| In this paper, I makes a new predator-prey model and improves the harvesting rate on the basis of actual production. Combined with economic benefit theory, the paper also builds the ecological economy system, which is described by the differential-algebraic system. Depend on stability theory, Hopf bifurcation theorem and the theory of differential-algebraic system, this paper studies the stability of the equilibrium and Hopf bifurcation, and discusses the center-focus problem about the second system. Furthermore, the paper picks out certain data to build specific image according to numerical simulation method by Matlab, and compares the results with theoretical ones. In a word, this paper focuses on testing the consistency of asymptotic behavior theoretically and practically.The first chapter is the introduction. It introduces the biomathematics’ research background, purpose and significance in the differential-algebraic systems. At the same time, we give the the main work in the paper and innovation points.Chapter Ⅱintroduces the equilibrium of dynamical system and its stability theory, Hopf bifurcation theorem, center stability theorem, the threshold theory and the some conclusions of differential algebraic system.The third chapter establish a predator-prey biological economic system with nonlinear harvesting rate for prey. Using the knowledge in chapter Ⅱ,we study the equilibrium stability and Hopf bifurcation and choose the economic profit as the bifurcation parameter.In Chapter Ⅳ, we establish a predator-prey biological economic system with threshold harvesting rate. As the above, we use the knowledge in chapter Ⅱto research the equilibrium stability and Hopf bifurcation and choose the economic profit as the bifurcation parameter. Then, the center-focus problem of the system is investigated.The fifth chapter is the summary of the article. Based on my paper, I give some improvement of work and expectation for the future research. |
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