Dynamic model of a nitrifying fixed bed column: Simulation of the biomass distribution of Nitrosomonas and Nitrobacter and of transient behaviour of the column

A dynamic model for a fixed bed nitrifying column with recirculation of the liquid and gas phases was developed. Liquid RTD experiments demonstrated that the liquid phase was perfectly mixed inside the column. Hete- rogeneity of biomass distribution on the solid phase (beads) was represented by an N-tanks in series model, and a back-mixing term was set to account for the well-mixed liquid phase throughout the column. In autotrophic conditions, competition for oxygen is the cause of the spatial segregation of the two species. Nitrosomonas is concentrated on beads at the bottom of the bed whereas Nitrobacter is more widely distributed. This is consistent with biomass distribution results reported by Cox et al. [17] in a nitrifying fixed bed column. Nitrification takes place at the bottom of the column, always in oxygen gas-liquid mass transfer limiting conditions. Nevertheless, considering the whole process, nitrification is complete (>98% of NH₃ oxidised) and there is no oxygen limitation (the outlet dissolved oxygen concentration is not limiting). The dynamic behaviour of the column, in conditions set up to avoid biofilm diffusion limitation, was simulated for different NH₃-load variations and oxygen shutdowns. The simulated behaviour of the column can be compared to results reported by Bazin et al. [16]. This confirms that the output transient nitrite peaks are higher when changes in the process conditions produce a rearrangement of biomass distribution in the fixed bed.     

Augmenting Epidemiological Models with Point-Of-Care Diagnostics Data

Although adoption of newer Point-of-Care (POC) diagnostics is increasing, there is a significant challenge using POC diagnostics data to improve epidemiological models. In this work, we propose a method to process zip-code level POC datasets and apply these processed data to calibrate an epidemiological model. We specifically develop a calibration algorithm using simulated annealing and calibrate a parsimonious equation-based model of modified Susceptible-Infected-Recovered (SIR) dynamics. The results show that parsimonious models are remarkably effective in predicting the dynamics observed in the number of infected patients and our calibration algorithm is sufficiently capable of predicting peak loads observed in POC diagnostics data while staying within reasonable and empirical parameter ranges reported in the literature. Additionally, we explore the future use of the calibrated values by testing the correlation between peak load and population density from Census data. Our results show that linearity assumptions for the relationships among various factors can be misleading, therefore further data sources and analysis are needed to identify relationships between additional parameters and existing calibrated ones. Calibration approaches such as ours can determine the values of newly added parameters along with existing ones and enable policy-makers to make better multi-scale decisions.     

内循环颗粒污泥床硝化反应器临界曝气强度的研究

内循环颗粒污泥床硝化反应器是一种新型高效硝化反应器 ,在反应器运行过程中 ,液体循环临界曝气强度和颗粒污泥流化临界曝气强度是两个重要操作参数。建立了升流区表观液速Ulr与曝气强度Ugr之间的关系 ,并测定了有关的模型参数 ,得到了具体的数学表达式 :U_lr=(2.613-0.024 )U^0.871_gr 0.276U^0.871_gr-0.28。根据该模型 ,计算得到的液体循环临界曝气强度为1.017cm/min ,颗粒污泥流化临界曝气强度为 2.662cm/min。实测结果证明 ,求得的两个临界曝气强度具有较高的准确性 ,能够用于指导内循环颗粒污泥床硝化反应器的操作优化.     

Aorta in vivo parameter identification using an axial force constraint

It was shown in a previous study by Stålhand et al. (2004) that both material and residual strain parameters for an artery can be identified noninvasively from an in vivo clinical pressure–diameter measurement. The only constraints placed on the model parameters in this previous study was a set of simple box constraints. More advanced constraints can also be utilized, however. These constraints restrict the model parameters implicitly by demanding the state of the artery to behave in a specified way. It has been observed in vitro that the axial force is nearly invariant to the pressure at the physiological operation point. In this paper, we study the possibility to include this behaviour as a constraint in the parameter optimization. The method is tested on an in vivo obtained pressure–diameter cycle for a 24-year-old human. Presented results show that the constrained parameter identification procedure proposed here can be used to obtain good results, and we believe that it may be applied to account for other observed behaviours as well.     

