Chaotic behaviour of a predator-prey system

Kooi, B.W. and Boer, M.P. 2003 Chaotic behaviour of a predator-prey system. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms 10: 259 - 272


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 Hénon map. The link is via a Poincaré map for a suitable chosen Poincaré plane where the predator attains an extremum. Global homoclinic bifurcations are associated with boundary and interior crises.

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