Criteria of global attraction in systems of delay differential equations with mixed monotonicity

n this paper we extend the classical “decomposing+embedding” method for systems of delay differential equations. Our extension has two advantages: (1) enhancing the range of applicability of the classical method and (2) providing delay dependent criteria of global attraction. The leading ideas are, on the one hand, the extension of the notion of dominance introduced first for difference equations and, on the other hand, the use of some monotone ideas developed by H.L. Smith and collaborators. We apply our results to some classical models of population dynamics, mainly models with patch structure.