Summary of the PhD-thesis by Martin Boer Tuesday 27th
2000, Email: mpboer@bio.vu.nl
The Dynamics of Tritrophic Food Chains
This thesis treats theoretical aspects of the dynamics of a tri-trophic food chain composed of a prey, a predator, and a top-predator. Several food chain models are investigated. All these models are described by a system of ordinary differential equations. Research in the last 10 years has shown that tri-trophic food chain models exhibit a rich set of dynamical behavior, including chaotic behavior and multiple attractors. The aim of the investigations that are presented in this thesis is to clarify some aspects of the complicated behavior of these tri-trophic food chain models.
One of the central themes of this thesis is the numerical computation of connecting orbits. The existence of these connecting orbits has important consequences on the dynamics of the tri-trophic food chain. Two types of connecting orbits that are found in tri-trophic food chain models are discussed. The first type is a homoclinic orbit of a saddle cycle. The other type is a heteroclinic orbit between a saddle equilibrium and a saddle cycle.
Another main topic is the computation of the basins of attraction of the tri-trophic food chain models. If the tri-trophic food chain model has two attractors, the domains of attraction are separated by the stable manifold of a limit cycle of saddle type. A continuation method is used to compute this separatrix. The separatrix can have a complex structure when the system has heteroclinic orbits.
Bifurcation analysis of the tri-trophic food chain model is the third central theme of this dissertation. The resulting bifurcation diagrams classify all possible modes of behavior of the system and transitions between them under parameter variations. One of the most remarkable transitions that can be observed in tri-trophic food chain models is the sudden disappearance of a chaotic attractor. This transition is related to a homoclinic bifurcation of a saddle cycle. At a homoclinic bifurcation the stable and unstable manifold of the saddle cycle have a tangency along a non-transversal homoclinic orbit.
A fourth central topic is to compare the dynamics of a continuous-time food chain model with the dynamics of a simple one-dimensional discrete-time system. This comparison of the dynamics of these two systems is motivated by the fact that the next-return map of the chaotic attractor of the tri-trophic food chain is essentially one-dimensional. It is shown that a large part of the complicated bifurcation structure of the tri-trophic food chain can be understood from the local and global bifurcations of a cubic map. The cubic map has two turning points, a maximum and a minimum. The dynamics of this cubic map are analysed in great detail and compared with the dynamical behavior of the original tri-trophic food chain model.
Chapter 1 gives an introduction in tri-trophic food chain dynamics. First, the dynamics of a Lotka-Volterra-based tri-trophic food chain model is described. This simple model can be analysed in great detail. Using this model some basic concepts of bifurcation theory are introduced. The drawbacks of the classic Lotka-Volterra model are discussed. Finally, a general formulation of the tri-trophic food chain models that are described in this thesis is given.
In chapter 2 a microbial food chain in the chemostat is presented. The model equations are derived from a dynamic energy budget model formulated at the individual level. This model exhibits a chaotic attractor. In some regions of the parameter space there are multiple attractors. The bifurcation diagrams of this food chain model show a complex cascade of intersecting tangent and flip bifurcations of limit cycles.
Chapter 3 treats the dynamics of the Monod-Herbert model of a microbial food chain in the chemostat. The Monod-Herbert model is a simplification of the dynamic energy budget model described in chapter 2. It is shown that the bifurcation structure of the food chain model has much in common with the bifurcation structure of a one-dimensional cubic map with two turning points. This map is used to explain how chaotic attractors are created and destroyed under variation of the bifurcation parameters. Two global bifurcation curves can be found for this tri-trophic food chain, a homoclinic bifurcation of a saddle cycle and a heteroclinic bifurcation of an equilibrium and a limit cycle. It is shown that a high concentration of input substrate can lead to extinction of the top-predator population.
Chapter 4 deals with the dynamics of a tri-trophic food chain model with logistic prey. The investigations that are presented in this chapter focus on the computation of global bifurcations. These bifurcations are associated with the appearance of heteroclinic and homoclinic orbits to a limit cycle of saddle type. It is shown that a homoclinic bifurcation curve bounds a region of extinction of the top-predator. This region of extinction is surrounded by a region of chaotic coexistence. It is shown that, if there are multiple attractors, the existence of heteroclinic orbits from an equilibrium to a periodic orbit has important consequences for the domains of attraction.
In chapter 5 the Monod model of a tri-trophic food chain in the chemostat is analysed. The Monod model is a further simplification of the Monod-Herbert model, as described in chapter 3. The advantage of the Monod model, from a mathematical point of view, is that it can be reduced to a three-dimensional system. The dynamics of the next-return map of a Poincaré section can be analysed in great detail. Numerical results strongly suggest that the two-dimensional invertible next-return map is almost one-dimensional. The analysis in this chapter explains why the complex bifurcation structure of a tri-trophic food chain model has essential features of the dynamics of a one-dimensional discrete time system.