Phytoplankton Model Research Articles

An efficient numerical approach to simulate NPZ and SIR biological models with diffusion

The two mathematical models of Nitrogen, Phytoplankton and Zooplankton (NPZ) and whooping cough SIR model that concerns with the susceptible, infected and recovered cases of the population are considered in this paper. To incorporate with the unidimensional movements of the three species in each of the NPZ and SIR models, the models are considered with diffusion terms. A numerical scheme based on the collocation of cubic B-splines is proposed to estimate the solutions of the considered NPZ and SIR models. The numerical results obtained are compared and found in good agreement with those already available in the literature. Due to unavailability of the analytic solutions of these models, such a numerical scheme can be of prime interest for biologists to interpret the results theoretically.

Taxonomic and functional spatial distribution model of phytoplankton in tropical cascading reservoirs

Taxonomic and functional spatial distribution model of phytoplankton in tropical cascading reservoirs

New Insights on Phytoplankton Communities as a Tool for the Sustainable Management of the Musselculture in a Mediterranean Coastal Site

A simulation of phytoplankton dynamics in the coastal shellfish farming site of Mar Piccolo in Taranto (Ionian, Mediterranean Sea) is here reported. This simulation is an element of a greater ecological model produced taking into account the System Approach Framework. The model represents a refinement of a previous phytoplankton tool aimed to better understand the complex mosaic of the ecosystem functioning by providing additional information to support mussel farming. In particular, throughout the years the increase of water temperature and intensive musselculture caused a phytoplankton size and biomass reduction. Results of our simulation evidenced that the phytoplankton tool reproduced the size-fractions (micro-, nano- and picophytoplankton) biomass growth and the mussel (Mytilus galloprovincialis) grazing with a good level of performance. This newly calibrated phytoplankton model aims to provide new insights on modelling scenarios for the sustainable production of mussels.

Hopf Bifurcation Analysis of a Diffusive Nutrient–Phytoplankton Model with Time Delay

In this paper, we studied a nutrient–phytoplankton model with time delay and diffusion term. We studied the Turing instability, local stability, and the existence of Hopf bifurcation. Some formulas are obtained to determine the direction of the bifurcation and the stability of periodic solutions by the central manifold theory and normal form method. Finally, we verify the above conclusion through numerical simulation.

Saturating relationship between phytoplankton growth rate and nutrient concentration explained by macromolecular allocation.

Phytoplankton account for about a half of photosynthesis in the world, making them a key player in the ecological and biogeochemical systems. One of the key traits of phytoplankton is their growth rate because it indicates their productivity and affects their competitive capability. The saturating relationship between phytoplankton growth rate and environmental nutrient concentration has been widely observed yet the mechanisms behind the relationship remain elusive. Here we use a mechanistic model and metadata of phytoplankton to show that the saturating relationship between growth rate and nitrate concentration can be interpreted by intracellular macromolecular allocation. At low nitrate levels, the diffusive nitrate transport linearly increases with the nitrate concentration, while the internal nitrogen requirement increases with the growth rate, leading to a non-linear increase in the growth rate with nitrate. This increased nitrogen requirement is due to the increased allocation to biosynthetic and photosynthetic molecules. The allocation to these molecules reaches a maximum at high nitrate concentration and the growth rate ceases to increase despite high nitrate availability due to carbon limitation. The produced growth rate and nitrate relationships are consistent with the data of phytoplankton across taxa. Our study provides a macromolecular interpretation of the widely observed growth-nutrient relationship and highlights that the key control of the phytoplankton growth exists within the cell.

