Mary Ballyk

Effects of random motility on microbial growth and competition in a flow reactor

The authors investigate the effects of random motility on the ability of a microbial population to survive in pure culture and to be a good competitor for scarce nutrient in mixed culture in a flow reactor model consisting of a nonlinear parabolic system of partial differential equations. For pure culture (*a single population), a sharp condition is derived which distinguishes between the two outcomes: (1) washout of the population from the reactor or (2) persistence of the population and the existence of a unique single-population steady state. The simulations suggest that this steady state is globally attracting. For the case of two populations competing for scarce nutrient, they obtain sufficient conditions for the uniform persistence of the two populations, for the existence of a coexistence steady state, and for the ability of one population to competitively exclude a rival. Extensive simulations are reported which suggest that (1) all solutions approach some steady state solution, (2) all possible outcomes exhibited by the classical competitive Lotka-Volterra ODE model can occur in the model, and (3) the outcome of competition between two bacterial strains can depend rather subtly on their respective random motility coefficients.

A model of microbial growth in a plug flow reactor with wall attachment

A mathematical model of microbial growth for limiting nutrient in a plug flow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each, and for the long term persistence of the bacteria in the reactor.

Exploitative competition in the chemostat for two perfectly substitutable resources

After formulating a general model involving two populations of microorganisms competing for two nonreproducing, growth-limiting resources in a chemostat, we focus on perfectly substitutable resources. León and Tumpson considered a model of perfectly substitutable resources in which the amount of each resource consumed is assumed to be independent of the concentration of the other resource. We extend their analysis and then consider a new model involving a class of response functions that takes into consideration the effects that the concentration of each resource has on the amount of the other resource consumed. This new model includes, as a special case, the model studied by Waltman, Hubbell, and Hsu in which Michaelis-Menten functional response for a single resource is generalized to two perfectly substitutable resources. Analytical methods are used to obtain information about the qualitative behavior of the models. The range of possible dynamics of model I of León and Tumpson and our new model is then compared. One surprising difference is that our model predicts that for certain parameter ranges it is possible that one of the species is unable to survive in the absence of a competitor even though there is a locally asymptotically stable coexistence equilibrium when a competitor is present. The dynamics of these models for perfectly substitutable resources are also compared with the dynamics of the classical growth and two-species competition models as well as models involving two perfectly complementary resources.

Microbial Competition in Reactors with Wall Attachment: A Mathematical Comparison of Chemostat and Plug Flow Models

Competition for nutrient and the ability of bacteria to colonize the gut wall are factors believed to play a role in the observed stability of the indigenous microbiota of the mammalian large intestine. These factors were incorporated into the two-strain continuous-stirred tank reactor (CSTR) model formulated and numerically investigated by Freter et al. In their model simulations, the reactor is parameterized using data for the mouse intestine. An invading bacterial strain is introduced into a CSTR that has already been colonized by a resident strain. The two strains compete for a single growth-limiting nutrient and for limited adhesion sites on the wall of the reactor. The mathematical model described in this paper is motivated in part by the CSTR model, but is based on the plug flow reactor (PFR). Parameter values and initial conditions are chosen so that the numerical performance of the PFR can be compared to that of the CSTR. In simulations bearing a remarkable qualitative and quantitative resemblance to those of the CSTR it is found that the invader is virtually eliminated, despite the fact that it has uptake rate and affinity for the wall identical to those of the resident. The PFR model is then parametrized using data for the human large intestine, and the two-strain simulations are repeated. Though obvious quantitative differences are noted, the more important qualitative outcome is preserved. It is also found that when three strains compete for a single nutrient and for adhesion sites there exists a steady-state solution characterized by the segregation of the bacterial strains into separate nonoverlapping segments along the wall of the reactor.

