## Differential and Integral Equations

### 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.

#### Article information

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

Dates
First available in Project Euclid: 6 May 2013

https://projecteuclid.org/euclid.die/1367846904

Mathematical Reviews number (MathSciNet)
MR1409931

Zentralblatt MATH identifier
0879.34047

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

#### Citation

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