Abstract
In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two constant values alternately. Employing the Poincaré map method, we show that the model has at most two T-periodic solutions when the release amount is not sufficient to eradicate the wild mosquitoes, and then obtain some sufficient conditions for the model to admit a unique or exactly two T-periodic solutions. In particular, we observe that the model displays bistability when it admits exactly two T-periodic solutions: the origin and the larger periodic solution are asymptotically stable, and the smaller periodic solution is unstable. Finally, we give two numerical examples to support our lemmas and theorems.
Highlights
Mosquito-borne diseases, such as dengue, malaria, Zika, yellow fever, West Nile fever, seriously imperil the health of people in tropical and sub-tropical areas
Most early studies on the interactive dynamics mainly concentrate on models where the number of sterile mosquitoes was assumed to be an independent variable satisfying an independent dynamical equation, see, for example, [20,21,22,23,24], whereas recent investigations have proposed a motivation-attention modeling idea where the number of sterile mosquitoes is treated as a control function instead of an independent variable, since the fundamental and only role of sterile mosquitoes is to mate with wild mosquitoes to induce the infertility of wild females, see, for example, [25,26,27,28,29,30,31,32,33,34,35,36,37,38]
Blood-feeding female mosquitoes are responsible for many life-threatening mosquitoborne diseases, and the most effective way to prevent and control mosquito-borne diseases is to control the vector mosquitoes, and many scholars have been focusing on this project [13,16,32,41,42,43,44]
Summary
Mosquito-borne diseases, such as dengue, malaria, Zika, yellow fever, West Nile fever, seriously imperil the health of people in tropical and sub-tropical areas. By regarding the number g(t) of released sterile mosquitoes at time t as an arbitrarily given nonnegative continuous function, the authors in [28] omitted the second equation in (1), and studied the global dynamics of wild mosquitoes by the following model dw dt ξw)w − μw. This treatment greatly simplifies the model and makes the analysis mathematically more tractable. When u ∈ (0, E1(c)), the following lemma implies that the model (3) and (4) has at most one T-periodic solution. The following lemma tells us that the model (3) and (4) has a unique T-periodic solution when u ∈ (E2, A).
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