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Ecological Modelling 151 (2002) 125–142

Optimization methodology for land use patterns usingspatially explicit landscape models

Ralf Seppelt a,*, Alexey Voinov b

a Department of En�ironmental System Analysis, Institute of Geoecology, Technical Uni�ersity of Braunschweig, Langer Kamp 19c,38106 Braunschweig, Germany

b Institute for Ecological Economics, Uni�ersity of Maryland Center for En�ironmental Sciences, P.O. Box 38, Solomons,MD 20688, USA

Received 26 March 2001; received in revised form 6 September 2001; accepted 11 October 2001

Abstract

Spatially explicit ecosystem models allow the calculation of water and matter dynamics in a landscape as functionsof spatial localization of habitat structures and matter input. For a mainly agricultural region we studied the nutrientbalance as a function of different management schemes. For this purpose we formulated optimization tasks. Thisrequired the definition of performance criteria, which compare economic aspects, such like farmer’s income fromharvest, with ecologic aspects, such like nutrient loss out of the watershed. The task was to calculate optimum landuse maps and fertilizer application maps maximizing the performance criterion. We developed a framework ofprocedures for numerical optimization in spatially explicit dynamic ecosystem simulation models. The results weretested using Monte-Carlo’s simulation, which based on different stochastic generators for the independent controlvariables. Gradient free optimization procedures (Genetic Algorithms) were used to verify the simplifying assump-tions. Parts of the framework offer tools for optimization with the computation effort independent of the size of thestudy area. As a result, important areas with high retention capabilities were identified and fertilizer maps were setup depending on soil properties. This shows that optimization methods even in complex simulation models can be auseful tool for a systematic analysis of management strategies of ecosystem use. © 2002 Elsevier Science B.V. Allrights reserved.

Keywords: Spatial modeling; Ecosystem management; Agroecosystem modeling; GIS

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1. Introduction

Simulation models are recognized as efficienttools for analysis of environmental processes, edu-cation, and decision making. The most fascinatingability of a model is the possibility to performnon-intrusive experiments over a system based onthe theoretical and practical knowledge aboutprocesses and interactions.

* Corresponding author. Tel.: +49-531-391-5608; fax: +49-531-391-8170.

E-mail addresses: [email protected] (R. Seppelt),[email protected] (A. Voinov).

0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.

PII: S0 304 -3800 (01 )00455 -0

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R. Seppelt, A. Voino� / Ecological Modelling 151 (2002) 125–142126

In this context every model consists of a simula-tor which represents the environmental processesand connects control parameters (forcing func-tions, inputs) to output variables (indicators). Theformer ones, the control parameters, define man-agement strategies in terms of the model vari-ables. The latter ones, output variables, are usedto describe the system behavior and to assess theconsidered management strategies.

To analyze the system and define most appro-priate management practices, a number of scenar-ios are formulated and then fed into the model.The output results are then compared to choosethe one that fits the requirements, the manage-ment goals best of all. The formulation of amanagement scenario, assessment and compari-son of results requires considerable efforts. In-stead of running numerous scenarios through themodel and then comparing the results (see forinstance Costanza et al., 2001), we may formulatea certain goal that we want the ecosystem toreach, and then let the computer sort through thenumerous parameter and pattern combinations toreach that goal. In this case the variations ofparameters and functions in the model input (con-trol) are performed automatically, as well as theprocessing of the output. The core of this processis the algorithm of numerical optimization, whichmakes the decision on how to define the nextscenario to analyze based on the available infor-mation about the results of previous model runs.This optimization procedure connects the scenar-ios, the simulation process and the performancecriterion. Algorithms of optimization are capableof performing a systematic search in the space ofcontrol variables to find an input vector whichcontrols the systems in the desired way, specifiedby the goal function.

In most cases to assess the outcome of a sce-nario in terms of environmental, economic, toxicor social aspects, more than one output variableneeds to be considered. To compare differentsimulation scenarios we then need to integrateoutput variables into a scalar value (or to analyzea multidimensional decision problem). This func-tion that we choose to integrate or aggregate theseveral output variables is called a goal functionor performance criterion and is a mathematical

formalization of the state of the system thatshould be maximized or minimized to reach thedesired state.

2. Spatial optimization in environmental modeling

2.1. State of the art

Table 1 summarizes recent publications on spa-tial ecosystem management based on simulationmodels. We look at the model structure, the con-trol variables, the goal function, the processes,and the spatial database used. The last columnlists the location of the study area, if any.

