The Unstirred Chemostat

The gradostat was an attempt to create a nutrient gradient in a piece of laboratory apparatus it was discussed in detail in Chapters 5 and 6. An alternative to the gradostat is to remove the well mixed hypothesis -that is, to think of the chemostat but without mixing the vessel. If the 

We do not give the derivation here, but the equations take the form 

The boundary conditions in (2.2) are fairly intuitive and appropriate for this type of equation. However, the boundary conditions are not defined in terms of the operating parameters of the simple chemostat. The problem will be considered in a heuristic way to see how the units compare between the simple chemostat and the chemostat without the assumption of well mixing. To keep matters simple, we work with the nutrient equation without consumption (equivalently, zero initial conditions for the microorganisms) the other cases will be clear by analogy. Under these circumstances, the simple chemostat takes the form 

The units of S are concentration, mass/volume = m/l. The total mass of substrate is KS, where V is the volume of the vessel if F is the flow rate (the rate of the pump operating the chemostat) then the parameter D is defined as F/V. Rewriting the equation just displayed for the mass of the substrate in the vessel yields 

Equation (2.4) states that the rate of change in mass is proportional to the difference between the incoming flux and the outgoing flux. 

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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... 

One of the basic assumptions in the unstirred chemostat was that the turnover rate was so slow that any transport (perhaps induced by pumps operating the chemostat) was negligible. If one thinks of a model for a flowing stream, or a lake with circulation, this assumption is unwarranted. Hence a mathematical analysis of the case where transport has been added to the model equations would be an important contribution. The steady-state case (with equal diffusion) was considered in [JW], but the dynamical model is the important one. 

HSW] S. B. Hsu, H. L. Smith, and P. Waltman (1994), Dynamics of competition in the unstirred chemostat, preprint. Department of Mathematics and Computer Science, Emory University, Atlanta. 

Delay models were discussed in Chapter 10. We repeat here that the most interesting problem is a modeling one. Since the problem is sensitive to how the delay is introduced, care must be taken in the modeling. A physical delay is caused by the physiology of the cell, so model equations must be modified to consider or approximate the cell physiology. Once a model is known, analysis of the corresponding system of equations (either functional differential equations or hyperbolic partial differential equations of a structured model) would be an important contribution. It is likely, however, that the delay will be state-dependent, and the theory for such equations is not well developed. A model with delays due to both cell physiology and diffusion in an unstirred chemostat would also be of interest. 

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