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A competition model with dynamically allocated inhibitor production

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Abstract

The chemostat is a basic model for competition in an open system and a model for the laboratory bio-reactor (CSTR). Inhibitors in open systems are studied with a view of detoxification in natural systems and of control in bio-reactors. This study allows the amount of resource devoted to inhibitor production to depend on the state of the system. The feasibility of one dependence is provided by quorum sensing. In contrast to the constant allocation case, a much wider set of outcomes is possible including interior, stable rest points and stable limit cycles. These outcomes are important contrasts to competitive exclusion or bistable attractors that are often the outcomes for competitive systems. The model consists of four non-linear ordinary differential equations and computer software is used for most of the stability calculations.

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... Although this theoretical prediction has been corroborated by the experiences of Hansen and Hubell [17], the biodiversity found in nature as well as in waste-water treatment processes and biological reactors are exceptions to this principle. Several authors [18][19][20][21][22][23][24][25][26] studied the inhibition as a factor in the maintenance of the diversity of microbial ecosystems: Can the production of internal inhibitors or the introduction of external inhibitors induce the stable coexistence of competitors in a chemostat-like environment? ...
... Let p c , x c and y c be defined by (23) and (25), respectively. System (19) has the following equilibria: The conditions of existence and stability of these equilibria are given in Table 3 ...
... Using these notations, we have the following description of existence and stability of the equilibria of (19). ...
Article
Understanding and exploiting the inhibition phenomenon, which promotes the stable coexistence of species, is a major challenge in the mathematical theory of the chemostat. Here, we study a model of two microbial species in a chemostat competing for a single resource in the presence of an external inhibitor. The model is a four-dimensional system of ordinary differential equations. Using general monotonic growth rate functions of the species and absorption rate of the inhibitor, we give a complete analysis for the existence and local stability of all steady states. We focus on the behavior of the system with respect of the three operating parameters represented by the dilution rate and the input concentrations of the substrate and the inhibitor. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species around a stable steady state and coexistence around a stable cycle. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.
... Different mechanisms of coexistence which were proposed in the literature are the intra-and interspecific competition [1,11,28,51], the flocculation [8,9,14,15] and the density-dependence [32][33][34][35]. Several mathematical models [4,12,13,[23][24][25]30] have attempted to understand the effects of an inhibitor on the competition and the coexistence of species in the chemostat. More precisely, competition models of two populations of microorganisms for a single nutrient have been studied with the presence of an inhibitor that affects the most stronger competitor while it is detoxified by the other competitor. ...
... We consider model (4). We make the following assumptions. ...
... 3. If Case 4 holds and there is no positive steady state, then E 2 is GAS for (4) in the interior of Ω. 4. E 0 is GAS for (4) in Ω if and only if µ i (S in , 0, 0) < D. ...
Article
This paper deals with a two-microbial species model in competition for a single-resource in the chemostat including general intra- and interspecific density-dependent growth rates with distinct removal rates for each species. In order to understand the effects of intra- and interspecific competition, this general model is first studied by determining the conditions of existence and local stability of steady states. With the same removal rate, the model can be reduced to a planar system and then the global stability results for each steady state are derived. The bifurcations of steady states according to interspecific competition parameters are analyzed in a particular case of density-dependent growth rates which are usually used in the literature. The operating diagrams show how the model behaves by varying the operating parameters and illustrate the effect of the intra- and interspecific competition on the disappearance of coexistence region and the occurrence of bi-stability region. Concerning the small enough interspecific competition terms, we would shed light on the global convergence towards the coexistence steady state for any positive initial condition. And, as far as the large enough interspecific competition terms, this system exhibits bi-stability with competitive exclusion of one species according to the initial condition.
... They considered the effect of the inhibitor on populations of the two organisms in a "well-stirred" bioreactor and proposed an ODE model. Mathematical analysis of this ODE model was presented in [Braselton & Waltman, 2001;Hsu et al., 2000;Hsu & Waltman, 1992;Lu & Hadder, 1998], and in the recent survey article [Hsu & Waltman, 2004]. ...
... Hsu and Waltman [1992] used the mathematical tools from nonlinear differential equations, particularly the theory of monotone dynamical systems, to obtain several global results. The other related mathematical analysis on this model can also be found in [Braselton & Waltman, 2001;Hsu et al., 2000;Hsu & Waltman, 2004;Lu & Hadder, 1998] and their references therein. ...
... In all of the simulations, the domain is divided uniformly into 40 cells. The parameters are chosen to be similar to those used in [Braselton & Waltman, 2001;Hsu et al., 2000;Hsu & Waltman, 1992]. Finally, the L 1 -norms of the components are plotted versus time in all figures. ...
Article
Full-text available
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.
... This coupled with the ability to build such devices has enabled the chemostat to take a central role in mathematical ecology. Many variants on the simple chemostat model have been analyzed including competition of species for common resources and the role of chemical inhibition, both internally and externally produced, on such competition [2,14,15]. ...
... Allelopathy is the production of a chemical by one species of microorganism to increase the mortality of a competitor. Several models exist representing allelopathic bacteria and the alleopathy has also been exhibited in species of algae [2,9,19,14,15,22]. Considered here is allelopathy in which the toxicity increases as the toxin producer becomes limited in nutrient [1]. ...
... (1. 6) There are four equilibia for System (1.6) 1) and N * (2) = 1 − R * (2) . The equilibrium E 3 is a special case as its existence requires R * (1) = R * (2) . ...
... Hsu and Waltman [16] considered mathematical model for two competing populations competing for a single resource where one of the competitors can produce a toxic chemical against its opponent at some cost to its reproductive abilities. Braselton and Waltman [2] investigated the role of density-dependent inhibitor production for two competing populations, one of which is toxic and a fraction of its potential growth is devoted to the production of toxin. Frank [13] also developed models for the dynamics of competition between producer and susceptible strains. ...
... It is reasonable to consider the toxic concentration as a separate entity in the model formulation. We would like to mention that the inclusion of this mechanism has already been applied in chemostat models [2,16]. In 2008, Chakraborty et al. [4] also investigated the effect of nutrient limitation on the toxin production by considering the toxin as a different variable. ...
... They also studied the effect of the initial concentration of C. polylepis on the growth response of H. triquetra in mixed cultures. They observed a decrease in the H. triquetra population on Day 4 in the culture with the highest initial C. polylepis concentration ( Figure 1); but, with the other two cultures which were initiated at lower C. polylepis concentrations, the decrease in H. triquetra numbers appeared some days later Degradation of toxic substances produced by the first species 0.3 d 2 Degradation of toxic substances produced by the second species 0.1 - (Days 6 and 7). They also found that irrespective of the initial concentration of C. polylepis, the decline in H. triquetra numbers always began when the C. polylepis concentration reached ca different initial concentrations of N 1 (3, 5, and 7 cells ml −1 ) keeping fixed the initial concentrations of the other variables ((N 2 , P 1 , P 2 ) ≡ (0.1 cells ml −1 , 0.1 μM, 0.1 μM)). ...
Article
Full-text available
In this paper, we study a two-species competitive system where both the species produce toxin against each other at some cost to their growth rates. A much wider set of outcomes is possible for our system. These outcomes are important contrasts to competitive exclusion or bistable attractors that are often the outcomes for competitive systems. We show that toxin helps to gain an advantage in competition for toxic species whenever the cost of toxin production remains within some moderate value; otherwise it may result in the extinction of the species itself.
... Considering two-species competition involving an inhibitor has been motivated by the work of Paul Waltman and collaborators who have investigated the corresponding question in a chemostat setting. We refer to [3] and the references therein. Their resulting system of four equations lacks strong monotonicity properties, hence the methods employed in the case of a chemostat are quite different from ours. ...
... It then follows that w = κb 3 ...
... , whereas E 2 (κ) ∈ (0, ∞) 3 for κ ∈ (0, 1 8 )∪( 1 4 , 3 8 ). Note that E 2 (0) = (1, 3 2 , 0) is an asymptotically stable stationary solution of (1.1) which corresponds to coexistence with (1, 3 2 ) globally within the framework of (1.2) by changingc 1 , but the example reveals the prospect of modeling κ as a function of u. ). Here Theorems 4.1 2) and 5.2 4) describe the asymptotic behavior of the solution flow for the respective κ. ...
