Growth bound and threshold dynamic for nonautonomous nondensely defined evolution problems

Abstract

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, X) , t∈ R↦ F(t) ∈ L(X, 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.