Characterization of compost biofiltration system degrading dichloromethane

The effects of acclimatization of microbial populations, compound concentration, and media pH on the biodegradation of low concentration dichloromethane emissions in biofiltration systems was evaluated. Greater than 98% removal efficiency was achieved for dichloromethane at superficial velocities from 1 to 1.5 m(3)/m(3). min (reactor residence times of 1 and 0.7 min, respectively) and inlet concentrations of 3 and 50 ppm Although acclimatization of microbial populations to toluene occurred within 2 weeks of operation start-up, initial dichloromethane acclimatization took place over a period of 10 weeks. This period was shortened to 10 days when a laboratory grown consortium of dichloromethane degrading organism, isolated from a previously acclimatized column, was introduced into fresh biofilter media. The mixed culture consisted to 12 members, which together were able to degrade dichloromethane at concentrations up to 500 mg/L. Only one member of the consortium was able to degrade dichloromethane were sustained for more than 4 months in a biofilter column receiving an inlet gas stream with 3 ppm(v) of dichloromethane acidification of the column and resulting decline in performance occurred when a 50-ppm(v) inlet concentration was used. A biofilm model incorporating first order biodegradation kinetics provided a good fit to observed concentration profiles, and may prove to be a useful tool for designing biofiltration systems for low concentration VOC emissions. (c) 1994 John Wiley & Sons, Inc.     

A Single Dynamic Metabolic Model Can Describe mAb Producing CHO Cell Batch and Fed-Batch Cultures on Different Culture Media

CHO cell culture high productivity relies on optimized culture medium management under fed-batch or perfused chemostat strategies enabling high cell densities. In this work, a dynamic metabolic model for CHO cells was further developed, calibrated and challenged using datasets obtained under four different culture conditions, including two batch and two fed-batch cultures comparing two different culture media. The recombinant CHO-DXB11 cell line producing the EG2-hFc monoclonal antibody was studied. Quantification of extracellular substrates and metabolites concentration, viable cell density, monoclonal antibody concentration and intracellular concentration of metabolite intermediates of glycolysis, pentose-phosphate and TCA cycle, as well as of energetic nucleotides, were obtained for model calibration. Results suggest that a single model structure with a single set of kinetic parameter values is efficient at simulating viable cell behavior in all cases under study, estimating the time course of measured and non-measured intracellular and extracellular metabolites. Model simulations also allowed performing dynamic metabolic flux analysis, showing that the culture media and the fed-batch strategies tested had little impact on flux distribution. This work thus paves the way to an in silico platform allowing to assess the performance of different culture media and fed-batch strategies.     

A one dimensional moving bed biofilm reactor model for nitrification of municipal wastewaters

This work presents a one-dimensional model of a moving bed bioreactor (MBBR) process designed for the removal of nitrogen from raw wastewaters. A comprehensive experimental strategy was deployed at a semi-industrial pilot-scale plant fed with a municipal wastewater operated at 10–12 °C, and surface loading rates of 1–2 g filtered COD/m² d and 0.4–0.55 g NH₄-N/m² d. Data were collected on influent/effluent composition, and on measurement of key variables or parameters (biofilm mass and maximal thickness, thickness of the limit liquid layer, maximal nitrification rate, oxygen mass transfer coefficient). Based on time-course variations in these variables, the MBBR model was calibrated at two time-scales and magnitudes of dynamic conditions, i.e., short-term (4 days) calibration under dynamic conditions and long-term (33 days) calibration, and for three types of carriers. A set of parameters suitable for the conditions was proposed, and the calibrated parameter set is able to simulate the time-course change of nitrogen forms in the effluent of the MBBR tanks, under the tested operated conditions. Parameters linked to diffusion had a strong influence on how robustly the model is able to accurately reproduce time-course changes in effluent quality. Then the model was used to optimize the operations of MBBR layout. It was shown that the main optimization track consists of the limitation of the aeration supply without changing the overall performance of the process. Further work would investigate the influence of the hydrodynamic conditions onto the thickness of the limit liquid layer and the “apparent” diffusion coefficient in the biofilm parameters.

