aegypti mosquitoes infected with the [*Wolbachia*]{} bacteria were initially released at a specific site. Hence, the infected female mosquitoes initially occupy only a small region, while the wild uninfected females are distributed over the whole area.

To model the spatial spreading of [*Wolbachia*]{} in the wild mosquito population and explore the possibility that the infection can indeed occupy the whole region, it is natural to consider system under the setting of a free boundary problem.

In this work, we consider the following free boundary problem in one-dimensional space: $$\label{Eq_ourmodel-two}
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial u}{\partial t}= d_{1}u_{xx}+u(b_{1}(x)-\delta_{1}(u+v))},&t>0, ~~ 0<x<h(t), \vspace{7 pt}\\
\displaystyle{\frac{\partial v}{\partial t}=d_{2}v_{xx}+v\left(\frac{b_{2}(x)v}{u+v}-\delta_{2}(u+v)\right),}&t>0,~~ x>0,\vspace{7 pt}
\\
\displaystyle{u_{x}(t,0)=v_{x}(t,0)=0, u(t,h(t))=0,}&t>0,\vspace{7 pt}\\
\displaystyle{h'(t)=-\mu u_{x}(t,h(t)),}&t>0, \vspace{7 pt}\\
 h(0)=h_{0},\vspace{7 pt}\\
u(0,x)=u_{0}(x),& x\in[0,h_{0}],\vspace{7 pt}\\
  v(0,x)=v_{0}(x),& x\in[0,+\infty).
\end{array}\right.$$ The equation governing the movement of the spreading front $x=h(t)$ is deduced in a manner similar to that in Section 1.3 of  [@Du2012]. It is known as the one-phase Stefan condition in the literature. This type of free boundary condition has been widely used in previous works such as [@caolizhao2017; @dingwwduyihong2017; @dingwwpengrui2017; @duyihongshou; @lifangDPP2016; @Lin2007A; @Linandzhu2016; @wuchanghong2016; @ShiguiRuan; @wangmingxinP-P2014; @weilzhoumaolin2016].

We shall consider system with constant birth rates $b_1$ and $b_2$ in Section \[specific model\] and consider space-dependent birth rates $b_1(x)$ 