In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of [Formula: see text]. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.

JEL Classification: 37G15; 34C23; 37C75; 34D45; Secondary: 92D40; 92D25We 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.

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