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Hopf and Bautin bifurcation in a tritrophic food chain model with Holling functional response types III and IV. (English) Zbl 1388.34047

Summary: 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 \(\mathbb R^3\). 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.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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