Singular perturbations of the Holling I predator-prey system with a focus

Abstract

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.