OIKOS 100: 209–222. 2003

Minireviews provides an opportunity to summarize existing knowledge of selectedecological areas, with special emphasis on current topics where rapid and significantadvances are occurring. Reviews should be concise and not too wide-ranging. All keyreferences should be cited. A summary is required.

MINI-REVIEW

Using pattern-oriented modeling for revealing hidden information:a key for reconciling ecological theory and application

Thorsten Wiegand, Florian Jeltsch, Ilkka Hanski and Volker Grimm

Wiegand, T., Jeltsch, F., Hanski, I. and Grimm, V. 2003. Using pattern-orientedmodeling for revealing hidden information: a key for reconciling ecological theoryand application. – Oikos 100: 209–222.

We suggest that the conscious use of information that is ‘‘hidden’’ in distinctstructures in nature itself and in data extracted from nature (=pattern) during theprocess of modeling (=pattern-oriented modeling) can substantially improve modelsin ecological application and conservation. Observed patterns, such as time-seriespatterns and spatial patterns of presence/absence in habitat patches, contain a greatdeal of data on scales, site-history, parameters and processes. Use of these dataprovides criteria for aggregating the biological information in the model, relates themodel explicitly to the relevant scales of the system, facilitates the use of helpfultechniques of indirect parameter estimation with independent data, and helps detectunderlying ecological processes. Additionally, pattern-oriented models produce com-parative predictions that can be tested in the field.We developed a step-by-step protocol for pattern-oriented modeling and illustrate thepotential of this protocol by discussing three pattern-oriented population models: (1)a population viability analysis for brown bears (Ursus arctos) in northern Spain usingtime-series data on females with cubs of the year to adjust unknown model parame-ters; (2) a savanna model for detecting underlying ecological processes from spatialpatterns of tree distribution; and (3) the incidence function model of metapopulationdynamics as an example of process integration and model generalization.We conclude that using the pattern-oriented approach to its full potential will requirea major paradigm shift in the strategies of modeling and data collection, and weargue that more emphasis must be placed on observing and documenting relevantpatterns in addition to attempts to obtain direct estimates of model parameters.

T. Wiegand and V. Grimm, Dept of Ecological Modelling, UFZ-Centre for En�iron-mental Res., PF 500136, DE-04301 Leipzig, Germany (towi@oesa.ufz.de). – F. Jeltsch,Inst. for Biochemistry and Biology, Uni�. of Potsdam, Maulbeerallee 2, DE-14469Potsdam, Germany. – I. Hanski, Dept of Ecology and Systematics, Di�ision ofPopulation Biology, P.O. Box 65 (Viikinkaari 1), FIN-00014 Uni�. of Helsinki,Finland.

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Accepted 15 August 2002

Copyright © OIKOS 2003ISSN 0030-1299

Over the past three decades, models have been increas-ingly used in applied ecology and, in particular, inconservation biology. For example, by the late 1970sresults from the island biogeographic theory werewidely applied in the design of nature reserves (Meffeand Carroll 1997). During the 1980s population viabil-ity analysis became an important tool in assessing thelikelihood of a population becoming extinct and PVAshave subsequently been used to aid management deci-sions about threatened species (Boyce 1992, Beissingerand Westphal 1998, Noon et al. 1999, Beissinger andMcCullough 2002). Since the early 1990s, the ideas ofmetapopulation dynamics have rapidly gained promi-nence in the literature and now form the conceptualframework for managing populations living in frag-mented landscapes (Levins 1970, Gilpin and Hanski1991, Hanski and Gilpin 1997, Hanski 1999). Since the1990s grid-based simulation models have increasinglybeen used to investigate long-term dynamics and man-agement of plant communities and animal populations(Wiegand et al. 1995, in press, Jeltsch et al. 1996, 1998,Jeltsch and Moloney 2002). Most recently, decisiontheory has been advanced as a tool to aid in reservedesign and in assigning relative ranking of managementoptions (Shea et al. 1998, Drechsler 2000, Possinghamet al. 2002).

Despite the growing use of models in conservationecology and the fact that many attempts have beenmade to compensate for the chronic lack of data byrelying heavily upon ecological theory and modeling(Doak and Mills 1994), the use of theory in makingactual management decisions about real species andcommunities has come under increasing attack (Caugh-ley 1994, Doak and Mills 1994, Harcourt 1995, Shea etal. 1998) and early enthusiasm has been tempered byseveral problems. Besides of unrealistically high expec-tations for the predictive power of population models(Wiegand et al. 1998, Burgman and Possingham 2000,Possingham et al. 2002), the application of theory isplagued by (1) the inherent lack and uncertainty ofdata, especially on endangered species, (2) but also bythe complexity of realistic simulation models, (3) prob-lems of error propagation, (4) missing criteria for ag-gregating the biological information, and (5) difficultiesin testing model predictions.

