We present a mechanical model for an oscillator with one degree of freedom under the influence of a flowing medium. Under fairly general conditions we show that the ensuing differential equation has at most two limit cycles and we give examples where exactly two limit cycles will occur. The implications of this result are that it is possible for a system of this kind to exhibit galloping even when the so-called Den Hartog criterion of local instability is not satisfied.

For a family of two-dimensional predator-prey models of Gause type, we investigate the simultaneous occurrence of a center singularity and a limit cycle. The family is characterized by the fact that the functional response is nonanalytical and ehibits group defense. We prove the eistence and uniqueness of the limit cycle using a new theorem for Liénard systems. The new theorem gives conditions for the uniqueness of a limit cycle which surrounds a period annulus. The results of this paper provide a mechanism for studying the global behavior of solutions to Gause systems through bifurcation of an integrable system which contains a center and a limit cycle.

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.

We study the occurrence of limit cyles in a model for two-species predator-prey interaction (referred to as Holling I). This model is based on the phenomenological fitting of the so-called functional response function to observed data in nature by a continuous, piecewise differentiable function. Of the original models presented by Holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet. We extend the work of Liu who showed that for certain values of parameters of the Holling I system at least two nested limit cycles will occur surrounding a focus. His case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities, i.e. a singular perturbation problem with slow-fast solutions. We prove that exactly two hyperbolic limit cycles occur after perturbation.

In this paper we study the generalized Gause model, with a logistic growth rate for the prey in absence of the predator, a constant death rate for the predator and for several different classes of functional response, all non-analytical. First we consider the piecewise-linear functional response of Holling type I, which essentially has a linear functional response on a bounded interval and a constant functional response for large enough prey density. Next we consider differentiable modifications of this type of functional response, one being a concave down function, the other one being a sigmoidal function. Our main interest is the number of closed orbits of the systems under consideration and the global stability of the system. We compare the generalized Gause model with a functional response that is non-analytical with the generalized Gause model with a functional response that is analytical (e.g., Holling type II or III) and show that the behaviour in the first case is more complicated. As examples of this more complicated behaviour we mention: the co-existence of a stable equilibrium with a stable limit cycle and the existence of a family of closed orbits.