Scope of ecosystem management problemsranges from forest management and timber har-vest (Loehle, 2000; Tarp and Helles, 1997) toagricultural problems (Nevo and Garcia, 1996;Seppelt, 2000; Makowski et al., 2000) to generalissues of land use change (Martınez-Falero et al.,1998), and to habitat suitability (Bevers et al.,1997). The models used differ in terms of mathe-matical structure. Modeling methodology rangesfrom aggregated dynamic models based on differ-ence equations of exponential growth (Bevers etal., 1997; Loehle, 2000) to complex models basedon systems of non-linear differential equations(Seppelt, 2000; Randhir et al., 2000). In terms ofoptimization methodology one can find a broadspectrum of approaches.

2.2. Problem classification

The last point is explained, if one analyzesenvironmental models in terms of mathematicalstructure. The optimization techniques in environ-mental modeling still differ considerably from oneproblem to another. This may be in contrast toengineering problems, where the optimizationmethods are usually the same as long as the modelstructure is defined by a system of ordinary differ-ential equations, (Bulirsch et al., 1993). This espe-cially so, when spatial problems are formulatedand the mathematical heterogeneity of the simula-tion model increases (Seppelt, 2000).

Complexity of an optimization task depends ontwo factors: the complexity of the ecosystem

R. Seppelt, A. Voino� / Ecological Modelling 151 (2002) 125–142 127

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model (number of state variables, degree of non-linearity etc.) and the spatial complexity (size ofstudy area, grid cell size, number of spatiallyinteracting processes). The more complex the sim-ulation model is and the more spatial relation-ships are considered, the lower are the chances tosucceed in the optimization. For such complexmodels, scenario analysis is usually the only feasi-ble method. Fig. 1 shows a diagram of thisrelationship.

Scenario analysis compares the outcome of agiven number of scenarios, which are identifiedwith possible management strategies. Optimiza-tion procedures perform a systematic search overthe whole control space automatically. Whereas,optimization tasks require much less effort inpreprocessing and formulating the numerous sce-nario options, their computational complexity isincredibly high. The models described in Table 1are considered in an optimization context. There-fore, they all required certain simplification oraggregation to allow optimization (Fig. 1). Thiswas achieved either by aggregation of the modelor aggregation of the study area, or both.

The only other option would be to improve theoptimization procedures. Apparently, there is notmuch that can be done in this field. Nevertheless,we have tried to combine some well known proce-dures of optimization in attempt to increase theiroverall efficiency.

3. Problem formulation

3.1. Study area

For the development of an appropriatemethodology for optimization procedures for spa-tial landscape models we focus on the problem ofoptimum land use patterns and optimum fertiliza-tion in a mainly agricultural region in SouthernMD, USA. We have focused our studies on theHunting Creek Watershed which is located en-tirely within Calvert County in MD, USA. The22.5 km2 large study area belongs to the drainagebasin of the Patuxent River (2356 km2) which isone of the major tributaries of the ChesapeakBay. Soil types are well drained, mostly severlyeroded soils that have a dominantly sandy clayloam to fine sandy loam subsoil (Soil Survey,Calvert County, MD, USA, July 1971, 76 pp.).The annual rainfall varies between 400 and 600mm. Main land use of the watershed are forestand agricultural habitats. Rapid populationgrowth, development and change in land use andland cover have become obvious features of thelandscape.

3.2. Modeling framework

The ecosystem model used for the regionalmodel covers the processes of hydrology (aboveground, unsaturated soil zone, groundwater),macrophytes and consumers as well as and nutri-ent cycling (Voinov et al., 2000).

A conventional approach is to regionalize themodel based on spatial aggregation to largerunits, called elementary landscapes, elementarywatersheds, elementary areas of pollution or hill-slopes (Beven and Kirkby, 1979; Krysanova et al.,1989; Band et al., 1991; Sasowsky and Gardner,1991). These units are considered homogeneousand form the basis for the hydrologic flow net-work. The smallest homogeneous unit may becalled ecotope (Naveh and Lieberman, 1984).

If we are to consider scenarios of land usechange, generated by the economic consider-ations, which were not envisioned in the design ofthe elementary spatial units this approach is inap-propriate. The boundaries between spatial units

Fig. 1. Concepts of regional optimization solutions as found inrecent literature.


are fixed and cannot be modified during thecourse of the simulation, which may be somewhatrestrictive.

