Algal Growth Model Research Articles

Asymptotic and Transient Dynamics of a Stoichiometric Algal Growth Model with Two-Stage Phosphate Uptake

Asymptotic and Transient Dynamics of a Stoichiometric Algal Growth Model with Two-Stage Phosphate Uptake

A high throughput array microhabitat platform reveals how light and nitrogen colimit the growth of algal cells

A mechanistic understanding of algal growth is essential for maintaining a sustainable environment in an era of climate change and population expansion. It is known that algal growth is tightly controlled by complex interactive physical and chemical conditions. Many mathematical models have been proposed to describe the relation of algal growth and environmental parameters, but experimental verification has been difficult due to the lack of tools to measure cell growth under precise physical and chemical conditions. As such, current models depend on the specific testing systems, and the fitted growth kinetic constants vary widely for the same organisms in the existing literature. Here, we present a microfluidic platform where both light intensity and nutrient gradients can be well controlled for algal cell growth studies. In particular, light shading is avoided, a common problem in macroscale assays. Our results revealed that light and nitrogen colimit the growth of algal cells, with each contributing a Monod growth kinetic term in a multiplicative model. We argue that the microfluidic platform can lead towards a general culture system independent algal growth model with systematic screening of many environmental parameters. Our work advances technology for algal cell growth studies and provides essential information for future bioreactor designs and ecological predictions.

Understanding Nutrient Loads from Catchment and Eutrophication in a Salt Lagoon: The Mar Menor Case

Eutrophication is a significant threat to aquatic ecosystems worldwide, and the Mar Menor hypersaline lagoon exemplifies a coastal lagoon at risk of algal blooms due to excessive nutrient loads, nitrogen, and phosphorus. These nutrients originate from various sources within the lagoon’s catchment area, including urban, agricultural, and livestock activities. Regular and occasional loads—during flood events—produce algal blooms that can significantly reduce the water oxygen content and cause massive mortalities, as observed in recent years. A daily algal growth model (Mmag) was developed to better understand the processes and determine key elements such as the phosphorus water–sediment interchange and deep water plants that effect the entire ecosystem and algal growth. The analysis developed can be applied in other wetlands around the world facing similar challenges. In the Mar Menor, both nitrogen and phosphorus have high relevance depending on the period of the year and the phosphorus legacy in the sediments. Floods are the main phosphorus input to the lagoon (80–90%), which goes to the sediment and is released after during the warm months in the following years. At the end of summer, phosphorus released from the sediment and the regular nitrate inputs to the lagoon increase the algal bloom risk. The good status of deep water plants, which reduces the phosphorus release in summer, is a key element to reduce the algal bloom risk. An integrated set of measures is required to reduce the mean chlorophyll to levels under 1 or 0.5 µgChla/L that can make the Mar Menor more robust and resilient.

Algae in a Drinking Water Reservoir: Development of an Air–Water–Algal Growth Model (AWAM) for Long-Term Prediction

The condition of the water in drinking water source environments is critical for public health. However, in recent years, the rapid growth of algae has become a significant threat to the safety of the drinking water supply. This study investigated the temporal trends and spatial differences in algae in the Nanwan Reservoir during 2022. Regression analysis using the least-squares method demonstrated that water temperature and initial biomass concentration were critical parameters that influenced the rate of algal growth. An air–water–algal growth model (AWAM) for algal growth prediction was developed using a 30-day forecast of air temperature, an air-to-water temperature extrapolating equation, a water temperature–algal growth relationship, and only four monthly measurements of algal concentration. The results demonstrated that the model accurately predicted algal growth in the next 30 days, with an R2 of 0.738, which aligned with the monitored results. Compared to the upstream Wudaohe River inflow point, algal growth in the drinking water intake area near the downstream dam was delayed by at least 30 days. By using the upstream inflow area as a reference point, the prediction period was extended to provide a 60-day early warning. The extended prediction period and the reduced need for monitoring data make the model more convenient for guiding the prevention and control of algal blooms in drinking water reservoirs.

