A Predator - Prey System Saddle-node Bifurcation

In the world of today, our ability of changing the nature is greatly heightened. Because of over-exploitation, some species are close to extinction or already extinct. We destroy the ecological balance seriously. Recognize the reality, in this paper, we investigate a mathematical model of a predator-prey system in which both predator and prey have their respective carry capacities and have a functional response of Holling type II. The existence of saddle-node bifurcation is verified. Consider the actual meaning of the system, we are mainly interested in the dynamical behaviour of the system in the first quadrant. In the first quadrant, the change of number and stability of equilibrium points are given. The boundedness of the system with given initial condition is studied. Combing the phase portraits, we discuss the biological significance of the model. At last, we find that the system may have different equilibrium state when the initial scale is different.