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

Boer, M.P., Kooi, B.W. and Kooijman, S.A.L.M. 2001.
Multiple attractors and boundary crises in a tri-trophic food chain.
*Math. Biosci.* **169**: 109 - 128
#### Abstract

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 Poincar e plane,
after the transient has died out, yield a two-dimensional Poincar e
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 enrichement of the
environment can even lead to extinction of the highest trophic level.