Josef Hofbauer: A Unified Approach to Persistence

Acta Applicandae Mathematicae 14 (1989), 11-22

Abstract. In this paper we present a simplified approach to persistence. We show in section 1 that essentially all known persistence theorems, including those of Butler, Freedman and Waltman [3,4], Gard and Hallam [9], Fonda [6], Hofbauer [11], Hutson [15] are all rather immediate consequences of a classical result of Ura and Kimura [20] on the flow near a compact invariant set, see chapter VI of Bhatia and Szegö [1].
(This result was independently discovered by Barnabas Garay : Uniform persistence and chain recurrence. J. Math. Anal. Appl. 139 (1989), 372-382.)

In section 2 we apply this to give a new and direct proof of the geometric permanence condition for Lotka-Volterra equations, and in section 3 we illustrate the whole approach with a concrete two-predator two-prey system.

Download: preprint dvi (sorry, no figures)
reprint (Please note that the above abstract is missing from the published paper)