Optimal Harvesting for a Population System Modelling Periodic Environment and Hierarchical Age-Structure
Ecological researches show that there exist dominance ranks of individuals in many species.Moreover,natural populations are actually subject to seasonal fluctu-ations which makes their habitats often undergoes some periodic changes.Motivated by these considerations,in this paper,we investigate the optimal harvesting problem for a hierarchical age-structured population system in a periodic environment.Here the objective functional represents the net economic benefit yielded from harvesting.Firstly,by means of frozen coefficients and fixed point theory we show that the state system is well posed if the reproducing number is less than one.Meanwhile,it is shown that the population density depends continuously on control parameters.Similarly,we show that the adjoint system is also well posed.Then,the optimality conditions given by the feedback forms of state variable and adjoint variable are obtained by using the adjoint system and tangent-normal cone techniques.The existence of optimal har-vesting policy is verified via Ekeland's variational principle and fixed point reasoning.Finally,we use numerical simulations to verify the main results and find other dynamic properties of the system.The results in this paper generalize and improve the previous related results.
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