Article Contents
Article Contents

# Some monotone properties for solutions to a reaction-diffusion model

• * Corresponding author: Rui Li
• Motivated by the recent investigation of a predator-prey model in heterogeneous environments [20], we show that the maximum of the unique positive solution of the scalar equation

$$$\begin{cases} \mu\Delta\theta+(m(x)-\theta)\theta = 0 \quad &\text{in} \quad \Omega,\\ \frac{\partial \theta}{\partial n} = 0 \quad &\text{on} \quad \partial\Omega \end{cases}$$$

is a strictly monotone decreasing function of the diffusion rate $\mu$ for several classes of function $m$, which substantially improves a result in [20]. However, the minimum of the positive solution of (1) is not always monotone increasing in the diffusion rate [15].

Mathematics Subject Classification: 34D23, 92D25.

 Citation:

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