     

On optimal selection of summary statistics for approximate Bayesian computation

How best to summarize large and complex datasets is a problem that arises in many areas of science. We approach it from the point of view of seeking data summaries that minimize the average squared error of the posterior distribution for a parameter of interest under approximate Bayesian computation (ABC). In ABC, simulation under the model replaces computation of the likelihood, which is convenient for many complex models. Simulated and observed datasets are usually compared using summary statistics, typically in practice chosen on the basis of the investigator's intuition and established practice in the field. We propose two algorithms for automated choice of efficient data summaries. Firstly, we motivate minimisation of the estimated entropy of the posterior approximation as a heuristic for the selection of summary statistics. Secondly, we propose a two-stage procedure: the minimum-entropy algorithm is used to identify simulated datasets close to that observed, and these are each successively regarded as observed datasets for which the mean root integrated squared error of the ABC posterior approximation is minimized over sets of summary statistics. In a simulation study, we both singly and jointly inferred the scaled mutation and recombination parameters from a population sample of DNA sequences. The computationally-fast minimum entropy algorithm showed a modest improvement over existing methods while our two-stage procedure showed substantial and highly-significant further improvement for both univariate and bivariate inferences. We found that the optimal set of summary statistics was highly dataset specific, suggesting that more generally there may be no globally-optimal choice, which argues for a new selection for each dataset even if the model and target of inference are unchanged.     

Two-zone model for stream and river ecosystems

A mechanistic two-zone model is developed to represent the food web dynamics of stream and river ecosystems by considering the benthic and nonbenthic (or water-column) zones as two separate, but interacting biotopes. Flow processes, solar radiation, and temperature are the dynamic external environmental drivers. State variables are defined to represent the hierarchical levels of detritus, limiting nutrient, vegetation, and invertebrates. The fish trophic level is included as a constant input parameter. Model parameters, constants, and boundary conditions are defined based on watershed as well as channel hydrology, stream geomorphology, and biological activities. Recent advances in ecological science and engineering are used in representing important biogeochemical processes. In particular, the turbulent diffusion, as well as sloughing or detachment, processes are defined based on these recent advancements. The two-zone model was evaluated for a gravel bed prealpine Swiss stream named River Necker with data for the study period of January 1992 through December 1994. The model was able to capture the general trends and magnitudes of the food web state variables. A comprehensive relative sensitivity analysis with five moment-based measures found that approximately 5% of the model parameters were important in predicting benthic vegetation. Results of sensitivity analysis guided the model calibration. Simulated benthic vegetation with the calibrated model, which was obtained by adjusting only four parameters, corresponded with observed data. Hydrology-dependent sloughing and detachment were dominant in determining the response of benthic vegetation and invertebrates. The proposed two-zone food web model is a potentially useful research tool for stream and river ecosystems.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

Modelling shellfish growth with dynamic energy budget models: an application for cockles and mussels in the Oosterschelde (southwest Netherlands)

Dynamic energy budget models for growth of individual cockles (Cerastoderma edule) and mussels (Mytilus edulis) are adjusted and calibrated to the Oosterschelde by formulating and parametrizing their functional responses using an extensive set of field observations. The resulting model predictions fit the observations satisfactorily. Results indicate that food quality and the importance of detritus as a food source are site-specific as well as species-specific. Despite these differences in their calibrated parameter values, both species show a very similar functional response. Compared with other systems, however, the functional responses of mussels in the present study are clearly higher than those of mussels in other systems. This may be explained by the absence of intra-specific competition in the measurement set-up that was used, and therefore supports the idea that the generally small functional response of M. edulis is caused by intra-specific competition.     

