Modeling And Computer Stimulation Of A Three-Life-History Stages Single-Species System With Cannibalism And Harvesting

China’s economy has developed rapidly in recent years. But it also brought up lots of problems. It especially cause many problems in the ecological environ-ment like the seriously destroyed environment and the over exploited resources and so on. People have to pay a heavy price for it, so we need to seek meth-ods for these issues continually. The population dynamics model can be used to help us to study and solve these problems. The regular of population and the state of stable steady are studied in this model. It is important in protecting the environment and developing and utilizing the population.There is a process that population’s growth from immature to mature, from mature to old. And population’s physiological function is different in immature, mature and the old, such as the rate of birth and death, the ability of competi-tion and prey. Therefore, it is more access to the real situation to consider the population with stage structure in the population dynamics model.How to develop and use the biological resources rationally? It needs us to study the development of biological resources. Through the establishing of the population model, qualitative and quantitative analyzing of the population model, predicting of the population changes and the impact on the population with people’s development and making the right optimal harvesting policy to make the biological resources to maintain the survival continually, and meeting human’s needs in the same time, these studies are very important in practice.In this paper, the works and the main conclusions are as follows:First, we introduce the background and significance of population dynamics, and summarize the model of Stage Structure Single population and its develop-ment and utilization in our country and abroad.Second, we improve the existed models and get a single cannibalistic popu- lation model with three-stage-structure in which the immature species are preyed by the mature as well as the old ones. We prove that the model has a positive so-lution. Then we give the non-negative equilibrium point. We obtain the sufficient conditions for the existence of a local asymptotical stability and global asymp-totical stability of the trivial equilibrium point as well as the sufficient conditions for the local asymptotical stability of positive equilibrium point.Third, on the basis of the improved model, we consider the model with capture immature species, mature species and old species respectively. Then we get equilibrium points and the sufficient conditions for the stability of equilibrium points, and make the optimal harvesting policy.Fourth, by using Maple mathematical software, we give a certain model parameters and initial values. In case of no-capture and capture, we draw a lot of images including vector fields, trajectories, trends and harvest. Then we calculate the numerical solutions. By analyzing the images and numerical solutions, we verify conclusions of the previously obtained.