encyclopedia of inland waters || modeling of lake ecosystems

Modeling of Lake EcosystemsL Hakanson, Uppsala University, Uppsala, Sweden

ã 2009 Elsevier Inc. All rights reserved.

Introduction: Basic Concepts andProblems inModeling of Lake Ecosystems

Every aquatic ecosystem is unique, but only a few arestudied ecologically in great detail. For reasonsrelated to the practical value of lakes, including fish-eries, recreation, and water supply, there are demandsfor analytical information or predictions concerninglakes for which no detailed studies are available.Ecosystem modeling is the main tool by which pre-dictions and analyses can be provided in such cases.Ecosystem modeling of lakes and other aquatic eco-

systems is important not only in practical mattersrelated to lake management or environmental remedia-tion, but also in demonstrating the ways in which mul-tiple factors act simultaneously on ecosystem featuressuch as community composition, biological productiv-ity, or biogeochemical processes. Therefore,modeling isuseful not only as a means of providing practical infor-mation for solving problems, but also in improving thebasic understanding of lakes as ecosystems.Lake ecosystem models are composed of linked

predictive equations incorporating multiple environ-mental variables. The environmental variables maybe physical, chemical, or biological, and they typi-cally are expressed as quantities per unit or volumeper unit area, or as fluxes (mass or energy per unittime). Relevant variables for lake ecosystem modelsinclude not only those applicable to the lake itself, butalso to the watershed from which the lake deriveswater and dissolved or suspended substances thataffect ecosystem processes (Figure 1).Environmental variables that are used in the lake

ecosystem models are of two types: those for whichsite-specific data must be available in support of mod-eling, and those for which typical or generic valuescan be used. Examples of site-specific variablesinclude the dimensions of the lake (e.g., meandepth), hydrologic features (e.g., hydraulic residencetime), and concentrations of dissolved or suspendedsubstances in the water (e.g., nutrients, organic mat-ter). Examples of variables for which typical orgeneric values can be used include rates of masstransfer from the water column to the sediment sur-face or from the sediment surface to water column.Variables that are specific to a given lake willbe referred to here as ‘lake-specific variables,’ but

sometimes are also referred to as ‘obligatory drivingvariables.’ Variables for which typical or genericinformation is adequate will be referred to here as‘generic variables.’

Repeated testing of the predictive power of models,as judged by actual ecosystem characteristics, hasshown which environmental variables must be site-specific and which can be generic. An importantresult of such experience with modeling over the lasttwo decades is that a number of environmental vari-ables that are very important to the predictive capa-bility of models can be generic.

Mathematically speaking, it is possible to createecosystem system models of great complexity. Experi-ence has shown, however, that complexity degradesthe predictive reliability of ecosystem models. There-fore, one important goal of modeling is to representthe relevant ecosystem processes in a manner that isboth realistic and simple. A combination of the princi-ple of simplicity with the principle of generic variableshas greatly simplified ecosystem modeling for lakes.

Once prepared as a set of linked equations basedon site-specific and generic variables, a model mustbe calibrated to make predictions for a specific lake.The calibration involves insertion of specific numericvalues for site-specific variables. Often the firstattempt at calibrating a model to make predictionsfor a specific lake shows that the model is makingbiased predictions. The modeler then attempts to findan error in calibration. If a suspected error is found,the calibration is adjusted until the model producespredictions showing low bias.

The calibration process involving adjustment ofcalibration to produce realistic results may be mis-leading. Because models contain coupled equations,many of which have mutual counteractive effects, it isalmost always possible to make a model produceresults of low bias by adjusting one or more of thecalibrated variables. Thus, an improved fit does notnecessarily indicate that the initial problem with themodel has been solved; it only shows that the modelhas been forced to produce a better prediction by anadjustment that may or may not be correct.