The poor quality of data used in most model applica-tions, although quite obvious, is frequently overlooked(Beissinger and Westphal 1998). It is not yet fullyaccepted that rarity of species itself precludes precisemeasurement of model parameters, and that in this casethere is no hope at all to accurately estimate parametervalues through direct measurement in the field. Becausetraditional measurement of parameter values reliesheavily on data obtained from a low hierarchic level ofpopulation dynamics, such as mortality rates, modelpredictions are usually quite sensitive to uncertainty inthe data. In principle, spatially-explicit and individual-

based simulation models are able to include manybiological details and may contain many parameters(DeAngelis and Mooij, in press). This can be a prob-lem, not only because of error propagation or lackingdirect estimates of model parameters, but also becausetheir complexity may prevent an exhaustive model anal-ysis (Beissinger and Westphal 1998). Models that aretoo complex may result because of missing criteria ofhow to aggregate biological information in the model(Grimm et al. 1996). On the other hand, models arecriticized for being poorly tied to applications (Caugh-ley 1994), mainly due to difficulties in adapting modelsof sufficient complexity to capture the relevant ecology.Again, we attribute these difficulties largely to missingcriteria of how to aggregate biological information inthe model (Grimm et al. 1996). As a consequence, themodel structure is often poorly adapted to the scales ofthe system investigated and models are difficult to test.One of our main concerns is that choosing an inade-quate model structure, introduced for reasons oftractability rather than biology (Wood 2001) or becauseof missing criteria for aggregating information in amodel, may cause a substantial loss or deformation ofthe original biological data. Especially under the cir-cumstances typical for conservation biology, wheremanagement problems force hasty decisions to be takenin spite of scarce data, we cannot afford such losses.

Mathematical modelers have increasingly used quan-titative statistical techniques to make rigorous inferencefrom biological pattern (Wood 1994, Harrison 1995,Hilborn and Mangel 1997, Johst and Brandl 1997,Lewellen and Vessey 1998, Blasius et al. 1999,Casagrandi and Gatto 1999, Doak and Morris 1999,Kendall et al. 1999, Bjørnstad et al. 1999, Briggs et al.2000, Claessen et al. 2000, Elliot et al. 2000, Turchin etal. 2000, Ellner et al. 2001, Fromentin et al. 2001,Turchin 2003). However, in conservation biology thepotentially rich source of data provided by patterns,such as time-series data (Wiegand et al. 1998), thatemerge as the high-level outcome of all processes ofpopulation dynamics and constraining factors (e.g.landscape structure, climate, management history) hasbeen recognized only sporadically and intuitively. Thepurpose of this paper is to make this intuition anexplicit modeling strategy that could be widely used. Byreflecting on what characterizes successful ecologicalmodels, Grimm et al. (1996) found that ‘‘the mostsuccessful models were those that took their orientationfrom [such] distinct patterns observed in nature’’. Such‘‘pattern-oriented models’’ (Grimm 1994, Grimm et al.1996) have several features in common: their structureis not arbitrary but is constrained by patterns of thereal system, they are explicitly linked to the relevantspatial and temporal scales of the system, and – mostimportantly – they are testable. Whereas Grimm et al.(1996) focused on model construction and theory, andWiegand et al. (in press) on uncertainty in spatially

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explicit population models, we here focus on applica-tion of pattern-oriented models and we generalize thepattern-oriented modeling approach into a systematicframework for exploiting the available biological dataat all steps of the modeling process: from the initialmodel construction to parameter estimation and todetection of deficiencies in the model structure andknowledge.

We provide a step-by-step protocol for applying thepattern-oriented approach and illustrate it by discussingthree pattern-oriented population models from appliedecology: (i) population viability analysis for brownbears (Ursus arctos) in northern Spain using time-seriesdata of females with cubs of the year for adjustingunknown demographic model parameters (Wiegand etal. 1998); (ii) a spatially explicit savanna model fordetecting underlying ecological processes from the spa-tial patterns of tree distribution (Jeltsch et al. 1999),and (iii) the incidence function model of metapopula-tion dynamics as an example for process integrationand model generalization (Hanski 1994, 1999). Wemake no attempt to review the literature on the role oftheory in ecological application and conservation, sincesuch reviews exist (Caughley 1994, Doak and Mills1994, Starfield 1997, Beissinger and Westphal 1998).

Pattern-Oriented Modeling

What is a pattern, and why do we need toconsider patterns?

Grimm et al. (1996) defined a ‘‘pattern’’ as a character-istic, clearly identifiable structure in nature itself or inthe data extracted from nature. A pattern is anythingthat goes beyond random variation and thus indicatesan underlying process that generates this pattern (Levin1992, Weiner 1995). Such patterns include time-seriesdata (Wiegand et al. 1998, in press), the distribution ofdispersal distances and the spatial pattern of speciesoccurrence in fragmented landscapes (Hanski 1994),wave-like pattern of the spread of rabies (Jeltsch et al.1997), the spatial pattern of savanna trees (Jeltsch et al.1999), and size-class distribution of Acacia trees (Wie-gand et al. 2000).

