The Optimal Harvesting Strategy Of Population System With Impulsive Harvest

In early studies of population dynamics, the ordinary differential equation mod-els are usually used to describe the dynamic behavior of the population systems, which has played an important role for solving many practical problems. However, ordinary differential equations can’t accurately describe the process with impulsive effects, in which the state variables are continuous between impulses and are dis-continuous at the impulses moments, impulsive differential equation can describe these problems better. The theory of impulsive differential is more complex, but are more important for studying many impulsive dynamics, especially for studying population systems involving impulsive effects.The study of impulsive differential equation started in1960’s, with the deep-ening of the theoretical research, the application research has gained great develop-ment. In recent years, impulsive differential systems are used widely to describe and solve the practical problems, such as, to study population dynamical systems, to con-trol pest ecosystems, and to optimize biological resources etc. However, biospecies resources is a kind of renewable resources, the problem that how to develop and utilize effectively the limited renewable resources and to achieve sustainable devel-opment has become a central problem for many mathematics, biologists and eco-nomics to consider. When people develop and utilize the species resources, ecological benefits and economy benefits usually are regarded as the principles of exploitation, which make it is possible for people to exploit and use biospecies resource perma-nently. In recent years, the study on the problem of exploitation and management of biospecies resource is a hot research topic, many scholars carry on a great deal of research works on different problems. At present, the optimal harvesting strategy for biological resources of population system usually is discussed by the method of mathematical analysis. We have noticed that the maximum principle of impulsive differential system is a powerful tool for solving the impulsive optimal problem. In this thesis, we mainly apply the maximum principle of impulsive differential system to study the optimal harvesting strategy of the population system with impulsive harvesting.The main results of this thesis are as follows: First, the optimal harvesting strategy of a class of single population system is studied. The population system is modeled by Gompertz equation with impul-sive harvesting within a finite time interval. By choosing the impulsive harvesting moments as control variables, the aim is to obtain the maximum stock level of the population at the end of given time period for given intensity and number of times of impulsive harvesting. By applying the maximum principle of impulsive differ-ential system, the condition that the optimal harvesting moment should satisfy is discussed and the optimal harvesting strategy is obtained for the case of long time period. Moreover, some analytical techniques are applied to study the most har-vesting times and the optimal harvesting strategy within given time interval, and some numerical simulations are also carried out.Secondly, a class of optimal harvesting problem of two population system with combine impulsive harvest is studied. Under the periodicity condition, both of two species are harvested with the same proportional harvest function at fixed moments. The purpose is to control the harvesting effort 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 of impulse differential system.