The Effect of Inhibitor on the Plasmid‐Bearing and Plasmid‐Free Model in the Unstirred Chemostat

Abstract

This paper deals with a chemostat model with an inhibitor in the context of competi- tion between plasmid-bearing and plasmid-free organisms. First, sufficient conditions for coexistence of the steady-state are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which represents the effect of the inhibitor, plays a very important role in deciding the number of the coexistence solutions. The results show that if μ is sufficiently large this model has at least two coexistence solutions provided that the maximal growth rate a of u lies in a certain range and has only one unique asymptotically stable coexistence solution when a belongs to another range. Finally, extensive simulations are done to complement the analytic results. The main tools used here include degree theory in cones, bifurcation theory, and perturbation technique. 1. Introduction. The chemostat is a common model in microbial ecology. It is used as an ecological model of a simple lake, as a model of waste treatment, and as a model for commercial production of fermentation processes. It is important in ecology because the parameters are readily measurable and, thus, the mathematical results are readily testable. For a general discussion of competitive systems see (29), while a de- tailed mathematical description of competition in the chemostat can be found in (30). Our study focuses on a chemostat model in the context of competition between plasmid-bearing and plasmid-free organisms. This issue has recently received consid- erable attention. The theoretical literature on this model includes Ryder and DiBiaso (25), Stephanopoulos and Lapidus (28), Hsu, Waltman, and Wolkowicz (17), Lu and Hadeler (22), Levin (20), Luo and Hsu (18), and Macken, Levin, and Waldstatter (23). In industry, genetically altered organisms are frequently used to manufacture a desired product, for instance, a pharmaceutical. The alteration is accomplished by introducing a piece of DNA into the cell in the form of a plasmid. The burden imposed on the cell by the task of production can result in the genetically altered (the plasmid-bearing) organism being a less able competitor than the plasmid-free organ- ism. Unfortunately, the plasmid can be lost in the reproductive process. Thus, it is possible for the plasmid-free organism to take over the culture. To avoid capture of the process by the plasmid-free organism, the obvious choice is to alter the medium in such a way as to favor the plasmid-bearing organism. An example of this would be to introduce an antibiotic into the feed bottle. See (10, 15, 16) for a detailed biological and chemical background. Models in this direction have been studied in Lenski and Hattingh (21), Hsu and Waltman (13, 15, 16), Hsu, Luo, and Waltman (12), Nie and Wu (24), and the references therein.