Effects of fast diffusion in the logistic equation with refuge

Abstract

This paper studies the behaviour of the positive solution of a logistic equation with respect to a space dependent diffusion rate. The equation also includes a refuge, a zone where the species grows freely. In contrast to the case of homogeneous diffusion coefficient, where the species dies for large diffusion regardless of the birth rate, we show that the species may die, persist or growth indefinitely, depending on the size of the birth rate, for a large increasing of this diffusion rate in a certain region of the space, even more, this growth causes blow up in this region as well as in the refuge.