Stability And Bifurcation Of A Predator-prey Biological Economic System With Nonlinear Harvesting Rate

In this paper, we study the stability and Hopf bifurcation of a predator-prey biological economic system with nonlinear harvesting rate through the use of differential algebraic system theory and Hopf bifurcation theory. Then, applying the Euler scheme to the system, a diference-algebraic system is obtained. The stability and Neimark-Sacker bifurcation of the discrete system are discussed by using the stability theory and Neimark-Sacker bifurcation theory. For continuous and discrete cases, the asymptotic behaviors of the systems are presented respectively by MATLAB numerical simulation. And their results are comparatively analyzed. The paper includes five chapters.The first chapter introduces the research background of this field, the modeling process of biological economic system and the practice meanings of the model. The main work and innovation of this paper are also pointed out.The second chapter is prearrangement knowledge. The related theoretical knowledge of the stability and Hopf bifurcation is introduced. We also introduce the related knowledge of the difference algebraic system and Neimark-Sacker bifurcation theory.The third chapter mainly studies the stability of the positive equilibrium point of a predator-prey biological economic system with nonlinear harvesting rate on the basis of the second chapter. Next, we choose economic revenue v as bifurcation parameter, the Hopf bifurcation of the system are studied.In fouth chapter, we first apply the Euler scheme to the system in the third chapter. Then we discuss the stability of positive fixed point of the obtained diference-algebraic system. And the Neimark-Sacker bifurcation of this system is studied. Here we still choose economic revenue v as bifurcation parameter.The final chapter gives a brief review to our problems studied, and future research work of the related area is introduced.