Periodic response to periodic forcing of the Droop equations for phytoplankton growth

The dynamics of a phytoplankton population growing in a chemostat under a periodic supply of nutrients is investigated with the model proposed by Droop. This model differs from the well-known Monod equations by incorporating nutrient storage by the cells. In spite of its nonlinearity and the time delays introduced by an internal nutrient pool, the model predicts a simple response to a periodic nutrient supply. The population is shown to oscillate with the same frequency as the forcing. To prove the existence of a periodic solution local and global bifurcation results are used. This work establishes a basis on which to evaluate experimental data against the model as a representation of the nutrient-phytoplankton interaction when nutrients fluctuate.     

Dynamic analysis of a microbial process: A systems engineering approach

A general mathematical model of the chemostat system is developed in order to define an experimental program of dynamic testing. A glucose-limited culture ofSaccharomyces cerevisiae was grown in a chemostat using chemically defined medium. The chemostat was perturbed from an initial steady state by changes in input glucose concentration, dilution rate, pH, and temperature. Dynamic responses of cell mass, glucose, cell number, RNA, and protein concentrations were measured. A number of simulation techniques were used in developing a dynamic mathematical model and in comparing the developed model with experimental data as well as the Monod model. The resulting model was found to be quantitatively accurate and superior to the Monod model. The developed model was interpreted in the light of cell physiology. Adjustment of intracellular RNA fraction was found to be rate limiting in acceleration of cell specific growth rate.     

The effect of short-term fluctuations in pH on NO−3 uptake and intracellular constituents in Skeletonemacostatum (Grev.) Cleve

Measurements of uptake rates, intracellular nitrogen pools, and other key intracellular constituents were made during exponential growth in Skeletonema costatum (Grev.) Cleve under varying pH levels. An understanding of the overall effects of extracellular pH on the above mentioned cellular parameters is crucial in order to ascertain the degree to which pH must be regulated and monitored in laboratory experiments with marine phytoplankton.It was found that uptake rates and intracellular pool sizes of NO^?₃ were directly influenced by the extracellular pH level, whereas, other cellular compounds remained relatively unchanged. Therefore, nitrogen uptake and intracellular nitrogen storage are dependent on key H⁺ and OH_? ion transport mechanisms that are associated with phytoplankton metabolism. These findings reiterate the fact that investigators examining nitrogen uptake and assimilatory mechanisms in marine phytoplankton must be conscious of cellular H ⁺ and OH^? fluxes that contribute to intracellular pH regulation and changes in extracellular pH levels, both of which interact to affect phytoplankton metabolic processes.     

Stoichiometry and food-chain dynamics

Traditional models of chemostat systems looking at interactions between predator, prey and nutrients have used only a single currency, such as energy or nitrogen. In reality, growth of autotrophs and heterotrophs may be limited by various elements, e.g. carbon, nitrogen, phosphorous or iron. In this study we develop a dynamic energy budget model chemostat which has both carbon and nitrogen as currencies, and examine how the dual availability of these elements affects the growth of phytoplankton, trophic transfer to zooplankton, and the resulting stability of the chemostat ecosystem. Both species have two reserve pools to obtain a larger metabolic flexibility with respect to changing external environments. Mineral nitrogen and carbon form the base of the food chain, and they are supplied at a constant rate. In addition, the biota in the chemostat recycle nutrients by means of respiration and excretion, and organic detritus is recycled at a fixed rate. We use numerical bifurcation analysis to assess the model's dynamic behavior. In the model, phytoplankton is nitrogen limited, and nitrogen enrichment can lead to oscillations and multiple stable states. Moreover, we found that recycling has a destabilizing effect on the food chain due to the increased repletion of mineral nutrients. We found that both carbon and nitrogen enrichment stimulate zooplankton growth. Therefore, we conclude that the concept of single-element limitation may not be applicable in an ecosystem context.     

