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     

Global dynamics of a class of complex valued neural networks

In this paper activation dynamics of a complex valued neural network has been studied. Sufficient conditions for global exponential stability of a unique equilibrium are obtained. Our results show that in the serial mode of operation, the network converges to a stable state.     

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.     

The general utility of a glycerophosphate-Tris buffered medium

An easily prepared medium, originally designed for the cultivation of lactic phages, has been found to have much wider application. Experience in its use over a ten year period with a range of physiologically diverse bacteria, for teaching and research are summarized and evaluated.     

The general utility of a glycerophosphate-Tris buffered medium

D ouglas , J., D hillon , J. & S mith , S.E. 1984. The general utility of a glycerophosphate-Tris buffered medium. Journal of Applied Bacteriology 56 , 321–326.
An easily prepared medium, originally designed for the cultivation of lactic phages, has been found to have much wider application. Experience in its use over a ten year period with a range of physiologically diverse bacteria, for teaching and research, are summarized and evaluated.     

Global stability in a chemostat with multiple nutrients

We study a single species in a chemostat, limited by two nutrients, and separate nutrient uptake from growth. For a broad class of uptake and growth functions it is proved that a nontrivial equilibrium may exist. Moreover, if it exists it is unique and globally stable, generalizing a result in [15]. supported in part by NSF grant DMS 0342153. supported in part by NSF grant DEB-0083566 and the Andrew W. Mellon Foundation. supported in part by USAF grant F49620-01-1-0063 and NSF grant CCR-0206789. Part of this work was carried out when P. De Leenheer was a post-doctoral fellow at DIMACS, Rutgers University, supported in part by NSF grant EIA03-331486 and USAF grant F49620-01-1-0063.     

Chaotic dynamics of a food web in a chemostat

We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.     

A method for the reconstruction of unknown non-monotonic growth functions in the chemostat

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be performed either continuously or step-wise. We present and compare both approaches. Even in the case of two species in competition, which is not directly accessible with our approach due to lack of controllability, feedback control improves identifiability of the non-dominant growth rate.     

Stochastic dynamics of the delayed chemostat with Levy noises

This paper formulates and studies a delayed ehemostat with Levy noises.Existence of the glohally positive solution is proved first by establishing suitable Lyapunov functions,and a further result on exact Lyapunov exponent shows the growth of the total concentration in the ehemostat.Then,we prove existence of the uniquely ergodic stationary distribution for a subsystem of the nutrient,based on this,a unique threshold is identified,which completely determines persistence or not of the microorganism in the ehemostat.Besides,recurrence is studied under special conditions in case that the microorganism persists.Results indicate that all the noises have negative effects on persistence of the microorganism,and the time delay has almost no effects on the sample Lyapunov exponent and the threshold value of the ehemostat.     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat

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.Research supported by the National Council of Science, Republic of ChinaResearch supported by National Science Foundation Grant, DMS-9204490Research supported by the Natural Science and Engineering Council of Canada. This author's contribution was made while on research leave visiting the Department of Ecology and Evolutionary Biology at Princeton University. She would especially like to thank Simon Levin for his guidance as well as for providing an exceptional working environment     

Population dynamics of a recombinant culture in a chemostat under prolonged cultivation

The dynamics of a chemostat culture of Escherichia coli K12 harboring plasmid pBR322 under prolonged cultivation with a nonselective complex medium were studied. The ability of the culture to form colonies on plates supplemented with different ampicillin concentrations was monitored. It was observed that almost all cells sampled were able to grow on a high concentration of ampicillin at the beginning of the experiment. However, a subpopulation which formed colonies on intermediate-concentration (500-1000 mg/L) plates, but not on a high-concentration (2000 mg/L) plate, was detected just before the appearance of the plasmid-free cells. As time progressed, the percentage of this subpopulation increased, reached a maximum, then decreased toward the end of experiment. At this time the culture was dominated by a subpopulation which could not form colonies on the 100 mg/L ampicillin plates. These results indicate that three major processes may occur in the chemostat: a gradual shift of the higher plasmid copy number population toward a relatively lower copy number population; the complete shedding of the plasmid due to faulty segregation of plasmids during cell division; and growth competition among the subpopulations. A previously derived model is extended to account for all subpopulations. The model agrees qualitatively with the experimental results.     