Such patterns represent high-level manifestations ofpopulation dynamic processes and are the outcome ofinterplay between demographic processes, dispersalcharacteristics and various constraining factors (e.g.management actions, or a climatic pattern). Therefore,empirically observed patterns contain a great deal ofinformation and memory about the history of the sys-tem. Because the patterns describe features of the sys-tem at a higher hierarchical level than are addressed bypossible model rules (e.g. individual-based rules vs pop-ulation level data), there are only limited possibilities toinclude this information directly into the rule set of a

model. Unfortunately, indirect methods of parameterestimation, a standard practice in many areas of sci-ence, which would allow the use of the ‘‘hidden’’ infor-mation are widely overlooked in conservation (but seeHanski 1994, Lindenmayer et al. 2000, McCarthy et al.2000, 2001). This is surprising since indirect methodsare commonly used in ‘‘hard sciences’’ such as physics,and even increasingly in mathematical biology (Burn-ham and Anderson 1998, Kendall et al. 1999, Wood2001, Turchin 2003). In conservation we could learnfrom this tradition to systematically use observed pat-terns for revealing information on processes andparameter values. Under circumstances of scarce data,the additional data provided by independent patternsmay be especially valuable and could greatly improvethe quality of model predictions and the general under-standing of the system. We therefore argue that there isan urgent need in ecological application and conserva-tion practice to consciously search for and measuresuch patterns, and there is an urgent need to developmethods for using these data.

The protocol for pattern-oriented modeling

The basic idea for using the data of observed patterns isto include in the model the constraining factors and theminimal set of processes that are necessary for repro-ducing the patterns and to systematically compare theobserved patterns with patterns produced by differentmodifications of the model (Fig. 1). In the following weformalize this idea to a four-step protocol of pattern-oriented modeling, which may be repeated in a cyclicmanner.

First step: aggregation of biological information andscales. Model construction should not only be guidedby the aim of the model and by the available knowl-edge, but also by patterns that can be detected in thesystem in question. The reason for this is that patternsare indicating both characteristic structures and pro-cesses of the system. By choosing a model structure thatin principle allows reproduction of the observed pat-terns, we achieve a structurally realistic model, i.e. amodel that contains key structural elements of the realsystem. Structural realism is achieved by the formula-tion of hypotheses on the structures and processesnecessary to reproduce the observed patterns. Thereby,hypotheses are automatically defined on the relevantspatial and temporal scales (Grimm et al. 1996). More-over, the observed patterns define a minimum level ofdetail that helps decide which processes should beincluded in the model. Given the aim of the modeling,one has to ask whether or not a particular process isindispensable for the creation of the pattern at the scaleof interest. In the context of conservation with limiteddata, it is important to focus on minimalistic models inwhich the number of parameters and processes are

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Fig. 1. Error propagation. (A) Non pattern-oriented modeling. Field investigations, additional information from literature, andguesses guide the construction of the model. Because poorly known parameters enter the model at a low hierarchical level, errorpropagation is unavoidable. (B) Pattern-oriented modeling. The hypotheses on parameters and processes are constrained by theobserved patterns, and comparison between the observed patterns and the patterns produced by the model restricts the range ofuncertain parameters and can be used for detecting the underlying processes that produce the observed patterns. The observedpatterns are compared with patterns produced by several alternative hypotheses on processes and/or a large number of plausibleparameter combinations. Because the high-level output of the model (the patterns), which contains relationships between themodel parameters, must match the observed pattern, error propagation does not occur. A helpful analysis of the model can beobtained by analyzing internal model relations (secondary predictions), and the model can be validated by comparing secondarypredictions with independent field data.

curtailed to what is absolutely necessary information.Otherwise the scarce information will not be sufficientto parameterize the model. However, one should avoidusing initially an exceedingly minimalistic model struc-ture, since this may bias model predictions. Formula-tion of alternative model structures – model variantswith nested complexity or model variants with alterna-tive hypotheses about the processes – provides a sys-tematic approach to finding an appropriate minimalisticmodel. If the simplest model does not reproduce anessential pattern it has to be replaced by a model witha more adequate structure, whereas alternative modelstructures that do not improve model performance canbe rejected (see below ‘‘cycles of pattern-oriented mod-eling’’, and step 1 of the savanna tree example).

Second step: determination of parameter �alues. Afterthe hypotheses on the processes have been formulatedand the appropriate modeling technique has been iden-tified, the values and the ranges of the model parame-ters have to be determined. This step does not differfrom any conventional parameterization of a popula-tion model.

Third step: systematic comparison between the ob-ser�ed patterns and the patterns predicted by themodel. In this step the model is applied with a largenumber of parameterizations and alternative hypothe-ses about the processes, and the predicted patterns aresystematically compared with the observed ones (Fig.1). The observed patterns acts as ‘‘filter’’ that sorts outimplausible parameter combinations and implausibleprocesses. Note that this technique is especially power-ful when multiple patterns are used (Kendall et al.

1999, Reynolds and Ford 1999, Ford 2000, Railsbackand Harvey 2002, Wiegand et al. in press). While itmight be relatively simple to reproduce one feature of asystem, the simultaneous fulfillment of several patternsdescribing different features of the system is by farnon-trivial. For a quantitative comparison, statisticalmethods are needed to decide whether or not a givenpattern matches the data (see e.g. Hilborn and Mangel(1997), Burnham and Anderson (1998), Wood (2001)for methods), and under which model the patterns aremost probable. In the time-series context, Kendall et al.(1999) and Turchin (2003) refer to such statistical de-scriptors collectively as ‘‘probes’’.

