cost case, $\kappa_1<\kappa_2$, represents a strong competition [@xriben1995]. Throughout this section, we always assume $u$ is a superior competitor. That is, the [*Wolbachia*]{} infection has a fitness benefit. The strong competition case is usually more complicated to be studied mathematically. To the best of our knowledge, results for competition models with a free boundary are very limited in strong competition case. Further details can be seen in [@Zhou1; @Zhou2].

We organize this section as follows. In subsection \[Preliminary\] we present some preliminary results, which play a role in proving our main results. Subsection \[the\_vanishing\] is devoted to the vanishing case. The invasion dynamics is studied in detail in Subsection \[the fitness benefit\]. A rough estimation of asymptotic spreading speed of [*Wolbachia*]{} invasion is given in Subsection \[the speed\].

Preliminary results {#Preliminary}
-------------------

Consider the system $$\label{eqjingchangyong}
\left\{\begin{array}{l}
\displaystyle{\frac{\partial u}{\partial t}=d_{1} u_{xx}(t,x)+u(t,x)(b_{1}-\delta_{1}u(t,x))},~~ t>0,~0<x<L,\vspace{7 pt}\\
\displaystyle{u_{x}(t,0)=u(t,L)=0},\qquad t\in (0, \infty).
\end{array}\right.$$ The following result holds.

\[le3.1\] Let $L^{*}=\displaystyle{\frac{\pi}{2}\sqrt{\frac{d_{1}}{b_{1}}}}$ and $\displaystyle{d_{*}=\frac{4b_{1}L^{2}}{\pi^{2}}}$. Then,

(i) if $L\leq L^{*}$, all positive solutions of tend to zero in $C([0, L])$ as $t \rightarrow +\infty.$

(ii) If $L> L^{*}$, there exists a unique positive stationary solution $\phi$ of such that all positive solutions of approach $\phi$ in $C([0, L])$ 