Global dynamics of a ratio-dependent predator-prey system

Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey interactions where pre dation involves serious searching processes. However. such models have set up a challenging issue regarding their dynamics near the origin since these models are not well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey system at the origin in the in terior of the first quadrant is studied. It is shown that the origin is ind eed a critical point of higher order. There can exist numerous kinds of top ological structures in a neighborhood of the origin including the parabolic orbits, the elliptic orbits, the hyperbolic orbits, and any combination of them. These structures have important implications for the global behavior of the model. Global qualitative analysis of the model depending on all pa rameters is carried out. and conditions of existence and non-existence of l imit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.