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}.
Categories: math.AP