Phytoplankton competition along a gradient of dilution rates

Natural phytoplankton from Lake Constance was used for chemostat competition experiments performed at a variety of dilution rates. In the first series at high Si:P ratios and under uniform phosphorus limitation for all species, Synedra acus outcompeted all other species at all dilution rates up to 1.6 d^-1, only at the highest dilution rate (2.0 d^-1) Achnanthes minutissima was successful. In the second series in the absence of any Si a green algal replacement series was found, with Mougeotia thylespora dominant at the lowest dilution rates, Scenedesmus acutus at the intermediate ones, and Chlorella minutissima at the highest ones. The outcome of interspecific competition was not in contradiction with the Monod kinetics of P-limited growth of the five species, but no satisfactorily precise prediction of competitive performance can be derived from the Monod kinetics because of insufficient precision in the estimate of k _s .     

Population dynamics and competition in chemostat models with adaptive nutrient uptake

 The standard Monod model for microbial population dynamics in the chemostat is modified to take into consideration that cells can adapt to the change of nutrient concentration in the chemostat by switching between fast and slow nutrient uptake and growing modes with asymmetric thresholds for transition from one mode to another. This is a generalization of a modified Monod model which considers adaptation by transition between active growing and quiescent cells. Global analysis of the model equations is obtained using the theory of asymptotically autonomous systems. Transient oscillatory population density and hysteresis growth pattern observed experimentally, which do not occur for the standard Monod model, can be explained by such adaptive mechanism of the cells. Competition between two species that can switch between fast and slow nutrient uptake and growing modes is also considered. It is shown that generically there is no coexistence steady state, and only one steady state, corresponding to the survival of at most one species in the chemostat, is a local attractor. Numerical simulations reproduce the qualitative feature of some experimental data which show that the population density of the winning species approaches a positive steady state via transient oscillations while that of the losing species approaches the zero steady state monotonically. Received 4 August 1995; received in revised form 15 December 1995     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator-prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator-prey system, Science 290 (2000), pp. 1358-1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Growth of Chlorella in a Nitrate-limited Chemostat

Chlorella pyrenoidosa was grown in a continuous-flow chemostat under nitrogen-limited conditions. The population density tended to oscillate very significantly. Net specific growth rate was only approximately a hyperbolic function of nitrate concentration in the chemostat. The best estimate of the half-saturation constant for nitrate is 6 mug of nitrogen per liter and it is unlikely that the value is greater than 14 mug per liter or 1 mum nitrate.The dry weight production of cells per unit of nitrogen taken up is a linearly decreasing function of the net specific growth rate with a maximum of 27.1 mg per mg N and a minimum of about 9 mg per mg N. Thus there is considerable storage of nitrogen at high growth rates. Both the dark respiration rate and the rate of photosynthesis at light saturation increase with increasing net specific growth rate.     

Bifurcation structure of a chemostat model for an age-structured predator and its prey

We model a chemostat containing an age-structured predator and its prey using a linear function for the uptake of substrate by the prey and two different functional responses (linear and Monod) for the consumption of prey by the predator. Limit cycles (LCs) caused by the predator's age structure arise at Hopf bifurcations at low values of the chemostat dilution rate for both model cases. In addition, LCs caused by the predator–prey interaction arise for the case with the Monod functional response. At low dilution rates in the Monod case, the age structure causes cycling at lower values of the inflowing resource concentration and conversely prevents cycling at higher values of the inflowing resource concentration. The results shed light on a similar model by Fussmann et al. [G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Crossing the Hopf bifurcation in a live predator–prey system, Science 290 (2000), pp. 1358–1360.], which correctly predicted conditions for the onset of cycling in a chemostat containing an age-structured rotifer population feeding on algal prey.     

Modeling of olive oil degradation and oleic acid inhibition during chemostat and batch cultivation of Bacillus thermoleovorans IHI-91

