Existence of a generalized Hopf bifurcation in the predator-prey model

Consider the predator-prey model given by Equation (Main), and suppose it admits a Hopf bifurcation as in the stated Hopf-bifurcation theorem for a given set of parameters. The coexistence equilibrium point is the equilibrium at which the two species coexist, and the first Lyapunov coefficient is the coefficient determining the local criticality of the Hopf bifurcation. Generalized Hopf-bifurcation conjecture. If the first Lyapunov coefficient becomes zero and the coexistence equilibrium point has a pair of purely imaginary eigenvalues, then the model undergoes a Bautin, or generalized Hopf, bifurcation. This conjecture identifies the codimension-two transition separating subcritical and supercritical Hopf-bifurcation branches; the supplied text provides no resolution, so its status remains open.