Bioaugmentation as a tool for improving the start-up and stability of a pilot-scale partial nitrification biofilm airlift reactor

The effectiveness of bioaugmentation in the improvement of the start-up of a biofilm airlift reactor to perform partial nitrification was investigated. Two identical biofilm airlift reactors were inoculated. The non-bioaugmented reactor (NB-reactor) was inoculated with conventional activated sludge, whereas the bioaugmented reactor (B-reactor) was seeded with the same conventional activated sludge but bioaugmented with nitrifying activated sludge from a pilot plant performing full nitritation under stable conditions (100% oxidation of influent ammonium to nitrite). The fraction of specialized nitrifying activated sludge in the inoculum of the B-reactor was only 6% (measured as dry matter). To simplify comparison of the results, operational parameters were equivalent for both reactors. Partial nitrification was achieved significantly faster in the B-reactor, showing a very stable operation. The results obtained by fluorescence in situ hybridization assays showed that the specialized nitrifying biomass added to the B-reactor remained in the biofilm throughout the start-up period.     

Modified CelliGen-packed bed bioreactors for hybridoma cell cultures

This study describes two packed bed bioreactor configurations which were used to culture a mouse-mouse hybridoma cell line (ATCC HB-57) which produces an IgG₁ monoclonal antibody. The first configuration consists of a packed column which is continuously perfused by recirculating oxygenated media through the column. In the second configuration, the packed bed is contained within a stationary basket which is suspended in the vessel of a CelliGen^™ bioreactor. In this configuration, recirculation of the oxygenated media is provided by the CelliGen Cell Lift impeller. Both configurations are packed with disk carriers made from a non-woven polyester fabric. During the steady-state phase of continuous operation, a cell density of 10⁸ cells per cm³ of bed volume was obtained in both bioreactor configurations. The high levels of productivity (0.5 gram MAb per 1 of packed bed per day) obtained in these systems demonstrates that the culture conditions achieved in these packed bed bioreactors are excellent for the continuous propagation of hybridomas using media which contains low levels (1 %) of serum as well as serum-free media. These packed bed bioreactors allow good control of pH, dissolved oxygen and temperature. The media flows evenly over the cells and produces very low shear forces. These systems are easy to set up and operate for prolonged periods of time. The potential for scale-up using Fibra-cel carriers is enhanced due to the low pressure drop and low mass transfer resistance, which creates high void fraction approaching 90% in the packed bed.     

The derivation of sink functions of wheat organs using the GREENLAB model

BACKGROUND AND AIMS: In traditional crop growth models assimilate production and partitioning are described with empirical equations. In the GREENLAB functional-structural model, however, allocation of carbon to different kinds of organs depends on the number and relative sink strengths of growing organs present in the crop architecture. The aim of this study is to generate sink functions of wheat (Triticum aestivum) organs by calibrating the GREENLAB model using a dedicated data set, consisting of time series on the mass of individual organs (the 'target data'). METHODS: An experiment was conducted on spring wheat (Triticum aestivum, 'Minaret'), in a growth chamber from, 2004 to, 2005. Four harvests were made of six plants each to determine the size and mass of individual organs, including the root system, leaf blades, sheaths, internodes and ears of the main stem and different tillers. Leaf status (appearance, expansion, maturity and death) of these 24 plants was recorded. With the structures and mass of organs of four individual sample plants, the GREENLAB model was calibrated using a non-linear least-square-root fitting method, the aim of which was to minimize the difference in mass of the organs between measured data and model output, and to provide the parameter values of the model (the sink strengths of organs of each type, age and tiller order, and two empirical parameters linked to biomass production). KEY RESULTS AND CONCLUSIONS: The masses of all measured organs from one plant from each harvest were fitted simultaneously. With estimated parameters for sink and source functions, the model predicted the mass and size of individual organs at each position of the wheat structure in a mechanistic way. In addition, there was close agreement between experimentally observed and simulated values of leaf area index.     