Article
Full-text available
The dynamics of the solution flow of a two-species Lotka-Volterra competition model with an extra equation for simple inhibitor dynamics is investigated. The model fits into the abstract framework of two-species com-petition systems (or K-monotone systems), but the equilibrium representing the extinction of both species is not a repeller. This feature distinguishes our problem from the case of classical two-species competition without inhibitor (classical case for short), where a basic assumption requires that equilibrium to be a repeller. Nevertheless, several results similar to those in the classical case, such as competitive exclusion and the existence of a "thin" separatrix, are obtained, but differently from the classical case, coexistence of the two species or extinction of one of them may depend on the initial conditions. As in almost all two species competition models, the strong monotonicity of the flow (with respect to a certain order on R 3) is a key ingredient for establishing the main results of the paper.
... In the sequel, many other species have been discovered to exhibit quorum sensing behavior, including major human pathogens such as Staphylococcus aureus and Pseudomonas aeruginosa. In the last several years, quorum sensing mechanism has received much attraction67891011, etc. Firstly, J.P. Braselton and P. Waltman [6] formulated a model with dynamically allocated inhibitor production and studied its qualitative properties. ...
... In the last several years, quorum sensing mechanism has received much attraction67891011, etc. Firstly, J.P. Braselton and P. Waltman [6] formulated a model with dynamically allocated inhibitor production and studied its qualitative properties. Then, to obtain a deeper understanding of how and when this mechanism works, J.D. Dockery and J.P. Keener [7] were devoted to developing and studying an ODE and a PDE mathematical models for quorum sensing in Pseudomonas aeruginosa. ...
... Subsequently, in 1927 and 1932 Kermack and McKendrick [15,16] respectively founded a SIR model and a SIS model by the method of compartment and proposed the theory of threshold which forms the foundation of the epidemic dynamics. It is in the middle of 20th, epidemic dynamic began to develop exponentially [12], and a tremendous variety of models have been formulated, mathematically analyzed and applied to infectious diseases, see (456789101117181920212223 and references therein). In the same time, lots of new factors such as passive immunity, gradual loss of vaccine and disease-acquired immunity, stages of infection, vertical transmission, disease vectors, macro-parasitic loads, age structure, social and sexual mixing groups, spatial spread, vaccination, quarantine, chemotherapy, etc. were involved in different models. ...
Article
A bacteria–immunity model with bacterial quorum sensing is formulated, which describes the competition between bacteria and immune cells. A distributed delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In this paper, we focus on a subsystem of the bacteria–immunity model, analyze the stability of the equilibrium points, discuss the existence and stability of periodic solutions bifurcated from the positive equilibrium point, and finally investigate the stability of the nonhyperbolic equilibrium point by the center manifold theorem.
... [17]). 0096 Many of the prior models similar to that analyzed here were tailored to represent allelopathic bacteria, with one of two competitors releasing a toxin increasing the mortality rate of the other [21,14,15,2]. In simple situations where the rate of poison production is proportional to either the density or the growth rate of the toxic species, there are three non-trivial outcomes [21,14,15]: ...
... Asymptotic outcomes were more complicated when poison production was assumed to be regulated physiologically in relation to the densities of both competitors [2,15], rather than depending only on the density or growth rate of the toxic species. In such complex cases, stable interior equilibria and limit cycles can occur, with long-term persistence of both populations for at least some initial conditions. ...
... The problem described by System (3) is well posed, since the first quadrant is positively invariant, and all solutions of System (3) are bounded, which is consistent with the biological interpretation of the model [2]. In order to analyze the behavior of System (3) we need to establish the existence and stability of the equilibria and then examine the global dynamics. ...
Article
Presented is a chemostat model in which one microbial population excretes a poison that increases the mortality of another, with toxin production increasing as the growth rate of the toxic species decreases. The model is intended to explore the role of allelopathy in blooms of harmful algae, such as red tide (Karenia brevis) and golden algae (Prymnesium parvum). This study introduces the model and its biological basis, and proceeds to the analysis of its asymptotic states. All theoretical results are supported by a set of numerical simulations. A discussion of biological conclusions and similarities to other mathematical models is also presented.
... This system was investigated in [42]. Since the level of inhibitor production depends on the organism being able to sense the state of the system, one must first answer as to a possible mechanism. ...
... This system was investigated in [42]. Since the level of inhibitor production depends on the organism being able to sense the state of the system, one must first answer as to a possible mechanism. This is possibly provided, although not yet established experimentally in this case, by the mechanism of quorum sensing. See Bassler [43] for a review. [42] considers two special cases that represent the extremes for reasonable functions k(x, y), ...
... For example, while the boundary rest points can be analyzed directly, the interior rest points require the solution of a fifth order polynomial. Thus the analysis in [42] proceeds by numerical computation using a general Mathematica notebook. The numerical examples show that a wide variety of dynamical systems can be achieved. ...
Article
Mathematical models of the effect of inhibitors on microbial competition are surveyed. The term inhibitor is used in a broad sense and includes toxins, contaminants, allelopathic agents, etc. This includes both detoxification where the inhibitor is viewed as a pollutant and control where the inhibitor is viewed as an aid to controlling a bioreactor. The inhibitor may be supplied externally or may be created as an anti-competitor toxin. This includes plasmid-bearing, plasmid-free competition. The literature is spread across journals in different disciplines and with different notation. The survey attempts to present the mathematical models and the results of the corresponding analysis within a common framework and notation. Detailed mathematical proofs are not given but the methods of proof are indicated, references cited, and the results presented in tables. Open problems are indicated where there is a gap in the theory.
... The chemostat is one of the standard models of an open system in ecology [1][2][3][4][5]. The monograph of Hsu and Waltman has various mathematical methods for analyzing chemostat models [4]. ...
... γ i , i=1,2 are the Yield constants. The constant k∈(0,1) represents the fraction of potential growth devoted to producing the inhibitor [3]. ...
... Now, many other species are observed to exhibit quorum sensing behavior, including major human pathogens such as Staphylococcus aureus and Pseudomonas aeruginosa. Quorum sensing has received more and more attention (see [4][5][6][7][8][9][10][11][12][13][14][15] and the references therein) and some models are formulated to investigate its effect on the transmission of disease. Braselton and Waltman [4] formulated the dynamically allocated inhibitor production. ...
... Quorum sensing has received more and more attention (see [4][5][6][7][8][9][10][11][12][13][14][15] and the references therein) and some models are formulated to investigate its effect on the transmission of disease. Braselton and Waltman [4] formulated the dynamically allocated inhibitor production. Dockery and Keener [5] were devoted to developing and studying an ODE and a PDE mathematical models for quorum sensing in Pseudomonas aeruginosa and found that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer. ...
Article
Full-text available
This paper formulates a delay model characterizing the competition between bacteria and immune system. The center manifold reduction method and the normal form theory due to Faria and Magalhaes are used to compute the normal form of the model, and the stability of two nonhyperbolic equilibria is discussed. Sensitivity analysis suggests that the growth rate of bacteria is the most sensitive parameter of the threshold parameter and should be targeted in the controlling strategies.
... Also among eukaryotic marine algae allelopathic potential has been recognized in a variety of species belonging to different taxonomic groups (recently reviewed by Cembella, 2003). Recently, the allelopathic competitions taking place in a chemostat type environment have been mathematically represented essentially according to two different types of models, respectively, adopted by Chattopadhyay ( Mukhopadhyay et al., 1998Mukhopadhyay et al., , 2003, and Waltman ( Smith and Waltman, 1995;Braselton and Waltman, 2001;Fergola et al., 2004Fergola et al., , 2006). In this last modeling approach a crucial role is played by the choice of functionals which describe the following four features: ...
... Experiments have also shown that the amount of allelochemicals produced by C. vulgaris is proportional to its own concentration. This information has been used to represent mathematically the mechanism of the allelochemicals production with a sort of energy conservation law, by assuming that the production of these toxic compounds has a cost which is paid by the reduction of the growing potential of the producer species ( Hsu and Waltman, 1998;Braselton and Waltman, 2001). ...
