MATHEMATICAL BIOSCIENCES doi:10.3934/mbe.2018054 AND ENGINEERING Volume 15, Number 5, October 2018 pp. 1181–1202 DYNAMICS OF DELAYED MOSQUITOES POPULATIONS MODELS WITH TWO DIFFERENT STRATEGIES OF RELEASING STERILE MOSQUITOES Liming Cai ∗ School of Mathematics and Statistics, Xinyang Normal University Xinyang 46400, China Shangbing Ai Department of Mathematical Sciences, University of Alabama in Huntsville Huntsville, AL 35899, USA Guihong Fan Department of Mathematics, Columbus State University, Columbus Georgia 31907, USA (Communicated by Xingfu Zou) Abstract. To prevent the transmissions of mosquito-borne diseases (e.g., malaria, dengue fever), recent works have considered the problem of using the sterile insect technique to reduce or eradicate the wild mosquito population. It is important to consider how reproductive advantage of the wild mosquito pop- ulation offsets the success of population replacement. In this work, we explore the interactive dynamics of the wild and sterile mosquitoes by incorporating the delay in terms of the growth stage of the wild mosquitoes. We analyze (both analytically and numerically) the role of time delay in two different ways of releasing sterile mosquitoes. Our results demonstrate that in the case of constant release rate, the delay does not affect the dynamics of the system and every solution of the system approaches to an equilibrium point; while in the case of the release rate proportional to the wild mosquito populations, the delay has a large effect on the dynamics of the system, namely, for some parameter ranges, when the delay is small, every solution of the system approaches to an equilibrium point; but as the delay increases, the solutions of the system exhibit oscillatory behavior via Hopf bifurcations. Numerical examples and bifurcation diagrams are also given to demonstrate rich dynamical features of the model in the latter release case. 1. Introduction. For more than a century, human beings have attempted to con- trol blood-feeding mosquitoes. This is because of the significant mortality and morbidity burden associated with mosquito-borne diseases (e.g., malaria, dengue fever, and West Nile virus), which are transmitted between humans via blood- feeding mosquitoes [7, 38]. Various control approaches have been explored, which include the development of more effective drug treatments, vaccines, and vector 2010 Mathematics Subject Classification. Primary: 92D25, 92B05, 34D20, 34A34. Key words and phrases. Mosquito population, stage-structure, stability, Hopf bifurcation. This research was supported partially the National Nature Science Foundation of China grant 11371305 and Nanhu Scholars Program for Young Scholars XYNU. * Corresponding author: limingcai@amss.ac.cn. 1181