Bifurcations in ecosystem models and their biological interpretation

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.