Article
In this paper the results of several new experiments concerning the allelopathic competition between the two algal species Chlorella vulgaris (C. vulgaris) and Pseudokirchneriella subcapitata (P. subcapitata) have been reported. They show that the growth rates of the two species are different and can be modeled by the Andrews function (P. subcapitata) and Michaelis–Menten one (C. vulgaris). They also prove that the two species have different yields and that allelochemicals produced by C. vulgaris (called chlorellin) produce inhibitory effects on P. subcapitata. These results have been used to validate a mathematical modeling approach widely applied in eco-toxicology. The validation test being based on the comparison between the experimental outcome of the competition and the possible dynamical behaviors exhibited by the mathematical model. The paper consists of two parts devoted to the experiments and to the mathematical model, respectively. The mathematical analysis of the experimental results allowed to compute the numerical values of some parameters appearing in the growth-rate functions of the two species, and in the function which represents the inhibitory effect of chlorellin. The stability properties of some biologically meaningful steady-state solutions have been investigated. Moreover, by means of numerical simulations, it has been shown that the outcome of the real competition is foreseen by our model, and it can be easily simulated, provided that suitable numerical values for the parameters and initial conditions are chosen.
... Analysis (mathematical and numerical) can permit a deeper understanding both by predicting new scenarios and by suggesting new experiments. In the context of population dynamics, the influence of toxicants on the growth of populations and on competition among species has been widely studied in the last few years (Hallam et al. 1983;Ma 1986, 1987a, b;Freedman et al. 1989;Smith and Waltman 1995;Hsu and Waltman 1998;Mukhopadhyay et al. 1998Mukhopadhyay et al. , 2003Braselton and Waltman 2001). In particular, importance has been given to the competition that takes place in a chemostattype environment. ...
... In particular, importance has been given to the competition that takes place in a chemostattype environment. Such competitions have been mathematically represented essentially according to two different types of modelling approaches, adopted by Chattopadhyay (Mukhopadhyay et al. 1998(Mukhopadhyay et al. , 2003, and Waltman (Smith and Waltman 1995;Braselton and Waltman 2001;Fergola et al. 2004Fergola et al. , 2006, respectively. The two models used here follow this latter approach and give a representation of the dynamics of the interference taking place between the two algal species. ...
Article
The role of extracellular fatty acids in the interference between two algae, Chlorella vulgaris Beijerink and Pseudokirchneriella subcapitata (Korshikov) Hindak, was assessed by the co-cultivation of the two selected strains, as well as by the chemical analysis of exudates from the culture media of single strain cultures. The effect of culture age and phosphate limitation was evaluated. The experiments showed that the composition and amount of fatty acids, released by C. vulgaris and by P. subcapitata, both in a batch and in a continuous monoculture, depend on the culture age and on the phosphate concentration in the culture medium. We also found that the amount of chlorellin generated in the two algae co-culture increased and was almost exclusively constituted by a mixture of C18 fatty acids. By using the evaluated concentrations of these fatty acids, an artificial chlorellin was prepared. The toxicity of this mixture to P. subcapitata appears to be similar to that of the natural chlorellin. For both algae, a stimulation of growth was observed at low concentrations of the natural chlorellin, whereas higher concentrations produced inhibitory effects on both species. However, P. subcapitata was much more sensitive than C. vulgaris. By using some of these new experimental results, two new mathematical models have been used to describe the toxicity of chlorellin to C. vulgaris and to the interference between C. vulgaris and P. subcapitata, respectively.
... In these situations there is a global cost but the collateral effect is to favor the emergence of new strategies that tend to increase the diversity and complexity in these systems. This is the case in some social and economic systems [1,2,3,4,5] and in certain ecological and biochemical systems [6,7,8,9,10]. For example, Axelrod [1,2] considers an evolutionary approach to social norms based on a n-person Prisoner's dilemma. ...
Preprint
Using a set of heterogeneous competing systems with intra-system cooperation and inter-system aggression, we show how the coevolution of the system parameters (degree of organization and conditions for aggression) depends on the rate of supply of resources. The model consists of a number of units grouped into systems that compete for the resource; within each system several units can be aggregated into cooperative arrangements whose size is a measure of the degree of organization in the system. Aggression takes place when the systems release inhibitors that impair the performance of other systems. Using a mean field approximation we show that i) even in the case of identical systems there are stable inhomogeneous solutions, ii) a system steadily producing inhibitors needs large perturbations to leave this regime, and iii) aggression may give comparative advantages. A discrete model is used in order to examine how the particular configuration of the units within a system determines its performance in the presence of aggression. We find that full-scale, one sided aggression is only profitable for less-organized systems, and that systems with a mixture of degrees of organization exhibit robustness against aggression. By using a genetic algorithm we find that, in terms of the full-occupation resource supply rate, the coevolution of the set of systems displays a variety of regimes. This kind of model can be useful to analyse the interplay of the cooperation/competition processes that can be found in some social, economic, ecological and biochemical systems; as an illustration we refer to the competition between drug-selling gangs.
... Hsu and Waltman [11] looked at a model of the effect of anti-competitor toxins on plasmid-bearing, plasmid-free competition, and also addressed competition in the chemostat when one competitor produces a toxin [9]. Hsu et al [10] studied competition in the presence of a lethal external inhibitor, Braselton and Waltman [2] developed a competition model with dynamically allocated inhibitor production, while Jianhua et al [20] addressed the effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat. These mathematical models of allelopathic inhibition in the chemostat were operated under constant input and dilution rates. ...
Article
Full-text available
We model a periodic chemostat with allelopathic growth inhibition. The operating parameters including the nutrient supply, washout rate and nutrient uptake function are allowed to be periodic functions of time with commensurate periods. We show that competitive exclusion always holds in a periodic chemostat with allelopathic growth inhibition. We demonstrate that the species with the smallest break-even concentration survives the competition for a single growth-limiting nutrient independent of the initial conditions. Using Matlab software, we carry out numerical simulations to confirm the theoretical findings.
... To be consistent with notation with [22] and [23], let S i (t), 1 ≤ i ≤3, denote the concentration of the three nutrients at time t, x 1 (t) the concentration of the organism susceptible to the toxin secreted by the organism with concentration x 3 (t), x 2 (t) the concentration of the organism susceptible to the toxin secreted by the organism with concentration x 1 (t), x 3 (t) the concentration of the organism susceptible to the toxin secreted by the organism with concentration x 2 (t). The concentrations of the toxin producing organisms are given by P i (t) (for x i ). ...
... In general, they use controlled ows to keep a population inside them growing at a speci c rate [17]. Much theoretical and experimental work has been done in these systems over the years, both in general [18][19][20][21] and for speci c instances, such as selection [22], predator-prey mechanics [23], competition [24,25] and many others [26,27]. ...
Thesis
Antibiotic resistance is on the rise throughout the world and poses an increasing risk to our health systems. Understanding how bacteria respond to drugs in complex environments can help us manage our current arsenal of drugs in a more effective way. Past work has primarily focused on mechanistic responses by bacteria that confer resistant to specific drugs, and this molecular level understanding is essential. Yet bacteria live in large and possibly heterogeneous populations, and it's often not clear how known molecular scale events lead to large scale behavior like survival or extinction. In this thesis I make use of quantitative in-vitro experiments and mathematical models to understand and predict the population level dynamics of bacterial communities in the presence of drugs. Traditional methods for investigations at this scale suffer from some experimental limitations which I am able to overcome using new custom hardware. Using such tools, my experiments can include both precise measurements of a bacterial population over time while also including precise control of the growth environment. I can use this control to respond to the population in some way, such as holding a size threshold for it, or use it to stress the population in a specific manner, such as using differing drug dosing protocols. This versatility has allowed me to perform new and interesting investigations about the bacterial population respond to drugs in various settings, three of such experiments make up this thesis. In the first chapter, I show that the efficacy of many common drugs is dependent upon the density of the bacterial population in E. faecalis. I am able to quantify the amount to which several common drugs inhibit the growth rate of a bacterial population at different densities within exponential phase growth. In general, if such a density dependence for a drug exists, the drug is less effective at growth inhibition the denser the population is. I also investigate the cause of this effect, and find that the resulting change in pH of the environment from cellular growth can explain the effect for a couple of drugs. These results are then used to create a mathematical model that shows that a treatment regime could possibly lead to treatment failure when populations are very dense. In the presence of resistant sub-populations this density-dependent inhibition then leads to counter-intuitive dynamics, and this is the subject of Chapter 2. Using ampicillin and constant drug influx, these counter-intuitive dynamics include preferential survival of low-density populations over high-density counterparts. Using a model to understand this system, I show that this result comes from the pH driven density effect, and disappears when pH modulation of the environment is not possible. Finally, I show how competitive suppression can limit growth of drug resistant populations for a sufficiently high-density population. Using sub-inhibitory amounts of drug, I show that a mixed population of bacteria can be held at a high density far past when a non-suppressed resistant sub-population would have completely taken over the population. par As a whole, my work in this thesis helps to underscore the importance of density-driven community level interactions in determining the fate of bacterial populations exposed to antibiotics.