In our approach the modeled landscape is parti-tioned into a spatial grid of square unit cells. Wepresent the landscape as a grid of relatively smallhomogeneous cells and run simulations for eachcell with relatively simple rules for material flux-ing between the cells (Sklar et al., 1985; Burke etal., 1990; Costanza et al., 1990; Engel et al., 1993;Maxwell and Costanza, 1995). This approach re-quires extensive spatial data sets and high compu-tational capabilities in terms of both storage andspeed. Spatial complexity increases cf. Fig. 1.However, the approach allows quasi-continuousmodifications of the landscape, where habitatboundaries may change in response to socioeco-nomic transformations. This is one of the prereq-uisites for spatial optimization analysis, since itallows one to modify the spatial arrangement ofthe model endogeneously, within the simulationprocedures. With this approach, the model buildson the format of a raster-based geographic infor-mation system (GIS), which is used to store all thespatially referenced data included in the model.

3.3. Model description

The model is designed to simulate a variety ofecosystem types using a fixed model structure foreach habitat type. The model captures the re-sponse of macrophyte and algal communities tonutrient concentrations, water and environmentalinputs. These processes are driven by hydrologicalalgorithms for upland, wetland and shallow-waterhabitats. It explicitly incorporates ecological pro-cesses that determine water levels or content ofsurface water and the saturated and unsaturatedsoil zone, plant production, nutrient cycling asso-ciated with organic matter decomposition andconsumer dynamics. Therefore, the simulationmodel for a habitat consists of a system of cou-pled non-linear ordinary differential equations,solved with a 1 day time step.

The model is hierarchical in structure, incorpo-rating the ecosystem-level unit model that is repli-cated in each of the unit cells representing thelandscape. Although, the same unit model runs in

each cell, individual models are parameterizedaccording to habitat type and georeferenced infor-mation for a particular cell. The habitat depen-dent information is stored in a parameterdatabase, which includes initial conditions, rateparameters, stoichiometric ratios, etc. The habitattype and other location-dependent characteristicsare referenced through links to GIS files.

The unit models in each cell exchange matterand information across space. Surface and subsur-face hydrology define the horizontal fluxes. Thisjoins the unit models together. The spatial hydrol-ogy module calculates the amount of water fluxedover the surface and in the saturated sediment.The fluxes are driven by cell-to-cell head differ-ences of surface water and saturated sedimentwater, respectively. Dissolved and suspended ma-terial (nutrients) is carried by water fluxes betweencells. At each time step, first the unit modelupdates the stocks within each cell due to verticalfluxing and then cells communicate to flux matterhorizontally, simulating flows and determiningecological condition across the landscape.

This model approach has been used to con-struct the Hunting Creek Model (HCM), a sub-watershed of the Patuxent. The local dynamics inthe HCM were similar to those developed inPatuxent Landscape Model (Voinov et al., 1999),but the spatial implementation, defined by theStudy Area, and the spatial resolution were differ-ent. By focusing on a smaller subwatershed, wecould make many more model runs, better cali-brate the model, and refine our understanding ofsome of the crucial ecological processes and spa-tial flows in the ecosystem. A rather small studyarea was especially essential for the optimizationprocedures that require numerous computer runsof the model.

3.4. General notations

The observed region is denoted by a set ofdiscrete grid points R={(i, j ), ni� i�Ni ; mj�j�Mj}. A grid cell of the map is denoted by z�R.Seven different land use type are considered: soy-beans, winter wheat, corn, fallow, forest, citiesand rural areas. c(z) denotes the land use (orhabitat type) in cell z. Controllable cells are L=


{soybeans, winter wheat, corn, fallow, forest}.Rc={z�R �c(z)�L} denotes the grid points withcontrollable cells c(z)�L. Further general nota-tions are:

is the yield of crop c (if any) har-H(c, z)�R

vested from cell zN(z, t)�R is the amount of nitrogen that es-

capes from cell z at time tis the amount of fertilizers ap-F(c, t)�R

plied for the habitat type c attime t

Without loss of generality we focus on theestimation of the total required amount of fertil-izer. As time of fertilization is a crucial parame-ter for reduction of nitrogen leached, we use thedistribution patterns recommended by the bestmanagement practices recommendations for thetemporal allocation of the total fertilizeramount. These are similar to the results de-scribed in Seppelt (1999) derived from solutionsof optimum control problems.