A new data assimilation method coupled with real-time detection of sensitive parameters: An example of phytoplankton modeling in lakes

数据同化是提升复杂机理过程模型精度的关键技术之一，而湖泊藻类模型的敏感参数具有随时间动态变化的特征，导致数据同化过程中无法精准更新某一时段的敏感参数，影响数据同化的模型精度提升效果.针对上述问题，本研究耦合了参数敏感性分析与集合卡尔曼滤波，研发了一种能够实时识别模型敏感参数的新型数据同化算法；为验证研发算法的效率，依托巢湖的高频水质自动监测数据，测试算法对藻类动态模型的精度提升效果.测试结果表明：研发算法能够精准跟踪模型敏感参数的动态变化，并根据监测数据实时更新模型敏感参数，实现了水质高频自动监测数据与藻类动态模型的深度融合，藻类生物量模拟精度提升了55%，即纳什系数（NSE）从0.49提升到0.76，模拟精度提升效果也显著优于传统数据同化算法（NSE=0.63）.研发算法可应用于其它水生态环境模型的数据同化，为水生态环境相关要素的精准模拟预测提供关键技术支撑.;Data assimilation is a critical method to improve the performance of complex process-based models. However, the sensitive parameters for lake models are generally changing over time. Therefore, it is challenging to accurately update the sensitive parameters for a specific period, which affects the performance of data assimilation. To address the problem, this study developed a new data assimilation method by coupling the methods of parameter sensitivity analysis and the Ensemble Kalman Filter. The new method aimed to identify the model's sensitive parameters in real time. To evaluate its performance, we collected the high-frequency water quality automatic monitoring data of Lake Chaohu, and investigated the performance improvement of a phytoplankton dynamic model using a new data assimilation method. Our investigation results showed that the developed method was able to identify the sensitive parameters of the model in each simulation period, and updated them based on the measured data to achieve better performance. The simulation accuracy of phytoplankton biomass increased by 55%, i.e., the Nash-Sutcliffe Efficiency (NSE) increased from 0.49 to 0.76. This performance is better than that of the traditional data assimilation method (NSE=0.63). The method can be applied to the data assimilation of other ecological and environmental models, technically supporting an accurate prediction of environmental and ecological factors.

Numerical investigation of fractional model of phytoplankton–toxic phytoplankton–zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

Intercomparison of Two Fluorescent Dyes to Visualize Parasitic Fungi (Chytridiomycota) on Phytoplankton

Fungal microparasites (here chytrids) are widely distributed and yet, they are often overlooked in aquatic environments. To facilitate the detection of microparasites, we revisited the applicability of two fungal cell wall markers, Calcofluor White (CFW) and wheat germ agglutinin (WGA), for the direct visualization of chytrid infections on phytoplankton in laboratory-maintained isolates and field-sampled communities. Using a comprehensive set of chytrid–phytoplankton model pathosystems, we verified the staining pattern on diverse morphological structures of chytrids via fluorescence microscopy. Empty sporangia were stained most effectively, followed by encysted zoospores and im-/mature sporangia, while the staining success was more variable for rhizoids, stalks, and resting spores. In a few instances, the staining was unsuccessful (mostly with WGA), presumably due to insufficient cell fixation, gelatinous cell coatings, and multilayered cell walls. CFW and WGA staining could be done in Utermöhl chambers or on polycarbonate filters, but CFW staining on filters seemed less advisable due to high background fluorescence. To visualize chytrids, 1 µg dye mL−1 was sufficient (but 5 µg mL−1 are recommended). Using a dual CFW–WGA staining protocol, we detected multiple, mostly undescribed chytrids in two natural systems (freshwater and coastal), while falsely positive or negative stained cells were well detectable. As a proof-of-concept, we moreover conducted imaging flow cytometry, as a potential high-throughput technology for quantifying chytrid infections. Our guidelines and recommendations are expected to facilitate the detection of chytrid epidemics and to unveil their ecological and economical imprint in natural and engineered aquatic systems.