The modeler’s tool for finding errors in calibrationis validation. Validation is the process by which themodeler uses a model that has been calibrated on one

441

Inflow Outflow

Surface water(epilimnion)

Deep water(hypolimnion)

Point source emissions Precipitation

Burial

Bioturbation

Diffusion

Mixing

Active sediments

Passive sediments

Compaction

Wave base

Sedimentation

Primary production Resuspension

Figure 1 Schematic illustration of fundamental transport processes affecting lakes.

442 Biological Integration _ Modeling of Lake Ecosystems

or a few ecosystems to make predictions on otherecosystems that were not involved in the calibrationprocess. In other words, it is a test of the modeloutside the framework within which the model wasfirst developed. If the model has been properly cali-brated, it will perform well on systems that were notused in the calibration process, provided that thesesystems have a general similarity to the ones that wereused in calibration.The application of a lake ecosystem model pro-

duces several indicators of the value of the model.First, the accuracy and precision of the model predic-tions during validation is a quantitative index of thevalue of the model. A second indicator is the practicalor fundamental importance of the model predictions;failure of a model to predict the most important typesof outcomes is an indicator of weakness in the model.A third index of value is the range of conditions overwhich it can be applied. Some models performwell under conditions very close to those used incalibration, but fail to predict conditions in lakesthat differ from those used in calibration. Morerobust models are more valuable. Finally, the mostsuccessful models are those that can be operatedsuccessfully on the basis of site-specific variables forwhich information is easily available. A demandfor obscure information reduces the value of a model.In emphasizing simplicity, as necessary in order

to conserve the predictive power of models, the mod-eler must avoid any attempt to represent mathemati-cally all of the possible connections of an ecosystemextending from a cellular level to whole organisms,populations, and communities. Themodelermust searchfor connections between abiotic variables and bioticresponses that are meaningful at the ecosystem level.One successful approach in this attempt at simplifica-tion is the development of quantitative relationships

betweenmass transport of environmentally importantsubstances and biologically driven ecosystem res-ponses. This concept may be called the ‘effect-load-sensitivity’ (ELS) approach to ecosystem modeling. Itis well suited to modeling that is directly relevant towater management. It is based on the principle thatlakes often have differing sensitivities to a given massloading of a contaminant. For example, lakes showinglow pH will sustain fish populations with higher mer-cury concentrations per unit mass than lakes of higherpH receiving the same mercury load per unit volumeor per unit area.

Classical Lake Modeling: TheVollenweider Approach

Richard Vollenweider first showed that phosphorusconcentrations of lakes could be predicted fromhydraulic residence time (water retention time) andthe mean concentration of phosphorus for rivers orstreams entering the lake using mass-balance model-ing. He then showed that characteristic abundancesof phytoplankton, as measured by concentrations ofchlorophyll in the water column, could be predictedfrom the modeled concentrations of phosphorus.Calibration of the relationship between phosphorusand chlorophyll awas accomplished by simple regres-sion analysis from field observations on numerouslakes. Vollenweider’s approach was of great practicalutility because it enabled lake managers for the firsttime to calculate howmuch reduction in the transportof phosphorus to a lake would be required in orderto reduce the growth of algae in the lake to an accept-able level for management purposes.

The Vollenweider approach, although very influen-tial, has shown some limitations. First, it does notdeal with temporal variations in phytoplankton

Biological Integration _ Modeling of Lake Ecosystems 443

biomass, and therefore fails to account for the excep-tional importance of peak algal biomass as contrastedwith average biomass. In addition, the Vollenweiderapproach, as originally constructed, failed to accountfor the substantial escape of phosphorus from sedi-ments in the most productive lakes (‘internal loadingof phosphorus’). Because such lakes can be essentiallyself-fertilizing, the predicted effects of controllingexternal sources of phosphorus may not be valid.Even so, the Vollenweider approach remains a key-stone for lake modeling of eutrophication becauseit shows the two key features of successful models:simplicity and use of readily defined, lake-specificvariables.Lake models deal differently with abiotic and biotic

variables. Mass fluxes of contaminants or nutrients,which carry units of gram per unit time, or theamounts or concentrations of any inorganic sub-stance, are generally dealt with through the use ofdifferential equations. Environmental variables thatare under direct biological control (bioindicators)must be treated differently because they do not reflectthe law of conservation of mass. For example,

