This is a little project, counting as 1/2 of a quiz. Its goal is to get you to look at a paper in the current literature (August 2006 Control Systems Magazines) that mentioned chemostats, so you see how people apply these things.

The paper is here.

Look at the equations in page 58 of the magazine, column 1. There are three variables, X, S, and P. This is because, besides the bacteria and the nutrient, the authors want to keep track of a product being produced by the bacteria.

In column two of the same page, the form of the "K(C)" term is show, which looks like Michaelis Menten, except that it depends on both the concentration of nutrient and of the product ("inhibitory " effects).

(1) Suppose that Pm and Ki are both very large (infinite), as discussed at the bottom of column 2 in page 58. Show that then we get the same model (ignoring the product) that we had.

Write down a little table "translating" notations: our F, V, Vmax, Km, C0, and alpha, versus their Sf, D, etc.

(2) Still assuming as in (1), there are exactly two steady states X1 and X2, except that now there is also a steady state value of the product. Find X1 and X2. (Give a formula. It is not difficult to do it in general, but if you prefer, take simple numerical values of your choice, for the parameters.)

(3) Answer: which of the machines, the one in Fig 3(a) or the one in Fig 3(b), measures P(t)?

(4) Explain (please, just one paragraph): what do the authors state about the number of possible equilibria if Ki and Pm are not very large? (You are not asked to do any mathematical calculations. Just keep reading and see what they say about equilibria. Explain in your own words.)