BioND — Dynamics of Biological Networks

Relating general and specific modeling in population dynamical systems

Dirk Stiefs
PhD thesis in natural sciences, Carl-von-Ossietzky Universität Oldenburg, 2009.
Published in Der Andere Verlag, Tönning, 2009.
ISBN 978-3-89959-908-4

Abstract

cover picture

Population dynamics are often investigated under the usage of simple mathematical models. The model properties of these models can be very sensitive to the mathematical formulation of the considered processes. A detailed derivation of these functional forms from field or lab experiments is in general difficult. However, in generalized modeling a further specification of the processes under consideration is avoided. Consequently, the analysis of these models allows to gain very generic system properties.

In the presented thesis a generalized modeling approach is used to analyze two new branches of theoretical population dynamics. On the one hand we investigate a generalized stoichiometric model that encounters the fact that producers have a rather variable nutrient content while consumers need a balanced diet of specific nutrients. On the other hand we analyze a generalized eco-epidemic model to show how a disease in a predator population can influence the predator-prey interactions. This research is based on bifurcation
theory.

Generalized and specific modeling approaches require different computation techniques to locate bifurcations in parameter space. An innovative technique to locate bifurcations in generalized models is introduced, that allows for an efficient computation of three dimensional bifurcation diagrams. The resulting bifurcation diagrams are partly combined with bifurcation diagrams of specific modeling approaches to demonstrate the interplay of generalized and specific modeling.

The analysis of the generalized stoichiometric model shows that, in conjunction to a variable efficiency of the biomass conversion, new dynamics appear. Moreover the analysis reveals a generic paradoxical effect of competition. It indicates that a change of the system which decreases the intra specific competition of producers tends to destabilize the system. Specific example models are used to illustrate these findings and their predictive capabilities. Investigating the generalized eco-epidemic model it shows that diseases in predator populations can in general cause quasi-periodic and chaotic dynamics. Subsequently the generalized analysis is used to locate these dynamics in a specific example model. This specific model allows to identify the size of the parameter regions with complex dynamics and to investigate exemplary routes to chaos.

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