Analysis of a continuously stirred two tank reactor cascade with Haldane kinetics

Biological reactors are employed in industrial applications to break down organic waste. We view the cascade of two open loop continuously stirred tank reactors with Haldane growth function as chemostats with bacterial inputs. A function of some of the reactor parameters is derived, the sign of which determines the maximum number of critical points a reactor can have. This allows us to determine the parameter combinations which ensure a reactor has only a single critical point for all bacterial removal rates (dilution rate plus death rate). Where a simple condition on the above function is confirmed to hold, if the first reactor in a cascade only has a single critical point for all bacterial removal rates then, the next reactor will also only have a single critical point for all bacterial removal rates. A global stability result is also given for some of these cases. A simple proof is given for the local stability of critical points of a reactor with a general class of bacterial growth functions, bacteria and substrate input, and a death rate. For the special case where the first reactor has zero bacteria input, we compare a two reactor cascade with a single reactor under various conditions, long and short residence times, and different death rates. This follows the pattern of similar papers that considered cascades using the Monod and Contois growth functions.