3.5. Global performance criteria

Our goal is to find out what is the optimumland use pattern and what should be the strat-egy of fertilizer application to reduce nutrientoutflow out of the watershed and increase yield?For a definition of a performance criterion onehas to take into account crop yield, fertilizerapplication and nutrient outflow. The first twofactors are easier to compare if we operate interms of prices. The revenue from the yield overthe whole study area is A=�z�R pH(c)H(c, z)where pH(c) is the current market price of cropc. The price of fertilizers applied is thenB=pF�z�R �1� t�T F(z, t), where pF is the unitprice of nitrogen fertilizer. Obviously A is to bemaximized while B is to be minimized, whichmeans that A−B is to be maximized. A−Bdenotes the ‘economic’ part of the goal function.

There are different ways of modeling the ‘eco-logical’ part of the performance criterion. Onepossibility is to take into account the totalamount of nutrients generated by all the cells in

the study area, C=�z�R �1� t�T N(z, t). This isthe distributed nutrient leaching. More realistic,and comparable to measurements of gaugingstations is to consider the amount of nitrogen inthe outlet cell of the watershed z0. This casetakes into account the compensation mecha-nisms of uptake along the pathways of nitrogenwhile it travels across the watershed and esti-mate the actual water quality in the river estu-ary: C=�1� t�T N(z0, t).

In both cases C is to be minimized. The cru-cial problem is to integrate the ‘ecological’ partC and the economic part A−B to a scalar goalfunction. For this purpose C is to be expressedin units that can be compared with the dollarmeasure that we derived in A−B. Without go-ing into any further details at this point, let ussimply assume a weighting coefficient �, and for-mulate the goal function as

J=A−B−�C. (1)

The optimization task is:

Task 1: Find maps c* and F* which maxi-mize J�max.

3.6. Local performance criteria

A second approach is to define the local per-formance criteria in each grid cell. This is struc-turally a different way, because it aims to mapthe regional goal function onto the processes ina grid cell. The basic idea is to set up character-istic functions, which are part of the perfor-mance criterion and are spatially dependent, aspart of the goal function.

Let us analyze, how the performance criterionfrom (Eq. (1)) may be reformulated in this con-text: the goal function is defined for every gridcell z, as a function of z. Let:

be the local profit fromA(z)=pH(c)H(c, z)crop yieldbe the local cost of fer-B(z)=pF�t F(z, t)tilizers applied, andbe the amount of nitro-C(z)=�t N(z, t)gen leaching producedlocally


A(z), B(z) and C(z) are now calculated for aspecific cell. They do not incorporate integrationover the entire study area. Based on this the localgoal function for every cell is then:

J(z)=A(z)−B(z)−�C(z). (2)

The optimization problem is now reduced tothe optimization of land use and fertilizer applica-tion for every grid cell. This approach neglectsany neighborhood effects.

Task 2: For each cell z�Rc estimate c*(z)�Land F*(z) which maximize: J(z)�max.

Obviously, the solutions from Task 2 are differ-ent from those of Task 1. The goal function ofTask 2 neglects any neighborhood relationships inthe control variable maps. The question is, howlarge are the differences in the solutions, or dothese solutions have more in common than onewould expect? Next chapter will show that thenumerical effort for the estimation of solutionsfor Task 2 is orders of magnitudes lower com-pared to Task 1.

However, we have to study the relationship ofthe global and the local solutions of the optimumcontrol problem. In Voinov et al. (2001) we as-sumed why it may be acceptable to do optimiza-tion in terms of Task 2: Task 2 formulates a worstcase scenario. Considering the nutrient outflowthe goal function from Task 1 takes into accountthe capability of the retention function of anecosystem, which models the throughflow of nu-trients more precisely. The goal function of Task2 on the other hand performs a worst case upperestimate of the net nutrient out flow. In thiscontribution we will study the methodologicalframework in more theoretical detail. Finally, weshould be aware of getting multiple solutions tothe problem.

4. Optimization methodology

4.1. Numerical effort

Task 1 and 2 can be classified a combinatorial

optimization problem. We are to sort through allthe possible combinations of six available land usetypes over the study area. Assuming a homoge-neous land use and discrete stages of possibletotal fertilizer input, for say six stages F�{0, 25,50, 75, 100, 150 kg/ha}, Task 2 leads to I2��F ��L �=36 combinations. Considering that no fer-tilization takes place for c�{forest, fallow} we getI2=26 combinations.

The number of possible combinations for Task1 depends on the size of the study area I1=�F ��L��Rc �. For example, for the Hunting Creek(Voinov et al., 2001) watershed, which is repre-sented by �Rc �=1681 controllable cells of 200×200 m2, there are I1=3.2×1064 different land usepatterns. For a smaller subwatershed that covers25% of the total area and �Rc �=483 controllablecells there are still I1=4.7×1053 possible land usepatterns.

