Dynamics of a physiologically structured population in a time-varying environment

Physiologically structured population models have become a valuable tool to model the dynamics of populations. In a stationary environment such models can exhibit equilibrium solutions as well as periodic solutions. However, for many organisms the environment is not stationary, but varies more or less regularly. In order to understand the interaction between an external environmental forcing and the internal dynamics in a population, we examine the response of a physiologically structured population model to a periodic variation in the food resource. We explore the addition of forcing in two cases: (A) where the population dynamics is in equilibrium in a stationary environment, and (B) where the population dynamics exhibits a periodic solution in a stationary environment. When forcing is applied in case A, the solutions are mainly periodic. In case B the forcing signal interacts with the oscillations of the unforced system, and both periodic and irregular (quasi-periodic or chaotic) solutions occur. In both cases the periodic solutions include one and multiple period cycles, and each cycle can have several reproduction pulses.     

Oscillatory behavior of Saccharomyces cerevisiae in continuous culture: I. Effects of pH and nitrogen levels

The appearance of sustained oscillations in bioreactor variables (biomass and nutrient concentrations) in continuous cultures of Saccharomyces cerevisiae indicates the complex nature of microbial systems, the inadequacy of current growth kinetic models, and the difficulties which may arise in bioprocess control and optimization. In this study we investigate continuous bioreactor behavior over a range of operating conditions (dilution rate, feed glucose concentration, feed ammonium concentration, dissolved oxygen, and pH) to determine the process requirements which lead to oscillatory behavior. We present new results which indicate that high feed ammonium concentrations may eliminate oscillations and that under oscillatory conditions ammonium levels are generally low and oscillatory as well. The effects of pH are complex and oscillations were only observed at pH values 5.5 and 6.5; no oscillations were observed at a pH of 4.5. Under our nominal operating conditions (feed glucose concentration 10 g/L, dilution rate 0.145 h(-1), feed ammonium concentration 0.0303M, dissolved oxygen level 50%, pH 5.5, and T = 30 degrees C) we found two possible final bioreactor states depending on the transient used to reach the nominal operating conditions. One of the states was oscillatory and characteristic of oxidative metabolism and the other was nonoscillatory and fermentative.     

Oscillatory behavior of Saccharomyces cerevisiae in continuous culture: II. Analysis of cell synchronization and metabolism

Sustained oscillations of biomass, ethanol, and ammonium concentrations, specific growth rate, and specific uptake rates of ethanol, ammonium, and oxygen were found in continuous cultures of Saccharomyces cerevisiae under controlled dissolved oxygen (DO), pH, and temperature conditions. The period of oscillations was approximately 2.5-3 h at a pH of 5.5 and 2-2.5 h at a pH of 6.5. Oscillations were observed only under conditions of low carbon (glucose below the minimum detectable level), nitrogen nutrient (ammonium concentration varied between 0.00001 and 0.0015M), and ethanol concentration (0.002-0.085 g/L) in the bioreactor.The oscillatory behavior at pH 5.5 was also characterized by partially synchronized cell growth and reproduction. Not only did the total percentage of budding cells oscillate with the same period as observed for the global biomass and nutrient concentrations, but the peaks in the individual subpopulations of initial budding, middle budding, and late budding cells appeared sequentially during the oscillation period. This provides strong evidence of the hypothesis that variations in metabolism during different periods in the cell cycle of a partially synchronized cell population are responsible for the observed oscillatory bioreactor behavior.The specific nutrient uptake rates for ammonium and oxygen as well as the net specific ethanol uptake rate oscillated with the same period as the biomass oscillations. These results show a dramatic increase in the ammonium and oxygen consumption rates prior to the initial budding of the synchronized subpopulation and a decrease in these rates during the late budding phase. At a pH of 5.5, the late budding phase is characterized by high specific ethanol productivity; however, the ethanol productivity lags the late budding phase at a pH pf 6.5. The observed time-varying metabolism in the oscillatory operating regime appears to be the result of the metabolic changes which occur during the cell cycle. Models which can predict the oscillatory biomass concentration and nutrient levels in this regime must be capable of predicting the concentrations and metabolic rates of the subpopulations as well.     

