A GENERALIZATION OF THE BUTLER-McGEHEE LEMMA AND ITS APPLICATIONS IN PERSISTENCE THEORY

The so-called Butler-McGehee lemma was first stated and proposed by Freedman and Waltman [11] to study persistence in three interacting predator-prey population models. Roughly speaking, the lemma says that if a trajectory, not on the stable manifold of a given isolated hyperbolic equilibrium P, has that equilibrium in its !-limit set, then its !-limit set also contains points on the stable and unstable manifolds of the equilibrium di↵erent from P. The lemma has been extended to di↵erent forms. The main purpose of this paper is to generalize one of the various formats of the Butler-McGehee lemma (Butler and Waltman [4]) in such a way as to encompass orbits from a set G rather than from a single point. An application to the uniform persistence of a class of dynamical systems which are not necessarily point dissipative is given.