4.2. Grid search strategy on local problem

A simulation period of 551 days is considered.It covers the growth periods of all crops, startingwith soybeans and ending with the harvest ofwinter wheat. Before planting and after harvestinga crop, fallow is assumed to be the land use typeof the cell. Twenty minutes processor time isrequired for one simulation run of the entiremodel on a Sparc Ultra 10 Workstation. Fromthis follows that a simple search through theentire control space of Task 1 is hardlyappropriate.

The first step is to solve Task 2. The approachmakes use of the ability of the landscape modelformulated within the framework of the SpatialModeling Environment (SME, see Maxwell andCostanza, 1995) to deal with spatially distributedinformation and to perform mathematical opera-tions on maps. The terms A(z), B(z) and C(z) ofthe local goal function can be formulated as mapsthat are calculated using the spatially explicitmodel.

The solution of Task 2 performs a grid searchthrough the entire control space assuming a ho-mogeneous land use and identical fertilizeramounts for each cell. A series of maps for differ-ent combinations of crops and fertilizer applica-


tion rates are generated and stored. MaximizingJ(z) for every grid cell depending on the pre-cal-culated maps of the goal functions A(z), B(z) andC(z) solves the optimization problem. The resultis a pair of land use and fertilizer maps thatoptimize the local performance criterion. This pairis then fed into a spatial simulation that is used tocalculate the global performance criterion Eq. (1).

4.3. Disturbing a solution: Monte-Carlo’ssimulation

One way of testing the results from the localoptimization approach is to run a Monte-Carlo’ssimulation based on the optimum land use andfertilization pattern. This Monte-Carlo’s simula-tion ‘disturbs’ the optimum solution stochasticallyand gives an estimate of how close the localsolution is to the unknown optimum.

For the given problem of land use and fertiliza-tion optimization the algorithm can be describedmathematically by a three stage stochastic pro-cess. Let Z1(z)� [0, 1] be a random variable. Anew land use is generated randomly if Z1(z)�p1

for cell z. This is done by a stochastic variableZ2�{corn, soybeans, wheat, fallow, forest} suchthat

P Z2(z)=c(z)�Z1(z)�p1 = f(c).

For every cell z the stochastically generatedland use c follows a distribution which is definedby a density function f(c). The density functionf(c) is constant for the entire region and may begenerated by a stochastic process, started beforegenerating the stochastic land use map. Or, it maybe derived from the distribution of a known landuse pattern, for instance the optimum solution.With this set up three different types of Monte-Carlo’s simulations can be distinguished:� p1=1, f(C) generated stochastically: land use

patterns are generated from scratch, withoutany knowledge about a possible land usepattern;

� p1=1, f(C) derived from known land use pat-terns such as historic data or optimum localsolutions: land use patterns are based onreallocations.

� p1=0.01x and f(C) derived from optimum so-

lution: land use patterns are disturbances ofoptimum solutions by a certain percentage x.

4.4. Genetic algorithm of global problem

As we have seen in the former sections theglobal optimization problem cannot be solved interms of a combinatorial optimization problem.The application of iterative procedures, which areusually based on gradient search algorithms maybe inappropriate since in many cases the deriva-tive cannot be estimated neither analytically nornumerically. This is because of the complexity ofthe ecosystem model and the combination of dis-crete and continuous control variables.

Genetic algorithms (GA) offer a solution to theoptimization problem based on the global perfor-mance criteria. The first step of GA is to define arepresentation of the control variables of the opti-mization problem to a genome. Based on the ideaof ‘survival of the fittest’ a stochastically gener-ated first population of a distinct number of indi-viduals runs through an evolutionary process. GAdetermines which individuals of a populationshould survive, which should reproduce andwhich should die. New individuals are createdbased on the operations of cross over, mutationand gene migration.

An application of GA to an optimization prob-lem requires three steps;1. Definition of a representation.2. Definition of the genetic operators.3. Definition of the goal function.

The goal function for the problem in hand isgiven by Eq. (1). Common libraries support thesecond step. We used a C++ Library of GeneticAlgorithm Components (GALib), see Wall (1996).The first step, the representation of the land useand fertilization patterns to a genome is defined asfollows� Each controllable cell z�Rc defines a single gene

gk, k=1, …, �Rc �. The genome has the length�Rc �. Urban and rural cells have no representa-tion in the genome as these are non-controllablecells.