Fluctuation of growth and photosynthetic characteristics in Prorocentrum shikokuense under phosphorus limitation: Evidence from field and laboratory

Exploring the complex interaction between algal growth and their photo-physiology is of significant interest as it can further the understanding of harmful algal blooms. To this end, we investigate variations in cell abundance and chlorophyll fluorescence parameters during the algal bloom formation of Prorocentrum shikokuense in the field and batch culture via the pulse amplitude modulated fluorometry. Furthermore, based on the interaction between algal growth and its photo-physiology status, we develop a novel algal growth model incorporating cell growth delay. The model parameters are estimated by fitting the experimental data of P. shikokuense and validated by the experimental data of Symbiodinium sp. The experimental results show that the growth status and photosynthetic parameters of algal cells fluctuate in both field and laboratory experiments and that the photosynthetic parameters have a faster response than growth parameters. According to the experimental and mathematical results, the time delay between the slow growth of algae and the rapid change of photosynthetic parameters may be a physiological mechanism leading to the fluctuations in algal growth. These results are significant for studying the relationship between phytoplankton growth dynamics and photosynthetic parameters and will help resource managers to predict and deepen the understanding of harmful algal blooms.

Integrated Model for Design, Optimization, Scale-Up, Culture Scheme, and Production Evaluation of a Bioreactor and Its Application for Improving Algal Biomass and Carbon Fixation in Bioreactors

A comprehensive numerical model system for the bioreactor design, structure and cultivation condition optimization, scale-up, and carbon fixation performance evaluation was proposed. Three basic models of computational fluid dynamics, particle and mass transfer, and two improved models of algae growth and light accumulation were integrated innovatively. The algae growth model directly predicted the algal production performance of the bioreactor, and the error was less than 15% compared with the experimental results. The light accumulation model was proposed to simulate the spatial distribution of light utilization with the changes of algal concentration and light intensity. The model system was applied to scale up the flat plate photobioreactor with inclined baffles, and the results showed that the photobioreactor with 9 cm light path had a maximum ideal areal productivity (505.94 g/m2) and better performance. Based on the light accumulation model, the strategy of changing the surface light intensity during the cultivation was proposed to maintain the light utilization and increase the production. Under the optimized surface light intensity, the maximum algal concentration and the average carbon fixation rate of the bioreactor were 43.41 and 248.35% higher than those of the constant light intensity. The proposed model system is a reliable and efficient experimental alternative tool for algal production evaluation and carbon fixation application of the bioreactor.

An echo state network based adaptive dynamic programming approach for time-varying parameters optimization with application in algal bloom prediction

The prediction of algal bloom is one of the important links in eutrophication prevention. Chlorophyll a concentration is the indicating variable of algal bloom, and its time series is non-stationary and non-linear, which brings challenges to its effective prediction. Although the current algae growth model (AGM) can directly describe the algal bloom dynamics, the fixed parameters limit the adaptability of the model. If the fixed parameters are dynamically adjusted, the trend of chlorophyll a concentration can be better captured. Therefore, the adaptive dynamic programming (ADP) approach is used to optimize the parameters of the AGM. The ADP contains an action network and a critic network by echo state network, where the action network is used to output the increment value of the fixed parameters, and the critic network is used to approximate the performance index function. In this paper, the input of the action network uses the time series features extracted by the relevant variables, so that the time-varying parameters of the AGM have better dynamic characteristics. We verify the effectiveness of the proposed model through the dataset of the North Canal and Taihu Lake, and the convergence analysis proves the theoretical reliability. In this way, the improved mechanism model with time-varying parameters not only maintains the better interpretability of the original AGM, but also further enhances the prediction accuracy and adaptability by extracting inherent interactive features from the relevant variables.

A model of algal growth depending on nutrients and inorganic carbon in a poorly mixed water column

A reaction-diffusion-advection model is proposed to describe the growth of algae depending on both nutrients and inorganic carbon in a poorly mixed water column. Nutrients from the water bottom and inorganic carbon from the water surface form an asymmetric resource supply mechanism for the algal growth. The existence and stability of semi-trivial steady state and positive steady state of the model are proved, and a threshold condition for the regime shift from extinction to survival of algae is established. The influence of environmental parameters on the vertical distribution of algae is investigated in the water column. It is shown that the vertical distribution of algae can exhibit many different profiles under the combined limitation of nutrients and inorganic carbon.

