Uniform persistence and net functions (Q1898805)

The authors discuss uniform persistence for retarded functional differential equations. For such equations, the phase space consists of some functions from \((- \infty, 0]\) into its range space \(X\). The approach developed in this paper, which is natural and useful, is to construct a set of Lyapunov functions \(\{V_1,\dots, V_p\}\) which are defined on \(X\) and determine a net, here a net is a partition \(\{X_1,\dots, X_{p+ 1}\}\) of \(X\) which is ordered by increasing time on trajectories: for \(k< p\), trajectories in \(X_k\) at some time cannot move into \(X_j\) for any \(j< k\) at any future time, and ultimately all trajectories lie in \(X_{p+ 1}\) with \(\text{dist}(X_{p+ 1}, \partial X)> 0\). Applications to a population dynamics model of Kolmogorov type with delays and to a simple food chain are provided to demonstrate the strength of the approach and the general results. More interesting applications can be found in other papers of the authors [Appl. Anal. 51, 197-210 (1993; Zbl 0822.92014); Differ. Integral Equations 6, 883-898 (1993; Zbl 0780.92019); Math. Biosci. 118, 197-210 (1993; Zbl 0806.92015)].