Nonmonotone Uptake Functions

An application of the LaSalle corollary yields the desired result.  

The biological conclusion is, of course, that differing removal rates do not alter competitive exclusion in the chemostat. One anticipates that a similar conclusion is true if the Michaelis-Menten dynamics is replaced by the general monotone term f,(S) used in Section 3. However, the Liapunov calculations depend on this form and the general question is still unresolved. 

The reader will have noticed that the Liapunov function used in the proof of the theorem was not obvious on either biological or mathematical grounds. Its discovery by Hsu greatly simplified and extended earlier arguments given in [HHW]. This is typical of applications of the LaSalle corollary. Considerable ingenuity, intuition, and perhaps luck are required to find a Liapunov function. 

It has been shown that competitive exclusion - that is, the extinction of all but one competitor - holds regardless of the number of competitors or the specific monotone functional response. If one restricts attention to the Michaelis-Menten functional response, then competitive exclusion has been shown even in the case of population-specific removal rates. 

There is evidence, however, that a monotone functional response may be inappropriate in some cases. A nutrient which is essential at low concentrations may be inhibiting (or, indeed, even toxic) at higher concentrations. Butler and Wolkowicz [BWol] consider this possibility their work has been recently extended in [ WuL]. We will describe some special cases of their work in terms of the unsealed system immediately preceding (3.1). Assume that the functional response fj satisfies (i), (ii), and (iv) of Section 3, but replace the strict monotonicity assumption (Hi) by 

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In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing death rates (replacing the parameter D by D, in the equations) does not change the competitive exclusion result. 

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