Persistence and permanence for a class of functional differential equations with infinite delay

TL;DR

This paper establishes broad, easily verifiable conditions for the persistence and permanence of solutions in a wide class of cooperative functional differential equations with infinite delay, relevant to mathematical biology.

Contribution

It introduces a general method for analyzing persistence and permanence in FDEs with infinite delay, applicable to both autonomous and non-autonomous systems, including many biological models.

Findings

Conditions for persistence are established for solutions with positive initial conditions.

Conditions for permanence are derived, ensuring long-term boundedness of solutions.

The method is applicable to a broad class of models, demonstrated with examples.

Abstract

The paper deals with a class of cooperative functional differential equations (FDEs) with infinite delay, for which sufficient conditions for persistence and permanence are established. Here, the persistence refers to all solutions with initial conditions that are positive, continuous and bounded. The present method applies to a very broad class of abstract systems of FDEs with infinite delay, both autonomous and non-autonomous, which include many important models used in mathematical biology. Moreover, the hypotheses imposed are in general very easy to check. The results are illustrated with some selected examples.