A generalization of the Butler-McGehee lemma and its applications in persistence theory

Abstract

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 $\omega$-limit set, then its $\omega$-limit set also contains points on the stable and unstable manifolds of the equilibrium different from $P.$ The lemma has been extended to different 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.