Prediction of maximum algal productivity in membrane bioreactors with a light-dependent growth model

Algal productivity in steady-state cultivation systems depends on important factors such as biomass concentration, solids retention time (SRT), and light intensity. Current modeling of algal growth often ignores light distribution in algal cultivation systems and does not consider all these factors simultaneously. We developed a new algal growth model using a first principles approach to incorporate the effect of light intensity on algal growth while simultaneously considering biomass concentration and SRT. We first measured light attenuation (decay) with depth in an indoor algal membrane bioreactor (A-MBR) cultivating Chlorella sp. We then simulated the light decay using a multi-layer approach and correlated the decay with biomass concentration and SRT in model development. The model was calibrated by delineating specific light absorptivity and half-saturation constant to match the algal biomass concentration in the A-MBR operated at a target SRT. We finally applied the model to predict the maximum algal productivity in both indoor and outdoor A-MBRs. The predicted maximum algal productivities in indoor and outdoor A-MBRs were 6.7 g·m−2·d−1 (incident light intensity 5732 lx, SRT approximately 8 d) and 28 g·m−2·d−1 (sunlight intensity 28,660 lx, SRT approximately 4 d), respectively. The model can be extended to include other factors (e.g., water temperature and carbon dioxide bubbling) and such a modeling framework can be applied to full-scale, continuous flow outdoor systems to improve algal productivity.

A dynamic thermal algal growth model for pilot-scale open-channel raceways

This study describes the makeup of and results from a simulation consisting of a lumped thermal model integrated with a dynamic algae growth model to simulate the microalgae productivity of an open-channel raceway cultivation system. The thermal model considers the dynamic effects of weather, light absorption, convective heat transfer, radiation heat transfer, conductive heat transfer, thermal capacitance, and water control strategies. The dynamic algae growth model solves a set of ordinary differential equations consisting of growth functions dependent on incident radiation, temperature, nutrient availability, basal metabolism, and losses due to dark- and photo-respiration. Relative errors in the predicted ash-free dry weight of Nannochloropsis oceanica are 12.5%, −6.1%, and 4.4% for three separate replicates of cultivation cycles (~4.2 m2) performed at AzCATI during the ATP3 Unified Field Studies in Fall, Spring, and Summer. This research demonstrates that thermal modeling is an essential contributor to the validation of microalgae growth models.

An Action Dependent Heuristic Dynamic Programming Approach for Algal Bloom Prediction With Time-Varying Parameters

Algal bloom is a nonlinear and time-varying process, which brings challenges for the accurate prediction. For the existing mechanism model of algae ignores the external key factors, we propose an algae growth model (AGM) optimized by action dependent heuristic dynamic programming (ADHDP). This model has the structure of information interaction with the outside, which can predict algal bloom with well adaptive ability. In this paper, chlorophyll-a concentration is used as the representative factor of algal bloom. We use ADHDP approach to map the external key factors to the time-varying parameters, so the AGM can be adjusted to realize the self-adaptive prediction with the changes in external environments. Compared with different prediction methods, the simulation result shows that the ADHDP-AGM prediction model can accurately predict the chlorophyll-a concentration under different data distributions. Moreover, the prediction process shows that the time-varying parameters in AGM conform to the evolution trend of chlorophyll-a concentration in fact, which further improves the interpretability of prediction model. It provides a new perspective for building a data-driven prediction model with clear physical significance, and makes the mechanism research and data science further fusion.

Threshold behavior in a stochastic algal growth model with stoichiometric constraints and seasonal variation

In this paper, we propose and investigate a stochastic algal growth model with the explicit incorporation of season-dependent light and nutrient availability. We first establish the threshold 〈λ〉ϖ that determines the persistence and extinction of the algae: when 〈λ〉ϖ>0, the algal will persist in mean, while when 〈λ〉ϖ<0, the algae will eventually die out. Then we prove the existence and global attractiveness of a positive stochastic periodic solution when 〈λ〉ϖ>0 by constructing suitable Lyapunov functions. Numerical simulations are carried out to substantiate our analytical results and investigate effective methods to control algal blooms.

Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model

By using bifurcation theory, we investigate the local asymptotical stability of non-negative steady states for a coupled dynamic system of ordinary differential equations and partial differential equations. The system models the interaction of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The asymptotic profile of positive steady states when the diffusion coefficients are sufficiently small or large are also obtained.

Dynamics of a periodic stoichiometric model with application in predicting and controlling algal bloom in Bohai Sea off China.

We develop a nonautonomous stoichiometric algal growth model incorporating a season-driven light intensity. We characterize the model dynamics by showing positive invariance, dissipativity, boundary dynamics, and internal dynamics. We use numerical simulations to uncover the impacts of the seasonal light intensity and the nutrient availability on the algal dynamics. We discuss two control methods, removing algae (RA) periodically and blocking nutrient (BN) input from rivers constantly, via ourmodeling approach. By comparison, the BNmethod is amore effective way to terminate algal bloom in Yellow Sea off China. The model dynamics can fit the Bohai Sea data well. Our model and analysis provide a possible explanation of seasonal algal bloom and give some measurements for controlling algal bloom in China's coastal regions.

Hybrid algal photosynthesis and ion exchange (HAPIX) process for high ammonium strength wastewater treatment

A hybrid algal photosynthesis and ion exchange (HAPIX) process was developed that uses natural zeolite (chabazite) and wild type algae to treat high ammonium (NH4+) strength wastewater. In the HAPIX process, NH4+ is temporarily adsorbed from the liquid, which reduces the free ammonia (FA) concentration below the inhibitory level for algal growth. The slow release of adsorbed NH4+ subsequently supports the continuous growth of algae. In this study, a HAPIX reactor reduced NH4+-N concentrations in centrate from an anaerobic digester from 1180 mg L−1 to below 10 mg L−1 without dilution. Chabazite doses of 60 g L−1 produced more algal biomass, with higher protein and starch contents, than doses of 150 g L−1 and 250 g L−1. Approximately 67–70% of fatty acids in the algal biomass harvested from HAPIX reactors were unsaturated. A mathematical framework that couples a homogeneous surface diffusion model with a co-limitation algal kinetic growth model reasonably predicted the algal biomass production and NH4+-N concentrations in the HAPIX reactors. The HAPIX process has the potential to serve a two-fold purpose of high NH4+-N strength wastewater treatment and agricultural or commercial biopolymer production.

Optimal control for micro-algae on a raceway model.

We apply numerical optimal control methods to an existing algae growth model with the aim to determine the best performance of the model under known conditions using a variety of decision variables. We transform the system of differential algebraic equations in the existing model to a system of ordinary differential equations which introduces dynamics for average light intensity and chlorophyll. In addition, we allow for variable nitrogen concentration of the inflow as well as variable initial nitrogen concentration of the raceway. Our main focus is on optimizing of the production of lipids. We calculate both open and closed loop optimal controllers and test their robustness. Finally, we also consider raceway depth as a decision variable. © 2017 American Institute of Chemical Engineers Biotechnol. Prog., 34:107-119, 2018.

Parameterization of a light distribution model for green cell growth of microalgae: Haematococcus pluvialis cultured under red LED lights

A light distribution model is a prerequisite for establishing an algal growth model. Based on approaches to account for light intensity, algal productivity models can be divided into three types, Type I models predict the rate of photosynthesis, Type II models computes productivity and Type III models considers the impacts of both light gradients and short light cycles. Among these, Type II models may offer the best compromise between accuracy and practicability for full-scale engineering applications. Addressing the need for examining indoor seed preparation of Haematococcus pluvialis under red LED irradiation, this study advanced a Type II model based on light distribution. The model was parameterized for green cell growth of Haematococcus pluvialis using the Gauss-Newton and bootstrap methods. The parameters k1 and k2 from the hyperbolic Beer-Lambert law of the two-flux model were 118.8±5.2m−1 and 0.25±0.04gL−1, with 95% confidence intervals of (111.6, 129.8) and (0.20, 0.33), respectively. The parameterized model was validated against experimental results, with the largest error and relative error of 0.08μmol/m2/s and 10.5%, respectively. Validation results demonstrate that the established model was reliable and could accurately predict the light intensity for indoor cultivation of Haematococcus pluvialis in the green stage. This study demonstrated that the effectiveness of using the bootstrap method to accurately estimate the parameters in these types of models. It provided information on model parameters, including standard deviation and 95% confidence intervals. This model also applied many more data points (89 in this study) to estimate parameters to reduce error. The results filled a technological gap for modeling the light distribution of indoor seed preparation of Haematococcus pluvialis under red LED lights. The approach can be used to estimate the values of parameters from parametric models for other similar applications using the bootstrap method.