The asymptotic behavior of a chemostat model with the Beddington-DeAngelis functional response

In this paper, the global asymptotic behavior of a chemostat model with Beddington-DeAngelis functional response is studied. The conditions for the global asymptotical stability of the model with time delays are obtained via monotone dynamical systems. Our results demonstrate that those time delays affect the competitive outcome of the organisms.     

Application of a noninhibitory growth model to predict the transient response in a chemostat

A method of adapting a kinetic model based on steady-state chemostat data to predict the transient performance of a chemostat culture is presented. The proposal provides for a time delay which can be considered equivalent to a period of reduced activity of the organism subsequent to the introduction of a step change in operating conditions. The adapted kinetic model gives substantially better performance in predicting the transient response of an experimental system than the unmodified kinetic model.     

Dynamics of chemostat culture:the effect of a delay in cell response

The effect of introducing a delay between a change in concentration of limiting nutrient and the response in division rate of the population is investigated. It is shown that this might or might not introduce a delay in the control of the chemostat system, depending upon how the uptake rate is assumed to double during the cell cycle. An area in the parameter space where autonomous oscillations will occur for one specific model is found analytically. This result is confirmed by computer simulations.     

Mitophagy, mitochondrial dynamics and the general stress response in yeast

Autophagy is a fundamental cellular process promoting survival under various environmental stress conditions. Selective types of autophagy have gained much interest recently as they are involved in specific quality control mechanisms removing, for example, aggregated proteins or dysfunctional mitochondria. This is considered to counteract the development of a number of neurodegenerative disorders and aging. Here we review the role of mitophagy and mitochondrial dynamics in ensuring quality control of mitochondria. In particular, we provide possible explanations why mitophagy in yeast, in contrast with the situation in mammals, was found to be independent of mitochondrial fission. We further discuss recent findings linking these processes to nutrient sensing pathways and the general stress response in yeast. In particular, we propose a model for how the stress response protein Whi2 and the Ras/PKA (protein kinase A) signalling pathway are possibly linked and thereby regulate mitophagy.     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat with inhibitions

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.Research Supported by the National Council of Science, Republic of China     

Global dynamics of a ratio-dependent predator-prey system

Recently, ratio-dependent predator-prey systems have been regarded by some researchers to be more appropriate for predator-prey interactions where predation involves serious searching processes. However, such models have set up a challenging issue regarding their dynamics near the origin since these models are not well-defined there. In this paper, the qualitative behavior of a class of ratio-dependent predator-prey system at the origin in the interior of the first quadrant is studied. It is shown that the origin is indeed a critical point of higher order. There can exist numerous kinds of topological structures in a neighborhood of the origin including the parabolic orbits, the elliptic orbits, the hyperbolic orbits, and any combination of them. These structures have important implications for the global behavior of the model. Global qualitative analysis of the model depending on all parameters is carried out, and conditions of existence and non-existence of limit cycles for the model are given. Computer simulations are presented to illustrate the conclusions.     

Transient response of Enterobacter aerogenes under a dual nutrient limitation in a chemostat.

Utilizing a chemostat with a dual nutrient limitation of nitrogen and phosphate, we examined the transient response of the culture following a pulse of one of the limiting nutrients (ammonia). This method provided quantitative evidence that cells can be grown under dual nutrient limitation. Furthermore, the pattern of response was consistent with the hypothesis that phosphate limitation restricts nucleic acid synthesis in the cell and that nitrogen limitation restricts protein synthesis. The net result is that under a phosphate limitation there is a restricted biosynthetic capacity which we feel is closely associated with the RNA content of the cell.     

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.