A Basic Chemostat -- Non-Dimensional Form
by Mike Martin

On this page, plots of solutions for a simple chemostat model that has been non-dimensionalized (variable transformations are utilized so that the number of parameters can be minimized).   In particular, the original basic chemostat model can be found at the following link. The original equations for the basic chemostat model are:

The variable transformations are

b = x no / β,   n = S no,   t = t* / D

The equations for this new system are:

Quite often the variable t* is simply written as t and the equations are frequently given by:

Note that some of the original parameters also have substations made for them:

m = kmax / D,   α = kn / no

Plots of the non-dimensionalized bacteria and nutrient concentrations are given below and the phase portrait with stream lines is below the concentration plots.   The user inputs values of the parameters m and α. Also, gird bounds, time bounds, and the initial conditions must be specified.



Enter in the value of the parameters and select the plotting bounds, then press the "Plot/Evaluate" button.

m =  

α =  


S (0) =  

x (0) =  


0   ≤   t   ≤  


  ≤   x   ≤  

  ≤   S   ≤  


References:

Wikipedia's Phase Portrait

MathWorld's Phase Portrait

DFIELD & PPLANE -- web utilities

Community of Ordinary Differential Equations Educators