Dynamics of continuous commensalistic cultures—I. Multiplicity and local stability of steady states and bifurcation analysis

Abstract

The population dynamics of continuous mixed cultures with pure commensalism or commensalism 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.