Modeling oscillatory behavior in a three variable model of continuous fermentation systems subject to product inhibition

Abstract

A 3-variable model of continuous fermentation processes characterized by product inhibition is studied. It is shown that if the cells to substrate yield is constant, the system can not have periodic solutions. If, on the other hand, the yield term is a variable function of substrate concentration, the model will exhibit oscillations in the cells, substrate and product concentrations in the form of Hopf bifurcation in the underlying system of 3 nonlinear, ordinary differential equations which comprises the model. As time progresses all trajectories in the 3 dimensional solution space tend toward a plane and thus the 3 variable system is equivalent to a model involving only 2 variables. With this result, we are able to obtain the stability condition of limit cycles bifurcating from a non-washout steady state of the 3-variable model, and at the same time rule out the possibility of chaotic trajectories through subsequent bifurcations.