The third step necessarily includes a critical assess-ment of whether or not the pattern is genuine (Grimmet al. 1996). A reproduction (or ‘‘explanation’’) of apattern with a model does not guarantee that the modelactually identified the processes or mechanisms respon-sible for the patterns in reality (Levin 1992, Moloney1993, Jeltsch et al. 1999). To minimize the risk ofmisinterpretation one has to ensure that predicted pat-terns that match the observed patterns are neither asingular output of the model nor that they can bereproduced with arbitrary combinations of parametersand/or processes. If the model is able to reproducemultiple patterns, each describing a different feature ofthe system, the risk that many different processes mayhave caused this particular combination of observedpatterns is lower than when relying only on one pattern(Kendall et al. 1999, Wiegand et al. in press). At thisstage, the modeler must adopt the attitude of an exper-imenter (Grimm 1999, 2002) and systematically investi-

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gate the behavior of the model with respect to thepatterns. To achieve a comprehensive understanding ofthe model, the experiments or scenarios included in thisstep are required to change processes or parametersbeyond their realistic ranges. The goal of this step is toidentify and understand the processes, or combinationsof processes and circumstances that lead to the specificpatterns observed in nature.

The third step necessarily includes a critical assess-ment of the ‘‘quality’’ of the patterns; that is themagnitude of error connected with data collection.Clearly, taking a purported pattern with possible ob-server bias and misinformed interpretation too seriouslymay bias the model results. For example, long timeseries data may be severely flawed if the data originatefrom counts or otherwise statistically questionable stud-ies. On the other hand, an attempt to adjust the simu-lated patterns to agree with the observed pattern moreclosely than stipulated by the measurement error in theobserved pattern would be inconsistent and erroneous.However, the assessment of the error of the observedpattern may guide the selection of an adequate criterionfor deciding when the simulated pattern matches theobserved one.

If the patterns are genuine and contain additionalinformation, the indirect and simultaneous parameteradjustment prevents error propagation, as may happenin conventional models based on point-estimates ofparameter values (Fig. 1). The degree to which theinitial uncertainty is reduced depends on the amount ofinformation that is carried by the observed patterns,and on how well the model describes the most impor-tant processes and the major constraints that generatethe observed patterns.

Fourth step: secondary predictions. Several inherentfeatures of pattern-oriented models, such as the linkageto relevant scales and the matching of realistic patternat a high hierarchical level, make it possible to deriveadditional relationships (secondary predictions, or an-cillary data, Turchin 2003). These relationships, whichare not identical with the initial patterns nor modelassumptions, but which arise out of the interactionsbetween the simulated processes, offer potentially valu-able possibilities for making predictions, for better un-derstanding the model performance, and for modelvalidation. To derive secondary model predictions onemay create a ‘‘virtual’’ ecologist or observer (Grimm etal. 1999) that scans the internal model performance,recording for example age structures, mortality rates,dispersal distances, population sizes, patch occupancy,extinction events, distribution of individuals in a land-scape, etc. The secondary predictions can be used formodel validation if field estimates are available. Alter-natively, they may also be used to further improve thedetermination of parameters and processes in the thirdstep of our protocol (i.e. they are treated as patterns,Wiegand et al., in press). However, in many cases it

may be interesting to check whether or not internalrelationships produced by the model, which are notprimarily envisaged as model output and are thus usu-ally not recorded, are within biologically realisticranges. Such tests can greatly improve the confidence inthe model and may help prevent unrealistic assump-tions from creeping in. These tests may also suggestnew patterns.

Cycles of pattern-oriented modeling. Ideally, the foursteps of our protocol are repeated in a cyclic manner(Thulke et al. 1999). In the first cycle the biologicalinformation is arranged in a logical manner and thedata and (preferably simple) hypotheses about the func-tioning of the system are integrated into a first model.Comparison between the predicted and observed pat-terns will usually point to sensitive parameters or pro-cesses, problems in the formulation of the processes, ormissing data. For example, a pattern may only bereproduced if a certain process is included. Such afinding may stimulate new field investigations for closerexamination of the process that appeared indispensablefor reproducing the pattern. On the other hand, if themodel structure is too simple the model will not be ableto reproduce one or more patterns. This failure of themodel, however, can be valuable in stimulating specificfield investigations on the missing processes and forderiving a more adequate model structure. In the sec-ond cycle the new information and hypotheses on alter-native model structures are included in the model,which is again tested by comparing predicted and ob-served patterns. Several feedback loops and steps be-tween field investigations, modifying hypotheses, andrunning the model may be required before an optimalmodel structure, that is, a minimal model that repro-duces all observed patterns, is identified and the mecha-nisms behind the observed patterns are wellunderstood.

Examples

Parameter estimation: the brown bear populationmodel and the temporal pattern of females withcubs

Wiegand et al. (1998) compiled the knowledge availableabout the western brown bear population in theCordillera Cantabrica, Spain, and performed a popula-tion viability analysis (PVA) to diagnose the currentstate of the population and to support management.The PVA required a close determination of demo-graphic model parameters. Model calibration was doneusing an independent data set with 14-yr data on thenumber of females with cubs.

The pattern. The pattern used in this example is theobserved 14-yr time series of females with cubs of theyear (COY index, Fig. 2), which provides data at

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Fig. 2. Observed and simulated patterns: the observed pattern(gray circles) and the two simulated patterns with the smallesterror � (solid line: �=1.42, dashed line: �=1.45), and rangeof the simulated patterns for the 90 best parameter combina-tions (minimal and maximal values; dotted lines).

space which yields a total of 9000 parameter combina-tions. Additionally, five alternative hypotheses of envi-ronmental variation were considered.