Olive oil degradation by the thermophilic lipolytic strain Bacillus thermoleovorans IHI-91 in chemostat and batch culture was modeled to obtain a general understanding of the underlying principles and limitations of the process and to quantify its stoichiometry. Chemostat experiments with olive oil as the sole carbon source were successfully described using the Monod chemostat model extended by terms for maintenance requirements and wall growth. Maintenance requirements and biomass yield coefficients were in the range reported for mesophiles. For a chemostat experiment at D = 0.3 h(-1) the model was validated up to an olive oil feed concentration of about 3.0 g L(-1) above which an inhibitory effect occurred. Further analysis showed that the liberated oleic acid is the main cause for this inhibition. Using steady-state oleic acid concentrations measured in chemostat experiments with olive oil as substrate it was possible to derive a kinetic expression for oleic acid utilization, showing that a concentration of 430 mg L(-1) leads to a complete growth inhibition. Oleic acid accumulation observed during batch fermentations can be predicted using a model involving growth-associated lipase production and olive oil hydrolysis. Simulations confirmed that this accumulation is the cause for the sudden growth cessation occurring in batch fermentations with higher olive oil start concentrations. Further, an oscillatory behavior, as observed in some chemostat experiments, can also be predicted using the latter model. This work clearly demonstrates that thermophilic lipid degradation by Bacillus thermoleovorans IHI-91 is limited by long-chain fatty acid beta-oxidation rather than oil hydrolysis.     

Examination of the role of detritus food quality,phytoplankton intracellular storage capacity,and zooplankton stoichiometry on planktonic dynamics

Nutrient stoichiometric ratios are primary driving factors of planktonic food web dynamics. Ecological stoichiometry theory postulates the elemental ratios of consumer species to be homeostatic, while primary-producer stoichiometry may vary with ambient nutrient availability. The notion of phytoplankton intracellular storage is far from novel, but remains largely unexplored in modeling studies of population dynamics. We constructed a seasonally-unforced, zero-dimensional, nutrient–phytoplankton–zooplankton–detritus (NPZD) model that considers dynamic phytoplankton phosphorus reserves and quasi-dynamic zooplankton stoichiometry. A generic food quality term is used to express seston biochemical composition, ingestibility, and digestibility. We examined the sensitivity of the planktonic food web patterns to light and nutrient availability, zooplankton mortality, and detritus food quality as well as to phytoplankton intracellular storage and zooplankton stoichiometry. Our results reinforce earlier findings that high quality seston exerts a stabilizing effect on food web dynamics. However, we also found that the combination of low algal and high detritus food quality with high zooplankton mortality yielded limit cycles and multiple steady states, suggesting that the heterogeneity characterizing seston nutritional quality may have more complicated ecological ramifications. Our numerical experiments identify resource competition strategies related to nutrient transport rates and internal nutrient quotas that may be beneficial for phytoplankton to persevere in resource-limiting habitats. We also highlight the importance of the interplay between optimal stoichiometry and the factors controlling homeostatic rigidity in zooplankton. In particular, our predictions show that the predominance of phosphorus-rich and tightly-homeostatic herbivores in nutrient-enriched environments with low seston food quality can potentially result in high phytoplankton abundance, high phytoplankton-to-zooplankton ratios, and acceleration of oscillatory dynamics. Generally, our modeling study emphasizes the impact of both intracellular/somatic storage and food quality on prey–predator interactions, pinpointing an important aspect of food web dynamics usually neglected by the contemporary modeling studies.     

Study of the dynamic behavior of the chemostat system

This paper deals with a theoretical study on the dynamic, character of the chemostat system. It. is primarily based on the Monod model for growth limitation, although certain more complex models are considered. Since the Monod model is described in terms of two variables, an analysis by use of a phase plane plot will show the various possible types of behavior theoretically expected for transient conditions of the system. In this paper it will be shown that the chemostat system might show an overshoot (or an underswing) with respect to changes in cell and substrate concentrations, depending on the extent to which the system might be disturbed from steady-slate conditions. Other types of transient behavior ran also be expected when one of the system parameters such as dilution rate or input substrate concentration is disturbed in a stepwise manner. The simple Monod chemostat model was found never to oscillate in either a damped or a sustained manner as has been experimentally reported. Discussion is included about the transient behavior of other chemostat models such as that involving a variable yield coefficient, i.e., including the effect of cell maintenance requirements.     

