Population Dynamics Behavior And Biological Resources, Optimal Development Strategy

This paper investigate mainly some problems about the Mathematical Ecology, they can be divided into two main parts: One is the population dynamics; Another is the optimal harvesting policy for renewable biological resources.This paper consists of five chapters.Chapter 1 is introduction, covers some preparations in Mathematics and Ecology. We will introduce some definitions in the Mathematics Ecology and the mathematical theories and methods. Methods for studding the population ecology or the more ordinary ecology are by setting up the mathematical modelings. We will give the examples of the single population models and the population with interaction models.Chapter 2 is the population dynamics that including the existence, bounded, stability and permanent(population goes to extinction or permanent). We study mainly the existence and stability of periodic solutions and almost periodic solutions for the population models by the Liapunov second method and the topological degree theorem(if population has dynamic balance or not).Chapter 3 is the optimal exploitation policy for the renewable biological resources. The optimal harvesting policy can be given to the manager who has the exploit arrangement of the population such as a single population and a stage-structure population. The approaches to the optimal harvesting policy are the Maximum principle and Bang-Bang singular control.Chapter 4 is the problem that the diffusion effects on the permanent and extinction of the population in the polluted environment. That to say, it is the significance of the effects of protective patch and conclude that it is effective for the conservation of population face to the polluted environment.Chapter 5 is the generalize Poincare sphere in R3 . That is , the methodto study the global structure of the orbit can be geralized in the space R3. We will give the example by the Lorenz equation .