Third step: systematic comparison between the ob-ser�ed pattern and the simulated pattern produced by themodel. The mean COY index obtained from a series of200 replicate simulation runs [COY(t)], and the ob-served number of females with cubs from 1982−1995[d(t)] were compared through the error measure�(COY)= (�years t(COY(t)−d(t))2/N)0.5, where N isthe number of years with data. The error � served as acriterion to distinguish between probable (� small) andless probable (� high) parameter combinations. Only intwo years was the confidence range of the patterngreater than �1 and the overall maximal measurementerror of the pattern yields ��0.9. Therefore, Wiegandet al. (1998) did not attempt to adjust simulated patternbelow the inherent error of the pattern of ��0.9.Simulations were performed for the 9000 possibleparameter combinations and five alternative hypothesesof environmental variation. Not all parameter combina-tions were able to reproduce the observed pattern offemales with cubs (Fig. 3). For parameter combinationsthat resulted in too high or too low overall rates ofincrease, differences in COY(t) and d(t) were marked,and even if the overall population trend was matched,‘‘wrong’’ parameter combinations could cause clear dif-ferences between the shape of the observed and thesimulated COY index (e.g. produce wrong minima andmaxima in the time series). The scenarios with low (andno) variation in adult mortality had minimal errors of��1.4, but high variation in adult mortality rate al-ways resulted in worse fits, with errors ��1.5 (Wie-gand et al. 1998). Thus, the authors rejected thehypothesis of high environmental variation in adultmortality rate, but not the hypothesis of high environ-mental variation in subadult mortality rate. Individualparameter values varied little among the best parametercombinations (Wiegand et al. 1998: Table 8), thusrejecting the possibility of several ‘‘optimal’’ modelparameterizations. Use of a second pattern, the numberof known mortalities from 1982–1994, further im-proved the quality of the fit from ��1.4 to ��1.0,but did not alter the parameter estimates. The proce-dure of indirect parameter estimation thus allowed thedetermination of credible reference parameter set.

Fourth step: secondary predictions. Because the ob-served pattern restricts the degree of freedom of popu-lation dynamics, the values of several unknownvariables of population dynamics (e.g. total populationsizes and numbers of independent females at differentyears, growth rates, mortality rates) could be deter-mined with simulation runs produced by parametercombinations that yielded a good match between theobserved and simulated patterns (Fig. 3A). Especiallythe estimate of the current number of independentfemales (25–26 in 1995) was an important reference

several levels: the COY index is related to the overallpopulation size, to the overall population trend, and todemographic parameters and environmental variation.

First step: aggregation of biological information andscales. A model able to reproduce a time-series of thenumber of females with cubs must necessarily be ademographic model that includes the processes of re-production, mortality and family break-up. Because thenumber of females with cubs was small (�11 in anyone year, Fig. 2), the individual-based approach is wellsuited in this case. Preliminary model runs showed thatthe COY index is critically influenced by environmen-tally caused variation in mortality rates, litter size andprobability for reproduction (Wiegand et al. 1998: Fig.4). Therefore, the criterion for including biological in-formation required consideration of environmentalvariation. Examination of data showed that environ-mental variation had indeed a significant impact on cubmortality and litter size, and the authors suspected thatsubadult and adult survival could also be influenced byenvironmental variation. To examine alternative hy-potheses on environmental fluctuations, the same pro-cedure of comparing observed and predicted patterns(step 3) was repeated for alternative scenarios of howenvironmental variation influenced adult and subadultmortality rates. Besides environmental variation, theCOY index may be influenced by the availability ofhabitat suitable for reproduction. Brown bear habitat islimited and may support no more than 18 breedingfemales in the same year, which is the sum of the largestnumber of females with cubs observed in any year ineach area. Therefore, density-dependent regulation wasconsidered by limiting the number of females that canbreed simultaneously to 18.

Second step: determination of parameter �al-ues. Following the data and literature analysis a totalof 8 model parameters remained uncertain, and theauthors selected for each of them 2, 3 or 5 probablevalues that entered into the final grid in parameter

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Fig. 3. Secondary prediction for the unknown simulated rate of population increase �sim during the study period 1982–1995.(A): The simulated rate of increase, �sim, is plotted for the 31% best parameter combinations, over the resulting error � betweenthe observed and the simulated COY index. The results are shown for the scenario with no environmental variation in adult andsubadult mortality rates (scenario S1 in Wiegand et al. 1998). The 90 ‘‘best’’ parameter combinations are located left from thevertical line ��1.51. The solid horizontal line gives the mean simulated rate of increase �sim obtained from the 90 bestparameter combinations and the dashed lines indicate the standard deviation (�sim=0.953�0.008). (B): Distribution of �sim forall 9000 parameter combinations (white bars), for the 31% best parameter combinations (gray bars), and for the 90 bestparameter combinations (white bars).

point for determining minimal viable population sizesand for assessing the risk of extinction for the brownbear population in the Cordillera Cantabrica (Wiegandet al. 1998). Because the COY index is also tied to theoverall population trend, it considerably restricts therange of the simulated rate of population increaseduring the study period 1982–1995 (Fig. 3A).