The periodically forced Droop model for phytoplankton growth in a chemostat

 It is proved that the periodically forced Droop model for phytoplankton growth in a chemostat has precisely two dynamic regimes depending on a threshold condition involving the dilution rate. If the dilution rate is such that the sub-threshold condition holds, the phytoplankton population is washed out of the chemostat. If the super-threshold condition holds, then there is a unique periodic solution, having the same period as the forcing, characterized by the presence of the phytoplankton population, to which all solutions approach asymptotically. Furthermore, this result holds for a general class of models with monotone growth rate and monotone uptake rate, the latter possibly depending on the cell quota. Received 10 October 1995; received in revised form 26 March 1996     

Growth kinetics of Pseudomonas putida G7 on naphthalene and occurrence of naphthalene toxicity during nutrient deprivation

The objectives of this work were (1) to demonstrate how the chemostat approach could be modified to allow determination of kinetic parameters for a sparingly soluble, volatile substrate such as naphthalene and (2) to examine the influence of the interactions of various nutrients on possible growth-inhibitory effects of naphthalene. Pseudomonas putida G7 was used as a model naphthalene-degrading microorganism. Naphthalene was found to be toxic to P. putida G7 in the absence of a nitrogen source or oxygen. The death rate of cells grown on minimal medium plus naphthalene and then exposed to naphthalene under anoxic conditions was higher than that observed under oxic conditions in the absence of a nitrogen source. The presence of necessary nutrients for the biodegradation of PAH compounds is indicated to be important for the survival of microorganisms that are capable of PAH degradation. The amounts of ammonia and oxygen necessary for naphthalene biodegradation and for suppression of naphthalene toxicity were calculated from growth yield coefficients. A chemostat culture of P. putida G7 using naphthalene as a carbon and energy source was accomplished by using a feed augmented with a methanol solution of naphthalene so as to provide sufficient growth to allow accurate evaluation of kinetic parameters. When naphthalene was the growth-limiting substrate, the degradation of naphthalene followed Monod kinetics. Maximum specific growth rate (micrometer) and Monod constant (Ks) were 0.627 +/- 0.007 h-1 and 0.234 +/- 0.0185 mg/L, respectively. The evaluation of biodegradation parameters will allow a mathematical model to be applied to predict the long-term behavior of PAH compounds in soil when combined with PAH transport parameters.     

Endogenous metabolism and the stability of microbial prey-predator systems

We formulate a variant of the "double Monod model" which takes explicit account of endogenous metabolism. Using parameter values appropriate to carbohydrate-limited substrate, bacterial prey, and protozoan predator, we study the stability of steady states under chemostat conditions. We conclude that the predator's endogenous metabolism may have a stabilizing effect at low dilution rates.     

Kinetics of phenol oxidation by washed cells

The uptake of phenol by pure cultures of Pseudomonas putida growing on phenol in continuous culture has been studied. The purpose of the experiments was to determine the kinetic parameters governing uptake of phenol by organisms growing on phenol in the high-conversion range by measuring uptake rates per unit biomass per unit time at various phenol concentrations. The microorganisms used were taken from a chemostat at residence times of 8, 5.25, 3.85, 3.2, 3, and 2.7h. The Monod–Haldane model and modifications of it were applied to the data and the best kinetic parameters were determined by nonlinear least-squares techniques. The best model was a two-parameters simplification of Monod–Haldane in which μ = K₁S/(K₂ + S²). The value of K₁ was found to increase monotonically with the value of phenol concentration in the original chemostat with an apparent induction “threshold” of 0.1 mg/L.     

Defined media optimization for growth of recombinant Escherichia coli X90

An optimized, defined minimal medium was developed to support balanced growth of Escherichia coli X90 harboring a recombinant plasmid. Foreign protein expression was repressed in these studies. A pulse injection technique was used to identify the growth responses to nutrients in a chemostat. Once the nutrients essential for growth had been identified, the yield coefficients for individual medium components. These yield coefficients were used to develop an optimized, glucose-limited defined minimal medium that supports balanced cell growth in chemostat culture. The biomass and substrate concentrations follow the Monod chemostat model. The maximum specific growth rate determined in a washout experiment is 0.87 h(-1) for this strain in the optimized medium. the glucose yield factor is 0.42 g DCW/g glucose and the maintenance coefficient is zero in the glucose-limited chemostat culture. (c) 1993 John Wiley & Sons, Inc.