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

TL;DR

This paper analytically proves the existence of a stable periodic orbit in a tritrophic food chain model through a triple Hopf bifurcation analysis, revealing complex oscillatory behaviors among three species.

Contribution

It provides the first analytical proof of a stable periodic orbit resulting from a triple Hopf bifurcation in a tritrophic food chain model.

Findings

Existence of three limit cycles via bifurcation

One limit cycle in the absence of top predator

A stable cycle in the coexistence region

Abstract

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.