Physiological response ofCandida tropicalis grown on n-paraffin to mixing in a tubular closed loop fermenter

Summary A special tubular closed loop fermenter was used in order to simulate the particular mixing condition of a large scale recycle fermenter.Some mixing parameters of the system are characerized.During continuous cultivation ofCandida tropicalis on n-paraffin as a substrate the biomass yield with respect to carbon and oxygen increased, when a controlled oxygen limit was imposed on the culture.Mixing in the closed loop fermenter generates undamped short period oscillations in the respiration activity, in the dissolved oxygen tension and in the actual ATP content of the culture. These oscillations likely represent oscillations of allosteric feedback loops which manifest themselves by some synchronising action of the particular environmental transients in the closed loop fermenter.     

Transitional steady-state investigations during aerobic-anaerobic transition of glucose utilization by Escherichia coli K-12.

Transitional steady-state investigations during changes in oxygen tension under aerobic and during aerobic-anaerobic transition conditions were carried out with the aim of finding an indicator system which separates the equilibrium from the non-equilibrium state. Of the parameters used i.e. biomass formation, CO2 production, Q02, NADH oxidase, succinate dehydrogenase, phosphofructokinase, glyceraldehyde-3 phosphate dehydrogenase, 6-phosphogluconate dehydrogenase and 2-oxoglutarate dehydrogenase, only the three enzymes requiring NADH or NADP for their function fulfilled the requirements. Biomass production and CO2 formation were useful only during the aerobic-anaerobic transition period. In each case the response was immediate and the indicator systems demonstrated that a new steady state of oxygen was always obtained after 11 h which, at the specific growth rate used, was equivalent to at least two volume replacements of the growth vessel.     

Steady state phytoplankton assemblages during thermal stratification in deep alpine lakes. Do they occur?

Phytoplankton seasonal and long-term succession can be described and functionally classified by associations similar as in terrestrial vegetation studies. Such a concept has to include 'climax' into pelagic succession which in turn leads to the question to what extent steady state assemblages occur and if during periods of dynamic equilibrium can be identified. Here we explore the situation with respect to the above question for deep, nutrient poor, alpine lakes in Austria. We first track the long-term development of phytoplankton biomass, their taxonomic structure and their relation to total phosphorus and chlorophyll-a as predictors of trophic state over the past 35 years. We then analyse this data set for coherent algal associations which can be ascribed to trait separated functional groups according to Reynolds et al. (2002). A three year period of stable environmental conditions has then be extracted from the progression of trophic state indices, having similar dominating species each year. These years were finally analysed for steady state conditions according to definitions given in Sommer et al. (1993). During thermal stratification, achievement of an equilibrium could be ruled out although coexistence of several dominating species lasted for several weeks. Habitat templates were constructed from environmental variables prior to biomass peaks for two species important in summer assemblages, the dinoflagellate Ceratium hirundinella and the diatom Fragilaria crotonensis. In summary, functional groups proved to be a valid and useful concept to describe species succession of phytoplankton in deep alpine lakes while pelagic climax is much less clear and steady state conditions were never met.     

Fishing,reproductive volume and regulation: population dynamics and exploitation of the eastern Baltic cod

The relative importance of exploitation rate and environmental variability in generating fluctuations of harvested populations is a key issue in academic ecology as well as population management. We studied how the eastern Baltic cod (Gadus morhua) is affected by fishing and environmental variation by using a newly developed single species state-space model. Survey data and auxiliary environmental data were used to estimate the model parameters. The model was then used to predict future development of the eastern Baltic cod under different fishing mortalities and abiotic conditions. Abiotic condition was represented by an index: reproductive volume which is the volume of water suitable (in terms of salinity and oxygen content) for the successful development of the early life stages of Baltic cod. The model included direct density dependence, fishing, and a lagged effect of reproductive volume. Our analysis showed that fishing rate is approximately three times more important than reproductive volume in explaining the population dynamics. Furthermore, our model suggests either under- or over-compensatory dynamics depending on the reproductive volume and long term catch levels. It follows that fishing can either reduce or increase temporal oscillations of the cod stock depending on whether the dynamics is over- or undercompensatory, respectively. The sustainable level of fishing rate is however dependent on reproductive volume. Our model predicts a dual role of fishing rate, stabilizing when reproductive volume is high and destabilizing when it is low. Exploitation rate may therefore increase or decrease the risk of the population of cod dropping below a given biomass reference point depending on the environmental conditions, which has practical implications for fisheries management.     

