Abstract algebraic-delay differential systems and age structured population dynamics

We consider the abstract algebraic-delay differential system,x′(t)=Ax(t)+F(x(t),a(t)),a(t)=H(xt,at). Here A   is a linear operator on D(A)⊂XD(A)⊂X satisfying the Hille–Yosida conditions, x(t)∈D(A)¯⊂X, and a(t)∈Rna(t)∈Rn, where X is a real Banach space. With a global Lipschitz condition on F and an appropriate hypothesis on the function H, we show that the corresponding initial value problem gives rise to a continuous semiflow in a subset of the space of continuous functions. We establish the positivity of the x-component and give some examples arising from age structured population dynamics. The examples come from situations where the age of maturity of an individual at a given time is determined by whether or not the resource concentration density, which depends on the immature population, reaches a prescribed threshold within that time.