Coexistence of species in a tritrophic food chain model with Holling functional response type IV

We determine conditions on the parameters of a tritrophic food chain model, implying the coexistence of the species. We consider that both predators corresponding to middle and top trophic levels have Holling functional responses type IV, and the prey at the lower trophic level has either linear or logistic growth rate. We prove that the differential system has an equilibrium point in which it exhibits a supercritical Hopf bifurcation independently of the growth rate of the prey. In the logistic case, we prove the existence of at least three equilibrium points in the positive octant and one of them exhibits a supercritical Hopf bifurcation.