Dancing between the devil and deep blue sea: the stabilizing effect of enemy-free and victimless sinks

Theoretical and empirical studies have shown that enemy–victim interactions in spatially homogenous environments can exhibit diverging oscillations which result in the extinction of one or both species. For enemy–victim models with overlapping generations, we investigate the dynamical implications of spatial heterogeneity created by enemy-free sinks or victimless sinks. An enemy-free sink is a behavioral, physiological or ecological state that reduces or eliminates the victim's vulnerability to the enemy but cannot sustain the victim population. For victims that move in an ideal-free manner, we prove that the inclusion of an enemy-free sink shifts the population dynamics from diverging oscillations to stable oscillations. During these stable oscillations, the victim disperses in an oscillatory manner between the enemy-free sink and the enemy-occupied patch. Enemy-free sinks with lower mortality rates exhibit oscillations with smaller amplitudes and longer periods. A victimless sink, on the other hand, is a behavioral, physiological or ecological state in which the enemy has limited (or no) access to its victims. For enemies that move in an ideal-free manner, we prove that victimless sinks also stabilize diverging oscillations. Simulations suggest that suboptimal behavior due to information gathering or learning limitations amplify oscillations for systems with enemy-free sinks and dampen oscillations for systems with victimless sinks. These results illustrate that the coupling of a sink created by unstable enemy–victim interactions and a sink created by unsuitable environmental conditions can result in population persistence at the landscape level.     

Ontogenetic symmetry and asymmetry in energetics

Body size ( $\equiv $ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.     

Multiple stable states and hysteresis in continuous, oscillating cultures of budding yeast

The conditions that precede the onset of autonomous oscillations in continuous yeast cultures were studied in three different types of experiments. It was found that the final state of the culture depended on the protocol used to start up the reactor. Batch cultures, switched to continuous operation at different stages of the batch growth curve, all exhibited similar dynamics-ethanol depletion followed by autonomous oscillations. Small perturbations of the distribution of states in the reactor, achieved by addition of externally grown cells, were able to quench the oscillatory dynamics. Reaching the desired operating point by slow dilution rate changes gave rise to different final states, two oscillatory states and one steady state, depending on the rate of change in dilution rate. The multiplicity of stable states at a single operating point is not explained by any current distributed model and points toward a segregated mechanism of these oscillations.     

Robust entrainment of circadian oscillators requires specific phase response curves

The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.     

Chaos and the (un)predictability of evolution in a changing environment

Among the factors that may reduce the predictability of evolution, chaos, characterized by a strong dependence on initial conditions, has received much less attention than randomness due to genetic drift or environmental stochasticity. It was recently shown that chaos in phenotypic evolution arises commonly under frequency‐dependent selection caused by competitive interactions mediated by many traits. This result has been used to argue that chaos should often make evolutionary dynamics unpredictable. However, populations also evolve largely in response to external changing environments, and such environmental forcing is likely to influence the outcome of evolution in systems prone to chaos. We investigate how a changing environment causing oscillations of an optimal phenotype interacts with the internal dynamics of an eco‐evolutionary system that would be chaotic in a constant environment. We show that strong environmental forcing can improve the predictability of evolution by reducing the probability of chaos arising, and by dampening the magnitude of chaotic oscillations. In contrast, weak forcing can increase the probability of chaos, but it also causes evolutionary trajectories to track the environment more closely. Overall, our results indicate that, although chaos may occur in evolution, it does not necessarily undermine its predictability.     

Involvement of a cell size control mechanism in the induction and maintenance of oscillations in continuous cultures of budding yeast

