Hopf Bifurcation Analysis and Control of a Ratio-Dependent Predator–Prey Model of Holling IV Type with Time Delayed Feedback

In present paper, the time-delayed feedback is coupled with a ratio-dependent predator–prey model of Holling ??? type. This predator-prey system can be seen as a human-controlled biological system. Regarding the delay as parameter, we investigate the existence of local Hopf bifurcations. By using the Hassard method and the center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcation. Finally, we give a numerical simulation, which indicates that when the delay passes through certain critical values, the positive equilibria is converted into a stable steady state. It means that we can control the stability of the equilibria by man-made control of the number of the predator with certain age.