### Abstract

Generally a predator-prey system is modelled by two ordinary differential equations which describe the rate of changes of the biomasses. Since such a system is two-dimensional no chaotic behaviour can occur. In the popular Rosenzweig-MacArthur model, which replaced the Lotka-Volterra model, a stable equilibrium or a stable limit cycle exist. In this paper the prey consumes a non-viable nutrient whose. dynamics is modelled explicitly and this gives an extra ordinary differential equation. For a predator-prey system under chemostat conditions where all parameter values are biologically meaningful, coexistence of multiple chaotic attractors is possible in a narrow region of the two-parameter bifurcation diagram with respect to the chemostat control parameters. Crisis-limited chaotic behaviour and a bifurcation point where two coexisting chaotic attractors merge will be discussed. The interior and boundary crises of this continuous-time predator-prey system look similar to those found for the discrete-time Henon map. The link is via a Poincare map for a suitable chosen Poincare plane where the predator attains an extremum. Global homoclinic bifurcations, are associated with boundary and interior crises

Original language | English |
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Pages (from-to) | 259-272 |

Journal | Dynamics of Continuous, Discrete and Impulsive Systems. Series B, Applications and Algorithms |

Volume | 10 |

Issue number | 1-3 |

Publication status | Published - 2003 |

### Keywords

- attractors
- crises

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## Cite this

Kooi, B. W., & Boer, M. P. (2003). Chaotic behaviour of a predator-prey system.

*Dynamics of Continuous, Discrete and Impulsive Systems. Series B, Applications and Algorithms*,*10*(1-3), 259-272.