Basic Equations of Growth

The contents of the feed bottle are pumped at a constant rate into the culture vessel the contents of the culture vessel are pumped at the same constant rate into the collection vessel. Let V denote the volume of culture vessel (V has units of P, where I stands for length), and let F denote the volumetric flow rate (F has units of l /t, where t is time). The concentration of the input nutrient, denoted by is kept constant. Concentration has units of mass//.  

The culture vessel is charged with a variety of microorganisms, so it contains a mixture of nutrient and organisms. The culture vessel is well 

We seek to write differential equations for this model, and begin by considering just one organism growing in the chemostat. (A more complete derivation can be found elsewhere see e.g. Herbert, Elsworth, and Telling [HET].) The rate of change of the nutrient can be expressed as 

Let S t) denote the concentration of nutrient in the culture vessel at time 1. Thus F5(/) denotes the amount of nutrient in the vessel at that time. The rate of change of nutrient is the difference between the amount of nutrient being pumped into the vessel per unit time and the amount of nutrient being pumped out of the vessel per unit time. If there were no organisms, and hence no consumption, then the equation for the nutrient would be 

The formulation of the consumption term, based on experimental evidence, goes back at least to Monod [Mol Mo2], The term takes the form 

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