Some monotone properties for solutions to a reaction-diffusion model

2018/09/18 by Li, Rui, Yuan, Lou · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.06519

Abstract

Motivated by the recent investigation of a predator-prey model in heterogeneous environments \citeLouYuan-WangBiao, we show that the maximum of the unique positive solution of the scalar equation \begincases μΔθ+(m(x)-θ)θ=0 \hspace0.5em amp;in\hspace0.5emΩ,
(∂ θ)/(∂ n)=0 \hspace0.5em amp;on\hspace0.5em∂Ω\endcases is a strictly monotone decreasing function of the diffusion rate μ for several classes of function m, which substantially improves a result in \citeLouYuan-WangBiao. However, the minimum of the positive solution of \eqrefeq:01 is monotone increasing under proper assumptions on the resource function, it is not always monotone increasing in the diffusion rate \citeHeXiaoqing-NiWeiMing2016.

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