Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-IV Functional Response

In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳ functional response are investigated in detail by using the analyses of qualitative method, bifurcation theory, and numerical simulation. The qualitative analyses and numerical simulation for the model indicate that it has a unique stable limit cycle. The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddlenode bifurcation, Hopf bifurcation, homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie. the Bogdanov-Takens bifurcation), and we show the existence of codimension three degenerated equilibrium and the existence of homoclinic orbit by using numerical simulation.