Hourly Prediction of Phytoplankton Biomass and Its Environmental Controls in Lowland Rivers

AbstractHigh‐resolution river modeling is valuable to study diurnal scale phytoplankton dynamics and understand biomass response to short‐term, rapid changes in its environmental controls. Based on theory contained in the Quality Evaluation and Simulation Tool for River‐systems model, a new river model is developed to simulate hourly scale phytoplankton growth and its environmental controls, thus allowing to study diurnal changes thereof. The model is implemented along a 62 km stretch in a lowland river, River Thames (England), using high‐frequency water quality measurements to simulate flow, water temperature, dissolved oxygen, nutrients, and phytoplankton concentrations for 2 years (2013–2014). The model satisfactorily simulates diurnal variability and transport of phytoplankton with Nash and Sutcliffe Efficiency (NSE) > 0.7 at all calibration sites. Even without high‐frequency data inputs, the model performs satisfactorily with NSE > 0.6. The model therefore can serve as a powerful tool both for predictive purposes and for hindcasting past conditions when hourly resolution water quality monitoring is unavailable. Model sensitivity analysis shows that the model with cool water diatoms as dominant species with an optimum growth temperature of 14°C performs the best for phytoplankton prediction. Phytoplankton blooms are mainly controlled by residence time, light and water temperature. Moreover, phytoplankton blooms develop within an optimum range of flow (21–63 m3 s−1). Thus, lowering river residence time with short‐term high flow releases could help prevent major bloom developments. The hourly model improves biomass prediction and represents a step forward in high‐resolution phytoplankton modeling and consequently, bloom management in lowland river systems.

Dynamical study of harmful algal bloom in Sundarban mangrove wetland with spatial interaction and competing effects

Sundarbans is the largest mangrove wetland ecosystem of the world with rich biodiversity suffering from deteriorated water quality due to excessive fertilization that leads to an uncontrolled increase in phytoplankton. Such eutrophication also changes the community structure and increases the harmful algal blooms (HABs). In this work, we propose an interacting population model for phytoplankton–zooplankton system in which the density of zooplankton is influenced by non-toxic phytoplankton (NTP) and toxin-producing phytoplankton (TPP) followed by Holling type II and Monod–Haldane (MH)-type functional response. The growth of zooplankton species is assumed to reduce due to toxic chemicals released by TPP population. The mathematical model of the proposed system includes the competition terms between TPP and NTP. System dynamics is studied in both cases, i.e., system with diffusion and without diffusion. For the non-diffusive system, we have investigated the condition for boundedness along with the existing criteria of all feasible equilibrium point. Stability analysis of the model system is carried out in detail for each equilibrium point. Forward and backward bifurcation diagrams are obtained for the temporal system in order to understand the behavior of different parameters that control the system dynamics. Theoretically, stability criteria and Turing instability of diffusive system are derived. In this study, we have taken a case of Sundarban mangrove wetland which is suffering from algal blooms due to the presence of toxic Dinoflagellates and Cyanophyceae. Our numerical investigation shows that the lower value of intraspecific interference of zooplankton promotes the complex spatiotemporal dynamics for the population of non-toxic, toxic phytoplankton and zooplankton. The higher value of inter-specific competition coefficient of NTP leads to reduction in zooplankton density that may cause bad health of the wetland system. This investigation renders the importance of diffusion in algal blooms by the occurrence of different Turing patterns and the role of time delay in destabilization of stationary points through the creation of limit cycles. We observed that the increasing value of diffusion coefficient of zooplankton and time allows the algal blooms to settle down from spot-strip mixture to spot patterns.

Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity.

In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

Modeled temperature dependencies of macromolecular allocation and elemental stoichiometry in phytoplankton

Warming oceans may affect how phytoplankton allocate nutrients to essential cellular processes. Despite the potential impact of such processes on future biogeochemical cycles, questions remain about how temperature affects macromolecular allocation and elemental stoichiometry within phytoplankton cells. Here, we present a macromolecular model of phytoplankton and the effect of increasing temperature on the intracellular allocation of nutrients at a constant growth rate. When temperature increases under nitrogen (N) and phosphorus (P) co-limitation, the model shows less investment in phosphorus-rich RNA molecules relative to nitrogen-rich proteins, leading to a more severe decrease in cellular P:C than N:C causing increased cellular N:P values. Under P limitation, the model shows a similar pattern, but when excess P is available under N limitation, we predict lowered N:P due to the effect of luxury uptake of P. We reflected our model result on the surface ocean showing similar latitudinal patterns in N:P and P:C to observation and other model predictions, suggesting a considerable impact of temperature on constraining the elemental stoichiometry in the ocean.