Error propagation. Fig. 3B shows clearly how thedata provided by the observed pattern prevents errorpropagation. Without considering the data contained inthe observed pattern, the simulated rate of increasevaries due to the initial uncertainty of the modelparameters between �sim=0.89 and 1.00 (Fig. 3B),while application of the indirect parameter estimationreduces the probable range of �sim to 0.953�0.008(Fig. 3A). Comparison with the results of the sensitivityanalysis performed in Wiegand et al. (1998) shows thatthis uncertainty is of the same magnitude as the uncer-tainty in �sim that arose out of the remaining uncer-tainty for the single most sensitive parameter, the adultmortality rate m5–16: ��sim=�m5–16�(m6–16)=0.037×0.512=0.0189 (see Wiegand et al. 1998: Table 12).These results are very good news and show explicitlythat indirect pattern-oriented parameter estimation canprevent error propagation and reduce the sensitivity ofthe model output to model parameters.

Detecting processes from pattern: the savannamodel and snapshot pattern of tree distribution

In the previous example the data of the observedpattern was used primarily for parameter estimation. Inthe present example the focus is to show how cyclicrepetition of the four step protocol facilitates the iden-

tification of the underlying processes that lead to theformation of the spatial pattern of savanna trees ratherthan on parameter values of processes that are alreadyknown. To investigate whether the tree population ofthe southern Kalahari Savanna is in a phase of decline,increase, or constancy with respect to tree abundanceJeltsch et al. (1999) applied methods of simulationmodeling and point pattern analysis to assess the poten-tial of a spatial pattern, snapshots of observed treedistributions, for identifying relevant pattern-generatingprocesses. They also investigated to what extent thecurrent tree distribution pattern is indicative or evendiagnostic of the dynamical status of the treepopulation.

The pattern. The pattern used in this example con-sisted of six maps of spatial distributions of trees, eachcovering an area of 50 ha, and digitized from aerialphotographs (Fig. 4A). Tree distribution patternsderived from aerial photographs and from simulationexperiments were analyzed and compared using Rip-leys’s L-function analysis (Bailey and Gatrell 1995), astatistical point pattern analysis (Jeltsch et al. 1999).

First step: aggregation of biological information andscales. Since the pattern in this example is spatiallyexplicit and based on individual trees, a combinedspatially-explicit and individual-based modeling ap-proach was chosen. The relevant scales are determinedby the spatial scale of the pattern (i.e. at least severalhectares) and the temporal scale of several tree genera-tions. Consequently, an area of 50 ha was subdividedinto a grid of 20,000 5×5m cells, an area large enoughto accommodate a mature tree canopy, and simulationruns encompassed a timespan of several thousandyears. A model capable of reproducing the spatialpattern of savanna trees, assumed to be in a state of

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Fig. 4. (A): An example of a snapshot pattern of tree distribution derived from an aerial photograph of a representative 50 haregion in the Kalahari Gemsbok National Park. (B): point-pattern analysis (L-value vs spatial scale given in grid-cell units) ofthe spatial distribution of trees shown in (A). The L-value (solid line) is given together with the 95% confidence interval (dashedlines). L-values within the range of the confidence interval indicate a random pattern. L-values above (below) the confidenceinterval indicate a significantly clumped (even spaced) distribution. (C): simulated spatial pattern of adult trees within a 50 haarea. (D): point-pattern analysis of the simulated pattern shown in (C). Modified after Jeltsch et al. (1999).

long-term coexistence with grasses, must contain theclassical key processes in savannas, namely competitionfor moisture, herbivory and grass fires (Walker et al.1981). However, in a first cycle of pattern-orientedmodeling Jeltsch et al. (1996) found that these keyprocesses were not sufficient to explain savanna-likecoexistence in their spatially explicit simulation model.Long-term coexistence occurred only with spatiallyscattered groups of trees, a spatial pattern which is notobserved in the southern Kalahari savanna. By intro-ducing hypothetical small-scale heterogeneities and dis-turbances, which caused differential probabilities of treeestablishment and survival, Jeltsch et al. (1998) ob-tained long-term savanna coexistence with realistic spa-tial patterns of tree distribution for a broad range ofparameter values. This finding gave rise to new fieldinvestigations that were aimed at identifying the miss-ing processes. Results of these field investigations andnew simulation experiments (the second cycle) showedthat seed dispersal in the dung of large herbivoresconsuming the pods of Acacia erioloba was the mostimportant process leading to micro-sites in the land-

scape with improved establishment and survivalchances for trees.

Second step: determination of parameter �alues. Thespatially explicit simulation model is based on theecological dynamics occurring in the South Africanpart of the southern Kalahari represented by the Kala-hari Gemsbok National Park (KGNP). Most parameterestimates are based on two decades of field research inthis area (see Jeltsch et al. 1996, 1998 for details).However, it was not possible from empirical studiesalone to identify the rate with which dung patches areformed. To estimate this unknown parameter Jeltsch etal. (1999) used a pattern-oriented approach. Here thepattern is the observed tree density in the KGNPtogether with the assumption that, historically, this areais in a state of long-term coexistence of trees andgrasses. Too high rates of annual seed patch formationlead to an unrealistic continuous increase in tree densi-ties whereas too low rates caused a rapid tree decline.With the determination of this unknown parameterrange it was possible to explore the initial question forthe dynamical status of the tree population using themore complex spatial pattern.