Nutrient loading o

Mass-balance model foInflow

Internal processes: sedimdiffusion, mixing, mineraliz

Watershed area spredicting nutrient

Nutrient conce

Bioindicator 1,Secchi depth

BioindicatoChlorophyll

Bioindicator 4,Macrophyte cover

Population, land

Figure 2 Basic elements in ELS modeling for aquatic eutrophicatio

regression analyses relating nutrient concentrations to bioindicatorssaturation in the deep-water zone and macrophyte cover).

nutrients present in a lake may not be completelyincorporated into phytoplankton biomass because ofthe presence of biological removal agents (grazers) orhydraulic removal of biomass (washout). Therefore,bioindicators are related to abiotic variables bymeans of empirical calibrations, which often involveregressions of the type used by Vollenweider.

ELS models combine a mass-balance approach tothe prediction of abiotic variables with an empiricalapproach such as regression analysis to relate bioin-dicators to an abiotic variables (Figure 2). In thisway, such a model uses the principle of mass conser-vation to calculate the load or concentration ofan important abiotic variable such as nutrient con-centration. The model then converts the abioticvariable into a biotic signal, as shown by one ormore bioindicators through an empirically estab-lished relationship between the abiotic variable andthe bioindicator. Thus, the ELS modeling approach,as pioneered by Vollenweider, can be used in predict-ing a wide range of bioindicator responses throughthe consistent use of a few simple model develop-ment principles.

f lake

r nutrientsOutflow

Surface water

Deep water

Sediments

entation, resuspension,ation

ub-model transport

ntrations

r 2,-a

Bioindicator 3,Oxygen saturationin deep-water zone

use

n studies and management utilizing mass-balance modeling and

(e.g., Secchi depth, chlorophyll a concentrations, oxygen


The Relevance of Scale to Modeling

Bioindicators, which typically are targets of highimportance for modeling, can be quantified overwide ranges of temporal and spatial scales. Modelsdesigned to produce predictions at different scalesmay produce qualitatively different kinds of results.It is critical for the modeler to select scales of spaceand time that are appropriate to the application.The flexibility in development of models of varying

scale to some extent is restricted by the availabilityand degree of uncertainty for all types of empiricaldata that support modeling. As shown in Figure 3,uncertainty increases as temporal scale increases(the same would be true of spatial scale), but theconfinement of modeling to short time scales pro-duces an impossible requirement for empirical doc-umentation. Therefore, optimal conditions formodeling involve intermediate scales of time andspace. In addition, focus on the ecosystem as anatural entity imposes certain spatial constraints.Modeling of ecosystems cannot be built up easilyfrom empirical data on specific organisms. There-fore, ecosystem modeling is facilitated by focus onbioindicators that have implications for the entireecosystem. For example, ecosystem modeling offood chain phenomena such as bioconcentrationof mercury can begin with a consideration of theabundance and type of top predators in a lake.The influence of top predators is spread throughthe lower links of the food chain, influencing trophicdynamics on an ecosystem basis.

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a re

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umbe

r ob

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atio

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Temporal

Figure 3 Illustration of factors controlling optimality of models in aq

variables show increasing coefficients of variation (CV) with increasin

data points needed to run or test a model) shows that data requiremrather than long time scales. The optimality curve illustrates the com

Recent Developments in Mass-BalanceModeling

The accidental release of large amounts of radio-cesium to the atmosphere from the Chernobyl reactorsite in the Ukraine during 1986 led to the transport ofsubstantial quantities of radiocesium to Scandinaviaand Europe. Although alarming from the viewpointof environment and human health, the Chernobylaccident allowed unprecedented mass-balance track-ing of an environmental constituent (cesium). Becauseradiocesium can be detected through its radioactiveemissions, the transport of even small amounts fromthe atmosphere to soil surface, and through the soilsurface to the drainage network and into food chainscould be studied on a large spatial scale across manydifferent types of aquatic ecosystems.

