Hopf Bifurcations in a Predator–Prey Model with an Omnivore

Abstract

We study limit cycle bifurcations in a three-dimensional Lotka–Volterra system which models a food web of three species, one of which is an omnivore. First, using a new approach based on the elimination theory of the computational algebra we find necessary and sufficient conditions for existence of a pair of pure imaginary eigenvalues for the Jacobian of the system at the stationary point with positive coordinates. Then it is shown that the system can have two small limit cycles bifurcating from the singular point.