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Third step: systematic comparison between the ob-ser�ed pattern and the pattern predicted by themodel. Simulations showed that the pattern of spatialtree distribution was not constant but changed in thecourse of time. Especially for long-lived plants, thissituation limits the amount of information that can beextracted from the pattern. Although there was vari-ability within and among runs, the patterns producedby the model were in general consistent with the actualpatterns observed at KGNP (cf. Fig. 4A and C, and 4Band D). However, more detailed investigations of sec-ondary model predictions facilitated by the pattern-ori-ented approach were necessary to identify the currentdynamical status of the tree population in the southernKalahari.

Fourth step: secondary predictions. The possibility toobserve the relevant ecological factors and processes inthe course of time allows for the examination of thefunctional relationship between the ecological state ofthe model system and the spatial pattern in tree distri-bution produced over time. Jeltsch et al. (1999) used avirtual observer that recorded during an individualmodel run, for 50 consecutive 10 yr windows, severalinternal model relations (e.g. number of years withmaximal topsoil and subsoil moisture, the number ofcells dominated by perennial grasses and herbs, themean number of tree seedlings, saplings, and adults).They showed that periods of tree decline caused bylower rainfall are characterized by reduced clumpingand an increase in randomness in the tree distributionpattern, and that both reduced fire frequency and in-creased self-thinning (caused by low rainfall) reduce thetendency of the tree distribution towards clumping.Thus, a spatial pattern that is characterized by a highdegree of clumping at different scales, as currentlyobserved in the southern Kalahari, is not in a dynamicstate of major decline nor in risk of extinction causedby reduced rainfall conditions.

General models for specific applications: theincidence function model and spatial pattern ofhabitat patch occupancy

The Incidence Function Model (IFM, Hanski 1994,1999) of metapopulation dynamics is perhaps the bestknown and most widely used pattern-oriented modelthat explicitly relies on indirect parameter estimation.The aim of the IFM is to construct a minimalisticdescription of metapopulation dynamics that maynonetheless provide guidance to quantitative questionsabout particular metapopulations (Hanski 1994).

The pattern. The pattern employed in the IFM isdefined by the spatially explicit layout of suitable habi-tat patches within unsuitable habitat matrix, includingpatch locations and sizes (or qualities), and one ormore snapshots of data on the presence/absence of the

species in these patches at stochastic equilibrium. If themodel is parameterized with just one snapshot of spa-tial data, as originally envisioned by Hanski (1994), allthe information on extinction and colonization pro-cesses is provided by the spatial pattern data. Specifi-cally, at equilibrium the pattern of patch occupancyreflects the reduced colonization rate of isolated patchesand the increased extinction rate of small populationsin small habitat patches.

First step: aggregation of biological information andscales. The patch occupancy pattern determines thelevel of detail in the model. Because only two states ofthe patches are considered, occupied or empty, themodel ignores all details of local dynamics (reproduc-tion and mortality) within single habitat patches. Toreproduce the pattern of patch occupancy, the IFMrelies on a description of local dynamics that onlyconsists of the patch-specific extinction probability anda patch-specific colonization probability of emptypatches. The key task is to make reasonable and practi-cal assumptions about how these two population pro-cesses depend on measurable landscape variables andsome life-history traits of the species.

The IFM assumes decreasing extinction probabilitywith increasing patch area, and a colonization probabil-ity that is dependent on patch connectivity to existingpopulations. The most difficult task is to define themeasure of patch connectivity, which summarizes theinformation on the spatial layout of habitat patchesand on the movement behavior of the species. Althoughit might be possible to derive a formula based on theactual movement behavior of individuals, Hanski(1994) used a simple phenomenological approach,which is consistent with the level of detail in the rest ofthe model. Note that the observed pattern (distancesamong the patches, patch areas, and the patch occu-pancies) enters the model at this step and ties the IFMexplicitly to the relevant spatial scales of the system.

Selection of the appropriate spatial scale has to bemade with relevant biological information. The patcharea and connectivity effects are likely to occur also ine.g. data on the instantaneous distribution of foragingindividuals among a set of resource patches, thoughthis pattern has nothing to do with population pro-cesses. Determining the appropriate population scalefor a particular organism is generally not that difficult,but it is essential that adequate biological knowledge isused in this step.

Second step: determination of parameter �al-ues. Because the IFM does not consider local dynam-ics, there are only five model parameters (x, y, e, �, andb) that have to be determined (Hanski 1994). Theparameters e and x describe the local extinction proba-bility and its scaling with patch area, y is a colonizationparameter, 1/� is the average migration distance, and bscales the emigration rate by patch area. Two of them(b, �) can often be estimated with independent data,

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and the data provided by a single snapshot of patchoccupancy are then used to estimate the remainingthree parameters (Hanski 1994, 1999, Moilanen andHanski 1998, Moilanen 1999). Alternatively, one mayuse occupancy data to estimate all the parameters. Inthe IFM, the third step (below) is part of actualparameter estimation.