... This theoretical prediction has been corroborated by the experiences of Hansen and Hubell [9], but the biodiversity found in nature as well as in waste-water treatment processes and biological reactors are exceptions to this principle. Several authors [3,5,15,16,17,19,29], and recently [2,6] studied the inhibition as a factor in the maintenance of the diversity of microbial ecosystems: Can the production of internal inhibitors or the introduction of external inhibitors induce the stable coexistence of competitors in a chemostat-like environment? ...
Article
A model of two microbial species in a chemostat competing for a single resource in the presence of an internal inhibitor is considered. The model is a four-dimensional system of ordinary differential equations. Using general growth rate functions of the species, we give a complete analysis for the existence and local stability of all steady states. We describe the behavior of the system with respect to the operating parameters represented by the dilution rate and the input concentrations of the substrate. The operating diagram has the operating parameters as its coordinates and the various regions defined in it correspond to qualitatively different asymptotic behavior: washout, competitive exclusion of one species, coexistence of the species, bistability, multiplicity of positive steady states. This bifurcation diagram which determines the effect of the operating parameters, is very useful to understand the model from both the mathematical and biological points of view, and is often constructed in the mathematical and biological literature.
... In (Lee & Lim 1999) the multiplicity and stability of a cheese pre-fermentation process steady states are investigated and the influence of changing process inputs is evaluated. The coexistence of two competing species is assessed in (Braselton & Waltman 2001). Volcke et al. (2006) study the existence, uniqueness and the stability of the equilibrium points of a process used in wastewater treatment for ammonium removal and evaluates the effect of changing parameter and input values on the process behaviour. ...
... For simplicity, we rest assume that k i (but k 1 . k 2 , and k 3 may have different values) is constant, but then generalize the approach followed by Braselton and Waltman [21], when k is not constant. ...
Article
Rock-Paper-Scissors is a game played by two players to deter-mine a single winner. Biological relationships of Rock-Paper-Scissors are documented. In this paper, we form a continuous model of Rock-Papers-Scissors in the chemostat that coincides with the biology of such relationships. The basic models that we develop coincide with the observed phenomena. Be-cause the model involves a system of seven nonlinear differential equations, global results are difficult to obtain. We present several numerical studies that are the result of a substantial number of numerical trials to illustrate the various possibilities that might occur in the context of the problem discussed here.
... Hsu, and Watman [9] looked at a model of the effect of anti-competitor toxins on plasmid-bearing, plasmid-free competition, and also addressed competition in the chemostat when one competitor produces a toxin [10]. Hsu et al [11]studied competition in the presence of a lethal external inhibitor, Braselton and Watman, [12] developed a competition model with dynamically allocated inhibitor production, while Jianhua et al [14] addressed the effect of inhibitor on the plasmid bearing and plasmid-free model in the unstirred chemostat. The inhibition included introduction of an external toxin or one of the species producing the toxin. ...
Article
Full-text available
A model addressing mutual inhibition in a periodic chemostat is presented in this paper. The operating parameters, including the nutrient uptake function, washout rate, and nutrient concentration are allowed to be periodic functions of time, with commensurate periods. It is shown that with mutual inhibition, competitive exclusion always holds in models that would allow coexistence without inhibition. We further show that initial conditions play a crucial role in determining which species survives. Simulations using MATLAB appear to confirm the predictions of the models. Some results from the simulations are presented graphically.
... As in [3], [4], [9], [10], letS = S/S (0 ...
Article
Full-text available
We consider a competition model for three different strains wild, mutant and variant bacteria induced by an inhibitor created by one of the strains. We obtain two steady state regimes and we show that is possible the persistence of the three bacteria in the system.
... Grover and Wang, 2013;Hsu and Waltman, 2004;Martines et al., 2009). Coexistence does not arise without additional complications, such as density-dependent regulation of toxin production by competitor populations (Braselton and Waltman, 2001), through quorum sensing for example. ...
Article
Allelopathy is added to a familiar mathematical model of competition between two species for two essential resources in a chemostat environment. Both species store the resources, and each produces a toxin that induces mortality in the other species. The corresponding model without toxins displays outcomes of competitive exclusion independent of initial conditions, competitive exclusion that depends on initial conditions (bistability), and globally stable coexistence, depending on tradeoffs between competitors in growth requirements and consumption of the resources. Introducing toxins that act only between, and not within species, can destabilize coexistence leading to bistability or other multiple attractors. Invasibility of the missing species into a resident's semitrivial equilibrium is related to competitive outcomes. Mutual invasibility is necessary and sufficient for a globally stable coexistence equilibrium, but is not necessary for coexistence at a locally stable equilibrium. Invasibility of one semitrivial equilibrium but not the other is necessary but not sufficient for competitive exclusion independent of initial conditions. Mutual non-invasibility is necessary but not sufficient for bistability. Numerical analysis suggests that when competitors display bistability in the absence of toxin production, increases in the overall magnitude of resource supply cause bistability to arise over a larger range of supply ratios between the two resources. When competitors display coexistence in the absence of toxin production, increases in overall resource supply destabilize coexistence and produce bistability or other configurations of multiple attractors over large ranges of supply ratios. The emergence of multiple attractors at high resource supplies suggests that blooms of harmful algae producing allelopathic toxins could be difficult to predict under such rich conditions.
... For example, this model assumes that toxin production increases as the producer species becomes more nutrient-limited, since this appears to be the case for some harmful algae [25,26]. In contrast, if toxin production is regulated by the population density of the producer, a complex set of outcomes is possible, including coexistence with the sensitive species at a stable equilibrium or at a stable limit cycle [30]. The model presented here assumes a particular form of spatial heterogeneity that is ''mild'' compared to some that have been explored: here organisms, nutrient, and toxin follow the exactly same transport processes. ...
Article
Presented is a two-vessel gradostat model in which one microbial species excretes a toxin that increases the mortality of another as both species compete for a limiting nutrient resource. The model is intended to explore the role of allelopathy in blooms of harmful algae, such as red tide (Karenia brevis) and golden algae (Prymnesium parvum). This study introduces the model and its biological basis, and proceeds to the analysis of its asymptotic states to delineate the ecological outcomes of competitive exclusion and coexistence. Theoretical results are supported by a set of numerical simulations. A discussion of biological conclusions and similarities to other mathematical models is also presented.
... In these situations there is a global cost but the collateral effect is to favor the apparition of new strategies that tend to increase the diversity and complexity in these systems. This is the case in some social and economic systems [1, 2, 3, 4, 5] and in ecological and biochemical sys- tems [6, 7, 8, 9, 10]. For example, Axelrod [1, 2] considers an evolutionary approach to social norms based on a n-person Prisoner's dilemma. ...
