PHD Thesis, Oldenburg University and Vrije Universiteit, Amsterdam

To cope with this problem models are developed and studied, generally based on mathematics. To ascertain whether the models that are employed are a good substitute for experiments, we first try to come up with models and experimental situations that are simple and comparable, before using more complex models. That requires making assumptions and simplifications, but also good analysis methods and techniques.

In my thesis we focus on food chain models. They consist of one or a few variables, where in general each variable represents one species' population. One species feeds upon one other species, forming a ``chain''. Because it is assumed that the environment is homogeneously mixed, ordinary differential equations can be employed in the description of the food chain models. The properties of the species are given with parameters, that remain fixed in time.

Simple descriptions, like that of the logistic growth (also known as the Verhulst-Pearl equation), display dynamics that can be properly understood, and that qualitatively remain the same under parameter variation. However, larger sets of ordinary differential equations, used to describe food chains, can display different types of dynamical behaviour under parameter change. Also, their dynamics can become very complex and difficult to understand. The transitions from one type of behaviour to another are usually associated with local bifurcations. The analysis methods to detect and continue these transition points in parameter space are referred to as local bifurcation analysis. Understanding how bifurcations occur in food chain models also increases our understanding of the types of behaviour generated by the models.

A relatively simply type of behavioural change occurs in the two-dimensional predator-prey Rosenzweig-MacArthur model, and is discussed in Chapter 2 of my thesis. When the carrying capacity of the system is low, the system exhibits a stable equilibrium, i.e., positive initial conditions converge to the same asymptotic population sizes. Under an increase of the carrying capacity this system's attractor undergoes a Hopf bifurcation, after which a stable limit cycle becomes the system's attractor. A further increase in carrying capacity generates cycles of larger amplitude, that have lower minimal values. Extinction through stochasticity becomes likely under an increase of carrying capacity, leading ecologists to refer to this phenomenon as the ``paradox of enrichment''. The paradox can be resolved by avoiding the Hopf bifurcation. In this chapter it has been shown what conditions need to be fulfilled in order to do this. Also, using a normalisation technique, it has been shown that many proposed mechanisms in the ecological literature fulfil these conditions.

More complex behaviour can be found in two-dimensional predator-prey Allee-models, that are discussed in Chapter 3 of my thesis. Instead of one system's attractor there are two attractors in the Allee-models: a trivial attractor where both populations are zero, and an internal attractor. The internal attractor is a stable equilibrium, that is replaced by a stable limit cycle after a Hopf bifurcation under a decrease of the predator mortality parameter. At an even lower mortality value, the limit cycle attractor disappears, and the multi-stability disappears with it. The only attractor left then is the trivial equilibrium, which is biologically interpreted as extinction of both species. This phenomenon, that a decrease in predator mortality leads to extinction of both species, is referred to a over-exploitation by the predator.

The transition from multi-stability to one equilibrium is not a local, but a global bifurcation, more specificly, a heteroclinic point-to-point connecting orbit. New techniques had to be developed to detect and continue the connecting orbit. It was found, that the Hopf bifurcation and the global bifurcation occur very close to each other, which previously lead other authors to conclude that no limit cycle attractor occur in Allee-models.

Global bifurcations of different type also occur in food chain models of more than two variables. For instance, in the three-dimensional Rosenzweig-MacArthur prey-predator-top-predator food chain model, chaos can occur at certain parameter values, while at the same time there exists a planar attractor where the top-predator is extinct. The basins of attraction of both attractors are separated by a stable manifold, that has a very complex shape. This separatrix is destroyed as the result of a global bifurcation, more precise, a point-to-cycle connection. The chaotic attractor is born through a series of local bifurcations, namely flip bifurcations. On the other hand, it disappears through a global bifurcation, in this case, a homoclinic cycle-to-cycle connection. Techniques to detect and continue point-to-cycle connections in three-dimensional food chain models have been developed and tested in Chapter 4 of my thesis. In Chapter 5 of my thesis these techniques have been extended to cope with cycle-to-cycle connections.

The techniques, developed in my thesis, have been applied in the final two Chapters. There we wanted to ascertain what type of global bifurcations occur in different types of food chain models, and what the biological consequences are of these global bifurcations.

In Chapter 6 of my thesis we have considered a small food chain of a nutrient, a prey and a predator in an aquatic environment, called a chemostat. Three scenario's have been evaluated. The first scenario is without toxicants. The second scenario includes a sublethal toxicant, that influenced both the predator and the prey, but only the growth of their populations. In the third scenario a sublethal toxicant affects only the prey species. It was found, that in the scenario without toxicant there is no multi-stability occurring. However, in the scenario's with toxicant multi-stability can occur. This means, that systems where a sublethal toxicant is introduced can start to display multi-stability. In that case parameter variations, like in the dilution rate or the nutrient availability, can possibly lead to global bifurcations occurring. Despite the non-lethal effects of the toxicant, the addition of these toxicants can lead to the extinction of one or several species, just like in the models discussed in Chapter 3.

In the last chapter of my thesis, Chapter 7, we evaluated several food chain models, and the global bifurcations that occur in them. It is found that there are similarities between the different types of global bifurcations that occur. In all evaluated cases multi-stability is replaced by only one type of stable behaviour after the occurrence of a homoclinic connecting orbit. One local bifurcation, the tangent bifurcation, can have the same effect. In the two-dimensional stoichiometric model the global bifurcation sometimes occurs because a tangent bifurcation occurs, and both events are coupled. However, in other cases the global and tangent bifurcation are not coupled, and then the occurrence of the global bifurcation and not the tangent results in the disappearance of one of the attractors. Although the global bifurcations occurring in the models in this chapter are not coupled to extinction, their occurrence still has significant consequences, for instance, hysteresis can occur.

In a three-dimensional food chain model it is found that the Shil'nikov bifurcation plays the organising role in the generation of chaos, and two chaotic attractors can exist simultaneously at one specific parameter set. There are also two cycle-to-cycle homoclinic connections that form the boundary for the chaotic attractors, where one connection bounds one attractor and the other connection bounds the other attractor. Both the global bifurcations intersect in parameter space, creating very rich state space dynamics at certain parameter values.

Chemostat experiments with bacteria and ciliates have made it plausible that local bifurcations, like the Hopf bifurcation, and also chaos can occur in small food chain systems. Because of this, I suggest it is worthwhile to consider the possibility of global bifurcations occurring in experimental set-ups, and perhaps real ecosystems. It is unlikely it will be possible to find them, since they occur at very specific parameter values, while experimental set-ups are noisy and hence it is hard to pin them at a specific parameter value. But sudden equilibrium size shifts or extinctions of populations could be explained by the occurrence of global bifurcations, especially when after a small parameter perturbation the old parameter setting is restored but a recovery of the old system does not occur. Further work could be aimed in this direction.

This is the symposium that concludes my project