Positive perturbation of operator semigroups: Growth bounds, essential compactness, and asynchronous exponential growth

If B is the generator of an increasing locally Lipschitz continuous integrated semigroup on an abstract L space X and C : D(B) → X perturbs B positively, then A = B+C is again the generator of an increasing I.L.c. integrated semigroup. In this paper we study the growth bound and the compactness properties of the C₀ semigroup S_o that is generated by the part of A in X_o = D(B). We derive conditions in terms of the resolvent outputs F(λ) = C(λ - B)^-1 for the semigroup S_o to be eventually compact or essentially compact and to exhibit asynchronous exponential growth. We apply our results to age-structured population models with additional structures. We consider an age-structured model with spatial diffusion and an age-size-structured model.