### 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
#### 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 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.