as $t \rightarrow +\infty$.

\(i) and (ii) follow from Propositions 3.1, 3.2 and 3.3 of  [@GUOLAOSHI].

We recall the following comparison principle.

\[le3.2\] Assume that $0\leq T_{0}<T<+\infty$ and $\bar{h},\underline{h}\in C^{1}([T_{0},T])$. Denote by $$G_{T}=\{(t,x)\in \mathbb{R}^{2}:t\in (T_{0},T],~~ x\in(0,\underline{h})\}$$ $$G^1_{T}=\{(t,x)\in \mathbb{R}^{2}:t\in (T_{0},T]~\text{ and }~ x\in(0,\overline{h})\}.$$ Let $$\underline{u}\in C(\overline{G_{T}})\cap C^{1,2}(G_{T}),\quad \bar{u} \in C(\overline{G^1_{T}})\cap C^{1,2}(G^1_{T})$$ and $$\bar{v},\underline{v} \in L^{\infty}\cap C([T_{0},T]\times[0,+\infty))\cap C^{1,2}((T_{0},T]\times[0,+\infty)).$$ Suppose that

\[eqbjdl\] $$\left\{\begin{array}{l}
\displaystyle{\frac{\partial \bar{u}}{\partial t}-d_{1} \bar{u}_{xx}\geq\delta_{1} \bar{u}(\kappa_{1}-\bar{u}- \underline{v}),}\qquad T_{0}<t\leq T ,~ 0<x<\bar{h}(t),\vspace{7 pt}\\

\displaystyle{\frac{\partial \underline{u}}{\partial t}-d_{1} \frac{\partial^{2}\underline{u}}{\partial x^{2}}\leq\delta_{1} \underline{u}(\kappa_{1}-\underline{u}-\bar{v}), }\qquad T_{0}<t\leq T , ~0<x<\underline{h}(t),\vspace{7 pt}\\

\displaystyle{\frac{\partial \bar{v}}{\partial t}-d_{2}\bar{v}_{xx}\geq\delta_{2}\bar{v}\left(\frac{\kappa_{2}\bar{v}}{\underline{u}+\bar{v}}-\bar{v}- \underline{u}\right),}\qquad T_{0}<t\leq T ,~ x>0,\vspace{7 pt}\\

\displaystyle{\frac{\partial \underline{v}}{\partial t}-d_{2}\frac{\partial^{2}\underline{v}}{\partial x^{2}}\leq \delta_{2}\underline{v}\left(\frac{\kappa_{2}\underline{v}}{\bar{u}+\underline{v}}-\underline{v}- \bar{u}\right),}\qquad T_{0}<t\leq T ,~x>0,
\end{array}\right.$$ $$\left\{\begin{array}{ll}
\underline{h}'(t)\leq -\mu \underline{u}_{x}(t,\underline{h}(t)),&T_{0}<t\leq T ,\vspace{7 pt}\\
\bar{h}'(t)\geq -\mu \bar{u}_{x}(t,\bar{h}(t)) , &T_{0}<t\leq T ,
\end{array}\right.$$\
and\
$$\left\{\begin{array}{ll}
\bar{u}_{x}(t,0)\leq0,~~ \bar{u}(t,\bar{h}(t))=0, &T_{0}<t\leq T, \vspace{7 pt}\\
\partial_{x}\underline{u}(t,0)\geq 0, ~~ \underline{u}(t,\underline{h}(t))=0,&T_{0}<t\leq T,\vspace{7 pt} \\
\bar{v}_{x}(t,0)\leq0,~~  \underline{v}_{x}(t,0)\geq 0,&T_{0}<t\leq T, \vspace{7 pt}\\
\underline{h}(T_{0})\leq h(T_{0})\leq \bar{h}(T_{0}), \vspace{7 pt}\\
\underline{u}(T_{0},x) \leq u(T_{0},x)\leq \bar{u}(T_{0},x), &0\leq x\leq  h(T_{0}),\vspace{7 pt}\\
\underline{v}(T_{0},x) \leq v(T_{0},x)\leq \bar{v}(T_{0},x),&x\geq 0.
\end{array}\right.$$

\
Let $(u, v, h)$ be the unique solution of . Then,\

(i) $h(t)\leq \bar{h}(t),$ $u(t,x)\leq \bar{u}(t,x)$ and $v(t,x)\geq \underline{v}(t,x)$ for all $(t,x)$ in $(T_{0},T]\times [0,+\infty).$\

(ii) $ h(t)\geq \underline{h}(t),$ $u(t,x)\geq \underline{u}(t,x)$ and $ v(t,x)\leq \bar{v}(t,x)$ for all $(t,x)$ in $(T_{0},T]\times [0,+\infty).$

The 