Optimal Control Of Gilpin-Ayala Population Harvesting System

Biological resources (such as forest resources, animal husbandry resources and fishery resources, etc.) are renewable resources. They can develop, breed and update to keep certain reserves according to their own characteristics with the aid of natural circulation. In the modern society, owing to the way of high input, high consump-tion and high pollution to seek rapid economic growth, the uptake of the biological resources of social production consumption ability has been far more than its own update cycle ability, which results in a depletion of resources. Therefore, the optimal management of renewable resource has great influence on the sustainable develop-ment. In recent years, the related problems about how to use the limited renewable resource to realize its sustainable development, has attracted the attention of many scholars. In some cases, people need to guarantee the ecological environment sus-tainable development, and base on that to pursue the maximizing the economic net income. In some other cases, it may be expected to obtain the maximum stock of the population at the end of the harvest season under the fixed harvest yield. The Common harvest strategies include continuous harvest and impulsive harvest, and the harvests that depend on the time. In order to study the influence of various key factors to the sustainable development of population system quantitatively, to exactly describe the various control strategies, and to evaluate their effectiveness, we need to establish the mathematical model of biological system under the background of the consumption of resources. Through the theoretical analysis of mathematical model, we can determine the relationship between the growth rule of the popula-tion resource and human intervention behavior. Further, we can evaluate, analyse and predict the change trends of population system in different parameter condi-tions. Finally, we can provide theoretical guidance for the management of renewable resources.In this thesis, we discuss the optimal control problem of a class of single popu-lation system with continuous harvest and impulsive harvest by using the extremum priciple and the basic theory of population dynamics. Our results enrich the ba-sic theory of population dynamics, and can help to make decision for the actual ecological problems.The main research contents and results in this thesis are as follows:(1) Continuous harvesting problem of Gilpin-Ayala equation under a Periodic environment is investigated. The main purpose is to study the influence of dif- ferent harvest yield to the system and obtain the optimal harvesting policy which maximizes the total harvest. Firstly the existence and stability of positive periodic solution is discussed and by choosing the harvest effort as control variable, the exact expressions of the optimal control strategy and the optimal profit are obtained by the maximum principle of differential system and some analysis techniques. Also, by the extremum principle and the steepest approximation principle, we study a maximum harvesting problem in given time range for a general nonautonomous Gilpin-Ayala system, the optimal harvesting function and the maximum amount of harvest are determined respectively for the different initial conditions.(2) A class of optimal impulsive control problem modeled by Gilpin-Ayala equa-tion is investigated. The species is harvested at fixed moments for economic profit under the periodicity condition. The main purpose is to research the influence of different impulsive harvesting effort to the system and obtain the optimal harvesting policy which maximizes the profit. We assume that profit is the difference between economic revenue and cost and the cost is proportional to the control variable. Firstly, it is showed that the system has a unique positive periodic solution which is globally asymptotically stable under the appropriate conditions. Further, the op-timal control strategy is characterized by the numerical equations by the extremum principle of impulsive differential system and some analysis techniques.(3) Assuming that a population growth follows the Gilpin-Ayala equation, the multiple linear impulsive harvest optimal policies for the maximum stock level of the population at the end of a harvesting season are investigated. Firstly, the necessary condition of the optimal harvesting moments is obtained by using the extremum principle of impulsive differential system and some analysis techniques. Moreover, for any given initial population and harvesting season length, we obtain the exact expression of the optimal harvest strategy when the stock level at the end of the season reaches its maximum. Further, we study the maximum number of impulsive harvest with same quantity of impulsive harvest during the exploitation period.