Abstract
In this paper, we study the existence and stability of a unique and exact two periodic orbits of an ordinary differential equation model for the interaction of wild and the released sterile mosquitoes, which consists of two sub-equations constantly switching each other. We first show that the model has at most two periodic solutions for all of the parameters and then surprisingly find that the model has a unique periodic solution if and only if the origin is unstable, and has exact two periodic solutions if and only if the origin is asymptotically stable provided the release amount does not exceed the release amount threshold value, respectively. We also give sufficient conditions for the existence of exact two periodic solutions and analyze the stability of each periodic solution.
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