Differential and Integral Equations

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

Shigui Ruan and Feng Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Differential Integral Equations, Volume 9, Number 6 (1996), 1321-1330.

First available in Project Euclid: 6 May 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C35
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 58F25 92D25: Population dynamics (general)


Yang, Feng; Ruan, Shigui. A generalization of the Butler-McGehee lemma and its applications in persistence theory. Differential Integral Equations 9 (1996), no. 6, 1321--1330. https://projecteuclid.org/euclid.die/1367846904

Export citation