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.

Location of integral curves, singular points, limit cycles (msc 34C05), Complex behavior, chaotic systems (msc 34C28), Numerical investigation of stability of solutions (msc 65L07), Population dynamics (general) (msc 92D25), Epidemiology (msc 92D30)
CWI
Department of Analysis, Algebra and Geometry [AM]

Kuznetsov, Y.A, & Rinaldi, S. (1995). Remarks on food chain dynamics. Department of Analysis, Algebra and Geometry [AM]. CWI.