Spontaneous oscillations occur in glucose-limited continuous cultures of Saccharomyces cerevisiae under aerobic conditions. The oscillatory behavior is detectable as a periodic change of many bioparameters such as dissolved oxygen, ethanol production, biomass concentration, as well as cellular content of storage carbohydrates and is associated to a marked synchronization of the yeast population. These oscillations may be related to a periodic accumulation of ethanol produced by yeast in the culture medium.The addition of ethanol to oscillating yeast cultures supports this hypothesis: indeed, no effect was observed if ethanol was added when already present in the medium, while a marked phase oscillation shift was obtained when ethanol was added at any other time. Moreover, the addition of ethanol to a nonoscillating culture triggers new oscillations. An accurate analysis performed at the level of nonoscillating yeast populations perturbed by addition of ethanol showed that both the growth rate and the protein content required for cell division increased in the presence of mixed substrate (i.e., ethanol plus limiting glucose). A marked synchronization of the yeast population occurred when the added ethanol was exhausted and the culture resumed growth only on limiting glucose. A decrease of protein content required for cell division was also apparent. These experimental findings support a new model for spontaneous oscillations in yeast cultures in which the alternative growth on limiting glucose and limiting glucose plus ethanol modifies the critical protein content required for cell division.     

Kinetic, dynamic, and pathway studies of glycerol metabolism by Klebsiella pneumoniae in anaerobic continuous culture: IV. Enzymes and fluxes of pyruvate metabolism

The activities of pyruvate kinase (PK), pyruvate: formate-lyase (PFL), pyruvate dehydrogenase (PDH), and citrate synthase (CS) involved in the anaerobic glycerol conversion by Klebsiella pneumoniae were studied in continuous culture under conditions of steady states and sustained oscillations. Both the in vitro and in vivo activities of PK, PFL, and PDH are strongly affected by the substrate concentration and its uptake rate, as is the in vitro activity of CS. The flux from phosphoenolpyruvate to pyruvate is found to be mainly regulated on a genetic level by the synthesis rate of PK, particularly at low substrate concentration and low growth rate. In contrast, the conversion of pyruvate to acetyl-CoA is mainly regulated on a metabolic level by the in vivo activities of PFL and PDH. The ratio of in vitro to in vivo activities is in the range of 1 to 1.5 for PK, 5 to 17 for PFL and 5 to 80 for PDH under the experimental conditions. The regulation of in vivo activity and synthesis of these enzymes is sensitive to fluctuations of culture conditions, leading to oscillations of both the in vitro and in vivo activities. In particular, PFL is strongly affected during oscillations; its average in vitro activity is only about half of its corresponding steady-state value under similar environmental conditions. The average in vitro activities of PDH and PK under oscillations are close to their corresponding steady-state values. In contrast to all other enzymes measured for the glycerol metabolism by K. pneumoniae PFL and PDH are more effectively in vivo utilized under oscillations than under steady state, underlining the peculiar role of pyruvate metabolism in the dynamic responses of the culture.     

Aerobic oxidation of hydrogen sulfide by Thiobacillus denitrificans

It has been demonstrated that Thiobacillus denitrificans may be readily cultured aerobically in batch and continuous flow reactors on H(2)S(g) under sulfide limiting conditions. Under these conditions sulfide concentrations in the culture medium were less than 1muM resulting in very low concentrations of H(2)S in the reactor outlet gas. Biomass yield under aerobic conditions was much lower than previously reported for anaerobic conditions, presumably because of oxygen inhibition of growth. However, biomass yield was not affected by steady state oxygen concentration in the range of 45muM-150muM. Biomass yield was also observed to be essentially independent of specific growth rate in the range of 0.030-0.053 h(-1). Indicators of reactor upset were determined and recovery from upset conditions demonstrated. Maximum loading of the biomass for H(2)S oxidation under aerobic conditions was observed to be 15.1-20.9 mmol/h/g biomass which is much higher than previously reported for aerobic conditions. Other aspects of the stoichiometry of aerobic H(2)S oxidation are also reported.     

A mathematical model of oxygen transport in intact muscle with imposed surface oscillations

A one-dimensional (1D) reaction-diffusion equation is presented to model oxygen delivery by the microcirculation and oxygen diffusion and consumption in intact muscle. This model is motivated by in vivo experiments in which oscillatory boundary conditions are used to study the mechanisms of local blood flow regulation in response to changes in the tissue oxygen environment. An exact periodic solution is presented for the 1D 'in vivo' model and shown to agree with experimental data for the case where the blood flow regulation system is not activated. Approximate low- and high-frequency solutions are presented, and the latter is shown to agree with the pure diffusion solution in the absence of sources or sinks. For the low frequencies considered experimentally, the 1D in vivo model shows that as depth increases: (i) the mean of tissue O(2) oscillations changes exponentially, (ii) the amplitude of oscillations decreases very rapidly, and (iii) the phase of oscillations remains nearly the same as that of the imposed surface oscillations. The 1D in vivo model also shows that the dependence on depth of the mean, amplitude, and phase of tissue O(2) oscillations is nearly the same for all stimulation periods >30s, implying that experimentally varying the forcing period in this range will not change the spatial distribution of the O(2) stimulation.     