Parameter stability of the functional-structural plant model GREENLAB as affected by variation within populations, among seasons and among growth stages

BACKGROUND AND AIMS: It is increasingly accepted that crop models, if they are to simulate genotype-specific behaviour accurately, should simulate the morphogenetic process generating plant architecture. A functional-structural plant model, GREENLAB, was previously presented and validated for maize. The model is based on a recursive mathematical process, with parameters whose values cannot be measured directly and need to be optimized statistically. This study aims at evaluating the stability of GREENLAB parameters in response to three types of phenotype variability: (1) among individuals from a common population; (2) among populations subjected to different environments (seasons); and (3) among different development stages of the same plants. METHODS: Five field experiments were conducted in the course of 4 years on irrigated fields near Beijing, China. Detailed observations were conducted throughout the seasons on the dimensions and fresh biomass of all above-ground plant organs for each metamer. Growth stage-specific target files were assembled from the data for GREENLAB parameter optimization. Optimization was conducted for specific developmental stages or the entire growth cycle, for individual plants (replicates), and for different seasons. Parameter stability was evaluated by comparing their CV with that of phenotype observation for the different sources of variability. A reduced data set was developed for easier model parameterization using one season, and validated for the four other seasons. KEY RESULTS AND CONCLUSIONS: The analysis of parameter stability among plants sharing the same environment and among populations grown in different environments indicated that the model explains some of the inter-seasonal variability of phenotype (parameters varied less than the phenotype itself), but not inter-plant variability (parameter and phenotype variability were similar). Parameter variability among developmental stages was small, indicating that parameter values were largely development-stage independent. The authors suggest that the high level of parameter stability observed in GREENLAB can be used to conduct comparisons among genotypes and, ultimately, genetic analyses.     

A comparative evaluation of models of cinnabar moth dynamics

Summary 1. A complex model of cinnabar moth dynamics proposed by Dempster and Lakhani (1979) with 23 parameters is reduced to a single equation with five parameters, and the behaviour of the reduced model shown to explain most features of the full model. 2. The efficiency of the full model is compared with the reduced model and with two even simpler models (the two parameter discrete logistic and a four parameter model based on a step-function for mortality) in their abilities to describe time series data of cinnabar moth population densities from Weeting Heath. Models with more parameters were not significantly better than few-parameter models in describing population trajectories. 3. Models that included a driving variable (in this case observed rainfall data) were no better at describing the data than simpler models without driving variables. It appears, therefore, that the routine inclusion of driving variables may be counterproductive, unless there is compelling empirical or theoretical evidence of their importance and the mode of action of the driving variables can be modelled mechanistically. For example, the regression model used to describe the relationship between rainfall and plant biomass in Dempster and Lakhani (1979), breaks down if rainfall is assumed to be constant, because there is no explicit model for the regulation of plant biomass. 4. The parameter values of the cinnabar-ragwort interaction suggest that cinnabar moth dynamics may be chaotic. Whether or not field data exhibit chaos or environmental stochasticity (or a mixture of both) is impossible to determine from inspection of time series data on population density. There is an urgent need for experimental and theoretical protocols to disentangle these two sources of population fluctuation.     

Model parameterization to represent processes at unresolved scales and changing properties of evolving systems

Modeling has become an indispensable tool for scientific research. However, models generate great uncertainty when they are used to predict or forecast ecosystem responses to global change. This uncertainty is partly due to parameterization, which is an essential procedure for model specification via defining parameter values for a model. The classic doctrine of parameterization is that a parameter is constant. However, it is commonly known from modeling practice that a model that is well calibrated for its parameters at one site may not simulate well at another site unless its parameters are tuned again. This common practice implies that parameter values have to vary with sites. Indeed, parameter values that are estimated using a statistically rigorous approach, that is, data assimilation, vary with time, space, and treatments in global change experiments. This paper illustrates that varying parameters is to account for both processes at unresolved scales and changing properties of evolving systems. A model, no matter how complex it is, could not represent all the processes of one system at resolved scales. Interactions of processes at unresolved scales with those at resolved scales should be reflected in model parameters. Meanwhile, it is pervasively observed that properties of ecosystems change over time, space, and environmental conditions. Parameters, which represent properties of a system under study, should change as well. Tuning has been practiced for many decades to change parameter values. Yet this activity, unfortunately, did not contribute to our knowledge on model parameterization at all. Data assimilation makes it possible to rigorously estimate parameter values and, consequently, offers an approach to understand which, how, how much, and why parameters vary. To fully understand those issues, extensive research is required. Nonetheless, it is clear that changes in parameter values lead to different model predictions even if the model structure is the same.     

