Multiple bifurcation in a predator–prey system with nonmonotonic predator response

Source abstract

Synopsis A model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.