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

Abstract

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.