� The location of the gene in the 1-dimensionalarray genome string is identified with the loca-tion of the cell in the grid map k�z= (i, j )�Rc.

��

�

��

�


Fig. 2. Framework for the estimation of the spatial optimum concept for optimization land use patterns and related control variablesfor Tasks 1 and 2.

� The land use type is a discrete attribute of acell or gene. The fertilization amount can beapproximated by a discrete value, see previoussection. The control variables are thereforedefined by an allele set L= (c, F(c)) for eachgene.Before starting the GA an initial population

has to be generated, usually performed by cloninga given individual. We used different populationsderived from the three stage stochastic processused for the Monte-Carlo’s-analysis. The initialpopulation therefore bases on a stochastic varia-tion of a local optimum land use and fertilizerpattern parameterized by the probability p1.

Based on the GALib-Library the global opti-mization problem is solved. For each generationthe algorithm creates an entirely new population

of individuals by selecting from the previous pop-ulation then mating to produce the new offspringfor the new population. Each new individual ofthe population requires a run of the entire spa-tially explicit simulation model. The ‘survival offittest’ strategy is implemented by evaluating thegoal function Eq. (1). This process continues untilthe stopping criteria are met (determined by theterminator).

Fig. 2 summarizes the methodological conceptdescribed in a flowchart. There are two mainbranches which are used. Branch 1 starts with theestimation of characteristic function maps, whichare the basis for local optimization. Branch 2shows, how to work on the optimization problemwithout any knowledge of an appropriate initialguess of the solution.


5. Results

5.1. Local optimization

We focus on the methodological aspects ofthe proposed framework. Results in terms oflandscape ecological issues will be shortly de-scribed in the following section. For detaileddiscussion of these results see also Voinov et al.(2001).

We start the analysis of the results withbranch 1 in the flow chart of Fig. 2. Homoge-neous control variables for each possible landuse type and a certain fertilizer amount are setup: c(z)=c0�L, F(z)=F0 for all z�Rc. Runningsimulations runs within SME for all possiblecombinations in the step ‘grid search’, we derivemaps A(z), B(z) and C(z) which are used toestimate the local optimum solution.

The estimation of local optimum land usemaps does not require any computational effort.It sorts through all possible combinations in themaps A(z), B(z) and C(z). This easily allowsparameter studies for the weighting parameter �.Fig. 3 shows the results of a parameter studyfor the Hunting Creek watershed in an aggre-gated way: the number of different land usetypes as a function of � is plotted. Fig. 4 showsthree maps of optimum land use derived out of

this parameter study: optimum land use patternsfor the �-values 0.0 (upper left), 0.1 (upperright), and 5.0 (lower left) are displayed.� A zero value of � leads to a pathologic solu-

tion: the optimum solution is to plant themost valuable crop in the entire study area.Only villages, urban area and open water re-mains the same (non-controllable cells).

� An increase of � introduces forest into theland use pattern. The more nutrient outflowis ‘punished’ by an increase of �, the higheris the fraction of forest in the study area.

� With an increase of � agricultural cellschange to crops with a better nutrient-up-take/yield-efficiency. This depends of the mar-ket prices of the crop.Due to the minimal computation effort, these

steps are embedded in the GIS Front end Ar-cView to offer simple result visualization. Fig. 4shows a screen copy of the ArcView displaywith a parameter study of the land use distribu-tion in the Hunting Creek Watershed.

5.2. Monte-Carlo’s simulations

An analysis of these results can be performedby a Monte-Carlo’s simulation. Fig. 2 showsMonte-Carlo’s simulations in two boxes in themid column:� Monte-Carlo’s simulation ‘from scratch’ cor-

respond to the first box of branch 2, comparefirst item in section above. It performs ananalysis of the variability of the entire pro-cess and sets up an initial population for thegenetic programming algorithm.

� The step after local optimization, which cor-responds to the reallocation p1=1 or the dis-turbance of an optimum solution p1�1, item2 and 3 of above section.Fig. 6 summarizes the results of a couple cof

different Monte-Carlo’s runs. We used a smallersub watershed for these analyses. This allowedus to generate more realizations. The firstcolumn in Fig. 6 shows histograms of the globalgoal function results according to Eq. (2). Thegoal function values are normalized by the goalfunction value derived by the local optimization.Values below unity denote simulation runs

Fig. 3. Results of parameter study of optimum land use maps.The number of forest, fallow, soybean, corn and wheat cells inthe Hunting Creek watershed are plotted against �.


