| To explain a kind of ecological phenomenon. Hanski et al.[5] proposed the following model which includes two kinds of functional response: Rolling type-II and Rolling type-Ill.In this paper, we give a qualitative analysis of the predator-prey model (1) depending on all parameters. However this system has set up a challenging issue regarding its dynamics near the origin since this system is not well defined there. In the second section, We apply the theories and ways given in [17] on the analysis of the equilibria of the nonlinear equations, and solve the problem successfully. When t→∞ or t →∞, all of the orbits tending to the origin are sketched in the phase portraits. Moreover, several sufficient conditions of non-existence of limit cycles for the model are given. Especially, when the isocline of x is monotone decreasing in 0 < x < 1, the svstem hasno limit cycle and is globally stable; Next, we construct a saddle bifurcation at the boundary equilibrium and a degenerated Bogdanov-Takens bifurcation at the interior equilibrium by choosing appropriate parameter values in the following two sections, where our work are based on the theory of central manifolds and normaltorms. We prove that is a codimention 3 focus-type equilibrium.System (6.1) will have two limit cycles at some appropriate bifurcation parameter values, and have homoclinic or double-homoclinic orbits at some other appropriate bifurcation parameter values; At last, we study the qualitative properties of the system at infinite in the Poincare Sphere. |
