Zero‐Hopf bifurcation in a family of tritrophic food chain model with Holling III–III functional response

In this paper, we apply the second‐order averaging theory for obtaining an explicit expression of the small amplitude periodic solution that bifurcates from a zero‐Hopf equilibrium point of a tritrophic food chain model. This model considers logistic growth rate for the lowest trophic level, Holling functional responses type III for the middle and for the highest level. We first prove that this model has a zero‐Hopf equilibrium point, after we show that from this equilibrium bifurcates a small limit cycle, and finally, we provide the explicit expression of the first two terms in the power series of this limit cycle. These differential systems for which the equilibrium point is non hyperbolic are not easy to study, in particular, if the equilibrium is zero‐Hopf. As far as we know, this is the first time that the averaging theory has been used to exhibit the first and second terms of the power series expansion of the analytical expression of a limit cycle that bifurcates from a zero‐Hopf equilibrium in the food chain models.