Note on some periodic optimal harvesting problems for age-structured population dynamics

We consider two optimal harvesting problems related to age-structured population dynamics with logistic term and time-periodic (of period T) control and vital rates. We derive the first order necessary optimality conditions for both problems. We prove that in the particular case of time-independent vital rates and under a natural hypothesis on the logistic term, the optimal control for the first problem is constant. The second problem concerns the constant harvesting efforts which act on subintervals of [0, T]. The maximizing here is treated with respect to the subinterval where the control acts. We find the directional derivative of the cost functional. Some final comments concerning the population dynamics with diffusion are given as well.