##
**Chaos in three species food chains.**
*(English)*
Zbl 0823.92030

Summary: We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model.

This work extends earlier numerical studies of a particular system by A. Hastings and T. Powell [Ecology 72, No. 3, 896-903 (1991)] by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by J. Guckenheimer and P. Holmes [Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983; Zbl 0515.34001)] on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.

This work extends earlier numerical studies of a particular system by A. Hastings and T. Powell [Ecology 72, No. 3, 896-903 (1991)] by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by J. Guckenheimer and P. Holmes [Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983; Zbl 0515.34001)] on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.

### MSC:

92D40 | Ecology |

37N99 | Applications of dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

### Citations:

Zbl 0515.34001
PDFBibTeX
XMLCite

\textit{A. Klebanoff} and \textit{A. Hastings}, J. Math. Biol. 32, No. 5, 427--451 (1994; Zbl 0823.92030)

Full Text:
DOI

### References:

[1] | Gilpin, M. E.: Spiral chaos in a predator-prey model. Am. Nat. 113, 306-308 (1979) |

[2] | Guckenheimer, J.: Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. (Appl. Math. Sci., Vol. 42) Berlin, Heidelberg, New York: Springer 1983. · Zbl 0515.34001 |

[3] | Hastings, A.: Powell, T.: Chaos in a three-species food chain. Ecology 72(3), 896-903 (1991) |

[4] | Klebanoff, A.: Chaos in three species food chains. PhD thesis. University of California, Davis, June 1992 · Zbl 0823.92030 |

[5] | Pimm, S. L.: Food Webs. London: Chapman and Hall 1982 |

[6] | Rosenzweig, M. L.: Exploitation in three trophic levels. Am. Nat. 107, 275-294 (1973) |

[7] | Steele, J. H.: Henderson, E. W.: The role of predation in planktaon models. J. Plankaton Res. 14(1) 157-172 (1992) |

[8] | Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. (Texts Appl. Math, Vol. 2) Berlin, Heidelberg, New York: Springer 1990 · Zbl 0701.58001 |

[9] | Wollkind, D. J.: Exploitation in three trophic levels: an extension allowing intraspecies carnivore interaction. Am. Nat. 110, 431-447 (1976) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.