Algal growth simulation in fluctuating water–sediment circumstances of the largest river-connected lake in China

In this paper, an improved simulation method was developed to investigate the mechanisms of algal growth in Poyang Lake, which is the largest freshwater lake and the most typical river-connected lake in China. In view of its frequent water and sediment exchange with the external rivers, the influences of flow disturbance and suspended sediment concentration on algal growth were quantitatively analyzed based on field investigation and laboratory experiment, and the controlling equations of l(u) and g(S) were established and embedded into the traditional algal growth model for more accurate simulation in Poyang Lake. The improved model was calibrated and validated against the consecutive field monitoring data from March to November in 2012. By simulation, the algal concentration fluctuation in a common-water year and the spatial distribution under two typical current structures: “Gravity Style” and “Jacking Style,” were quantitatively estimated. The results showed that the improved model gave a good approximation of the algal growth processes under the varied water–sediment environment in Poyang Lake. Algal concentrations in the middle area were evidently higher than that in the south and north lake area, while the variation trends were basically the same. Due to the slower current, the algal concentration under “Jacking Style” flow structure was averagely increased by 18.8 % than that under the “Gravity Style.”

Modeling and Optimization of Algae Growth

Microalgae is a promising source of renewable biofuels, and optimization and control of the biomass growth stage can make techno-economic improvements. This work explores the development of an algae growth model from first-principles, which includes the impact of natural vagaries of weather (and such) associated with production in an open system. Consequently, the process simulation is stochastic as well as fundamental; it returns a distribution of results representing the day-to-day realizations from natural variability. It also expresses day-to-night variation in the cycling solar energy. The simulation is then used to optimize pond design and management (the growth time, raceway depth, pH control, etc.) to improve profitability. Since the simulation is stochastic, nonlinear, and with multiple optima, a multiplayer direct search optimization technique with steady state convergence criteria is used for optimization. Conclusions are that (1) accounting for natural variation in the optimization leads to no...

Energy Productivity of the High Velocity Algae Raceway Integrated Design (ARID-HV)

Abstract. The original Algae Raceway Integrated Design (ARID) raceway was an effective method to increase algae culture temperature in open raceways. However, the energy input was high and flow mixing was poor. Thus, the High Velocity Algae Raceway Integrated Design (ARID-HV) raceway was developed to reduce energy input requirements and improve flow mixing in a serpentine flow path. A prototype ARID-HV system was installed in Tucson, Arizona. Based on algae growth simulation and hydraulic analysis, an optimal ARID-HV raceway was designed, and the electrical energy input requirement (kWh ha -1 d -1 ) was calculated. An algae growth model was used to compare the productivity of ARID-HV and conventional raceways. The model uses a pond surface energy balance to calculate water temperature as a function of environmental parameters. Algae growth and biomass loss are calculated based on rate constants during day and night, respectively. A 10 year simulation of DOE strain 1412 (Chlorella sorokiniana) showed that the ARID-HV raceway had significantly higher production than a conventional raceway for all months of the year in Tucson, Arizona. It should be noted that this difference is species and climate specific and is not observed in other climates and with other algae species. The algae growth model results and electrical energy input evaluation were used to compare the energy productivity (algae production rate/energy input) of the ARID-HV and conventional raceways for Chlorella sorokiniana in Tucson, Arizona. The energy productivity of the ARID-HV raceway was significantly greater than the energy productivity of a conventional raceway for all months of the year.