SOME MONOTONE PROPERTIES FOR SOLUTIONS TO A REACTION-DIFFUSION MODEL

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 {mu Delta theta + (m(x) - theta)theta = 0 in Omega, partial derivative theta/partial derivative n = 0 on partial derivative Omega (0,1) is a strictly monotone decreasing function of the diffusion rate it for several classes of function m, which substantially improves a result in [20]. However, the minimum of the positive solution of (0.1) is not always monotone increasing in the diffusion rate [15].