Some monotone properties for solutions to a reaction-diffusion model

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}.