Analytical Study of a Triple Hopf Bifurcation in a Tritrophic Food Chain Model

We provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain. The method used consists in analyzing a triple Hopf bifurcation. For some values of the parameters three limit cycles bear via this bifurcation. One is contained in the plane where the top predator is absent. Another one is not contained in the domain of interest where all variables are positive. The third one is contained where the three species coexist.