\\
\widetilde{v}_{x}(t,h_{0})=\max\left\{\kappa_{2},\|v_{0}\|_{L^{\infty}([0,+\infty))}\right\},& t>0.
\end{array}\right.$$

It follows from the comparison principle that $$u(T_{1},x)\geq\widetilde{u}(T_{1},x)~\text{ for all }~(t,x)\in[0,+\infty)\times[0,h_{0}].$$ Clearly, $\widetilde{u}(T_{1},x)$ is independent of $\mu$. Now, we consider the following system. $$\label{eqth4.4y1}
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial \underline{u}}{\partial t}- d_{1}\underline{u}_{xx}=\delta_{1}\underline{u}[\kappa_{1}-\kappa_{2}-\underline{u}]},& t>T_{1},~~ 0<x<\underline{h}(t),\vspace{7 pt}\\
\underline{u}_{x}(t,0)=0=\underline{u}(t,\underline{h}(t)), &t>T_{1},\vspace{7 pt}\\
\underline{h}^{'}(t)=-\mu\underline{u}_{x}(t,\underline{h}(t)), &t>T_{1},\vspace{7 pt}\\
\underline{u}(T_{1},x)=\widetilde{u}(T_{1},x),& x\in [0,h_{0}],\vspace{7 pt}\\
\underline{h}(T_{1})=h_{0}.
\end{array}\right.$$ By Lemma \[le3.2\], we know that $\underline{h}(t)\leq h(t)$ for $t>T_{1}$. It follows from [@duyihongshou Lemma 3.7] that $\underline{h}_{\infty}=+\infty$ if $\mu\geq\bar{\mu},$ where $$\displaystyle{\bar{\mu}:=\max \left(1,\frac{\|\widetilde{u}(T_{1},x)\|_{\infty}}{\kappa_{1}-\kappa_{2}} \right)\frac{d_{1}(h_0^*-h_{0})}{\displaystyle{\int^{h_{0}}_{0}\widetilde{u}(T_{1},x)dx}}}.$$ This implies that $h_{\infty}=+\infty$.

By Theorems \[th4.3\] and \[th4.4\], we can also derive spreading criteria in terms of the diffusion coefficient $d_1$, for any fixed $h_0$.

\[th4.6\] Let $\displaystyle{d^*_{1}=\frac{4\delta_{1}(\kappa_{1}-\kappa_{2})h^{2}_{0}}{\pi^{2}}}$, where $h_{0}$ is any prefixed positive constant. Then, spreading occurs provided that either

1.  $0< d_{1}\leq d^*_{1}$

2.  [or]{}

3.  $d_{1}> d^*_{1}$ and $\mu\geq\bar{\mu}$.

Our next result is a criterion on “vanishing".

\[th4.5\] Assume that $$\displaystyle{h_{0}<\frac{\pi}{2}\sqrt{\frac{d_{1}}{b_{1}}}=\frac{\pi}{2}\sqrt{\frac{d_1}{\delta_1\kappa_1}}<h_0^*}.$$ Then, there exists $\underline{\mu}>0$ such that $h_{\infty}<+\infty,$ whenever $\mu\leq\underline{\mu}$.

Consider the following problem $$\label{eqth4.5y2}
\left\{\begin{array}{ll}
\bar{u}_{t}-d_{1}\bar{u}_{xx}=\bar{u}(b_{1}-\delta_{1}\bar{u}),&t>0,~~ 0<x<\bar{h}(t),\vspace{7 pt}\\
\bar{u}_{x}(t,0)=0,\bar{u}(t,\bar{h}(t))=0,&t>0,\vspace{7 pt}\\
\bar{h}'(t)=-\mu u_{x}(t,\bar{h}(t)),&t>0,\vspace{7 pt}\\
u(0,x)=u_{0}(x),~~ \bar{h}(0)=h_{0},&x\in[0,h_{0}].
\end{array}\right.$$ Lemma \[le3.2\] applies and yields that $$h(t)\leq \bar{h}(t)~ \text{ and }~u(t,x)\leq \bar{u}(t,x)~\text{ for }~t>0 \text{ and }0\leq x\leq h(t).$$ Furthermore, by Lemma 3.8 of [@duyihongshou], there exists $\underline{\mu}>0$ such that $\bar{h}_{\infty}<+\infty$ in the case $\mu \leq \underline{\mu}$, 