L Hakanson, Uppsala University, Uppsala, Sweden © 2009 Elsevier Inc. All rights reserved.
Introduction: Basic Concepts and Problems in Modeling of Lake Ecosystems
Every aquatic ecosystem is unique, but only a few are studied ecologically in great detail. For reasons related to the practical value of lakes, including fisheries, recreation, and water supply, there are demands for analytical information or predictions concerning lakes for which no detailed studies are available. Ecosystem modeling is the main tool by which predictions and analyses can be provided in such cases.
Ecosystem modeling of lakes and other aquatic ecosystems is important not only in practical matters related to lake management or environmental remediation, but also in demonstrating the ways in which multiple factors act simultaneously on ecosystem features such as community composition, biological productivity, or biogeochemical processes. Therefore, modeling is useful not only as a means of providing practical information for solving problems, but also in improving the basic understanding of lakes as ecosystems.
Lake ecosystem models are composed of linked predictive equations incorporating multiple environmental variables. The environmental variables may be physical, chemical, or biological, and they typically are expressed as quantities per unit or volume per unit area, or as fluxes (mass or energy per unit time). Relevant variables for lake ecosystem models include not only those applicable to the lake itself, but also to the watershed from which the lake derives water and dissolved or suspended substances that affect ecosystem processes (Figure 1).
Environmental variables that are used in the lake ecosystem models are of two types: those for which site-specific data must be available in support of modeling, and those for which typical or generic values can be used. Examples of site-specific variables include the dimensions of the lake (e.g., mean depth), hydrologic features (e.g., hydraulic residence time), and concentrations of dissolved or suspended substances in the water (e.g., nutrients, organic matter). Examples of variables for which typical or generic values can be used include rates of mass transfer from the water column to the sediment surface or from the sediment surface to water column. Variables that are specific to a given lake will be referred to here as 'lake-specific variables,' but sometimes are also referred to as 'obligatory driving variables.' Variables for which typical or generic information is adequate will be referred to here as 'generic variables.'
Repeated testing of the predictive power of models, as judged by actual ecosystem characteristics, has shown which environmental variables must be site-specific and which can be generic. An important result of such experience with modeling over the last two decades is that a number of environmental variables that are very important to the predictive capability of models can be generic.
Mathematically speaking, it is possible to create ecosystem system models of great complexity. Experience has shown, however, that complexity degrades the predictive reliability of ecosystem models. Therefore, one important goal of modeling is to represent the relevant ecosystem processes in a manner that is both realistic and simple. A combination of the principle of simplicity with the principle of generic variables has greatly simplified ecosystem modeling for lakes.
Once prepared as a set of linked equations based on site-specific and generic variables, a model must be calibrated to make predictions for a specific lake. The calibration involves insertion of specific numeric values for site-specific variables. Often the first attempt at calibrating a model to make predictions for a specific lake shows that the model is making biased predictions. The modeler then attempts to find an error in calibration. If a suspected error is found, the calibration is adjusted until the model produces predictions showing low bias.
The calibration process involving adjustment of calibration to produce realistic results may be misleading. Because models contain coupled equations, many of which have mutual counteractive effects, it is almost always possible to make a model produce results of low bias by adjusting one or more of the calibrated variables. Thus, an improved fit does not necessarily indicate that the initial problem with the model has been solved; it only shows that the model has been forced to produce a better prediction by an adjustment that may or may not be correct.
The modeler's tool for finding errors in calibration is validation. Validation is the process by which the modeler uses a model that has been calibrated on one
Point source emissions Precipitation
Point source emissions Precipitation
or a few ecosystems to make predictions on other ecosystems that were not involved in the calibration process. In other words, it is a test of the model outside the framework within which the model was first developed. If the model has been properly calibrated, it will perform well on systems that were not used in the calibration process, provided that these systems have a general similarity to the ones that were used in calibration.
The application of a lake ecosystem model produces several indicators of the value of the model. First, the accuracy and precision of the model predictions during validation is a quantitative index of the value of the model. A second indicator is the practical or fundamental importance of the model predictions; failure of a model to predict the most important types of outcomes is an indicator of weakness in the model. A third index of value is the range of conditions over which it can be applied. Some models perform well under conditions very close to those used in calibration, but fail to predict conditions in lakes that differ from those used in calibration. More robust models are more valuable. Finally, the most successful models are those that can be operated successfully on the basis of site-specific variables for which information is easily available. A demand for obscure information reduces the value of a model.
In emphasizing simplicity, as necessary in order to conserve the predictive power of models, the modeler must avoid any attempt to represent mathematically all of the possible connections of an ecosystem extending from a cellular level to whole organisms, populations, and communities. The modeler must search for connections between abiotic variables and biotic responses that are meaningful at the ecosystem level. One successful approach in this attempt at simplification is the development of quantitative relationships between mass transport of environmentally important substances and biologically driven ecosystem responses. This concept may be called the 'effect-load-sensitivity' (ELS) approach to ecosystem modeling. It is well suited to modeling that is directly relevant to water management. It is based on the principle that lakes often have differing sensitivities to a given mass loading of a contaminant. For example, lakes showing low pH will sustain fish populations with higher mercury concentrations per unit mass than lakes of higher pH receiving the same mercury load per unit volume or per unit area.
