Several Bifurcation Mechanisms for Limit Cycles in a Predator–Prey System

The research presented in this paper compares the occurrence of limit cycles under different bifurcation mechanisms in a simple system of two-dimensional autonomous predator–prey ODEs. Surprisingly two unconventional approaches, for a singular system and for a system with a center, turn out to produce more limit cycles than the traditional Andronov–Hopf bifurcation. The system has a functional response function which is a monotonically increasing cubic function of x for 0 ≤ x≤ 1 where x represents the prey density, and which is constant for x> 1. It acts as a proxy for investigating more general systems. The following results are obtained. For the Andronov–Hopf bifurcation the highest order of the weak focus is 2 and at most 2 small-amplitude limit cycles can be created. In the center bifurcation cases are shown to exist with at least 3 limit cycles. In the singular perturbation cases are shown to exist with at least 4 limit cycles and in some cases an exact upper bound of 2 limit cycles is obtained. Finally we indicate how the conclusions can be extended to more general systems. We show how an arbitrary number of limit cycles can be created by choosing an appropriate functional response function and growth function for the prey. One special situation is the system with group defense: the three bifurcation mechanisms typically produce less limit cycles if a group defense element is included.