Washout Equilibrium Research Articles

Dynamics of a stochastic turbidostat model with sampled and delayed measurements.

In this paper, a stochastic turbidostat model with controllable output is established by using piecewise constant delayed measurements of the substrate concentration. We commence by proving the existence and uniqueness of the global positive solution of the stochastic delayed model. Then, sufficient conditions of extinction and stochastic strong permanence of the biomass are acquired. In quick succession, we investigate the stochastic asymptotical stability of the washout equilibrium as well as the asymptotic behavior of the random paths approaching the interior equilibrium of its corresponding deterministic model by employing the method of Lyapunov functionals. Numerical and theoretical findings show that the influence of environmental random fluctuations on the dynamics of the model may be more pronounced than that of time delay.

The general chemostat model with multiple nutrients and flocculating agent: From deterministic behavior to stochastic forcing transition

Quantifying the effects of environmental perturbations on ecosystems is essential to understanding many rules in nature, such as the survival and extinction of populations. In this paper, a class of deterministic chemostat models with perfectly complementary nutrients, flocculating agent and general functional response functions is investigated in details as well as its stochastic version. For the deterministic model, the well-posedness of the solutions, the dissipativeness and the existence of the equilibria (forward/backward bifurcation) are discussed. Sufficient conditions are given for the global stability of the boundary equilibrium (the washout equilibrium). If the basic reproduction number is greater than one, the model is uniformly persistent, and an explicit expression for the estimation of the ultimately lower bound of microorganism population is given by using different techniques from the standard persistence theory. For the stochastic model, the existence of an invariant probability measure is analyzed by establishing two thresholds which automatically become the basic reproduction number when stochastic perturbations are not considered. Furthermore, sufficient conditions are given for the extinction of microorganism populations. The numerical simulations reveal that noise excitation may make the prediction and control of the evolutionary trend of microorganism population more difficult and complex.

Dynamics and Hopf Bifurcation Analysis of a Chemostat Model with Modified Growth Rate and Variable Yield Coefficient

The objective of this study is to analyze a chemostat model of very simple type with the Haldane expression of growth rate and a variable yield coefficient. The proposed modified model is analyzed qualitatively and quantitatively. Analytic conditions for stability and optimality are determined for washout and no washout equilibrium solutions. One of the main focuses of the study is to determine parameter values for which Hopf Bifurcations occur in a bioreactor. It has been shown that the maximum stable non-washout equilibrium exits at a residence time under suitable parameter values. Hopf bifurcation is observed at three different conditions of the parameters.

Thermodynamic Inhibition in a Biofilm Reactor with Suspended Bacteria.

We formulate a biofilm reactor model with suspended bacteria that accounts for thermodynamic growth inhibition. The reactor model is a chemostat style model consisting of a single replenished growth promoting substrate, a single reaction product, suspended bacteria, and wall attached bacteria in the form of a bacterial biofilm. We present stability conditions for the washout equilibrium using standard techniques, demonstrating that analytical results are attainable even with the added complexity from thermodynamic inhibition. Furthermore, we numerically investigate the longterm behaviour. In the computational study, we investigate model behaviour for select parameters and two commonly used detachment functions. We investigate the effects of thermodynamic inhibition on the model and find that thermodynamic inhibition limits substrate utilization/production both inside the biofilm and inside the aqueous phase, resulting in less suspended bacteria and a thinner biofilm.

Dynamics and productivity of microalgae in presence of predators

Abstract This paper is concerned with the dynamics and the productivity of biomass in a continuous microalgae culture contaminated by predators. We propose a general model describing a cultivation system where photolimited microalgae are grazed by zooplankton. We state necessary and sufficient conditions for the existence of a coexistence equilibrium. If this equilibrium exists, then microalgae and predators converge either to it or to a globally asymptotically stable periodic solution. In absence of the coexistence equilibrium, any solution approaches either an equilibrium characterized by the presence of algae and the absence of predators, or the washout equilibrium. This description of the dynamics allows to define the microalgae productivity in the long-term (i.e. when an attracting set is reached). With the help of numerical simulations, we show that operating the system at optimal (constant) dilution rate triggers the extinction of predators.

Dynamical Analysis of a Stochastic Delayed Two-Species Competition Chemostat Model

In this paper, we consider a stochastic delayed two-species competition chemostat model with the Monod growth response function. First, we verify that there is a unique global positive solution for any given initial conditions for this stochastic delayed system. Second, we give the dynamical behavior of the solution of stochastic delay system around the washout equilibrium of deterministic system; moreover, we discuss the competition exclusion and coexistence of microorganisms $$x_1$$ and $$x_2$$ . Finally, computer simulations are carried out to illustrate the obtained results; in addition, results show that time delay has critical effects on the survival of the microorganisms.

Stochastic modeling of algal bloom dynamics with delayed nutrient recycling.

Using the discrete Markov chain, in this paper we develop a stochastic model for algal bloom, in which white noise terms are introduced to describe the effects of environmental random fluctuations and time delay to account for the time needed in the conversion of detritus into nutrient. For the proposed model, we firstly discuss the well-posedness, namely the existence and uniqueness of the global positive solution. Then, it is followed by seeking the sufficient conditions for the stochastic stability of its washout equilibrium. Then by using Fourier transform method, the spectral densities of the nutrient and the algae population are estimated. Finally, we show that larger noise can make the algae population extinct exponentially with probability one. Our theoretical and numerical results suggest that the environmental random fluctuations may have more significant influences on the dynamics of the model than the delay. These findings are helpful for a better understanding of the formation mechanism of algal blooms.

