Stability And Hopf Bifurcation Of A Ratio-dependent Predator-prey System

In recent years, the ratio-dependent theory is a very important theory put forward by mathematical biologists and focused by many scholars. In this dissertation, the problem of the local Hopf bifurcation of the positive equilibrium in the ratio-dependent predator-prey dynamical systems with the stage-structures of the prey and the second functional response function is mainly considered. And it is verified employing the nonlinear dynamical software,WinPP. The results of the numerical simulation employing the dynamical software WinPP are in good agreement with those of the theoretical analysis. It verifies the correctness of theoretical analysis methods in this dissertation. The results of this dissertation may have very important revelations for the biological control.In Chapter 1, some fundamental theories which are employed in this dissertation , i.e., the theories of stability and Hopf bifurcation, are presented.In Chapter 2, the problem of the local Hopf bifurcation of the positive equilibrium in the ratio-dependent predator-prey model with the stage-structures of the prey and the second functional response function is considered. First, the existence of the local Hopf bifurcation is analyzed by the theory of Hopf Bifurcation Theorem. Then by the Center Manifold Theorem and the Normal Form theory, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are calculated. Employing the nonlinear dynamical software WinPP, the correctness of the above theoretical results are verified.