BioND — Dynamics of Biological Networks

Instabilities and pattern formation in simple ecosystem models

Martin Baurmann, Thilo Gross, and Ulrike Feudel
BioGeoChemistry of tidal flats, FZ Terramare Berichte 12, 22-28, 2003.

Abstract

Bifurcation diagram

Models of chemical and biological processes in the pelagic as well as in the benthic part of the Wadden Sea are very complex and highly nonlinear. As known for a long time nonlinearities can give rise to sudden changes in the behaviour of a system. Such changes are caused by instabilities which can occur if environmental parameters cross certain threshold values (bifurcations). From an ecological point of view such instabilities may endanger the ecosystem since the new state of the ecosystem after the bifurcation can be characterised by different properties, i. e. a different composition of the species community. Moreover, nonlinear systems can exhibit different states for a given set of parameters. This phenomenon is called multistability. If several different states coexist then it crucially depends on the initial conditions which of these states is finally realised. The aim of our studies is to analyse the stability properties of simplified models for particular processes in the pelagial and in benthos. Our main goal is twofold: On the one hand we try to identify threshold values for the parameters which correspond to instabilities leading to a new kind of system behaviour. On the other hand our studies shall contribute to the understanding of the underlying mechanisms of possible instabilities in ecosystems. In particular we study models for food chains in the pelagic system and a minimal model for the interaction of nutrient and bacteria in the sediment. The first model class in the pelagial consists of ordinary differential equations, that means we assume a well mixed system. Therefore, the transitions from one behaviour to another one are always assumed to be homogeneous in space. In the sediment we have to deal with partial differential equations, thus the investigation is focussed on the interaction of biological growth processes and physical transport processes. In this part of the work we are mainly interested in the formation of spatially inhomogeneous distributions of nutrients and microorganisms depending on depth in the sediment. Before we present the results of our analysis of models describing key processes in the pelagic and in the benthic system, we give a short survey of the transitions which may occur in ecological systems. Furthermore, we introduce some of the methods we have developed to study transition phenomena in different systems.

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