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Zbl 0844.92025
Kuznetsov, Yu.A.; Rinaldi, S.
Remarks on food chain dynamics. (English)
[J] Math. Biosci. 134, No.1, 1-33 (1996). ISSN 0025-5564

Summary: The main modes of behavior of a food chain model composed of logistic prey and Holling type II predator and superpredator are discussed. The study is carried out through bifurcation analysis, alternating between a normal form approach and numerical continuation. The two-parameter bifurcation diagram of the model contains Hopf, fold, and transcritical bifurcation curves of equilibria as well as flip, fold, and transcritical bifurcation curves of limit cycles. The appearance of chaos in the model is proved to be related to a Hopf bifurcation and a degenerate homoclinic bifurcation in the prey-predator subsystem. The boundary of the chaotic region is shown to have a very peculiar structure.

MSC 2000:: *92D40 Ecology
34C23 Bifurcation (periodic solutions)
34C05 Qualitative theory of some special solutions of ODE
34C30 Manifolds of solutions of ODE
34C37 Homoclinic and heteroclinic solutions of ODE
37N99 Applications of dynamical systems

Keywords: normal forms; food chain model; logistic prey; Holling type II predator; superpredator; two-parameter bifurcation diagram; equilibria; limit cycles; chaos; Hopf bifurcation; degenerate homoclinic bifurcation; prey-predator subsystem

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