Unstirred Chemostat Research Articles

Uniqueness and stability for coexistence solutions of the unstirred chemostat model

This article deals with the uniqueness and stability of coexistence solutions of a basic N-dimensional competition model in the unstirred chemostat by Lyapunov–Schmidt procedure and perturbation technique. It turns out that if the parameter G ≠ 0, which is given in Theorem 1.1, this model has a unique coexistence solution provided that the maximal growth rates a, b of u, v, respectively, lie in a certain range. Moreover, the unique coexistence solution is globally asymptotically stable if G > 0, while it is unstable if G < 0. In the later case, the semitrivial equilibria are both stable.

Coexistence of an unstirred chemostat model with Beddington–DeAngelis functional response and inhibitor

This paper deals with a N -dimensional competition model between plasmid-bearing and plasmid-free organisms in the unstirred chemostat, which incorporates the Beddington–DeAngelis functional response and inhibitor. By the application of degree theory in cones, bifurcation theory and perturbation technique, we obtain the uniqueness, multiplicity and stability of the coexistence solutions of the steady-state system when the effects of the inhibitor is large enough.

On a system of reaction–diffusion equations arising from competition with internal storage in an unstirred chemostat

In this paper we study a system of reaction–diffusion equations arising from competition of two microbial populations for a single-limited nutrient with internal storage in an unstirred chemostat. The conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient. The reduced system (limiting system) generates a strongly monotone dynamical system in its feasible domain under a partial order. We construct suitable upper, lower solutions to establish the existence of positive steady-state solutions. Given the parameters of the reduced system, we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors. The primary conclusion is that the survival of species depends on species's intrinsic biological characteristics, the external environment forces and the principal eigenvalues of some scalar partial differential equations. We also lift the dynamics of the limiting system to the full system.

Global dynamics of the periodic un-stirred chemostat with a toxin-producing competitor

This paper is concerned with the un-stirred chemostat with a toxin-producingcompetitor. The novelties of the modified model are the periodicity appearingin the boundary conditions, the different diffusive coefficients of the nutrientand the microorganisms, and some kinds of death rates. Both uniform persistenceand global extinction of the microorganisms are established under suitableconditions in terms of principal eigenvalues of scalar periodic paraboliceigenvalue problems. Our result implies that the toxin inhibits the sensitivemicroorganism indeed. The techniques includes the theories of asymptotic periodicsemi-flows, uniform persistence and the perturbation of global attractor.

Is Storage an Adaptation to Spatial Variation in Resource Availability?

When individuals store resources acquired while moving through a spatially variable habitat, a form of population structure arises. The theoretical consequences of this process for resource competition are studied for phytoplankton species consuming a single nutrient resource, using a Lagrangian modeling approach. Each competitor population is divided into many subpopulations that move through two model habitats with gradients in nutrient availability: an unstirred chemostat and a partially mixed water column. The results provide little indication that resource storage contributes to competitive fitness in the scenarios analyzed. Superior competitors usually reduce the limiting nutrient to low concentrations at steady state or sometimes have high maximal growth rates. Resource storage enhances competitive fitness in temporally variable habitats where encounters with rich nutrient pulses are strongly periodic. However, in purely spatially variable habitats where encounters with rich nutrient patches are random, resource storage does not appear to provide much benefit, at least for passively moving organisms that cannot control their location.

Positive solutions of a competition model for two resources in the unstirred chemostat

This paper deals with a competition model between two species for two growth-limiting and perfectly complementary resources in the unstirred chemostat. The main purpose is to determine the exact range of the parameters of two species so that the system possesses positive solutions, and to investigate multiple positive steady states of the system. The main tools used here include the monotone methods and the topological fixed point theory developed by Amann.

A food chain model for two resources in un-stirred chemostat

This paper deals with a multiple food chain model in un-stirred chemostat, where the prey feeds on two growth-limiting, nonreproducing resources. The conditions for existence of positive steady states are established. The global bifurcation of positive solutions is considered also. Furthermore, we obtain the conditions under which the predator possesses uniformly strong persistence or goes to extinction. The techniques used in this paper include the degree theory, the global bifurcation theory, the semigroup theory, and the maximum principle.

Characterization of positive solutions of the unstirred chemostat with an inhibitor

This paper deals with the unstirred chemostat model in the presence of an internal inhibitor. The main purpose of the paper is to determine the exact range Λ of the maximal growth ( a , b ) of two species where the system possesses positive solutions. It turns out that Λ is a connected unbounded region in R + 2 , whose boundary consists of two monotone nondecreasing curves Γ 1 : a = H 1 ( b ) and Γ 2 : b = H 2 ( a ) . For every ( a , b ) inside Λ the system has positive solutions and for ( a , b ) outside Λ there exists no positive solution. The functions H 1 ( b ) and H 2 ( a ) are constructed in terms of the limit of the corresponding time-dependent solution with a specific initial function. In particular, it is also shown that the system has at least two positive solutions in certain subregion of Λ .

Asymptotic behavior of an unstirred chemostat model with internal inhibitor

This paper deals with a chemostat model with an internal inhibitor. First, the elementary stability and asymptotic behavior of solutions of the system are determined. Second, the effects of the inhibitor are considered. It turns out that the parameter μ, which measures the effect of the inhibitor, plays a very important role in deciding the stability and longtime behavior of solutions of the system. The results show that if μ is sufficiently large, this model has no coexistence solution and one of the semitrivial equilibria is a global attractor when the maximal growth rate a of the species u lies in certain range; but when a belongs to another range, all positive solutions of this model are governed by a limit problem, and two semitrivial equilibria are bistable. The main tools used here include monotone system theory, degree theory, bifurcation theory and perturbation technique.

