A note on permanence of nonautonomous cooperative scalar population models with delays

For a large family of nonautonomous scalar-delayed differential equations used in population dynamics, some criteria for permanence are given, as well as explicit upper and lower bounds for the asymptotic behavior of solutions. The method described here is based on comparative results with auxiliary monotone systems. In particular, it applies to a nonautonomous scalar model proposed as an alternative to the usual delayed logistic equation.