Classical Lake Modeling: The Vollenweider Approach
Richard Vollenweider first showed that phosphorus concentrations of lakes could be predicted from hydraulic residence time (water retention time) and the mean concentration of phosphorus for rivers or streams entering the lake using mass-balance modeling. He then showed that characteristic abundances of phytoplankton, as measured by concentrations of chlorophyll in the water column, could be predicted from the modeled concentrations of phosphorus. Calibration of the relationship between phosphorus and chlorophyll a was accomplished by simple regression analysis from field observations on numerous lakes. Vollenweider's approach was of great practical utility because it enabled lake managers for the first time to calculate how much reduction in the transport of phosphorus to a lake would be required in order to reduce the growth of algae in the lake to an acceptable level for management purposes.
The Vollenweider approach, although very influential, has shown some limitations. First, it does not deal with temporal variations in phytoplankton biomass, and therefore fails to account for the exceptional importance of peak algal biomass as contrasted with average biomass. In addition, the Vollenweider approach, as originally constructed, failed to account for the substantial escape of phosphorus from sediments in the most productive lakes ('internal loading of phosphorus'). Because such lakes can be essentially self-fertilizing, the predicted effects of controlling external sources of phosphorus may not be valid. Even so, the Vollenweider approach remains a keystone for lake modeling of eutrophication because it shows the two key features of successful models: simplicity and use of readily defined, lake-specific variables.
Lake models deal differently with abiotic and biotic variables. Mass fluxes of contaminants or nutrients, which carry units of gram per unit time, or the amounts or concentrations of any inorganic substance, are generally dealt with through the use of differential equations. Environmental variables that are under direct biological control (bioindicators) must be treated differently because they do not reflect the law of conservation of mass. For example, nutrients present in a lake may not be completely incorporated into phytoplankton biomass because of the presence of biological removal agents (grazers) or hydraulic removal of biomass (washout). Therefore, bioindicators are related to abiotic variables by means of empirical calibrations, which often involve regressions of the type used by Vollenweider.
ELS models combine a mass-balance approach to the prediction of abiotic variables with an empirical approach such as regression analysis to relate bioin-dicators to an abiotic variables (Figure 2). In this way, such a model uses the principle of mass conservation to calculate the load or concentration of an important abiotic variable such as nutrient concentration. The model then converts the abiotic variable into a biotic signal, as shown by one or more bioindicators through an empirically established relationship between the abiotic variable and the bioindicator. Thus, the ELS modeling approach, as pioneered by Vollenweider, can be used in predicting a wide range of bioindicator responses through the consistent use of a few simple model development principles.
Population, land use
i Watershed area sub-model ^predicting nutrient transport
Nutrient loading of lake
Nutrient loading of lake
Bioindicator 1, Secchi depth fBioindicator 2' IjChlorophyll-a
Bioindicator 4, Macrophyte cover
^Bioindicator 3, A Oxygen saturation in deep-water zone
Figure 2 Basic elements in ELS modeling for aquatic eutrophication studies and management utilizing mass-balance modeling and regression analyses relating nutrient concentrations to bioindicators (e.g., Secchi depth, chlorophyll a concentrations, oxygen saturation in the deep-water zone and macrophyte cover).
Bioindicators, which typically are targets of high importance for modeling, can be quantified over wide ranges of temporal and spatial scales. Models designed to produce predictions at different scales may produce qualitatively different kinds of results. It is critical for the modeler to select scales of space and time that are appropriate to the application.
The flexibility in development of models of varying scale to some extent is restricted by the availability and degree of uncertainty for all types of empirical data that support modeling. As shown in Figure 3, uncertainty increases as temporal scale increases (the same would be true of spatial scale), but the confinement of modeling to short time scales produces an impossible requirement for empirical documentation. Therefore, optimal conditions for modeling involve intermediate scales of time and space. In addition, focus on the ecosystem as a natural entity imposes certain spatial constraints. Modeling of ecosystems cannot be built up easily from empirical data on specific organisms. Therefore, ecosystem modeling is facilitated by focus on bioindicators that have implications for the entire ecosystem. For example, ecosystem modeling of food chain phenomena such as bioconcentration of mercury can begin with a consideration of the abundance and type of top predators in a lake. The influence of top predators is spread through the lower links of the food chain, influencing trophic dynamics on an ecosystem basis.
The accidental release of large amounts of radio-cesium to the atmosphere from the Chernobyl reactor site in the Ukraine during 1986 led to the transport of substantial quantities of radiocesium to Scandinavia and Europe. Although alarming from the viewpoint of environment and human health, the Chernobyl accident allowed unprecedented mass-balance tracking of an environmental constituent (cesium). Because radiocesium can be detected through its radioactive emissions, the transport of even small amounts from the atmosphere to soil surface, and through the soil surface to the drainage network and into food chains could be studied on a large spatial scale across many different types of aquatic ecosystems.
