerical simulations.

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