$\tau>0$ depending only on $M_{1}$, $M_{2}$ and $M_{3}$ such that the solution of with the initial time $T_{max}-\frac{\tau}{2}$ can be extended uniquely to the time $T_{max}+\frac{\tau}{2}$, which contradicts the definition of $T_{max}.$ Thus, $T_{max}=+\infty$ and the proof is complete.

The special case of constant birth rates {#specific model}
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System was investigated in [@huangmugen1; @huangmugen2] for two disjoint cases. Namely, the fitness benefit case and the fitness cost case. Define $\kappa_1$ and $\kappa_2$ as $\kappa_{1}=b_{1}/\delta_{1}$ and $ \kappa_{2}=b_{2}/\delta_{2}$. [*Wolbachia*]{} is said to have the fitness benefit if $\kappa_1>\kappa_2$, which means that the local area is more (or at least equally) favourable for infected mosquitoes. The fitness cost case is represented by $\kappa_1<\kappa_2$, see [@BOZHENG].

In this section, we assume that $b_{i}(x)=b_{i} $ for $i=1,2$, where $b_{i}$ are positive constants. In other words, we have the constant-coefficient free boundary problem given by$$\label{Eq_ourmodel}
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial u}{\partial t}= d_{1}u_{xx}+u(b_{1}-\delta_{1}(u+v)),}&t>0, ~~0<x<h(t), \vspace{7 pt}\\
\displaystyle{\frac{\partial v}{\partial t}=d_{2}v_{xx}+v\left(\frac{b_{2}v}{u+v}-\delta_{2}(u+v)\right),}&t>0,~~ x>0,\vspace{7 pt}
\\
u_{x}(t,0)=v_{x}(t,0)=0, u(t,h(t))=0,&t>0,\vspace{7 pt}\\
h'(t)=-\mu u_{x}(t,h(t)),&t>0,\vspace{7 pt}\\  h(0)=h_{0},\vspace{7 pt}\\
u(0,x)=u_{0}(x), ~~ x\in[0,h_{0}], & \vspace{7 pt}\\v(0,x)=v_{0}(x), ~~ x\in[0,+\infty).
\end{array}\right.$$ System is essentially a competition model. For the fitness benefit case, $\kappa_1>\kappa_2$, $u$ is the so-called superior competitor and $v$ the inferior competitor (see [@dusupandinf]). For the fitness 