Optimal Control For Some Time-varing Population Dynamics With Diffusion

The study of the optimal control of population dynamics has important economic value and practical meaning. Recently people more and more are concerned of the control problem of time-varing population dynamics Lotka-Mckendick model with age-dependent and spatial diffusion. On the basis of the functional analysis and distributed parameter systems, the paper considers optimal control for a nonlinear population dynamics with state observation, optimal harvesting for predator-prey system of three species and n-dimensional food chain model.On the basis of presented results, we take an important factor that the natural death-rate and fertility-rate of an individual depends on the total population size at the time into account in the article, and investigate the optimal control with state observation. Firstly according to preliminary knowledge in chaper2, we establish existence result of optimal control by the Banach fixed point theory using the normal cone. At the same time, the paper analyzes optimal harvesting for predator-prey system of three species, which is controlled by fertility, the existence and uniqueness of solution for the system are proven using the comparison principle of population dynamics and the Banach fixed point theory, and also get existence result of optimal harvesting and necessary optimal conditions. Finally, we extend the three species system to the n dimensional chain food population dynamics and discuss its optimal harvesting, we obtain the similar result.