Multiple attractors and boundary crises in a tri-trophic food chain.

The asymptotic behaviour of a model of a tri-trophic food chain in the chemostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differential equations. Mass conservation makes it possible to reduce the dimension by 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincaré plane, after the transient has died out, yield a two-dimensional Poincaré next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single curve in the intersection plane. This motivated the study of a one-dimensional bi-modal, non-invertible map of which the graph resembles this curve. We will show that the bifurcation structure of the food chain model can be understood in terms of the local and global bifurcations of this one-dimensional map. Homoclinic and heteroclinic connecting orbits and their global bifurcations are discussed also by relating them to their counterparts for a two-dimensional map which is invertible like the next-return map. In the global bifurcations two homoclinic or two heteroclinic orbits collide and disappear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition there is a saddle cycle. The stable manifold of this limit cycle forms the basin boundary of the interior attractor. We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle equilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic attractor disappears suddenly when a bifurcation parameter is varied. Thus, similar to a tangent local bifurcation for equilibria or limit cycles, this homoclinic global bifurcation marks a region in the parameter space where the top-predator goes extinct. The 'Paradox of Enrichment' says that increasing the concentration of nutrient input can cause destabilization of the otherwise stable interior equilibrium of a bi-trophic food chain. For a tri-trophic food chain enrichment of the environment can even lead to extinction of the highest trophic level.