On the stability of periodic solutions for a product inhibition model of continuous biological reactors.

Abstract

A three-variable product-inhibition model of a continuous fermentation process is considered. It is shown that as time progresses all solution trajectories in the three-dimension solution space approach a plane and thus the three-variable system is equivalent to a model involving only two variables. With this result, we are able to obtain the stability condition for limit cycles bifurcating from a non-washout steady state of the three-variable model, and at the same time rule out the possibility of developing chaotic trajectories through subsequent bifurcations.Oscillatory behavior frequently occurs in both batch and continuous fermentation processes (6,14). In batch cell growth data, we often observe what appears to be random deviations about a smooth exponentially increasing function. For most design and control applications such a smoothed function describes the system adequately. However, scaleup from the small batch fermentor to large commercial batch and continuous reactors utilizing microorganisms is still very much an art. As Tanner suggested in his recent work (14), part of the difficulty in scaling up is perhaps directly attributable to the neglect of such small deviations in the process of designing the large vessel. He further stated that if small irregularities in the cell growth were not just random deviation from a smoothed function, then they must be reproducible, lead to recognizable patterns, and ultimately make sense in terms of the underlying biochemical events among the growing microorganisms.Attempts along this line have been made by Lenbury et al. (9,11) who showed that the yield expression must depend on both the cell and substrate levels for oscillations to occur in both the substrate and the cell concentrations.With continuous fermentation, on the other hand, oscillations in the ex-* Address reprint requests to: