A single consumer model with Neumann boundary condition.

This paper is concerned with the positive solution of a scalar problem dΔw+a(x)−g(x)w2=0inΩ,∂w∂ν=0on∂Ω.We show that the maximum and the minimum of the solution w are monotone with respect to the diffusion rate d. We also prove that there exists a threshold value d0 such that the problem admits a unique positive solution when d > d0 and not the other way around. [ABSTRACT FROM AUTHOR]