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Using a set of heterogeneous competing systems with intra-system cooperation and inter-system unfair competition, we show how the co- evolution of the system parameters (degree of cooperation and unfair competition) depends on the external supply of resources. This kind of interaction is found in social, economic, ecological and biochemi- cal systems; as an illustration we consider the competition between drug-selling gangs. The model consists of a set of units (individuals, machines or enzymes) grouped in a number of systems (organizations, factories or glycosomes), each one composed by a fixed number of units that can be organized in three configurations: isolated (monomers), cooperating in couples (dimers), and cooperating in groups of four (tetramers). The units working in cooperating configurations increase their ability to obtain the resources (customers, raw material or sub- strates). The supply of resources can be polluted by the systems through inhibitors. When an unit absorbs an inhibitor, its function is blocked during a period of time. When the blocked unit belongs to dimers or tetramers, all units in the group are also inhibited to acquire the resource. Two parameters characterize each system: the fraction of monomers and the range of the average production in which the system is allowed to produce inhibitors. By using a genetic algorithm, we observe that the evolution of the parameters of the systems main- tains its long term average values for low and high supply rates, but tends to display global evolutive transitions when the supply of raw material lies between abundance and scarcity.
... For example, if species are sensitive to mortality induced by the same toxin that they produce, there can be coexistence at equilibrium [6]. When the toxin production rate is regulated by the densities of competitors (rather than constant as assumed here), then coexistence at equilibrium or in a limit cycle can arise [2]. Moreover, such coexistence attractors can be bistable with one of the system's semitrivial equilibria. ...
Article
This study presents a mathematical model of two species competing in a chemostat for one resource that is stored internally, and who also compete through allelopathy. Each species produces a toxin to that increases mortality rate of its competitor. The two species system and its single species subsystem follow mass conservation constraints characteristic of chemostat models. Persistence of a single species occurs if the nutrient supply of an empty habitat allows it to acquire a threshold of stored nutrient quota, sufficient to overcome loss to outflow after accounting for the cost of toxin production. For the two-species system, a semitrivial equilibrium with one species resident is unstable to invasion by the missing species according to a similar threshold condition. The invader increases if acquires a stored nutrient quota sufficient to overcome loss to outflow and toxin-induced mortality, after accounting for the cost of the invader's own toxin production. If both semitrivial equilibria for the two-species system are invasible then there is at least one coexistence equilibrium. Numerical analyses indicate another possibility: bistability in which both semitrivial equilibria are stable against invasion. In such a case there is competitive exclusion of one species, whose identity depends on initial conditions. When there is a tradeoff between abilities to compete for the nutrient and to compete through toxicity, the more toxic species can dominate only under nutrient-rich conditions. Bistability under such conditions could contribute to the unpredictability of toxic algal blooms.
... However, if strong enough, the toxin may cause complete wash-out of bacteria, fluctuations (limit cycles) or additional bistabilities. This is similar to what has been seen for other chemostat models with inhibitors, [4,10,21]. The effects of a strong toxin are such that the system can have two stable coexistence steady states. ...
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Anaerobic digestion has been modeled as a two-stage process using coupled chemostat models with non-monotone growth functions, [9]. This study incorporates the effects of an external toxin. After reducing the model to a 3-dimensional system, global stability of boundary and interior equilibria is proved using differential inequalities and comparisons to the corresponding toxin-free model. Conditions are given under which the behavior of the toxin-free model is preserved. Introduction of the toxin results in additional patterns such as bistabilities of coexistence steady states or of a periodic orbit and an interior steady state.
... Moreover, several mechanistic models have been analyzed to understand the effect of inhibition, either from external or from internal sources (review, Hsu and Waltman, 2004). It has been shown mechanistically that toxin production might be helpful in some cases for coexistence: e.g., when the toxin production is a plasmidencoded trait (Hsu and Waltman, 2004), or when toxin production is regulated in relation to competitor density, as might happen through quorum sensing mechanisms (Braselton and Waltman, 2001). Recent results suggest that, by modulating the top-down effects of grazer zooplankton, the inhibitory effects of TPP can drive the planktonic non-equilibrium (Roy et al., 2006). ...
Article
Without the top-down effects and the external/physical forcing, a stable coexistence of two phytoplankton species under a single resource is impossible — a result well known from the principle of competitive exclusion. Here I demonstrate by analysis of a mathematical model that such a stable coexistence in a homogeneous media without any external factor would be possible, at least theoretically, provided (i) one of the two species is toxin producing thereby has an allelopathic effect on the other, and (ii) the allelopathic effect exceeds a critical level. The threshold level of allelopathy required for the coexistence has been derived analytically in terms of the parameters associated with the resource competition and the nutrient recycling. That the extra mortality of a competitor driven by allelopathy of a toxic species gives a positive feed back to the algal growth process through the recycling is explained. And that this positive feed back plays a pivotal role in reducing competition pressures and helping species succession in the two-species model is demonstrated. Based on these specific coexistence results, I introduce and explain theoretically the allelopathic effect of a toxic species as a ‘pseudo-mixotrophy’—a mechanism of ‘if you cannot beat them or eat them, just kill them by chemical weapons’. The impact of this mechanism of species succession by pseudo-mixotrophy in the form of alleopathy is discussed in the context of current understanding on straight mixotrophy and resource-species relationship among phytoplankton species.
... In article [2], a mathematical analysis is given to a competition model in a wellmixed chemostat with the equation the ...
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Taking the spatial diffusion into account, we consider a reaction-diffusion system that models three species on a growth-limiting, nonreproducing resources in an unstirred chemostat. Sufficient conditions for the existence of a positive solution are determined. The main techniques is the Leray-Schauder degree theory.
... For numerical simulations, we chose the following values for the additional parameters: η = 0.2 and P 0 = 0.5. Two aspects of (21) are worth mentioning when compared to (3). The bifurcation diagrams (Fig. 6) show the variable x 1 in dependence of a 1 . ...
Article
Classical chemostat models assume that competition is purely exploitative and mediated via a common, limiting and single resource. However, in laboratory experiments with pathogens related to the genetic disease Cystic Fibrosis, species specific properties of production, inhibition and consumption of a metabolic by-product, acetate, were found. These assumptions were implemented into a mathematical chemostat model which consists of four nonlinear ordinary differential equations describing two species competing for one limiting nutrient in an open system. We derive classical chemostat results and find that our basic model supports the competitive exclusion principle, the bistability of the system as well as stable coexistence. The analytical results are illustrated by numerical simulations performed with experimentally measured parameter values. As a variant of our basic model, mimicking testing of antibiotics for therapeutic treatments in mixed cultures instead of pure ones, we consider the introduction of a lethal inhibitor, which cannot be eliminated by one of the species and is selective for the stronger competitor. We discuss our theoretical results in relation to our experimental model system and find that simulations coincide with the qualitative behavior of the experimental result in the case where the metabolic by-product serves as a second carbon source for one of the species, but not the producer.
Article
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.
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The quorum sensing mechanism is considered, an epidemical model which reflecting the competition between bacteria and immune system with general action law is formulated. Then, the existence and asymptotical stability of equilibria are discussed, respectively. Especially, by means of the center manifold theorem, the stability of the degenerate equilibria is proved. At last, numerical simulation is employed to verify our theory results, and which provides theory and numerical basis to control and prevent the epidemic disease.
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Quorum sensing, a widespread phenomenon in bacteria that is used to coordinate gene expression among local populations, intervenes in the competition between bacteria and the immune system. The domain of attraction of the bacteria-free equilibrium results from a linear matrix inequality optimization with a multivariate polynomial objective under constraints. The Bogdanov-Takens singularity and bifurcation, including a saddle-node bifurcation, a Hopf bifurcation, and a homoclinic bifurcation, are obtained from normal form theory. The normal form of a bifurcation is a simple dynamical system which is equivalent to all systems exhibiting this bifurcation.
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These proceedings are reporting on the conference "Math Everywhere", a successful event celebrating a leading scientist, promoting ideas he pursued and sharing the open atmosphere he is known for. The areas of the contributions are the following - Deterministic and Stochastic Systems. - Mathematical Problems in Biology, Medicine and Ecology. - Mathematical Problems in Industry and Economics. © Springer-Verlag Berlin Heidelberg 2007. All rights are reserved.
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A 4-equation delay differential system representing a bacterial allelopathic competition is analyzed. A distributed delay term models a linear quorum-sensing mechanism which regulates the delayed allelochemicals’ production process. The proved qualitative properties of the solutions are positivity, boundedness, global existence in the future, and uniqueness. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. The global asymptotic stability of a biologically meaningful steady-state solution is proved by constructing a suitable Lyapunov functional.