Fig. 4. Local optimum land use maps for �=0.0 (a), 0.1 (b) and 5.0 (c).

where the global goal function returns valuesbelow the local optimization. Values above unitydenote that the local solution was improved. Thesecond column shows the histogram of the yieldin the study area in U.S. $/m2 for comparison.

Row one can be identified as Monte-Carlo’ssimulation from scratch: the distribution of landuse types f(c) as well as the land use type and thefertilizer amount for each grid cell are generatedby random. The second row derived f(c) from thelocal optimization distribution. Row three andfour of Fig. 3 disturb the local optimization landuse pattern with the probability p1=0.1 and 0.01.Obviously, a stochastic generation of land usepatterns hardly identifies a map, which is opti-mum in terms of the performance criterion. Fromthe last two rows of Fig. 6 one can derive, that anincrease in the probability p1 immediately leads tosolutions which are far away from the solutionfrom local optimization. The local optimization

approach seems to be very close to a globaloptimum. However, there are land use patternswhich are ‘better’ in terms of the global optimiza-tion. Are these land use patterns the ones thattake into account neighborhood relationships ofthe control variables?

5.3. Statistical analysis

What are the driving parameters for the opti-mum solutions? The question can be answered bya simple bivariate correlation analysis based onthe resulting land use and fertilizer maps and thespatial input data of the model. Table 2 summa-rizes the results of four correlation analysis stud-ies of the optimum solutions to the set of weights�=0.0, 0.1, 0.2 and 1.0. However, the sample sizeis high (1690) we used non-parametric correlationaccording to Spearman–Rho (Davis, 1984) be-cause a normal distribution of the resulting


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parameters cannot be assumed. Input data mapsare the soil and the elevation maps. Parameters ofthe soil map are porosity (m3 pore space per m3

sediment), infiltration rate (m/day), field capacity(m pore space per m sediment), percolation rate(m/day), horizontal hydrologic conductivity (1/day). From the elevation map the aspects andslope are derived using GIS functions.

One general result is, that the fertilizer applica-tion always correlates with the habitat type: eachcrop gets its specific optimum amount of fertilizer.

Setting �=0.0 neglects any ecological issues ofagricultural production and fertilization. The op-timum solution is a land use map with the mostvaluable crop and a high fertilizer amount. Thecorrelation analysis shows, that almost none ofthe important parameters for nutrient transport insoil are responsible for the land use map.

An increase of the �-value shows significantcorrelation to the parameters of the soil map. For�=1.0(z) and F(z) show significant correlationwith all parameters of the soil map. Weak correla-

Fig. 5. Screen shot of ArcView front end with the local optimization project.


tions are also identified to the parameters of theelevation map, especially to aspect and slope.This shows that spatial relationships (to neigh-borhood cells) find their interpretation in the lo-cally optimum solution from the spatially explicitmodel!

Note that due to the large sample size, a smallsample set is sufficient for a significant correla-tion. In general all correlation values are low.However, statistical analysis gave a validation ofthe derived results.

5.4. Genetic algorithms

The only way to answer the question about theimportance of neighborhood relationships in thecontrol variable maps is to set up an optimizationprocedure which can uses Eq. (2) for assessment.As we have seen from the results of the Monte-Carlo’s simulation, the application of GA basedon an initial population from scratch will fail,because the variability of a population fromscratch is much too broad. GA needs too muchiterations to converge to the solution we derivedby the local optimization approach (compare toFig. 2 upper right).

We used a smaller subwatershed of the Hunt-ing Creek for a detailed study of this behavior.Fig. 5 displays the convergence process of a GArun from scratch and the GA run from localoptimum (smaller graph). Note, it took 300 gen-erations (4500 simulation runs) to achieve 80% ofthe goal function value. For this reason the localoptimum solution is used for the generation ofthe initial population, see Fig. 2 (lower part, midcolumn).

Configuring parameters of GA algorithms ismore an art than a science (Wall, 1996). Fromthe former results we can generate certain rulesfor the parameterization: a too broad variety inthe initial population takes us away from theoptimum. We set p1=0.01 for the stochastic gen-eration of the initial population. This has to beassured within each step when generating a newpopulation: mutation probability should be muchsmaller than p1, cross-over population should beequal to zero. To enable modifications in thepopulation we set migration probability to a high

value (0.9). The graph in Fig. 7(b) shows theresults of the GA run started from the localoptimum solution. The GA process clearly sepa-rates from the initial population and improvesthe optimum solution by 2%.

