  Motivated by the recent investigation of a predator-prey model in
heterogeneous environments \cite{LouYuan-WangBiao}, we show that the maximum of
the unique positive solution of the scalar equation
\begin{equation}\label{eq:01}\begin{cases}
\mu\Delta\theta+(m(x)-\theta)\theta=0 \hspace{0.5em}
&\text{in}\hspace{0.5em}\Omega,\\ \frac{\partial \theta}{\partial n}=0
\hspace{0.5em} &\text{on}\hspace{0.5em}\partial\Omega \end{cases}\end{equation}
is a strictly monotone decreasing function of the diffusion rate $\mu$ for
several classes of function $m$, which substantially improves a result in
\cite{LouYuan-WangBiao}. However, the minimum of the positive solution of
\eqref{eq:01} is monotone increasing under proper assumptions on the resource
function, it is not always monotone increasing in the diffusion rate
\cite{HeXiaoqing-NiWeiMing2016}.