AN EXAMINATION OF THE THRESHOLDS OF ENRICHMENT: A RESOURCE-BASED GROWTH MODEL

A model of single-species growth in the chemostat on two non-reproducing, growth-limiting, noninhibitory, perfectly substitutable resources is considered. The medium in the growth vessel is enriched by increasing the input concentration of one of the resources. Analytical methods are used to determine the effects of enrichment on the asymptotic behaviour of the model for different dilution rates. It is shown that there exists a threshold value for the dilution rate which depends on the maximal growth rate of the species on each of the resources. Provided the dilution rate is below the threshold, enrichment is beneficial in the sense that the carrying capacity of the environment is increased, regardless of which resource is used to enrich the environment. When the dilution rate is increased beyond the threshold, it becomes important to consider which resource is used for enrichment. For one of the resources it is shown that, while moderate enrichment can be beneficial, sufficient enrichment leads to the extinction of the microbial population. For the other resource, enrichment leads from washout or initial condition dependent outcomes to survival, and is thus beneficial. There are important implications of these results to the management of natural aquatic ecosystems. For example, while enrichment may be beneficial to the microbial species during the summer months, it can lead to their decimation during spring run-off, when the natural dilution rate is higher.

Classical and resource-based competition: a unifying graphical approach

A graphical technique is given for determining the outcome of two species competition for two resources. This method is unifying in the sense that the graphical criterion leading to the various outcomes of competition are consistent across most of the spectrum of resource types (from those that fulfill the same growth needs to those that fulfill different needs) regardless of the classification method used, and the resulting graphs bear a striking resemblance to the well-known phase portraits for two species Lotka–Volterra competition. Our graphical method complements that of Tilman. Both include zero net growth isoclines. However, instead of using the consumption vectors at potential coexistence equilibria to determine input resource concentrations leading to specific competitive outcomes, we introduce curves bounding the feasible set (the set where the resource concentrations of any equilibrium solution must be located). The washout equilibrium (corresponding to the supply point) occurs at an intersection of curves defining the feasible set boundary. The resource concentrations of all other equilibria are found where zero net growth isoclines either intersect each other inside the feasible set or they intersect the feasible set boundary. A species has positive biomass at such an equilibrium only if its zero net growth isocline is involved in such an intersection. The competitive outcomes are then determined from the position of the single species equilibria, just as in the phase portrait analysis for classical competition (rather than from information at potential coexistence equilibria as in Tilman’s method).

The Biofilm Model of Freter: A Review

R. Freter et al (1983) developed a simple chemostat-based model of competition between two bacterial strains, one of which is capable of wall-growth, in order to illuminate the role of bacterial wall attachment on the phenomenon of colonization resistance in the mammalian gut. Together with various collaborators, we have re-formulated the model in the setting of a tubular flow reactor, extended the interpretation of the model as a biofilm model, and provided both mathematical analysis and numerical simulations of solution behavior. The present paper provides a review of the work in [5, 6, 4, 52, 50, 36, 33, 32, 35, 34].

Stabilization of chemostats using feedback linearization

Stabilization via feedback linearization of two-species competition models based in the chemostat is considered. Competition is for a single growth-limiting resource in the first model, and for two perfectly substitutable resources in the second. The dilution rate and the input concentration of one nutrient are taken as input controls in each model. The main issues we investigate are limitations on stabilizable states and stabilizer design due to the structure of the linearized system. It is shown that determination of feasible stabilizable states requires computation of an equilibrium submanifold in the state space. Once this is accomplished, the potential exists for failure of a proportional-derivative stabilizer strategy due to singular structure in the resulting closed-loop dynamics caused by failure of the linearizing coordinate transformation to be a global diffeomorphism. This is exhibited explicitly in the three-dimensional model, and it is shown numerically that similar phenomena persist in four dimensions.

Stabilization of chemostats using feedback linearization and reduction of dimension

Stabilization via feedback linearization of models of competition between two species of microroganisms for two essential resources based in the chemostat is considered, extending previous work recently done by the authors; see [1], [14] and references therein. We show that though the full four-dimensional system is not stabilizable due to the dynamical properties of the system, it is possible to achieve the goal in modified form by pursuing a process of dimensional reduction prior to feedback linearization that results in replacing the four dimensional analysis with one in three dimensions. This technique has appeared in the literature applied to a similar system in the seminal papers of Hoo and Kantor [7]. However, in that case it appears that the authors thought of the method as a matter of convenience, and apparently did not realize that their original (higher dimensional) system was not stabilizable without utilizing the reduction procedure. (That is, that the reduction process was necessary in order to achieve the stabilization goal.) In this paper we show how the problem and its solution are very similar for both our model and that of Hoo and Kantor. This suggests that the dimensional reduction method could be rather generally applicable.