The Standard Gradostat

Note that, for the first result, the equation / (S) = /y(S) has at most one positive solution, and that the additional hypotheses imply uniqueness and global stability. Distinct Michaelis-Menten functions automatically satisfy all but the last hypothesis, which holds for almost all A. The second result shows that if the aforementioned equation can have two distinct solutions, yet / / , then an unstable positive rest point exists for some two-vessel gradostat. 

The boundary conditions follow naturally from the conventions Uq = = 0, and similarly for v, which say that there are no microorganisms in the two reservoirs. They are justified by the agreement between the numerically computed rest points of (5.1) and the solutions of (5.2) obtained by using singular perturbation theory (as in [S9]). 

Consider first the single-population equilibrium for u. Setting t = 0 in (5.2) leads to the boundary value problem 

A positive solution u must be concave on 0 x 1, since Uxx 0. The function 

With u given approximately by (5.5) for small e, the nutrient is given by 

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It is possible to obtain interesting information on how each microbial population is distributed among the n vessels of the standard gradostat of Figure 1.1 at equilibrium if the number of vessels is large. The approach is to pass to a continuum limit. This section is devoted to an informal presentation of the results (following [S9J). 

All of the open problems for the standard gradostat system of Chapter 6 are open problems for the unstirred chemostat model discussed in Chapter 10. It can be shown [HSW] that the dynamics of the unstirred chemostat system mirror those of the gradostat in the sense that there is an order interval, bounded by two (possibly identical) positive rest points, that attracts all solutions. Furthermore, an open and dense set of initial data generates solutions that converge to a stable rest point. The question of the uniqueness of the interior rest point is a major open problem. Another is how to handle the case where the diffusion coefficients of the competitors and nutrient are distinct. Although there must still be conservation of total nutrient, it is no longer a pointwise conservation relation and the reduction to two equations is not clear. Even if accomplished, it may be difficult to exploit. If one is forced to analyze the full... 

Figure 1.1. The standard n-vessel gradostat. The left vessel labeled is a reservoir containing nutrient at concentration C is an overflow vessel, and D denotes the dilution rate. All vessels have the same volume.

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