What are the changes compared to the localoptimum solution? We compiled two maps toshow the differences in each grid cell based onthe best generation of GA set up by 15 individu-als. Fig. 8 shows the results. The map in Fig. 8(a)shows the average fertilizer difference. The mapin Fig. 8(b) presents the modified land use cells.

Only a small number of cells are changed: 43out of 513 cells (8%). The distribution of land usetypes stayed the same. The global optimizationby GA performed a reallocation of habitats. Weexpected a couple of solutions, which can im-prove the local solution. A complex problem likethis might have multiple solutions. This wouldhave led to various realizations (of habitat andfertilization maps) which change different cellscompared to the initial solution from the localoptimization. However, the map (b) shows thatnearly all of the individuals in the best generationmodified the same grid cells. Most of themodified cells belong the class where more then13 out of 16 individual changed the cell in thesame way.

An explanation why these are the crucial cellsis hard to derive by statistical approaches, how-ever that would allow further improvement ofspatial optimization. Due to a broad spectrum ofinput data/maps and due to a very complex net-work of dynamic and spatial process in the simu-lation model, no significant correlation can beidentified. Using principle component analysis wewere not able to reduce state space for analysis.One possible approach is to perform a bivariatecorrelation of c(z0) and F(z0) to the neighbor-hood cells, (north-east, north, …, south-westc(z0)). Two topics make us expect illusory corre-lations: first, analysis of the entire region, usingevery grid cell (and its neighbors) as a repetitionjust gives a general answer, which is mainlydriven by a global aspect of the global slope ofthe catchment. Second, SME uses linkages be-tween cells which are no direct neighbors for thehydrological sub module (Voinov et al., 1999).


Fig. 6. Resulting distributions of goal function values (left column) and total Harvest biomass (right column) of Monte-Carlo’ssimulations for different stochastic processes (rows).

6. Outlook and discussion

Note that the only simplification, which hasbeen introduced to derive a solution to this verycomplex optimization task, was to assume a weakdependency of the single parameters of the con-trol variable maps upon the neighborhood. Thisallows us to set up a characteristic function foreach grid cell of the model. In this contributionwe focus on simple goal functions which assessthe state of a certain grid cell only. However, thisapproach may be extended by the integration ofthe neighborhood of a given grid cell. For in-stance, if R(z0)={z �z borders z0} denotes theneighborhood of z0, the proposed framework iscapable of dealing with goal functions defined onR(z0). This allows the assessment of habitat struc-

tures in the goal functions. Habitat suitability ofcertain species may be assessed with thisextension.

The core question which lays behind the consid-eration of neighborhood relationships of the con-trol input in ecosystems models, is how muchdoes the synergistic effect of land use changematter for a given watershed? It is an excitingquestion to be investigation in the framework ofoptimization, see for example (Muller, 1998).

Building upon the background of the SpatialModeling Environment the proposed frameworkis capable of answering different questions oflandscape management. Run time solutions foroptimization problems can be derived (for exam-ple in the ArcView front end) if it is possible to setup a local optimization criterion and if a global


Fig. 7. Development of global performance criteria values during Genetic programming process: (a) the GA process ‘from scratch’;(b) the GA process started from a local optimum solution with a stochastically ‘disturbed’ of ‘mutated’ population.

Fig. 8. Analysis of GA results. The maps show the difference from one generation (15 individuals) of the GA to the local optimummaps for the sub watershed: (a) fertilizer amounts; (b) this maps counts how may individuals of the population have a different landuse type than the local optimum solution. The small map in the center displays the extent and location of the subwatershed withinthe Hunting Creek watershed.


solution is not required. In this case the computa-tion effort does not significantly depend on thesize of study area. On the other hand, the use ofGA for global optimization offers the ability ofparallelization in the optimization process. Insummary, this framework seems to be applicableto a large class of problems in landscape ecology.It offers an integration of simulation models andenvironmental management.

Acknowledgements

This research has been supported by the Ger-man Research Foundation (Deutsche Forschungs-gemeinschaft DFG) SE 796-1/1 and a grant fromthe U.S. Environmental Protection Agency’s Sci-ence to Achieve Results (STAR) program(R827169). We thank Thomas Maxwell for sev-eral upgrades in SME that were required forlinkages to the optimization tools. Additionalthanks are due to Dagmar Sondgerath for hervaluable assistance concerning the statistical anal-ysis. Results in this contributions were obtainedusing the GALib genetic algorithm package, writ-ten be Matthew Wall at the Massachusetts Insti-tute of Technolgy.

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