as $d\rightarrow 0$.\

3.  For any fixed $h_{0}>0$, there exists $d=d^{*}>0$ such that\

    -   $\lambda_{1}(d)<0$ for $0<d<d^{*}$,\

    -   $\lambda_{1}(d)>0$ for $d>d^{*}$, and\

    -   $\lambda_{1}(d)=0$ for $d=d^{*}$.

\[the case about l\] Assume that holds, where $B(x)$ is replaced by $b(x)$. Then, $\lambda_{1}=\lambda_{1}(h_{0})$ has the following properties:

1.  $\lambda_{1}(h_{0})$ is monotone decreasing with respect to $h_{0}$.\

2.  $\lambda_{1}(h_{0})\rightarrow+\infty$ as $h_{0}\rightarrow0$ and $\displaystyle{\lim_{h_{0}\rightarrow+\infty}\lambda_{1}(h_{0})<0}$.\

3.  For any fixed $d>0$, there exists $h_{0}=h^{*}_{0}>0$ such that\

    -   $\lambda_{1}(h_{0})>0$ for $0<h_{0}<h^{*}_{0}$,\

    -   $\lambda_{1}(h_{0})<0$ for $h_{0}>h^{*}_{0},$\

    -   $\lambda_{1}(h_{0})=0$ for $h_{0}=h^{*}_{0}$.

For the reader’s convenience, we also recall some facts related to the following problem $$\label{Eq_ourmodelstationary1}
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial v}{\partial t}=d_{2}v_{xx}+v\left(b_{2}(x)-\delta_{2}v\right),}&t>0, ~~ x>0,\vspace{7 pt}\\
v_{x}(t,0)=0, &t>0,\vspace{7 pt}
\\
v(0,x)=v_{0}(x), &x\in[0,+\infty).
\end{array}\right.$$ The proof of the next lemma follows from Lemma 5.2 and Lemma 6.2 of [@zhouxiaowenzi].

\[the long time behavior\] Assume that $b_{2}(x)$ satisfies , where $B(x)$ is replaced by $b_2(x)$. Let $v(t,x)$ be the unique solution of (\[Eq\_ourmodelstationary1\]) with an initial condition $$v_{0}\in C^{2}[0,\infty) \cap {L}^{\infty}[0,\infty)\text{ and }v_{0}>0.$$ Then, $$\lim_{t\rightarrow+\infty}v(t,\cdot)=\phi_{v*}\text{ uniformly in any compact subset of } [0,\infty),$$ where $\phi_{v*}$ is the unique positive solution of the following elliptic problem $$\label{Eq_ourmodelstationary2}
\left\{\begin{array}{ll}
d_{2}v_{xx}+v\left(b_{2}(x)-\delta_{2}v)\right)=0,&x>0,\vspace{7 pt}\\
v_{x}(0)=0.
\end{array}\right.$$

Sharp criteria for spreading and vanishing {#inhom_criteria}
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Let us first consider the 