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 chem ostat is analysed in detail. The Monod growth model is used for all trophic levels, yielding a non-linear dynamical system of four ordinary differenti al equations. Mass conservation makes it possible to reduce the dimension b y 1 for the study of the asymptotic dynamic behaviour. The intersections of the orbits with a Poincare plane, after the transient has died out, yield a two-dimensional Poincare next-return map. When chaotic behaviour occurs, all image points of this next-return map appear to lie close to a single cu rve in the intersection plane. This motivated the study of a one-dimensiona l 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 und erstood in terms of the local and global bifurcations of this one-dimension al map. Homoclinic and heteroclinic connecting orbits and their global bifu rcations are discussed also by relating them to their counterparts for a tw o-dimensional map which is invertible like the next-return map. In the glob al bifurcations two homoclinic or two heteroclinic orbits collide and disap pear. In the food chain model two attractors coexist; a stable limit cycle where the top-predator is absent and an interior attractor. In addition the re is a saddle cycle. The stable manifold of this limit cycle forms the bas in boundary of the interior attractor, We will show that this boundary has a complicated structure when there are heteroclinic orbits from a saddle eq uilibrium to this saddle limit cycle. A homoclinic bifurcation to a saddle limit cycle will be associated with a boundary crisis where the chaotic att ractor disappears suddenly when a bifurcation parameter is varied. Thus, si milar to a tangent local bifurcation for equilibria or limit cycles, this h omoclinic global bifurcation marks a region in the parameter space where th e top-predator goes extinct. The 'Paradox of Enrichment' says that increasi ng the concentration of nutrient input can cause destabilization of the oth erwise stable interior equilibrium of a bi-trophic food chain. For a tri-tr ophic food chain enrichment of the environment can even lead to extinction of the highest trophic level. (C) 2001 Elsevier Science Inc. All rights res erved.