The radiocesium studies produced new understand-ing of rates and mechanisms for the transport not justof radiocesium, but also for many other substances,including contaminants and nutrients relevant to themodeling of lake ecosystems. This improved level ofunderstanding has lead to increased sophisticationof mass-balance prediction in the lake ecosystemmodels (Figure 2).

Sedimentation is the name for flux carrying massfrom water to sediments. Return of mass from sedi-ments to the water column can occur either throughresuspension, which is generated by turbulence at thesediment–water interface, or by diffusion across con-centration gradients at the sediment–water interface.In addition, some mass is buried through the

CV

Optimality

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scale

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uatic ecology. The curve marked CV shows that most lake

g temporal spacing of samples. The curve marked N (number of

ents are much higher for models designed to predict over shortbined effect of CV and N.


accumulation of mass on the bottom of the lake, andsome leaves the lake through its outflow. The fluxesillustrated in Figure 2 form the core of the mass-balance modeling of lakes.Mass-balance modeling as illustrated in Figure 2,

when improved through the new insights derivedfrom the Chernobyl contamination, show very highpredictive capabilities. For example, modeling ofradiocesium concentrations from 23 very differentEuropean lakes accounted for 96% of variance andshowed a slope relating protected to observe concen-trations of 0.98. Such high predictability in mass-balance modeling is extremely useful.Figure 4 shows modeling of phosphorus in support

of management through use of LakeMab, a lake

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Lake bullaren (intermediate s

TP

con

cent

ratio

n (µ

g/L)

0

M1 13

(b)

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Harp lake (small, deep and o

TP

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Lake balaton (large, shallow a

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TP

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g/L)

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M(a)1 13

Figure 4 Example of modeling results for (a) very large, shallow an

which is one of moderate size (in this study) andmesotrophic and (c) H

figures give modeled TP concentrations as well as observed long-ter

ecosystem model, as guided by insight gained throughthe Chernobyl disaster. The results shown in themodel were achieved with no recalibration.

Improvement in Modeling Practices

Abbreviations are used extensively in ecosystem mod-eling. Modern abbreviation systems follow rules thatare intended to simplify the use of mathematical ter-minology. Contrary to custom in mathematics andphysics, Greek letters are avoided in favor ofmnemonic letter combinations. Consistency is usedfor like measures. For example, length measures areconsistently designated as L, and subclassified with a

Observed

odeled

ize and mesotrophic)

onths25 37 49

Modeled

ligotrophic)

onths25 37 49

nd eutrophic)

ed

onths25 37 49

d eutrophic Lake Balaton, Hungary, (b) Lake Bullaren, Sweden,

arp Lake, Canada, which is very small, deep and oligotrophic. The

m median values.


subscript. For example, Lmax for maximum length.Fluxes from one compartment to another of an ecosys-tem are designated by subscripts: Fab means flux fromcompartment A to compartment B.Different forms of a particular substance that is

being modeled may be distinguished by use of a dis-tribution coefficient in a model. For example, a coef-ficient may differentiate between dissolved andparticulate fractions in a water column. Such a dis-tinction is important functionally because the particu-late fraction can settle to the sediments, whereas thedissolved fraction cannot. Coefficients are also givenfor sedimentation into deep water as opposed to shal-low water where resuspension is more likely. Overall,the use of distribution coefficients facilitates model-ing 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 ofsurface waters with deep waters, and conversion fromorganic 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 accumu-lation. Symbolic notation references these compart-ments. For example, FSWDW indicates flux fromsurface water to deep water.

