First of all, it seems important at this stage to present the way we interpret the evolution of the population in the chemostat. We want to impose to our model the fact that the rate of production has to be proportional to the existing population and to the amount of available resources.
Hence a logistic model of the population, where c denotes the concentration of cells in the medium:
Then, we need to add the death term, and a dilution term cause by the renewal of the chemostat. Finally we get :
where cmax is the carrying capacity for cell growth, and
the dilution rate, d the death rate constant