Microbial models with data‐driven parameters predict stronger soil carbon responses to climate change

Long‐term carbon (C) cycle feedbacks to climate depend on the future dynamics of soil organic carbon (SOC). Current models show low predictive accuracy at simulating contemporary SOC pools, which can be improved through parameter estimation. However, major uncertainty remains in global soil responses to climate change, particularly uncertainty in how the activity of soil microbial communities will respond. To date, the role of microbes in SOC dynamics has been implicitly described by decay rate constants in most conventional global carbon cycle models. Explicitly including microbial biomass dynamics into C cycle model formulations has shown potential to improve model predictive performance when assessed against global SOC databases. This study aimed to data‐constrained parameters of two soil microbial models, evaluate the improvements in performance of those calibrated models in predicting contemporary carbon stocks, and compare the SOC responses to climate change and their uncertainties between microbial and conventional models. Microbial models with calibrated parameters explained 51% of variability in the observed total SOC, whereas a calibrated conventional model explained 41%. The microbial models, when forced with climate and soil carbon input predictions from the 5th Coupled Model Intercomparison Project (CMIP5), produced stronger soil C responses to 95 years of climate change than any of the 11 CMIP5 models. The calibrated microbial models predicted between 8% (2‐pool model) and 11% (4‐pool model) soil C losses compared with CMIP5 model projections which ranged from a 7% loss to a 22.6% gain. Lastly, we observed unrealistic oscillatory SOC dynamics in the 2‐pool microbial model. The 4‐pool model also produced oscillations, but they were less prominent and could be avoided, depending on the parameter values.     

Does the Structure–Function Model GREENLAB Deal with Crop Phenotypic Plasticity Induced by Plant Spacing? A Case Study on Tomato

BACKGROUND AND AIMS: Plant growth models able to simulate phenotypic plasticity are increasingly required because (1) they should enable better predictions of the observed variations in crop production, yield and quality, and (2) their parameters are expected to have a more robust genetic basis, with possible implications for selection of quantitative traits such as growth- and allocation-related processes. The structure-function plant model, GREENLAB, simulates resource-dependent plasticity of plant architecture. Evidence for its generality has been previously reported, but always for plants grown in a limited range of environments. This paper aims to test the model concept to its limits by using plant spacing as a means to generate a gradient of competition for light, and by using a new crop species, tomato, known to exhibit a strong photomorphogenetic response. METHODS: A greenhouse experiment was carried out with three homogeneous planting densities (plant spacing = 0.3, 0.6 and 1 m). Detailed records of plant development, plant architecture and organ growth were made throughout the growing period. Model calibration was performed for each situation using a statistical optimization procedure (multi-fitting). KEY RESULTS AND CONCLUSIONS: Obvious limitations of the present version of the model appeared to account fully for the plant plasticity induced by inter-plant competition for light. A lack of stability was identified for some model parameters at very high planting density. In particular, those parameters characterizing organ sink strengths and governing light interception proved to be environment-dependent. Remarkably, however, responses of the parameter values concerned were consistent with actual growth measurements and with previously reported results. Furthermore, modifications of total biomass production and of allocation patterns induced by the planting-density treatments were accurately simulated using the sets of optimized parameters. These results demonstrate that the overall model structure is potentially able to reproduce the observed plant plasticity and suggest that sound biologically based adaptations could overcome the present model limitations. Potential options for model improvement are proposed, and the possibility of using the kernel algorithm currently available as a fitting tool to build up more sophisticated model versions is advocated.     

Studying and approximating spatio-temporal models for epidemic spread and control

A class of simple spatio-temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice-based susceptible-infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short-range interaction representing the secondary infection of susceptibles by infectives within the population. Recent data-modelling studies have suggested that the above model may describe the spread of aphid-borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using Cellular-Automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean-field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long-run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.