Dynamic analysis of a mosquito population model with a stage structure and periodic releases of sterile males

This paper focuses on the key issues of mosquito population control, particularly exploring the impact of periodic releases of sterile males in the population model with a stage structure. We construct and analyze a model that includes only sexually active sterile mosquitoes in the dynamic interaction system. We focus on the system's dynamical behaviors under two scenarios: when the sexual lifespan $\overline{T}$ equals the release period $T$ of sterile mosquitoes, and when $\overline{T}$ is less than $T$. In the first scenario, we explore the existence and stability of equilibria, identifying a pivotal threshold $m^{\ast }$ that determines the requisite release amount. In the second scenario, we convert the problem into an impulsive switched system and derive sufficient conditions for the local asymptotic stability of the extinction equilibrium. We also establish the existence of positive periodic solutions using the geometric method of differential equations and the fixed point theorem. Our conclusions show that the relationship between the sexual lifespan and release period of sterile mosquitoes significantly impacts the stability of the mosquito population. Additionally, our numerical simulations not only corroborate but they also complement our theoretical findings.