Stationary distribution of a stochastic hybrid phytoplankton model with allelopathy

This research proposes and delves into a stochastic competitive phytoplankton model with allelopathy and regime-switching. Sufficient criteria are proffered to ensure that the model possesses a unique ergodic stationary distribution (UESD). Furthermore, it is testified that these criteria are sharp on certain conditions. Some critical functions of regime-switching on the existence of a UESD of the model are disclosed: regime-switching could lead to the appearance of the UESD. The theoretical findings are also applied to research the evolution of Heterocapsa triquetra and Chrysocromulina polylepis.

Climate change in a conceptual atmosphere–phytoplankton model

Abstract. We develop a conceptual coupled atmosphere–phytoplankton model by combining the Lorenz'84 general circulation model and the logistic population growth model under the condition of a climate change due to a linear time dependence of the strength of anthropogenic atmospheric forcing. The following types of couplings are taken into account: (a) the temperature modifies the total biomass of phytoplankton via the carrying capacity; (b) the extraction of carbon dioxide by phytoplankton slows down the speed of climate change; (c) the strength of mixing/turbulence in the oceanic mixing layer is in correlation with phytoplankton productivity. We carry out an ensemble approach (in the spirit of the theory of snapshot attractors) and concentrate on the trends of the average phytoplankton concentration and average temperature contrast between the pole and Equator, forcing the atmospheric dynamics. The effect of turbulence is found to have the strongest influence on these trends. Our results show that when mixing has sufficiently strong coupling to production, mixing is able to force the typical phytoplankton concentration to always decay globally in time and the temperature contrast to decrease faster than what follows from direct anthropogenic influences. Simple relations found for the trends without this coupling do, however, remain valid; just the coefficients become dependent on the strength of coupling with oceanic mixing. In particular, the phytoplankton concentration and its coupling to climate are found to modify the trend of global warming and are able to make it stronger than what it would be without biomass.

Towards modeling growth rates of cyanobacteria in the Baltic Sea

Potential influences of various environmental factors (light conditions, mixed layer depth, water temperature) on growth rates of different cyanobacteria in the Baltic Sea (BS) are discussed. Growth rates were modeled as a function of quanta absorbed by phytoplankton. This in turn depends on phytoplankton exposition to light, which we simulated to represent realistic conditions encountered in the Baltic Sea in summer. We considered eight realistic scenarios (two cases of water clarity and four depths of the mixed layer). According to our calculations, exposition of phytoplankton to photosynthetically available radiation (PAR) in the BS can change about five times in case of the same solar surface insolation and water turbidity, solely due to changes in the mixed layer depth from 2 to 20 m. When we additionally accounted for differences in water turbidity, phytoplankton PAR exposition changed by one order of magnitude and this resulted in one order of magnitude differences in absorbed quanta for the same species of phytoplankton. Knowledge of single-species absorption characteristics allowed us to show that under the same water column conditions, the number of absorbed quanta varies significantly between different species of cyanobacteria, because of differences in their absorptive properties. The effectiveness of light absorption translated to different growth rates estimated for each phytoplankton species. Our research shows that further progress in phytoplankton modeling requires incorporating in the models improved information about optical properties, growth rates, and temperature and nutrient dependencies for different phytoplankton types.