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The role of extracellular fatty acids in the interference between two algae, Chlorella vulgaris Beijer-ink and Pseudokirchneriella subcapitata (Korshikov) Hin-dak, was assessed by the co-cultivation of the two selected strains, as well as by the chemical analysis of exudates from the culture media of single strain cultures. The effect of culture age and phosphate limitation was evaluated. The experiments showed that the composition and amount of fatty acids, released by C. vulgaris and by P. subcapitata, both in a batch and in a continuous monoculture, depend on the culture age and on the phosphate concentration in the culture medium. We also found that the amount of chlorellin generated in the two algae co-culture increased and was almost exclusively constituted by a mixture of C18 fatty acids. By using the evaluated concentrations of these fatty acids, an artificial chlorellin was prepared. The toxicity of this mixture to P. subcapitata appears to be similar to that of the natural chlorellin. For both algae, a stimulation of growth was observed at low concentrations of the natural chlorellin, whereas higher concentrations produced inhibitory effects on both species. However, P. subcapitata was much more sensitive than C. vulgaris. By using some of these new experimental results, two new mathematical models have been used to describe the toxicity of chlorellin to C. vulgaris and to the interference between C. vulgaris and P. subcapitata, respectively.
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The dynamics of some algal allelopathic competitions taking place in a laboratory chemostat-like environment are analyzed. Our main results concern the global asymptotic stability of some biologically meaningful steady-state solutions and have been obtained by means of limiting systems theory and construction of suitable Liapunov functions. In particular one of these results completely explains the outcome of several experiments recently performed and illustrated in Fergola et al. (Ecol Model 208:205–214, 2007) on the competition between C. vulgaris and P. subcapitata. Numerical simulations confirm the analytical results and show that saddle-node bifurcation phenomena can appear.
Conference Paper
An allelopathic competition between two populations of microorganisms, taking place in a chemostat-like environment, is analyzed. The allelochemicals production by one of the two species is supposed delayed. The same allelochemical compound is also introduced as an external input concentration. Meaningful steady-state solutions and their stability properties are analyzed. The survival of the producing species is, in particular, studied in the special case of an exponential delay kernel.
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A model for three competing bacterial strains that incorporates mutation and/or phenotypic switching is studied. We consider three different strains: wild, mutated and phenotypic bacteria generated by an inhibitor introduced in the environment. Our model considers that all new phenotypic bacteria are sensitive to the inhibitor and there is no phenotypic replication. Two steady state regimes are identified, finding that the strain surviving is the one arising from mutation of the wild strain. The model may also show three steady state regimes with the persistence of the three bacteria in the system.
Chapter
A new mathematical model for an algal allelopathic competition is analyzed. The competition takes place in a chemostat-like environment, in the presence of a further concentration input of the same allelochemical compound produced by one of the two species. Steady-states and their asymptotic stability properties are investigated in the two different cases: in the case of an instantaneous constant internal production of allelochemicals and in the case of an instantaneous production linearly increasing with the concentration of the producing species. Some numerical simulations, obtained by means of Mathematica and performed by using recent experimental data are presented, which confirm the allelopathic nature of the competition.
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Using a set of heterogeneous competing systems with intra-system cooperation and inter-system aggression, we show how the coevolution of the system parameters (degree of organization and conditions for aggression) depends on the rate of supply of resources. The model consists of a number of units grouped into systems that compete for the resource; within each system several units can be aggregated into cooperative arrangements whose size is a measure of the degree of organization in the system. Aggression takes place when the systems release inhibitors that impair the performance of other systems. Using a mean field approximation we show that i) even in the case of identical systems there are stable inhomogeneous solutions, ii) a system steadily producing inhibitors needs large perturbations to leave this regime, and iii) aggression may give comparative advantages. A discrete model is used in order to examine how the particular configuration of the units within a system determines its performance in the presence of aggression. We find that full-scale, one sided aggression is only profitable for less-organized systems, and that systems with a mixture of degrees of organization exhibit robustness against aggression. By using a genetic algorithm we find that, in terms of the full-occupation resource supply rate, the coevolution of the set of systems displays a variety of regimes. This kind of model can be useful to analyse the interplay of the cooperation/competition processes that can be found in some social, economic, ecological and biochemical systems; as an illustration we refer to the competition between drug-selling gangs.
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Allelopathy is the chemical inhibition of one species by another. Bacteriocins, which are toxins produced by bacteria to inhibit the growth of closely related species, are a particular type of allelopathy that is of special interest because of the importance of bacteriocins in the food industry and in the development of vaccines. We form a model of this situation in the chemostat by incorporating parameters that measure relatedness and mutation rates as well as the cost of toxin production into standard competition models. Numerically, we show that depending upon growth rates and toxin sensitivity, coexistence of competitors may or may not occur.
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Competing species use a variety of strategies to gain an advantage over a competitor. We show that a desirable auxotrophic mutant can sometimes gain a growth advantage over its parental (or, wild-type) organism by using an offensive inhibitory or lethal strategy against the parental organism that lower’s the parental organism’s growth rate. Our numerical results indicate that inhibitive offensive strategies can stabilize a system while lethal offensive strategies can destabilize a system. Thus, even though a mutant may have a lower growth rate and/or higher metabolic needs than the parental organism, it may gain an advantage over the parental organism if it can limit the parental organism’s growth allowing it to coexist with the parental organism.
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A standard predator-prey model is modified to incorporate a mechanism of attack by the original predator against the prey species via the production of a toxin. Those members of the prey species affected by the toxin produced by the predator are potential prey for the predator. The model is expanded to incorporate additional predator and prey. We suggest that introducing additional prey or predators of the original predator may assist in its control.
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In this paper, we study some general models suggested to describe the effects of chemical compounds produced by an algal population on its survival in a chemostat-like environment. The conditions for its persistence and extinction are found. In particular, in the first model we make very general assumptions to represent the uptake, the regulative and the inhibiting functions, and analyze its global stability completely. In the second one we specify the first two functions and leave general the third one. Here the regulative function has different property from that in the first model, and a saddle-node bifurcation phenomenon occurs. In addition, according to the experimental data reported in DellaGreca et al. [2010. Fatty acids released by Clorella vulgaris and their role in interference with Pseudokirchneriella subcapitata: experiments and modelling. J. Chem. Ecol. 36, 339-349], we present a further model in which a new inhibiting function gives rise to a complex dynamics. The three models exhibit different dynamical behaviors, in particular the number of positive equilibria associated with each model varies resulting one, two and three, respectively. We also point out that the main differences exhibited by these models result from different specializations of the regulative and the inhibiting functions.
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The dynamics of a differential functional equation system representing an allelopathic competition is analyzed. The delayed allelochemical production process is represented by means of a distributed delay term in a linear quorum-sensing model. Sufficient conditions for local asymptotic stability properties of biologically meaningful steady-state solutions are given in terms of the parameters of the system. A global asymptotic stability result is also proved by constructing a suitable Lyapunov functional. Some simulations confirm the analytical results.
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The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.
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A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models are relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local (using index arguments). This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the case of uninhibited specific growth rate.
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We analyse the population dynamics of two strains of bacteria (Escherichia coli): one strain produces a toxin (called colicin) that increases the mortality of a colicin-sensitive strain in the neighbourhood, but does not harm the colicin-producing strain itself. On the other hand, in the absence of colicin in the environment, the colicin-sensitive strain enjoys a higher population growth rate. The model is closely related to the evolutionary dynamics of social interaction. It has been established previously that a perfectly mixing population shows bistability; that is, whichever strain dominates initially tends to defeat the other. On the other hand, empirical and computer simulation results in lattice structured populations show neither co-existence nor bistability of the two strains. In this paper, we analyse the lattice model based on pair approximation (forming a system of ordinary differential equations of global densities and local densities), and by the direct c omputer simulation of the spatial stochastic model. Both the pair approximation dynamics and the computer simulation show that, for most regions of parameter values, one strain defeats the other, irrespective of the initial abundance. However, the pair approximation analysis also suggests a relatively narrow parameter region of bistability, which should disappear when the model is considered on a lattice of infinitely large size. The biological implications of the results and the relationship of the present model with other models of the evolution of spite or altruistic behaviours are discussed. This suggests that the dynamics reflected by the spatially explicit lattice model may be sufficient, perhaps even necessary, to support the evolution of colicin.
