Chemical Engineering Science, Vol. 41. No. IO. pp. 2605-2616, 1986. OOO-2509186 $3.00+0.“” Printed in Great Britain. Pergamon Journals Ltd. DYNAMICS OF CONTINUOUS COMMENSALISTIC CULTURES-I. MULTIPLICITY AND LOCAL STABILITY OF STEADY STATES AND BIFURCATION ANALYSIS SATISH J. PARULEKART and HENRY C. LIM School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received 19 December 1984) Abstract-The population dynamics of continuous mixed cultures with pure commensalism or commen- salism plus competitive assimilation are investigated in detail. As many as seven steady states are possible in these systems when the growth processes of the two species are inhibited by the substrates they prey on (self- inhibition). The seven states comprise of a complete washout state, two partial washout states and four coexistence states. A priori information about the number and types of steady states possible. for a given set of parameters is obtained by dividing the entire multi-dimensional parameter space into several regions. Up to three steady states can be locally, asymptotically stable in these systems. The conditions under which transition takes place from a stable steady state to an unstable steady state (and vice versa) (static and Hopf bifurcation points) have been identified with the application of static and Hopf bifurcation theories. The necessary and sufficient conditions for existence of periodic solutions have been derived. It is shown that bifurcation to periodic solutions is possible for only two steady states. 1. INTRODUCTION Mixed microbial cultures have been extensively used in manufacturing of food products such as cheese, yoghurt and pickles; fermentations of wines, beer and whisky; and pharmaceutical processes such as trans- formation of steroids (Ryu et al., 1969; Tseng and Phillips, 1981). An excellent review of the various classes of interactions that may exist in such mixed cultures has been given by Bungay and Bungay (1968). One such class of interactions is known as commen- salism. It is a class of binary microbial population interactions in which one population (the commensal population) depends for its growth on a product produced by the other population (the host popu- lation), while the host population is not affected substantially by the growth of the commensal popu- lation (Bungay and Bungay, 1968; Chiu et al., 1972; Fredrickson and Tsuchiya, 1977; Lee et al., 1976; Stephanopoulos, 1981; Tseng and Phillips, 1981). Several engineering studies on commensalistic sys- tems have been reported in the literature (Chao and Reilly, 1972; Harrison, 1976; Howell ef al., 1973, Lee et al., 1976; Megee et al., 1972; Miura et al., 1978, 1980; Shindala et al., 1965; Tseng and Phillips, 198 1; Wang et al., 1979). These systems can be divided further into two types: (I) pure commensalistic systems; and (II) systems with commensalism and competitive assimi- lation. The reaction scheme for the first type of systems can be expressed as The primary substrate A is consumed for the growth of the host population Band for the production of species C which serves as substrate for the growth of com- mensal population D. Population D thus depends for its growth on population B whereas the reverse is in general not true. Type II systems are those in which both the host and commensal populations compete for the same substrate. Whereas the host population requires only the common substrate for its growth, the commensal population requires a growth factor (re- leased by the host population) for its growth in addition to the substrate. The reaction scheme is represented as B (host population) A \ C (growth factor) (substrate) D (commensal population) P (metabolite) Lumped kinetic models due to Monod (1949) and B (host population) D (commensal population) A L c * p (1) (primary substrate) (secondary substrate) (metabolite) ~To whom correspondence should be addressed at Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. 2605