The radiocesium studies produced new understanding of rates and mechanisms for the transport not just of radiocesium, but also for many other substances, including contaminants and nutrients relevant to the modeling of lake ecosystems. This improved level of understanding has lead to increased sophistication of mass-balance prediction in the lake ecosystem models (Figure 2).
Sedimentation is the name for flux carrying mass from water to sediments. Return of mass from sediments to the water column can occur either through resuspension, which is generated by turbulence at the sediment-water interface, or by diffusion across concentration gradients at the sediment-water interface. In addition, some mass is buried through the
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w s 200
Figure 3 Illustration of factors controlling optimality of models in aquatic ecology. The curve marked CV shows that most lake variables show increasing coefficients of variation (CV) with increasing temporal spacing of samples. The curve marked N (number of data points needed to run or test a model) shows that data requirements are much higher for models designed to predict over short rather than long time scales. The optimality curve illustrates the combined effect of CV and N.
accumulation of mass on the bottom of the lake, and some leaves the lake through its outflow. The fluxes illustrated in Figure 2 form the core of the massbalance modeling of lakes.
Mass-balance modeling as illustrated in Figure 2, when improved through the new insights derived from the Chernobyl contamination, show very high predictive capabilities. For example, modeling of radiocesium concentrations from 23 very different European lakes accounted for 96% of variance and showed a slope relating protected to observe concentrations of 0.98. Such high predictability in massbalance modeling is extremely useful.
Figure 4 shows modeling of phosphorus in support of management through use of LakeMab, a lake ecosystem model, as guided by insight gained through the Chernobyl disaster. The results shown in the model were achieved with no recalibration.
Abbreviations are used extensively in ecosystem modeling. Modern abbreviation systems follow rules that are intended to simplify the use of mathematical terminology. Contrary to custom in mathematics and physics, Greek letters are avoided in favor of mnemonic letter combinations. Consistency is used for like measures. For example, length measures are consistently designated as L, and subclassified with a
Lake balaton (large, shallow and eutrophic)
Lake bullaren (intermediate size and mesotrophic)
Lake bullaren (intermediate size and mesotrophic)
Harp lake (small, deep and oligotrophic)
Figure 4 Example of modeling results for (a) very large, shallow and eutrophic Lake Balaton, Hungary, (b) Lake Bullaren, Sweden, which is one of moderate size (in this study) and mesotrophic and (c) Harp Lake, Canada, which is very small, deep and oligotrophic. The figures give modeled TP concentrations as well as observed long-term median values.
subscript. For example, Lmax for maximum length. Fluxes from one compartment to another of an ecosystem are designated by subscripts: Fab means flux from compartment A to compartment B.
Different forms of a particular substance that is being modeled may be distinguished by use of a distribution coefficient in a model. For example, a coefficient may differentiate between dissolved and particulate fractions in a water column. Such a distinction is important functionally because the particu-late fraction can settle to the sediments, whereas the dissolved fraction cannot. Coefficients are also given for sedimentation into deep water as opposed to shallow water where resuspension is more likely. Overall, the use of distribution coefficients facilitates modeling of the important mass-flux processes within lakes: for example, sedimentation from water to sediments, resuspension from the sediments back to the water, diffusion from the sediment to the water, mixing of surface waters with deep waters, and conversion from organic to inorganic forms.
The model shown in Figure 2 is typical in its use of spatial compartments: surface water, deep water, zones of erosion and transport, and zones of accumulation. Symbolic notation references these compartments. For example, FSWDW indicates flux from surface water to deep water.
Management questions related to productivity, community composition, and concentrations of biomass commonly require modeling of food web interactions. Figure 5 shows a typical application. For most lake ecosystem modeling, several functional groups are included in the model: predatory fish, prey fish, benthic animals, predatory zooplankton, herbivorous zooplankton, phytoplankton, bacterioplankton, ben-thic algae, and macrophytes. The groups are functionally linked. For example, predatory fish eat prey fish, which consume both herbivorous and predatory zooplankton as well as benthic animals. Changes in compartments through time are achieved by use of ordinary differential equations used at weekly or monthly time scales. Calibrated versions of such models produce predictions of typical patterns. Lakes responding to unusual influences such as contamination will show deviations from these patterns, which would indicate a need for data collection and empirical analysis.
Concepts that are fundamental to modeling of food webs include rates of consumption by predators, metabolic efficiencies, turnover rates for biomass in a compartment, selectivity of feeding, and migration rates (especially for fish).
Recent improvements in modeling suggest that the lake ecosystem modeling will be more useful and more broadly applicable than in the past. Therefore, models dealing with issues important to management may be more extensively incorporated into university training programs and used in support of decision making. It will always be true, however, that each model has a specific domain of use, outside which its use will not be appropriate. Maybe the most crucial aspect for future model development, and hence also for our understanding of the structure and function of aquatic ecosystem, has to do with the access of reliable empirical data from different types of lakes. This is time-consuming and expensive work, which should have a high, and not a low, priority in spite of the fact that it may not always be regarded as very glamorous work.
Zoobenthos^^^^ Predatory Zooplankton
Sediments Benthic algae Macrophytes Herbivorous Zooplankton _
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