We derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems. This result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation. In particular, we characterize the bifurcation values from infinity to positive steady states. The stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed. © 1996 Academic Press, Inc.