Persistence in food webs—I Lotka-Volterra food chains

A unified approach to persistence

“…Choose a compact isolating neighborhood U of M. w(x) of any z ~ M must meet X \ U if neither (1) nor (2) is violated. Then X \ U is a weakly attracting region for X \ M. Its forward invariant closure 7+(X \ U) is then still compact and contains an attractor for X \ M whose dual repeller is M. [ As shown by Fondu [6] this condition readily implies those given earlier by Gard and Hallam [9] in terms of 'persistence functions' and Hofbauer [11] and Hutson [15] in terms of 'average Ljapunov functions' which are more accessible to concrete applications, since they involve only conditions for x E M: COROLLARY 2. If P is differentiable a/ong orbits then condition (b) can be replaced by (b') There is a continuous function ~b : X ~ 1R, such that P(z) >_ P(z)~b(z) /'or a/l • E X, and for each ~ E M there is a time T > 0 such that…”

Uniformly persistent systems

“…Generally speaking the term persistence is given to systems in which strictly positive solutions do not approach the boundary of the nonnegative cone as t -* oo. Various precise definitions of persistence have been given: a version of (weak) persistence [5,6,7] applied when it is required only that positive solutions do not asymptotically approach the boundary as t -» oo; persistence [3,4] means that each strictly positive solution is eventually at some positive distance from the boundary; uniform persistence, also called cooperativeness or permanent coexistence [10,12], means that strictly positive solutions are eventually uniformly bounded away from the boundary. Weak persistence has a drawback in that it guarantees only that extinction is not certain.…”

Uniform persistence and flows near a closed positively invariant set

“…The approach of using Liapunov-like functions has appeared in various forms, for example, Gard and Hallam (1979), Hoibauer (1980), Hutson (1984), Gard (1987), Fonda (1988), Hofbauer and So (1989), Fernandes and Zanolin (1989), and Freedman and Ruan (1994). The nicest statement is due to Fonda (1988), who stated the result in terms of repellers.…”