Competition between plasmid-bearing and plasmid-free organisms in a chemostat with nutrient recycling and an inhibitor

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.     

Nonlinear control for algae growth models in the chemostat

This paper deals with output feedback control of phytoplanktonic algae growth models in the chemostat. The considered class of model is of variable yield type, meaning that the ratio between the environmental nutrient absorption rate and the cells’ growth rate varies, which is different from classical bioprocesses assumptions. On the basis of weak qualitative hypotheses on the analytical expressions of the involved biological phenomena (which guarantee robustness of the procedure toward modeling uncertainties) we propose a nonlinear controller and prove its ability to globally stabilize such processes. Finally, we illustrate our approach with numerical simulations and show its benefits for biological laboratory experiments, especially for ensuring persistence of the culture facing classical experimental problems.     

Stoichiometric food quality and herbivore dynamics

Herbivores may grow with nutrient or energy limitation, depending on food abundance and the chemical composition of their food. We present a model that describes herbivore growth as a continuous function of two limiting factors. This function uses the synthesizing unit concept, has the hyperbolic Monod model as a limiting case, and has the same number of parameters as the Monod model coupled to Liebig's discontinuous minimum rule. We use the model to explore nutrient-limited herbivore growth in a closed system with algae, Daphnia and phosphorus as the limiting nutrient. Phosphorus in algae may substantially influence Daphnia growth. This influence changes over time and is most pronounced when algae and Daphnia populations fluctuate strongly. Relative to classic models that only consider food quantity as a determinant of Daphnia growth, our model shows richer dynamical behaviour. In addition to the standard positive equilibrium, which may be stable or unstable depending on nutrient availability, a new positive equilibrium may arise in our model when mortality rates are relatively high. This equilibrium is unstable and reduces the likelihood of long-term persistence of Daphnia in the system.     

Persistence in the chemostat

Sufficient conditions are obtained for persistence in chemostat models for interactions of a limiting nutrient (or substrate) and two populations, and for two limiting complementary substrates and a single population. The results of Freedman and Waltman are extended for the three interacting predator-prey populations.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Persistence of bacteria and phages in a chemostat

The model of bacteriophage predation on bacteria in a chemostat formulated by Levin et al. (Am Nat 111:3–24, 1977) is generalized to include a distributed latent period, distributed viral progeny release from infected bacteria, unproductive adsorption of phages to infected cells, and possible nutrient uptake by infected cells. Indeed, two formulations of the model are given: a system of delay differential equations with infinite delay, and a more general infection-age model that leads to a system of integro-differential equations. It is shown that the bacteria persist, and sharp conditions for persistence and extinction of phages are determined by the reproductive ratio for phage relative to the phage-free equilibrium. A novel feature of our analysis is the use of the Laplace transform.     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

Bacteriophage and Bacteria in a Flow Reactor

The Levin-Stewart model of bacteriophage predation of bacteria in a chemostat is modified for a flow reactor in which bacteria are motile, phage diffuse, and advection brings fresh nutrient and removes medium, cells and phage. A fixed latent period for phage results in a system of delayed reaction-diffusion equations with non-local nonlinearities. Basic reproductive numbers are obtained for bacteria and for phage which predict survival of each in the bio-reactor. These are expressed in terms of physical and biological parameters. Persistence and extinction results are obtained for both bacteria and phage. Numerical simulations are in general agreement with those for the chemostat model.     

Persistence times of populations with large random fluctuations

The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.     

Models of competition in the chemostat with instantaneous and delayed nutrient recycling

Two models for competition of two populations in a chemostat environment with nutrient recycling are considered. In the first model, the recycling is instantaneous, whereas in the second, the recycling is delayed. For each model an equilibrium analysis is carried out, and persistence criteria are obtained. This paper extends the work done by Beretta et al. (1990) for a single species.Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC A4823Research carried out at the University of Alberta while on a Canada-China Scholarly Exchange Program     

A simple model to describe transient differences between cell number and biomass growth rates of Escherichia coli

Information on the response of a microbial culture to dynamic environmental conditions is necessary for the design of transient operation processes. However, most attempts at modelling culture response have been directed at describing the steady-state behavior. Thus, there is a need for adequate dynamic models for process design. Simulations of nutrient shifts were completed using a "single-cell" model for Escherichia coli. It was discovered that the specific mass growth rate and the specific number of cells growth rate were different under transient conditions, whereas at steady state (balanced growth) these rates are equivalent. Using these observations, a simple delay model to describe the transient behavior of the two growth rates is formulated and tested. The model contains as state variables only the readily measurable macroscopic quantities (biomass, cell number, and limiting nutrient). This model agreed well with the predictions of the single-cell model.     