Modeling of Food Webs Based onFunctional Groups

Management questions related to productivity, com-munity composition, and concentrations of biomasscommonly require modeling of food web interac-tions. Figure 5 shows a typical application. For mostlake ecosystem modeling, several functional groupsare included in the model: predatory fish, prey fish,benthic animals, predatory zooplankton, herbivorous

Predatory fish

Prey fish

Zoobenthos

Sediments Benthic algae Macrophytes

Primary producers

Figure 5 Illustration of a typical lake food web including nine groupmacrophytes, herbivorous zooplankton, predatory zooplankton, zoob

zooplankton, phytoplankton, bacterioplankton, ben-thic algae, and macrophytes. The groups are func-tionally linked. For example, predatory fish eat preyfish, which consume both herbivorous and predatoryzooplankton as well as benthic animals. Changes incompartments through time are achieved by use ofordinary differential equations used at weekly ormonthly time scales. Calibrated versions of suchmodels produce predictions of typical patterns.Lakes responding to unusual influences such as con-tamination will show deviations from these patterns,which would indicate a need for data collection andempirical analysis.

Concepts that are fundamental to modeling of foodwebs include rates of consumption by predators,metabolic efficiencies, turnover rates for biomass ina compartment, selectivity of feeding, and migrationrates (especially for fish).

The Future of Lake Ecosystem Modeling

Recent improvements in modeling suggest that thelake ecosystem modeling will be more useful andmore broadly applicable than in the past. Therefore,models dealing with issues important to managementmay be more extensively incorporated into universitytraining programs and used in support of decisionmaking. It will always be true, however, that eachmodel has a specific domain of use, outside whichits use will not be appropriate. Maybe the most cru-cial aspect for future model development, and hencealso for our understanding of the structure and func-tion of aquatic ecosystem, has to do with the access ofreliable empirical data from different types of lakes.This is time-consuming and expensive work, whichshould have a high, and not a low, priority in spite ofthe fact that it may not always be regarded as veryglamorous work.

Predatory zooplankton

Herbivorous zooplankton

Secondaryproducers

Decomposers

Phytoplankton Bacterioplankton

s of organisms (phytoplankton, bacterioplankton, benthic algae,enthos, prey fish and predatory fish).


Glossary

Calibration – Adjustment of variables within a modelin order to achieve minimum bias.

Effect-load-sensitivity model – A quantitative modelthat uses information on mass balance of nutrientsor toxins to predict important bioindicators such asfish production or algal biomass.

Food-web model – A model whose main objective isto predict the interactions of primary producers,herbivores, and carnivores within a lake ecosystem.

Generic variable – A rate or quantity used in model-ing that is typical of most ecosystems or of a groupof ecosystems.

Modeling scale – Resolution of a model expressed interms of dimensions of time or space.

Site-specific variable – A rate or quantity used inmodeling that is unique to a particular ecosystem.

Validation – Use of information not used in a modelcalibration to determine whether the calibration hasbeen successful.

See also: Biological Interactions in River Ecosystems;Ecological Zonation in Lakes; Eutrophication of Lakesand Reservoirs; Nitrogen; Phosphorus; Regulators ofBiotic Processes in Stream and River Ecosystems.

Further Reading

Hakanson L and Peters RH (1995) Predictive Limnology – Meth-ods for Predictive Modelling. Amsterdam: SPB Academic Pub-lishers. p. 464.

Hakanson L (2000) Modelling Radiocesium in Lakes and CoastalAreas – New Approaches for Ecosystem Modellers. A Textbookwith Internet Support. Dordrecht: Kluwer Academic. p. 215.

Hakanson L and Boulion V (2002) The Lake Foodweb –ModellingPredation and Abiotic/Biotic Interactions. Leiden: Backhuys

Publishers. p. 344.

Monte L (1996) Collective models in environmental science.Science of the Total Environment 192: 41–47.

Odum EP (1971) Fundamentals of Ecology, 3rd edn. p. 574

Philadelphia: W.B. Saunders. p. 574.OECD (1982) Eutrophication of waters. Monitoring, assessment

and control. Paris: OECD. p. 154.

Vollenweider RA (1968) The scientific basis of lake eutrophication,with particular reference to phosphorus and nitrogen as eutrophi-cation factors. Tech. Rep. DAS/DSI/68.27. Paris: OECD. p. 159.

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