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We demonstrate that in liquid cultures, defined in this study as a mass habitat, the outcome of competition between Escherichia coli that produce an antibacterial toxin (colicin) and sensitive E. coli is frequency dependent; the colicinogenic bacteria are at an advantage only when fairly common (frequencies in excess of 2 X 10(-2)). However, we also show that in a soft agar matrix, a structured habitat, the colicinogenic bacteria have an advantage even when initially rare (frequencies as low as 10(-6)). These contrasting outcomes are attributed to the colicinogenic bacteria's lower intrinsic growth rate relative to the sensitive bacteria and the different manner in which bacteria and resources are partitioned in the two types of habitats. Bacteria in a liquid culture exist as randomly distributed individuals and the killing of sensitive bacteria by the colicin augments the amount of resource available to the colicinogenic bacteria to an extent identical to that experienced by the surviving sensitive bacteria. On the other hand, the bacteria in a soft agar matrix exist as single-clone colonies. As the colicinogenic colonies release colicin, they kill neighboring sensitive bacteria and form an inhibition zone around themselves. By this action, they increase the concentration of resources around themselves and overcome their growth rate disadvantage. We suggest that structured habitats are more favorable for the evolution of colicinogenic bacteria.
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Populations of Escherichia coli initiated with a single clone and maintained for long periods in glucose-limited continuous culture, become polymorphic. In one population, three clones were isolated and by means of reconstruction experiments were shown to be maintained in stable polymorphism, although they exhibited substantial differences in maximum specific growth rates and in glucose uptake kinetics. Analysis of these three clones revealed that their stable coexistence could be explained by differential patterns of the secretion and uptake of two alternative metabolites acetate and glycerol. Regulatory (constitutive and null) mutations in acetyl-coenzyme A synthetase accounted for different patterns of acetate secretion and uptake seen. Altered patterns in glycerol uptake are most likely explained by mutations which result in quantitative differences in the induction of the glycerol regulon and/or structural changes in glycerol kinase that reduce allosteric inhibition by effector molecules associated with glycolysis. The evolution of resource partitioning, and consequent polymorphisms which arise may illustrate incipient processes of speciation in asexual organisms.
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1. The simple chemostat 2. The general chemostat 3. Competition on three trophic levels 4. The chemostat with an inhibitor 5. The simple gradostat 6. The general gradostat 7. The chemostat with periodic washout rate 8. Variable yield models 9. A size-structured competition model 10. New directions 11. Open questions Appendix A. Matrices and their eigenvalues Appendix B. Differential inequalities Appendix C. Monotone systems Appendix D. Persistence Appendix E. Some techniques in nonlinear analysis Appendix F. A convergence theorem.
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An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.
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The concept of persistence reflects the survival of all components of a model ecosystem. Most of the results to date are restricted to ordinary differential equations or to dynamics on locally compact spaces. The concept is investigated here in the setting of a C0C^0 -semigroup which is asymptotically smooth. Since the equations of population dynamics often involve delays or diffusion this seems the appropriate setting. Conditions are placed on the flow on the boundary which, given the presence of a global attractor provided by the assumption of dissipativeness and asymptotic smoothness, are necessary and sufficient for persistence.
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A model of the chemostat with an external nutrient and an external inhibitor is considered. A preliminary analysis reduces the problem to a three-dimensional competitive system. The theory of monotone flows is applied to obtain several global results. Global results fail when questions of multiple limit cycles cannot be answered. An example of an attracting limit cycle is given. The chemostat with inhibitor can model competition between two populations of microorganisms, where one strain is resistant to an antibiotic or competition in detoxification, a system where one strain can take up the pollutant while the other is inhibited by it.
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The use of selective media is important in bioreactor problems. While the selective component of the medium (an antibiotic, for example) may be added from an external source, there seems to be an advantage from having the selective process generated internally. Models for two common ways of achieving selective media are considered. In the first, proposed originally by Sardonini and DiBiasio, the plasmid-free organism is auxotrophic for a metabolite which is produced by the plasmid bearing organism in excess. For this model we are able to characterize completely the global behavior of solutions, completing that theory. In the second, the plasmid-bearing organism devotes a portion of its resources to producing a toxin to the plasmid-free organism. Such a model was proposed by Chao and Levin and the model considered here is a slight variant of theirs. Again, the global asymptotic behavior of the model as a function of the parameters is obtained.
Article
A model of competition in the chemostat with an inhibitor is combined with a model of competition in the chemostat between plasmid-bearing and plasmid-free organism to produce a model that more closely approximates the way chemostat-like devices are used in biotechnology. The asymptotic behavior of the solutions of the resulting system of nonlinear differential equations is analyzed as a function of the relevant parameters. The techniques are those of dynamical systems although perturbation techniques are used when the parameter reflecting plasmid-loss is small.
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A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models were relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local. The rigorous global analysis was done in a paper of Hsu, Waltman and Wolkowicz in the case of the uninhibited specific growth rates. This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the cases of combinations of inhibited and uninhibited specific growth rates.
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We introduce a general model for the dynamics of a single plasmid type and a single bacterial cell type, following Stewart and Levin (1977) in subdividing the population into plasmid-bearing and plasmid-free cells. For the particular case of mortality being a linear function of population sizes, we demonstrate the existence of multiple stable states and threshold values, thus illustrating that, in some cases, the establishment of a population of plasmids may depend on introduction of plasmids at sufficiently high levels. We also analyze in detail, for a general monotonically increasing mortality function, the epidemiological case in which plasmids confer a net cost on their hosts, and demonstrate that it is possible for such plasmids to become established. Stewart and Levin previously demonstrated this effect in a more restricted model.
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The usual models of the chemostat assume that the competition is purely exploitative, the competition is only through the consumption of the nutrient. However, it is known that microorganisms can produce toxins against its competitors. The basic experiments are due to Chao and Levin. In this work, we consider a model of competition in the chemostat of two competitors for a single nutrient where one of the competitors can produce a toxin against its opponent at some cost to its reproductive abilities. We give a complete characterization of the outcome of this competition in terms of the relevant parameters in hyperbolic cases. In three of four cases, the asymptotic results are global.
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The problem of plasmid stability and strain reversion in recombinant cultures is investigated through a complete stability analysis of a plasmid-bearing, plasmid-free mixed culture growing in a chemostat. Using a general method based on the index theory of a singular point, the complete stability portrait of all competitive interactions is obtained which can occur under all possible mutual dispositions of the specific growth rate curves and chemostat dilution rate. It is found that such a mixed culture can coexist in a chemostat only if there is a range of substrate concentrations where the plasmid-bearing cells grow at a specific rate which is larger than the specific growth rate of the plasmid-free cells. Realistic further genetic modifications that could possibly yield a culture with such properties are discussed.
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Quorum sensing, or the control of gene expression in response to cell density, is used by both gram-negative and gram-positive bacteria to regulate a variety of physiological functions. In all cases, quorum sensing involves the production and detection of extracellular signalling molecules called autoinducers. While universal signalling themes exist, variations in the design of the extracellular signals, the signal detection apparatuses, and the biochemical mechanisms of signal relay have allowed quorum sensing systems to be exquisitely adapted for their varied uses. Recent studies show that quorum sensing modulates both intra- and inter-species cell-cell communication, and it plays a major role in enabling bacteria to architect complex community structures.
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There are many situations in which the direction and intensity of natural selection in bacterial populations will depend on the relative frequencies of genotypes. In some cases, this selection will favour rare genotypes and result in the maintenance of genetic variability; this is termed stabilizing frequency-dependent selection. In other cases, selection will only favour genotypes when they are common. Rare types cannot invade and genetic variability will not be maintained; this is known as disruptive frequency-dependent selection. Phage-mediated selection for bacteria with novel restriction-modification systems is frequency-dependent and stabilizing. In mass culture, selection for the production of toxins and allelopathic agents is likely to be frequency-dependent but disruptive. This also occurs in selection favouring genes and transposable elements that cause mutations. Here I review the results of theoretical and experimental studies of stabilizing and disruptive frequency-dependent selection in bacterial populations, and speculate on the importance of this kind of selection in the adaptation and evolution of these organisms and their accessory elements (plasmid, phage and transposons).
