The Optimal Control Problem Of Some Population System With Harvest

The sustainable development of resources means that resources can be used forever, the optimal management of renewable resources has great influence on the sustainable development. In recent years, the study about optimal harvest problem of population systems has attracted wide attention of scholars. For some problems, it is expected that to keep maximal sustainable yield or economic benefits for a long time, but in other cases, the purpose may be to obtain the maximal storage level at the end of the time period under the fixed harvest yield condition. It is important to note that the economic benefits, environmental protection and sustainable develop-ment should all be taken into account in the process of population resources being exploited. That’s to say, we should not only exploit, but also should pay attention to protect these resources, which requires scientific and effective management.In this thesis, we discuss the optimal control problem of two classes of single population system with impulsive harvest and a class of three population system with continuous harvest by using the extremum principle of pulse and continuous differential system and the basic theory of population dynamics. The results provide a theoretical basis for maintaining ecological balance and optimal management of the population resources.The main research contents and results in this thesis are as follows:(1) A class of optimal impulsive control problem modeled by periodic logistic equation with impulsive harvest is studied. Under the periodicity condition, the species is harvested at fixed moments for economic profit, the harvesting function is supposed to include both cases of the proportional and constant harvesting. The purpose is to control the harvesting effort and harvesting constants as to maximize the profit which is the difference between economic revenue and cost. The existence of an optimal impulsive harvest control is showed and the exact expression of the optimal control policy is obtained by the maximum principle for optimal impulse control problem and some analytical skills.(2) The maximum storage problem of a class of population system with impul-sive harvest is studied. The population growth rule is described by the logistic model and the harvesting function involves both proportional and constant harvesting. In a given time period, we choose appropriate moments to harvest the specie impul-sively, the influence of different impulsive harvesting moments to the development of population system is investigated, and the optimal harvesting policy maximizing the population level at terminal time of exploitation period is obtained. Firstly, the necessary condition for the optimal harvesting moments is obtained using the maximum principle of impulsive differential system. And if the time period is long enough, the optimal harvesting strategy with multiple impulsive harvesting is dis-cussed. Furthermore, the optimal harvesting problem for any given time period and initial value is researched.(3) A class of three population prey-predator system with a cooperative popula-tion of the prey is established and studied. For the ecological balance and economic benefit, the prey and predator population are harvested. The dynamics and the optimal harvesting policy of the system are investigated. Firstly, the existence and stability of equilibrium solutions of the system are analyzed, the results indicate that unreasonable harvesting can lead the resources to extinct. Further, under the sustainable development requirement, with the goal of maximizing the long-term ecological and economic benefits, the optimal control problem is proposed and the optimal harvesting policy is obtained by using Pontryagin’s maximum principle.