Litterfall and forest floor dynamics in Eucalyptus pilularis forests

Abstract Calculations relating the input of litterfall to litter or forest floor mass in forests generally assume that the forest floor reaches an equilibrium state. Based on this assumption, a decomposition factor (k) can be calculated. In the present paper, this basic assumption is questioned and the implications considered. Data on litterfall and forest floor from blackbutt (Eucalyptus pilularis) regrowth forests and plantations were collated from publications and the authors' studies to evaluate both assumptions and relationships. Blackbutt grows over a wide environmental range but its main distribution is in mild temperate to subtropical conditions. Data were from single‐plot studies, sequential studies and chronosequences in both plantations and native regrowth forests. Stands ranged in age from 3 years to maturity in the case of pure, or almost pure blackbutt stands. The forest floor biomass increased up to 12.3 tha^?1 at 33 years of age with no evidence of steady state. Litterfall increased up to 7.8 t ha^?1 year^?1 and was correlated with crown biomass. Regrowth stands were relatively undisturbed and more than 20 years of age, and litterfall ranged from 4.1 to 11.6 tha^?1 year^?1 and was correlated with stand basal area. Forest floor mass in regrowth forests was variable between the different aged stands but did not exceed 18 tha^?1, and there was no evidence that steady state was achieved. The forest floor mass was related to, and approximately 1.7 times the input of litterfall. Although the assumption of steady state was not valid, a k' factor was estimated that related input to forest floor mass and this was relatively constant across all stands and correlated with generalized environmental data. Although assumptions of forest floor equilibrium cannot be supported for E. pilularis, it still should be possible to predict forest floor mass and decomposition from stand conditions and general environmental data.     

Evaluation of growth yield of Spirulina (Arthrospira) sp. in photoautotrophic, heterotrophic and mixotrophic cultures

In microbial cultures, both cellular growth rate and yield (defined as the degree of substrate conversion into the biomass) are important. Although effect of culture conditions on growth kinetics has been well documented for various microbial strains, there is almost no literature concerning the effect of environmental conditions on growth equilibrium, expressed as biomass yield coefficients from substrate. The present paper discusses the effect of culture conditions: irradiance (physical substrate) and glucose concentration (chemical substrate) on biomass yield coefficients from two chemical substrates: glucose and nitrate-nitrogen in photoautotrophic, heterotrophic and mixotrophic culture of blue-green alga Spirulina (Arthrospira) sp. The efficiency of substrates incorporation into the biomass can be precisely determined only if the elemental composition of the biomass is known. The experimental results showed that culture conditions had a substantial influence on biomass yield coefficients (biomass yield from glucose and nitrate-nitrogen). It was found that, the increase of irradiance favoured increase of biomass yield coefficient from both, glucose and nitrate-nitrogen. However, in the case of yield from nitrogen in mixotrophic culture, the effect was opposite. The effect of glucose concentration was different: the higher the initial glucose concentration, the lower the biomass yield coefficients from chemical substrates.     

Stability and change of phytoplankton communities in a highly dynamic environment—the case of large,shallow Lake Balaton (Hungary)

Time series data of key environmental variables (water temperature, global radiation, vertical light attenuation, internal P load) and biomass of four colour classes of photosynthetically active algae were collected during 2003 and 2004 with daily resolution. Using these data, seasonal patterns of phytoplankton were analyzed as a function of the dynamic environment. Abstraction of the environmental state as a point in multi-dimensional space was used to identify habitat templates of bloom-forming groups and derive an indicator of environmental stability/physical disturbance. These templates were synthesized into a simple threshold model that sufficiently simulated development and collapse of various blooms. Blooms were, however, rare events related to specific environments with strong, unidirectional forcing. Tentative quantification of disturbance and compositional stability/community change allowed discriminating disturbance-driven changes and autogenic succession with reasonable success. The two processes were found to be equally important in shaping the composition and biomass of phytoplankton.