Modelling and analysis of Limnothrissa miodon population in a Lake

Abstract A mathematical model of nutrients, phytoplankton, zooplankton and Limnothrissa miodon is formulated, analysed and simulated. The model is analyzed to gain insight into the qualitative features of the equilibrium states, which enable us to determine their stability. We employ analytical and numerical techniques to investigate the impact of nutrients and intra-specific competition on the population density of Limnothrissa miodon. Stability analysis results agree with the simulations in that the coexistence equilibrium is locally asymptotically stable provided certain conditions are met. The coexistence equilibrium is globally-asymptotically stable if a certain condition is met. The inflow rate of the nutrients has a positive effect on the coexistence equilibrium, whereas the Limnothrissa miodon intra-specific competition, has a negative effect on the coexistence equilibrium. Theoretical and numerical simulations show that the nutrients inflow rate is key to the productivity of the water body and that the population of Limnothrissa miodon will continue to thrive in the water body as long as the nutrient inflow rate, is greater than some threshold value.

Construction of lyapunov function using gradient method to stability analysis of the nitrogen-phosphate-phytoplankton-sediment interaction model

The significant growth in catch fisheries and aquaculture production has enhanced the world’s capacity to consume diverse and nutritious food; however, this fast-growing industry has resulted in the environmental disturbance, especially caused by fish farming. The nutrient was generated from fish excretion and unfed pellet in the form of dissolved organic matter, especially as nitrogen and phosphate particulates. Therefore, applying mathematical models is crucial to understand and minimize the impact on the water ecosystem, thus maximizing productivity. This paper aims to research to analyze the global and stability of the equilibrium point the dynamical system that water and sediment in aquaculture Integrated Multi-Trophic Aquaculture (IMTA) system. Analysis stability global has been proved by constructing the Lyapunov function. The gradient method is used to construct the Lyapunov function in the most general form. The method is based on the assumption of a variable gradient function by determining V and ∇V. In the Lyapunov method construction, if the function V(x) would be definite positive, whereas derivative V(x) would be a definite negative. Models of nitrogen, phosphate, phytoplankton, and sediment system exhibited globally asymptotically stable. This implies that fish farming using the IMTA system on Karimunjawa island was still considerably under normal condition.

Novel approach to Design and Analysis of Water Resources Development Model in Terms of Phytoplankton Population Dynamics

Phytoplankton is one of the key markers of ecosystem health. In this article we explore a phytoplankton model described by a system of differential equations. The analytical design of aggregate regulators method is applied to the model in order to investigate its behaviour. The development of algal blooms and their diminution to the normal state of a water body are simulated. The results of simulations align well with the expected ecosystem behaviour.

Complex plankton dynamics induced by adaptation and defense

In this paper, we investigate a model for phytoplankton–zooplankton interaction and incorporated the adaptation (dormancy of the predators such as resting eggs) and defense. The dormant stage is the better equipment to withstand unpleasant environmental conditions than active ones. The ability to defense from predator attack is an important trait shaping prey population dynamics. We study how (i) adaptations allow an organism to be successful in a particular harsh environment, (ii) toxin spread surrounding the water surface provide a defense? Analytically, we study the local stability condition of the model system. To understand the effect of adaptation and defense on plankton dynamics, we have plotted the bifurcation diagram, time series and spatiotemporal pattern. Our numerical investigation reveals that the adaptation can suppress the fluctuation in population density and system shows a transient complex spatiotemporal pattern which is either a mixture of spatially periodic steady states or traveling/standing waves by increasing the time and space.

The Extinction of a Non-Autonomous Allelopathic Phytoplankton Model with Nonlinear Inter-Inhibition Terms and Feedback Controls

A non-autonomous allelopathic phytoplankton model with nonlinear inter-inhibition terms and feedback controls is studied in this paper. Based on the comparison theorem of differential equation, some sufficient conditions for the permanence of the system are obtained. We study the extinction of one of the species by using some suitable Lyapunov type extinction function. Our analyses extend those of Xie et al. (Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Advances in Difference Equations, 2016, 2016, 258) and show that the feedback controls and toxic substances have no effect on the permanence of the system but play a crucial role on the extinction of the system. Some known results are extended.