HOPF-FOLD BIFURCATION AND FLUCTUATION PHENOMENA IN A DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR-PREY MODEL

A class of delayed ratio-dependent Gause-type predator-prey model is considered. We study the eigenvalue problem for the linearized system at the coexisting equilibrium. For a critical case when the characteristic equation has a single zero root and a simple pair of pure imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. We analyzed the influence of the time delay on the Hopf-Fold bifurcation and showed the occurrence of quasi-periodic motion and bursting behavior. This phenomenon is in line with the seasonal variation law of the population.