On a diffusive logistic equation

We study a reaction-diffusion version on all of R-N of the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values. Our results which are obtained by the construction of sub- and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady-state distributions of population. (C) 1998 Academic Press.