Study of a delayed mosquito population suppression model with stage and sex structure

In this paper, we develop a new two-sex mosquito population suppression model including stage structure and a reproduction delay. Sterile mosquitoes are introduced to suppress the development of wild mosquito populations and a special function Ms(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_s(t)$$\end{document} is used as a control function to describe the number of sterile mosquitoes in the field. Firstly, the dynamic behaviors of the system when the control function is a constant function, a general continuous function and a periodic pulse function are analyzed theoretically, and the existence and stability of the equilibria of the system are determined. In particular, the conditions for the global stability of the wild mosquito-extinction equilibrium, that is, the conditions for the successful suppression of wild mosquitoes, are found. Then, a series of numerical simulations are carried out which on the one hand verify the theoretical results obtained, and on the other hand supplement the imperfections of the theoretical study. Finally, a brief conclusion is given, and the focus of further research is pointed out.