We propose a general framework for simultaneously calculating the threshold value for population growth and determining the sign of the growth bound of the evolution family generated by the problem below dv(t) dt = Av(t) + F(t)v(t) − V(t)v(t), where A : D(A) ⊂ X → X is a Hille-Yosida linear operator (possibly unbounded, non-densely defined) on a Banach space (X, ∥ • ∥), and the maps t ∈ R → V(t) ∈ L(X 0 , X), t ∈ R → F(t) ∈ L(X 0 , X) are p-periodic in time and continuous in the operator norm topology. We give applications of our approach for two general examples of an age-structured model, and a delay differential system. Other examples concern the dynamics of a nonlocal problem arising in population genetics and the dynamics of a structured human-vector malaria model.