invade host population easily driving the host population to decline. Successful [*Wolbachia*]{} invasion in [*Aedes aegypti*]{} has been observed by Xi et al. in the laboratory caged population within seven generations [@Wolbachia9].

By releasing [*Aedes albopictus*]{} mosquitoes infected with [*Wolbachia*]{} bacteria into the wild, it is expected that over a long time period, the wild [*Aedes aegypti*]{} mosquito population would decline drastically and hopefully be completely replaced by infected mosquitoes so that the mosquito-borne infectious diseases such as Zika, Dengue fever and Chikungunya would be eradicated. To qualitatively examine if [*Wolbachia*]{} can effectively invade the wild uninfected mosquito population, Zheng, Tang and Yu [@BOZHENG] considered the following model: $$\label{eq zhengbotang}
\left\{\begin{array}{ll}
\displaystyle{\frac{{\rm d} u}{d t}}= u[b_{1}-\delta_{1}(u+v)]&\hbox{for }~t>0,\vspace{7 pt} \\
\displaystyle{\frac{{\rm d} v}{d t}}=v\left[\displaystyle{\frac{b_{2}v}{u+v}}-\delta_{2}(u+v)\right]&\hbox{for }~t>0,
\end{array}\right.$$ where $u$ denotes the number of reproductive infected insects and $v$ denotes uninfected ones, $b_{1}$ and $b_{2}$ denote half of the constant birth rates for the infected and uninfected insects respectively. The parameter $\delta_{1}$ (resp. $\delta_{2}$) denotes the density-dependent death rate for the infected (resp. uninfected) population. The birth rate of uninfected mosquitoes is diminished by the factor $\frac{v}{u+v}$ due to the sterility caused by cytoplasmic incompatibility (CI) for mating between infected males and uninfected 