B.W. (Bob) Kooi

Dept. Theoretical Biology, Biological Laboratory, Vrije Universiteit,
De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands

kooi@bio.vu.nl

http://www.bio.vu.nl/thb/personnel/members/kooi.html

 

 

                          Food web models as dynamical systems

 

Bifurcation analysis is a systematic approach to investigate the dependence of the dynamics of a system on its parameters. The parameter space is divided into regions with qualitatively different long-term dynamic behaviour. Hence, the boundaries of these regions mark points where this behaviour changes. For example, a point attractor can become a limit cycle. To calculate the position of these boundaries, continuation techniques  which are implemented in a number of free available computer packages can be used.

 

Ordinary differential equations describe the temporal change in biomass of populations and nutrient densities, that make up a food web. Then, bifurcation points can be interpreted biologically. For instance, a transcritical bifurcation often indicates the situation where invasibility by a top-predator becomes possible, while a Hopf bifurcation marks the onset of sustained oscillatory behaviour.

 

The literature mentions the analysis of rather simple types of food web models using numerical bifurcation techniques. The population processes such as growth and death, have been modelled based on various degree of biological detail. A variety of food web interaction structures have been considered where the trophic interactions have been described mathematically by Lotka-Volterra, Holling or other type of functional response. For interaction with the environment immigration, emigration and harvesting are important. However, bifurcation analysis allows  the study of more complicated and biologically realistic tropic interactions, such as competition, omnivory, mutualism and symbiosis.

 

We will give an overview of the various numerical bifurcation techniques including the use of maps with the study of chaotic behaviour of continuous-time systems. Global bifurcations appear to play an important role with suddenly disappearance of chaotic behaviour under parameter variation. With the interpretation of the results the relationship with modelling and biological realism is emphasised.