Long-run Behaviors Of Several Nonlinear Discrete Size-structured Population Models

In order to study the evolution and control problems of a biological population,we often establish some mathematical models for the population based on somesuitable assumptions. Doing so transforms a problem associated with a biologicalpopulation into a mathematical one. On one hand, this approach can take the greatadvantages of the existing mathematical methods. On the other hand, it can also getsome new conclusions which cannot be obtained by the traditional field analysis andexperimental research. So mathematical modeling plays a very important role in thepopulation dynamics. Compared with continuous type, discrete models is naturalmathematical object since they model ecological situations better. In1945, H. Leslieput forward an important discrete population model. From then on, there are manyscholars who pay more attention to this topic. By discrete models we can not onlypredict the survival trends of the population, but also obtain some insights about howto conserve or explore population resources, such as harvesting or logging.In the thesis, we study mainly three types of discrete population models, whichare of linear, nonlinear matrix forms, and two-patch one, respectively. Equilibria andtheir stability constitute the former, while the latter is devoted to the investigation ofthe boundedness and long-term behaviors. By the use of matrix theory and numericalanalysis tools, we obtain some new results, which provide us a theoretical foundationfor the practical applications.The chapter2proposes and deals with a generalized Leslie model, existence andstability of equilibrium are studied, and long-run behaviors analyzed.The chapter3sees a nonlinear population model with a typical fecundityfunction displaying intra-population competition or crowding. We carefully analyze the condition and distribution of equilibrium, and determine their stability. Andfinally we present some numerical examples.The chapter4is concerned with a two-patch model of species. We divide theindividuals in each patch into three groups, in which the first one is non-reproductive.The model involves the diffusion between the two patches and the various growthcases. We demonstrate the boundedness of population distribution, and derive theconditions for equilibrium of zero.