Mathematical model of transitional processes in the chemostat culture of microorganisms

A mathematical model of the growth of the cell culture was developed. The model takes into account changes of the levels of the enzymes which define the metabolism rate, transport of the substrate into the cell, regeneration of the donors of energy. The model is based on the proposition that the rate of overall protein synthesis in the cell is defined by the concentration of a few aminoacids limiting the growth. The chemostat culture of the methanol-assimilating yeast was used as the object of modelling. The model allows to explain the experimental kinetics of alterations in cell number (biomass) and other measurable characteristics of the culture during the transient process when the dilution rate was changed.     

Global dynamics of a chemostat competition model with distributed delay

 We study the global dynamics of n-species competition in a chemostat with distributed delay describing the time-lag involved in the conversion of nutrient to viable biomass. The delay phenomenon is modelled by the gamma distribution. The linear chain trick and a fluctuation lemma are applied to obtain the global limiting behavior of the model. When each population can survive if it is cultured alone, we prove that at most one competitor survives. The winner is the population that has the smallest delayed break-even concentration, provided that the orders of the delay kernels are large and the mean delays modified to include the washout rate (which we call the virtual mean delays) are bounded and close to each other, or the delay kernels modified to include the washout factor (which we call the virtual delay kernels) are close in L ¹-norm. Also, when the virtual mean delays are relatively small, it is shown that the predictions of the distributed delay model are identical with the predictions of the corresponding ODEs model without delay. However, since the delayed break-even concentrations are functions of the parameters appearing in the delay kernels, if the delays are sufficiently large, the prediction of which competitor survives, given by the ODEs model, can differ from that given by the delay model. Received: 9 August 1997 / Revised version: 2 July 1998     

A stoichiometric organic matter decomposition model in a chemostat culture

Biodegradation, the disintegration of organic matter by microorganism, is essential for the cycling of environmental organic matter. Understanding and predicting the dynamics of this biodegradation have increasingly gained attention from the industries and government regulators. Since changes in environmental organic matter are strenuous to measure, mathematical models are essential in understanding and predicting the dynamics of organic matters. Empirical evidence suggests that grazers’ preying activity on microorganism helps to facilitate biodegradation. In this paper, we formulate and investigate a stoichiometry-based organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. We determine the criteria for the uniform persistence and extinction of the species and chemicals. Our results show that (1) if at the unique internal steady state, the per capita growth rate of bacteria is greater than the sum of the bacteria’s death and dilution rates, then the bacteria will persist uniformly; (2) if in addition to this, (a) the grazers’ per capita growth rate is greater than the sum of the dilution rate and grazers’ death rate, and (b) the death rate of bacteria is less than some threshold, then the grazers will persist uniformly. These conditions can be achieved simultaneously if there are sufficient resources in the feed bottle. As opposed to the microcosm decomposition models’ results, in a chemostat culture, chemicals always persist. Besides the transcritical bifurcation observed in microcosm models, our chemostat model exhibits Hopf bifurcation and Rosenzweig’s paradox of enrichment phenomenon. Our sensitivity analysis suggests that the most effective way to facilitate degradation is to decrease the dilution rate.

     

Delayed feedback control for a chemostat model

In this paper, we discuss asymptotic properties and numerical simulations of a chemostat model with delayed feedback control. A chemostat model with two organisms can be made coexistent by feedback control of the dilution rate which depends affinely on the concentrations of two organisms [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48]. Then the coexistence takes its simplest form; the equilibrium point in the non-negative orthant is globally asymptotically stable. We show that stability of the equilibrium point is changed by 'time-delay' caused in controlling the dilution rate after measuring the concentrations of two organisms.     

由微分方程所描述的微生物连续培养动力系统(Ⅰ)

陆续介绍微生物连续培养(Chemostat)的基本原理,以单种微生物连续培养模型为基础,较详细地介绍几类由微分方程所描述的微生物连续培养动力系统模型,涉及的问题有解的稳定性,系统的持久性,周期解和Hopf分支等．     

Physiological, biochemical, and mathematical studies of micro-aerobic continuous ethanol fermentation by Saccharomyces cerevisiae. III: mathematical model of cellular energetics and catabolism

Mathematical models of the catabolic pathways, the utilization and waste of ATP, and the factors affecting yeast growth in a micro-aerobic chemostat are presented. The models incorporate the intracellular metabolite and enzyme activity assays performed in Part II to explain the unusual macroscopic chemostat behaviors reported in Part I. The catabolic model successfully predicts a maximum in the specific ethanol productivity as a function of the intracellular ATP concentration. The ATP balance model enables the prediction of the intracellular ATP concentration and the ATP yield for given dissolved oxygen concentrations. Finally, in the context of a growth model, singularity theory provides a framework to explain the transition observed in Part I between hysteresis and the monotonic biomass versus oxygenation profiles in response to changes in the nutrient composition. The models serve to organize data and to concretely express proposed metabolic mechanisms and cause-effect hypotheses. The model is only applicable to the micro-aerobic and excess glucose conditions encountered in this study.