Global attractivity in concave or sublinear monotone infinite delay differential equations

We study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities. We analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way, for which we prove the existence of a globally asymptotically stable minimal set given by a 1-cover of the base flow. We apply these results to the description of the long term dynamics of a nonautonomous model representing a stage-structured population growth without irreducibility assumptions on the coefficient matrices.