Permanence criteria for Kolmogorov systems with delays

TL;DR

This paper establishes criteria for the permanence of delayed Kolmogorov systems, extending existing results to nonautonomous cases with delays, and providing delay-independent algebraic conditions for system stability.

Contribution

It extends Jansen's permanence results to delayed nonautonomous Kolmogorov systems, offering delay-independent criteria for stability and permanence.

Findings

Provided sufficient conditions for compact uniform attractors.

Extended Jansen's results to delayed nonautonomous systems.

Derived delay-independent algebraic conditions for permanence.

Abstract

In this paper, a class of Kolmogorov systems with delays are studied. Sufficient conditions are provided for a system to have a compact uniform attractor. Then Jansen's result (J. Math. Biol. Vol. 25 (1987) 411-422) for autonomous replicator and Lotka-Volterra systems has been extended to delayed nonautonomous Kolmogorov systems with periodic or autonomous Lotka-Volterra subsystems. Thus, simple algebraic conditions are obtained for partial permanence and permanence. An outstanding feature of all these results is that the conditions are irrelevant of the size and distribution of the delays.