Food web consist of a number of species living in a region having different kind of inter- and intraspecific interactions. The dynamics of each biotic and abotic species is a result of processes such as reproduction, growth and mortality. Furthermore the interaction between different species are important for instance predator-prey interaction, but also competition for nutrients or light. These processes fix the energy and material flow through biota. The studied models are based on the Dynamic Energy Budget theory. The resulting mathematical models are used to study these flows which are related to structure and composition of food webs.
Bifurcation theory can be used to study how the long-term dynamics of food webs depends on parameter variation. The parameter space is subdivided into regions with different long-term dynamics such as equilibria or limit cycles. With application to food web models the bifurcation diagrams give insight into the conditions in which a species can invade an established food web. With high nutrient levels food webs show often oscillatory behaviour, a phenomenon known as nutrient enrichment. the region is marked by a Hopf bifurcation.
It is well known that food web models predict often complex behaviour including chaos and crises where chaotic bahaviour disappears abruptly when a parameter is varied. In Martin Boer's pictures these complex dynamics is demonstrated for the tri-trophic food chain in the chemostat. Whether chaos occurs in nature or not, models can predict chaos so there is a need for methods to detect it and under which conditions it occurs. With crises global bifurcations are important. Continuation of these global bifurcations in the parameter space mark regions where chaos is to be expected. Other global bifurcations mark regions in the parameter space where invasion of a population into an existing food web leads to replacement or coexistence. Shil'nikov bifurcation which occur in some food webs described by more than two ode's, form the `skeleton' in the parameter space for chaotic dynamics.
Small-scale (number of populations less than, say, 7) food web models such as food chains with simple population interactions have been studied a lot. Here we will study more realistic food web medium-scale models (number of populations greater than 7 less than, say, 20) using bifurcation theory including the study of local as well as global bifurcations.
Only the very simple models can be studied analytically, in general numerical techniques have to be used. In order to calculate the equilibria as set of nonlinear equations have to be solved. Limit cycles are calculated by solving a boundary value problems. These invariant sets are calculated varying one or more parameters. During this process critical points are detected generally based on criteria where the eigenvalues of the Jacobian matrix (equilibria) cross the imaginary axis or where the eigenvalues of the Monodromy matrix (limit cycles then called Floquet multipliers). There are a number of computer packages available (Locbif, Auto, Content) to do numerical bifurcation analysis. Visit also the site by Hinke Osinga with a description of software for Dynamical System Theory: Bifurcation Software. For local bifurcations these packages work fine. For global bifurcations only a limited number of programs are available ( HomCont). In this project algorithms to calculate and continue global bifurcations homoclinic and heteroclinic will be studied and implemented in MATLAB Continuation Toolbox of Yuri Kuznetsov.
see for VU-papers also Publications
Downloads: Matlab code to calculate one-parameter bifurcation diagrams
VU group- Publications