BioND — Dynamics of Biological Networks

Population Dynamics: General results from local analysis

Thilo Gross
PhD thesis in natural sciences, Carl-von-Ossietzky Universität Oldenburg, 2004.
Published in Der Andere Verlag, Tönning, 2004.
ISBN 3-89959-249-2


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Mathematical models are frequently used to investigate the dynamics of interacting populations. In these models the biological interactions are usually described by simple mathematical functions. However, simple functions can hardly capture the complexity of biological interactions. The use of simple mathematical functions may therefore result in ecological models of limited validity. This risk can be avoided if general models are used. In general models the mathematical functions that describe the interaction between populations are not specified. As a result a single general model describes a whole class of similar systems at once.

In this thesis a new approach to the formulation and analysis of general models is presented. This approach is used to formulate and study models of general food chains and food webs.

This work illustrates that general models can be analysed in the framework of local bifurcation theory. In particular the computation of Hopf bifurcations provides much information about the local and global dynamics of general models. In order to compute Hopf bifurcations efficiently a special mathematical technique, the method of resultants, is derived.

Application of the method of resultants and the general modelling approach reveals a new, general solution to the famous paradox of enrichment. It is shown that enrichment always destabilizes certain ecological models if the interaction is described by simple functions. However, a large class of more complex functions exists which result in more complex model behaviour. If these functions are used in a model enrichment may have a stabilizing effect. In this way general models explain why enrichment does not generally lead to instability in experiments. Another question which is much debated is whether chaotic dynamics is possible in ecological systems. While chaos is observed in many models it is believed to vanish if certain biological details are taken into account. The investigation of general models reveals that chaotic parameter regions generally exist in systems with more than three trophic levels. This result holds for a large class of models regardless of the specific details.

The investigation of ecological food webs reveals that a large class of food webs behaves qualitatively similar to food chains. For many purposes it is therefore sufficient to model these food webs as food chains. Furthermore, it is shown that the strong form of the famous competitive-exclusion principle only occurs because of certain degeneracies that exist in simple ecological models.