The operating diagram of a model of two competitors in a chemostat with an external inhibitor

Substrate inhibition can produce coexistence and limit cycles in the chemostat model with allelopathy

In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. By including both monotone and non-monotone growth functions (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.

Operating diagrams for a three-tiered microbial food web in the chemostat

In this paper, we consider a three-tiered food web model in a chemostat, including chlorophenol, phenol, and hydrogen substrates and their degraders. The model takes into account the three substrate inflowing concentrations, as well as maintenance, that is, decay terms of the species. The operating diagrams give the asymptotic behavior of the model with respect to the four operating parameters, which are the dilution rate and the three inflowing concentrations of the substrates. These diagrams were obtained only numerically in the existing literature. Using the mathematical analysis of this model obtained in our previous studies, we construct the operating diagrams, by plotting the curves that separate their various regions. Hence, the regions of the operating diagrams are constructed analytically and there is no requirement for time-consuming algorithms to generate the plots, as in the numerical method. Moreover, our method reveals behaviors that have not been detected in the previous numerical studies. The growth functions are of Monod form with the inclusion of a product inhibition term. However, our method applies for a large class of growth functions. We construct operating diagrams with and without maintenance showing the role of maintenance on the stability of the system.

DYNAMICAL BEHAVIOR OF STOCHASTIC COMPETITION BETWEEN PLASMID-BEARING AND PLASMID-FREE ORGANISMS IN A CHEMOSTAT MODEL

In this paper, a model of stochastic competition between plasmid-bearing and plasmid-free organisms in a chemostat is investigated. First, we show that there is a unique global positive solution for the stochastic system. Second, by employing stochastic Lyapunov functions, Itô formula, strong law of large number and some other important inequalities, stochastic characteristics of the stochastic competition chemostat model are studied such as the stochastic asymptotic behaviors of the system. Finally, some numerical simulations are given.

Competition in the chemostat with an undesirable lethal competitor

In this study, we compare the effects of competitors in a chemostat when one of the competitors is lethal to the other. The first competitor ("the mutant") is the desired organism because it provides a benefit, such as a substance that is harvested. However, when the mutant undergoes cell division the result may return to the original ("wild type") organism that produces a substance ("toxin") that is lethal to the mutant. We introduce an external inhibitor that negatively affects the growth of the wild type organism but that does not affect the mutant. The goal is for the mutant to dominate in the competition while co-existing with its wild type relative that is controlled. In this manner, we hope that understanding the dynamics of the system will help in designing methods to control the purity of the harvesting vessel without having to periodically restart the process more than necessary. We show that it is possible for co-existence in which the undesirable wild-type coexists with the mutant. However, it is also possible to destabilize the system and cause the extinction of the mutant.