Competition on Three Trophic Levels

The previous two chapters showed that competitive exclusion holds under a variety of conditions in the chemostat and its modifications. In this chapter it will be shown that if the competition is moved up one level - if the competition occurs among predators of an organism growing on the nutrient - then coexistence may occur. The fact that the competitors are at a higher trophic level allows for oscillations, and the coexistence that occurs is in the form of a stable limit cycle. Along the way it will be necessary to study a three-level food-chain problem which is of interest in its own right it is the chemostat version of predator-prey equations. The presentation follows that of [BHWl]. 

Although most of the results can be established with mathematical rigor, there are some elusive problems. These center around the possibility of multiple limit cycles and the difficulty of determining the stability of such limit cycles. At this point one must simply make a hypothesis and resort to numerical evidence in any specific case. Determining the number of limit cycles is a deep mathematical problem, and even in very simple cases the solution is not known. Hilbert s famous sixteenth problem, concerning the number of limit cycles of a second-order system with polynomial right-hand sides, remains basically unresolved. In principle, the stability of a limit cycle can be determined from the Floquet exponents (see Section 4), but this is a notoriously difficult computation - indeed, generally an impossible one. 

Throughout the chapter, one assumes that the equilibria and periodic orbits that occur are hyperbolic. This means that local stability is determined from the linearization. Of course one knows this only after making the linearized computations. It is simply that nothing can be said in the 

The new mathematics that is introduced here is elementary bifurcation theory, in particular, bifurcation from a simple eigenvalue. Although the necessary theorems will not be proved, the material will be discussed in some detail. 

---