### Bifurcations in ecosystem models and their biological interpretation

Kooi, B. W. and Boer, M. P. 2001
Bifurcations in ecosystem models and their biological interpretation.
*Applicable Analysis* **77**: 29 - 59
#### Abstract

The use of bifurcation analysis for the study of the long-term
dynamic behaviour of food webs is discussed. Food webs are
collections of populations with often very complicated
interactions. The non-viable nutrients are consumed by populations
which are consumed by other populations at higher trophic levels,
except for the top-predator species. In the most simple setting
the dynamics of each population is described mathematically by an
ordinary differential equation (ode). In this paper we give
an overview of local as well as global bifurcations found for a
simple food web model. With chaotic behaviour all points in a
next-return map appear to lie close to single curve. A cubic map
with two critical points is studied and the results obtained are
used to clarify global bifurcations of the ode system.
These global bifurcations are homoclinic bifurcations of a saddle
limit cycle and a heteroclinic bifurcation from the equilibrium to
the limit cycle. The homoclinic bifurcation of the saddle limit
cycle is associated with a boundary crisis where a chaotic
attractor is abruptly destroyed by a collision with the saddle
limit cycle. The complex geometry of the basin of attraction of
positive attractors is related to the existence of heteroclinic
orbits. Furthermore, a homoclinic bifurcation for the saddle-focus
equilibrium forms a `skeleton' for the period doubling and tangent
bifurcations of a limit cycle. Most results were already
discussed in earlier papers; in this paper we emphasize the
biological interpretations. For example, the transcritical
bifurcations will be associated with invasion of species into an
ecosystem.