Article
Populations of Escherichia coli, initiated with a single clone and maintained for long periods in glucose-limited continuous culture, developed extensive polymorphisms. In one population, examined after 765 generations, two majority and two minority types were identified. Stable mixed populations were reestablished from the isolated strains. Factors involved in the development of this polymorphism included differences in the maximum specific growth rate and in the transport of glucose, and excretion of metabolites by some clones which were utilized by minority clones.
Article
We present a continuous time model of the dynamics of two species competing for a single limiting resource in the presence of a substance that inhibits the growth of one of the species. Resource and inhibitor are both derived from external sources. These inputs, and all other model parameters, are assumed to be constant in space and time. There exist conditions that permit the stable coexistence of the competitors, provided that the sensitive species is more efficient in exploiting the limiting resource, and the resistant species removes the inhibitor from the environment. There exists a subset of these conditions wherein the sensitive species can become established if and only if the resistant species is already established. If the resistant species does not remove the inhibitor from the environment, then coexistence of sensitive and resistant species is structurally unstable. If the resistant species produces the inhibitor, then coexistence is dynamically unstable. We review several studies of bacterial competition in the presence of antibiotics that support these conclusions.
Article
When microbial strains compete for the same limiting nutrient in continuous culture, resource-based competition theory predicts that only one strain will survive and all others will die out. The surviving strain expected from theory will be the one with the smallest subsistence or "break-even" concentration of the limiting resource, a concentration defined by the J parameter. This prediction has been confirmed in the case of auxotrophic bacterial strains competing for limiting tryptophan. Because the value of J can be measured on the strains grown alone, the theory can predict the qualitative outcomes of mixed-growth competition in advance of actual competition.
Article
Populations of microorganisms inhabiting a common environment complete for nutrients and other resources of the environment. In some cases, the populations even excrete into the environment chemicals that are toxic or inhibitory to their competitors. Competition between two populations tends to eliminate one of the populations from their common habitat, especially when competition is focused on a single resource and when the populations do not otherwise interact. However, a number of factors mitigate the severity of competition and thus competitors often coexist.
Article
This review explores features of the origin and evolution of colicins in Escherichia coli. First, the evolutionary relationships of 16 colicin and colicin-related proteins are inferred from amino acid and DNA sequence comparisons. These comparisons are employed to detail the evolutionary mechanisms involved in the origin and diversification of colicin clusters. Such mechanisms include movement of colicin plasmids between strains of E. coli and subsequent plasmid cointegration, transposition- and recombination-mediated transfer of colicin and related sequences, and rapid diversification of colicin and immunity proteins through the action of positive selection. The wealth of information contained in colicin sequence comparisons makes this an ideal system with which to explore molecular mechanisms of evolutionary change.
Article
In this paper, we consider competition between plasmid-bearing and plasmid-free organisms with nutrient recycling and an inhibitor in a chemostat-type systems. We discuss the cases where the nutrient is supplied at a constant rate and the nutrient supply is time-dependent. For each case, we obtain criteria for the boundedness of solutions and persistence.
Article
Microorganisms are engaged in a never-ending arms race. One consequence of this intense competition is the diversity of antimicrobial compounds that most species of bacteria produce. Surprisingly, little attention has been paid to the evolution of such extraordinary diversity. One class of antimicrobials, the bacteriocins, has received increasing attention because of the high levels of bacteriocin diversity observed and the use of bacteriocins as preservatives in the food industry and as antibiotics in the human health industry. However, little effort has been focused on evolutionary questions, such as what are the phylogenetic relationships among these toxins, what mechanisms are involved in their evolution, and how do microorganisms respond to such an arsenal of weapons? The focus of this review is to provide a detailed picture of our current understanding of the molecular mechanisms involved in the process of bacteriocin diversification.
Article
Producing toxic chemicals to suppress both the growth and survivorship of local competitors is called allelopathy; some strains of the bacteria Escherichia coli produce a toxin (named colicin) which may kill colicin-sensitive neighbors while they themselves are immune. In a previous paper, the competitive outcome between colicin-producing and colicin-sensitive strains was shown to differ between a spatially structured and a completely mixed population. In this paper, we analyze the role of a third, "colicin-immune," strain, which does not produce colicin but is immune to it. Without spatial structure, the colicin-immune strain suppresses the colicin-producing strain and enables the colicin-sensitive strain to win. In a spatially structured population, modeled as a reaction-diffusion system, we examine the speed of boundaries between areas dominated by different strains in traveling waves and the events after the collision of two such boundaries. The colicin-immune strain passes through the area dominated by the colicin-sensitive strain and drives the colicin-producing strain to extinction. Subsequently the colicin-sensitive strain occupies the whole population.
Article
The study considers two organisms competing for a nutrient in an open system in the presence of an inhibitor (or toxicant). The inhibitor is input at a constant rate and is lethal to one competitor while being taken up by the other without harm. This is in contrast to previous studies, where the inhibitor decreases the reproductive rate of one of the organisms. The mathematical result of the lethal effect, modeled by a mass action term, is that the system cannot be reduced to a monotone dynamical system of one order lower as is common with chemostat-like problems. The model is described by four non-linear, ordinary differential equations and we seek to describe the asymptotic behavior as a function of the parameters of the system. Several global exclusion results are presented with mathematical proofs. However, in the case of coexistence, oscillatory behavior is possible and the study proceeds with numerical examples. The model is relevant to bioremediation problems in nature and to laboratory bio-reactors.
Article
The asymptotic behavior of solutions of a model for competition between plasmid-bearing and plasmid-free organisms in the chemostat with two distributed delays and an external inhibitor is considered. The model presents a refinement of the one considered by Lu and Hadeler [Z. Lu, K.P. Hadeler, Model of plasmid-bearing plasmid-free competition in the chemostat with nutrient recycling and an inhibitor, Math. Biosci. 167 (2000) p. 177]. The delays model the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition. Furthermore, it is assumed that there is inter-specific competition between the plasmid-bearing and plasmid-free organisms as well as intra-specific competition within each population. Conditions for boundedness of solutions and existence of non-negative equilibrium are given. Analysis of the extinction of the organisms, including plasmid-bearing and plasmid-free organisms, and the uniform persistence of the system are also carried out. By constructing appropriate Liapunov-like functionals, some sufficient conditions of global attractivity to the extinction equilibria are obtained and the combined effects of the delays and the inhibitor are studied.
Article
A simple operational strategy is shown to offer a viable means of enhancing plasmid stability in chemostat systems where plasmid loss is a common problem. Feedback control can be used to stabilize coexistence states, which are naturally unstable in the system investigated, and thus gurantee retention of the plasmid-carrying strain. The strategy exploits the normally undesirable characteristics of substrate inhibited growth kinetics, and is illustrated with specific reference to methanolutilizing organisms. Since the methodology may be easily implemented in practice, it offers an alternative to costly environmental methods such as antibiotic addition.
ˆ Table[ss5x[{aa, bb, g}, {s0, d}, {m1, a1, g1}, {m2, a2, g2} (m1*s)/(a1 + s))/g) + s*(bb*
  • Irpx
irpx [{aa_, bb_, g_}, {s0_, d_}, {m1_, a1_, g1_}, {m2_, a2_, g2_}]: ˆ Module[{t1, t2, t3, t4, t5, t6, x1, y1, p1}, t1 ˆ Table[ss5x[{aa, bb, g}, {s0, d}, {m1, a1, g1}, {m2, a2, g2}, n], {n, 1, 5}]; t2 ˆ Select[t1, Head[#1] ˆ ˆ ˆ Real && #1 > 0 &]; x1 ˆ (a2*d*(-bb + (-d + (m1*s)/(a1 + s))/g) + s*(bb*(-d + m2) + (d*(-d + (m1*s)/(a1 + s)))/g))/ (a2*d + (d + (-1 + aa)*m2)*s);
Structured habitats and the evolution of anti-competitor toxins in bacteria
  • Chao