Stability Analysis Of Two Classes Of Predator Models

Biological mathematics is an emerging interdisciplinary subject developed in20thcenturybetween biological and mathematics. In recent years, the research of the theory on ecologyresearch becomes a hot issue. The most notable two problems are the stability analysis of modelswith harvest rate and constant prey refuge. People should have quantitative analysis to thedevelopment and change of the population, through the establishment of the models with harvestrate and prey refuge, people should judge pay how much capture amount of effort can we obtainthe maximum yield and satisfy people’s needs, while we can keep all kinds of groups of commonto save and won’t die out. Therefore the research of population biology is of great significance tokeep nature biological diversity and maintain the sustainable development of human resources.In this paper, we consider two system’s dynamic behavior, the structure of this paper arearranged as following:In Chapter1, we introduce the outline of the background and history of bi omathematics, including the development tendency and the hot issues in current research.In Chapter2, we put forward a prey model with harvest. Through defining the model againand roughly by three kinds of circumstance discussion, we give the asymptotic behavior at theorigin. Finally, the model will appear Hopf branch when the parameter in a certain range.Chapter3is devoted to study a general predator-prey model incorporating a constant preyrefuge with Hassell-Varley type functional response. Some conditions of the uniform persistenceare obtained. And a Lyapunov function is constructed to show the global stability of the positiveequilibrium.Finally, we summarize our main results in chapter4.The original and creative idea in our paper is: different from other model to the discussionabout the origin, through defining the model at the origin again and the polar form, we analyzedthe characteristic and give global stability results. As for the model incorporating a prey refuge,is not a certain number and it has a value range, this gives the qualitative analysis that cannotform a fixed number of population of biological.