Commensalism I. C.'s continued

We now need to calculate the initial conditions at steady state for the concentrations of yeast and bacteria. The differential equation for sugar concentration comes from a simple mass balance with terms for addition of sugar, washout of sugar due to pumping, consumption by yeast, and consumption by bacteria. The only unknowns after our previous calculation of S are the concentrations of yeast and of bacteria. Similarly, the differential equation for vitamin concentration comes from a mass balance this time with no addition term but instead a production term based on yeast concentration. The other terms are for washout and for consumption by the bacteria. Thus there are two equations and two unknowns, yeast concentration and bacteria concentration. Of course, each differential equation is set equal to zero for steady state. Solving simultaneously gives the remaining initial conditions. The algebra is not shown but should not be too difficult a challenge for students. A typical computer simulation run is shown in the next figure:

The competition in which the bacterium would take over the culture with complete wash out of the yeast is short lived because pumping dilutes the added vitamin to restore the commensalism. If the vitamin were added to the reservoir instead of to the medium in the bioreactor, the competition would continue. 
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  • Computerized analysis of an ecosystem
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    while on sabbatical leave at ESB, Porto, Portugal, 1996