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

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.

Asymptotic Behavior on a Competition Model Arising from an Unstirred Chemostat

This paper is concerned with the unstirred chemostat model with two-species and one non-reproducing resource. The global attractivity of the positive steady-state solutions of the original system is established. Moreover, the effects of the growth rate on the unique positive equilibrium of the single population model are studied.

A SYSTEM OF REACTION–DIFFUSION EQUATIONS IN THE UNSTIRRED CHEMOSTAT WITH AN INHIBITOR

A system of reaction–diffusion equations is considered in the unstirred chemostat with an inhibitor. Global structure of the coexistence solutions and their local stability are established. The asymptotic behavior of the system is given as a function of the parameters, and it is determined when neither, one, or both competing populations survive. Finally, the results of some numerical simulations indicate that the global stability of the steady-state solutions is possible. The main tools for our investigations are the maximum principle, monotone method and global bifurcation theory.

A Mathematical Model of Competition for Two Essential Resources in the Unstirred Chemostat

A mathematical model of competition between two species for two growth-limiting, essential (complementary) resources in the unstirred chemostat is considered. The existence of a positive steady-state solution and some of its properties are established analytically. Techniques include the maximum principle, the fixed point index, and numerical simulations. The simulations also seem to indicate that there are regions in parameter space for which a globally stable positive equilibrium occurs and that there are other regions for which the model admits bistability and even multiple positive equilibria.

Coexistence solutions for a reaction–diffusion system of un-stirred chemostat model

In this paper, we study a reaction–diffusion system of multiple food chain model, where two predators feed on a single prey growing in an un-stirred chemostat. The conditions for the coexistence of steady states are determined. The main technique used here is the degree theory in cones.

Coexistence for a resource-based growth model with two resources

We investigate the coexistence of positive steady-state solutions to a parabolic system, which models a single species on two growth-limiting, non-reproducing resources in an un-stirred chemostat with diusion. We establish the existence of a positive steady-state solution for a range of the parameter (m; n), the bifurcation solutions and the stability of bifurcation solutions. The proof depends on the maximum principle, bifurcation theorem and perturbation theorem.

Predator–prey in an unstirred chemostat with periodical input and washout

Predator–prey in an unstirred chemostat with periodical input and washout

A System of Resource-Based Growth Models with Two Resources in the Unstirred Chemostat

Models of single-species growth in the unstirred chemostat on two growth-limiting, nonreproducing resources are considered. For the case of two complementary resources, the existence and uniqueness of a positive steady-state solution is established. It is also proved that the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values. For the case of two substitutable resources, the existence of a positive steady-state solution is determined for a range of the parameter (m, n). Techniques include the maximum principle, monotone method and global bifurcation theory. The longtime behavior of the corresponding limiting system is given for a range of (m, n). In the special case of m=n, the uniqueness and global attractivity of the positive steady-state solution of the original system is established.

Competition in the Unstirred Chemostat with Periodic Input and Washout

The model of an unstirred chemostat is generalized to that of a chemostat with time-dependent input/washout rates. The novelty of the new model is that time periodicity appears in the boundary conditions. The asymptotic dynamics of the competition between two microbial populations is determined in terms of the corresponding period map, which is shown to preserve the standard competitive ordering. It is shown that the dynamics of competition is similar to that of a chemostat with constant boundary conditions. Simple criteria for coexistence versus competitive exclusion are presented. 1. Introduction. The chemostat represents a basic model of an open system in microbial ecology. In its simplest form, it consists of three vessels. The rst, called the feed bottle, contains medium with all of the nutrients needed for growth in sur- plus except one, which hereafter is simply called the nutrient. The contents of the feed bottle are pumped at a constant rate into the second vessel, called the culture vessel or bioreactor. The culture vessel is charged with one or more populations of microorganisms. The contents of the culture vessel are pumped into the remaining vessel, called the over∞ow vessel, at a constant rate, keeping the volume of the reactor constant. The organisms compete for the nutrient in a purely exploitative manner. Basic assumptions include that the vessel is well mixed and that all other parame- ters (pH, temperature, etc.) are strictly controlled. The ∞ow rate is assumed to be sucient to preclude wall growth or the accumulation of metabolic products. Let S(t) denote the concentration of the nutrient in the culture vessel and xi(t), i =1 ; 2, denote the concentration of the competitors. LetS 0 denote the concentration of the input nutrient and let D denote the dilution rate (∞ow rate/volume). If growth is assumed to be proportional to consumption then the basic equations take the form

Coexistence in the unstirred chemostat

We consider a rather general system of reaction diffusion equations containing multiple parameters and prove existence of positive solutions for some sets of parameters. The problem models long term coexistence of competing microorganisms in an unstirred chemostat with diffusion and flow. The proof, as in previous work, depends on a bifurcation argument using a Crandall-Rabinowitz theorem. Since stability of these steady states, as well as the more general question of persistence is unresolved, we include some numerical results which shed light on these questions.

A remark on the global asymptotic stability of a dynamical system modeling two species competition

1. IntroductionIn this paper we shall consider a strongly order preserving semiflowmodeling the two species competition in population biology. To motivate theprincipal result, we consider three examples of competitive dynamical systemsin an infinite dimensional space which is a product space. The first exampleis the model of the unstirred chemostat with equal diffusions [6]