Dynamic behaviors of a turbidostat model with Tissiet functional response and discrete delay

In this paper, dynamic behaviors of a turbidostat model with Tissiet functional response, linear variable yield and time delay are investigated. The existence and boundedness of solutions, the local asymptotic stability of its equilibria and the phenomenon of Hopf bifurcation for this system are considered. Using the Liapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Furthermore, based on some knowledge of limit set, we show the necessary and sufficient conditions of permanent of the turbidostat model. Finally, numerical simulations are offered to support our results.

Asymptotic behavior of a Lotka–Volterra food chain stochastic model in the Chemostat

ABSTRACTThis article studies the asymptotic behavior of a stochastic Chemostat model with Lotka–Volterra food chain in which the dilution rate was influenced by white noise. The long-time behavior of the model is studied. Using Lyapunov function and Itô's formula, we show that there is a unique positive solution to the system. Moreover, the sufficient conditions for some population dynamical properties including the boundedness in mean and the stochastically asymptotic stability of the washout equilibrium were obtained. Furthermore, we show how the solutions spiral around the predator-free equilibrium and the positive equilibrium of deterministic system. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.

Stochastic Sensitivity Analysis for a Competitive Turbidostat Model with Inhibitory Nutrients

A stochastic model of turbidostat in which two microorganism species compete for an inhibitory growth-limiting nutrient is considered. In the deterministic case, the model has rich dynamics: a coexistence equilibrium and the washout equilibrium can be simultaneously stable, and a stable limit cycle may exist. In the stochastic case, a phenomenon of noise-induced extinction occurs. Namely, the stochastic trajectory near the deterministic coexistence equilibrium will tend to the washout equilibrium. Based on the stochastic sensitivity function technique, in this paper, we construct the confidence ellipse and then estimate the critical value of the intensity for noise generating a transition from coexistence to extinction. We also construct the confidence band to find the configurational arrangement of the stochastic cycle.

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake and a time delay

In this paper, we consider a simple chemostat model with inhibitory exponential substrate uptake and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Using the fluctuation lemma, the sufficient condition of the global asymptotic stability of the positive equilibrium [Formula: see text] is obtained. Numerical simulations are also performed to illustrate the results.

The Biomass Control in a Biochemical Reactor Process with Time-Varying Dilution Rate: A Mathematical Study

In this paper, a continuous bioprocess model with single species for an inhibitory growth-limiting nutrient is considered, where the dilution rate is time varying and controlled by a critical cell concentration in the bioreactor. It is shown that the model has rich dynamics. A coexistence equilibrium and the washout equilibrium can be asymptotically stable simultaneously so that coexistence may depend on initial conditions.

Qualitative analysis of the chemostat model with variable yield and a time delay

In this paper, we consider the chemostat model with inhibitory exponential substrate, variable yield and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. The Hopf bifurcation of solutions to the system is studied. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Based on some known techniques on limit sets of differential dynamical systems, we show that, for any time delay, the chemostat model is permanent if and only if only one positive equilibrium exits.

Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation on the Dilution Rate

We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise. The long time behavior of the system is studied. Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in the mean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results.

Long time behaviour of a stochastic model for continuous flow bioreactor

The global dynamics of a deterministic model in wastewater treatment has been investigated in Zhang (J Math Chem 50:2239–2247, 2012). The stochastic version, which can be used for continuous flow bioreactor and membrane reactor is presented in this study. Precisely, we assume there is some uncertainty in the part describing the recycle, which results in a set of stochastic differential equations with white noise. We first show that the stochastic model has always a unique positive solution. Then long time behavior of the model is studied. Our study shows that both the washout equilibrium and non-washout equilibrium are stochastically stable. At the end, we carry out some numerical simulations, which support our theoretical conclusions well.

Correlation among DCE‐MRI measurements of bone marrow angiogenesis, microvessel density, and extramedullary disease in patients with multiple myeloma

Correlation among DCE‐MRI measurements of bone marrow angiogenesis, microvessel density, and extramedullary disease in patients with multiple myeloma

Classical and resource-based competition: a unifying graphical approach

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

Competition in a turbidostat for an inhibitory nutrient†

A model of competition in a turbidostat between two species for an inhibitory growth-limiting nutrient is considered. It is shown that the model has rich dynamics. A coexistence equilibrium and the washout equilibrium can be asymptotically stable simultaneously so that coexistence may depend on initial conditions. Under certain conditions, periodic coexistence of the two species occurs. There is a possibility that two species coexist, whereas one species dies out in the absence of its rival.

A delayed chemostat model with general nonmonotone response functions and differential removal rates

A chemostat model with general nonmonotone response functions is considered. The nutrient conversion process involves time delay. We show that under certain conditions, when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations, the competitive exclusion principle holds. In the case of insignificant death rates, the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times (see [H.M. El-Owaidy, M. Ismail, Asymptotic behavior of the chemostat model with delayed response in growth, Chaos Solitons Fractals 13 (2002) 787–795; H.M. El-Owaidy, A.A. Moniem, Asymptotic behavior of a chemostat model with delayed response growth, Appl. Math. Comput. 147 (2004) 147–161; S. Yuan, M. Han, Z. Ma, Competition in the chemostat: convergence of a model with delayed response in growth, Chaos Solitons Fractals 17 (2003) 659–667]). However, the proofs are all incorrect. In this paper, we provide a correct proof that also applies in the case of differential death rates. In addition, we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium, thus showing the outcome can be initial condition dependent. Moreover, we show that when the species specific death rates are included, damped oscillations may occur even when there is no delay. Thus, the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments.