Mathematical Models And Dynamic Analysis Of Management And Development In Single Population Resource

Along with the socio-economic development, the ecological environment for human survival has been continuously destroyed. The natural resource is confronted with being exhausted because of human over-exploitation, which threatens seriously the safety of human survival. The grim situation of gradual deterioration of the environment restricts socio-economic development severely. In recent years, dynamics of the mathematical models of the group's exploitation and management has been developed quickly which plays a guiding role in the use of biological resources with the long-term eco-efficiency and economic benefits. Based on this case, this article establishes mathematical models and uses the methods of mathematical analysis to study exploitation and management of the single population and obtains the following four major conclusions at last.Firstly, when the population is developed in a continuous time, and the growth of this population is continuing, this article establishes a harvesting model of population growth with logistic rule and has a detailed analysis on the explored behavior with constant capacity of harvesting. Using the qualitative theory of differential equations, it analyzes optimal sustainable output and maximum economic benefits. It obtains the level of optimal sustainable output and largest economies profits and discusses this level how to affect the population, which provides a theoretical basis for the continuous exploitation of the population.Secondly, when the population is developed in a continuous time, and the growth of this population with age structure is discrete. It establishes Single-population model .It divides this single-population into four age structures, namely infancy, youth, middle age or old age. It analyzes the stability of this model thoroughly, finds the necessary and sufficient conditions for critical stability of the model, gets solutions of limit state in the critical stability of the balance and obtains the biological significance of the critical stability.Thirdly, when the development of population is not managed, it establishes the feedback harvesting model of population growth with Ricker rule. In this model, it assumes the exploitation is unlimited and the price changes with the relationship between supply and demand changes. The qualitative theory of differential equations is used to discuss existence and stability of the equilibrium and non-existence of the limit cycles. It explains the results from the perspective of ecology and economics, researches the related issues on the exploitation of biological resources and provides the necessary theoretical basis on the actual exploitation and management of biological resources.Fourthly, when the population exploitation has pulse effect .It establishes a pulse capture model of population growth with Gompertz rule. It analyzes maximum sustainable output of human harvesting from two aspects, namely constant harvesting rate and constant harvesting capacity. With the knowledge of dynamics system it obtains the expression of maximum sustainable output and analyzes the relations between the largest harvesting output of population and maximum sustainable output which determine whether the development of population can be sustainable. At the same time, the article compares constant harvesting rate with constant harvesting capacity. It finds that the exploitation of constant harvesting capacity is superior to constant harvesting rate.Finally, it puts forward some ideas and thought of development and management of population. The author of this article hopes that the human not only consider the recent socio-economic benefits, but also take into account the long-term ecological and economic benefits when they make use of biological resources, that is, use biological resources constantly.