Chaotic behaviour of a predator-prey system

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