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1995•Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the NetherlandsOpen access

Remarks on food chain dynamics

Yuri A. Kuznetsov, S. Rinaldi

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Abstract

The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. 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. AMS Subject Classification (1991): 34C05, 34C28, 65L07, 92A15 Keywords & Phrases: Food chain, mathematical models, bifurcations, normal forms, chaos Note: The paper is submitted for publication to Mathematical Biosciences 1.

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The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. 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. AMS Subject Classification (1991): 34C05, 34C28, 65L07, 92A15 Keywords & Phrases: Food chain, mathematical models, bifurcations, normal forms, chaos Note: The paper is submitted for publication to Mathematical Biosciences 1.

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Available abstract

The main modes of behavior of a food chain model, composed of logistic prey and Holling type II predator and superpredator, are discussed in this paper. 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. AMS Subject Classification (1991): 34C05, 34C28, 65L07, 92A15 Keywords & Phrases: Food chain, mathematical models, bifurcations, normal forms, chaos Note: The paper is submitted for publication to Mathematical Biosciences 1.

Key concepts: Bogdanov–Takens bifurcation, Homoclinic bifurcation, Transcritical bifurcation, Bifurcation diagram, Mathematics, Biological applications of bifurcation theory, Saddle-node bifurcation, Period-doubling bifurcation

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