Third step: systematic comparison between the patternproduced by the model and the obser�ed pattern. Theoriginal parameter estimation method for the IFM doesnot require simulations since the model is formulated ina closed mathematical form (Hanski 1994). Theparameter values are estimated by regressing the trans-formed model-predicted incidences against the empiri-cally observed patch occupancies (Hanski 1994, 1999).The logistic regression model that contains the observedpattern represents a very explicit connection betweenthe IFM and the empirical field studies (Hanski 1999).

Given the structural assumptions of the IFM, asingle pattern of patch occupancy provides, in principle,sufficient information to infer the values of the parame-ters characterizing the metapopulation processes (Han-ski 1994). However, including the ‘‘rescue effect’’ intothe model leads to a model in which the parameters eand y occur in a product. In this case, some additionalinformation is needed to separately estimate the valuesof these parameters. For example, one may estimate theminimum patch area, or one may use informationabout population turnover rate between two or moretime steps (Hanski 1994).

The original parameter estimation method based onlogistic regression (Hanski 1994) ignores spatial andtemporal correlations in patch occupancy and assumesmetapopulation quasi-equilibrium and constant colo-nization probabilities. Moilanen (1999) has developed aparameter estimation method that is not affected bythese problems. Moilanen’s approach however requiresdata on population turnover, that is, several snapshotsof patch occupancy. On the other hand, it is clear thatif temporal as well as spatial data are available, oneshould employ all available information in parameterestimation. A useful feature of Moilanen’s (1999), andMoilanen’s (2000) approach is that one may make aconscious decision as to which kind of information touse in parameter estimation. Moilanen (1999) foundthat the value of parameter �, which scales distancedependence of migration, was generally estimated wellwith his Monte Carlo method. This is a good exampleof how an observed pattern can be used effectively andmay replace costly studies for determining singleparameter values.

Fourth step: secondary predictions. The use of a vir-tual observer offers numerous possibilities for testingthe model with secondary predictions. For instance,Moilanen et al. (1998) tested the IFM parameterizedfor the American pica in two different ways usingsecondary model predictions. First, they constructed a

measure that characterizes the dynamic behavior of thesimulated and empirically observed sequences of patchoccupancy, and checked whether the measure calcu-lated from the empirically observed sequences of patchoccupancy patterns falls within the 95% confidencelimits of the predicted distribution. Because the modelwas parameterized using only spatial data on patchoccupancy, a second test was made by comparing thepredicted number of turnover events, extinctions andcolonizations, with the observed ones.

Because the spatial layout of the patch network isexplicitly considered, the IFM can be used as a practi-cal tool for management purposes. The parameter esti-mates allow numerical simulation of the dynamics ofthe focal species in any system of habitat patches withknown values of patch sizes and patch locations, andstarting from any configuration of occupied patches(for an extensive review see Hanski 1999). It shouldalso be noted that, although one has to make theassumption of metapopulation quasi-equilibrium forthe purpose of parameter estimation, the model canalso be used to make predictions about the transient(non-equilibrium) dynamics. For instance, one may as-sess the likelihood of colonization of a patch networkfrom one or a few (possibly introduced) local popula-tions (Hanski 1999).

The IFM has been used as a component in a recentindividual-based model of the evolution of migrationrate (Heino and Hanski 2001). In this model, the long-term dynamics predicted by the individual-based modelwere constrained to match the predictions of the IFMby comparing patterns of habitat patch occupancy andtemporal turnover of local populations. The actualmigration of individuals was modeled with anothersubmodel, the ‘‘Virtual Migration’’ (VM) model (Han-ski et al. 2000), which can be considered as a pattern-oriented model of mark-release-recapture data (thepattern consisting of the dependence of emigration andimmigration rates on habitat patch area and connectiv-ity). A great advantage of this model construction isthat most model parameters could be rigorously esti-mated and the individual-based model was focused onsituations that occur in real metapopulations (Heinoand Hanski 2001).

Discussion

In this article, we summarize and conceptualize ourexperience concerning population modeling within acommon framework, the pattern-oriented modelingstrategy, and we advocate this approach as a possibleway out of the problems described in the introduction.Our framework is related to modeling strategies thathave been developed to study population cycles andfluctuations using mathematical models (Kendall et al.1999, Turchin 2003), and to an approach for construc-

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tion and analysis of ‘‘mechanistically-rich’’ simulationmodels (DeAngelis and Mooij, in press). Basically, pat-tern-oriented modeling is a strategy for optimally ex-ploiting the available biological data in all the steps ofthe modeling process: from the initial model construc-tion to parameter estimation and detection of deficien-cies in the model structure and in our knowledge.However, one may argue that the general features of ourprotocol are by no means new and that the attempt toexplain pattern is the basic program of science. Clearly,scientists have always tried to identify patterns, and byexplaining their creation they could reveal essentialproperties of the system in question. For example,prominent cases in natural sciences where patternshelped to revealed the ‘‘essence’’ of a system includeclassical mechanics (Kepler’s laws), quantum mechanics(atomic spectra), cosmology (red shift), molecular genet-ics (Chargaff’s rule), and mass extinctions (Iridium layerat Cretaceous boundary). On the other side of thespectrum, even a simple regression model can be seen asa pattern-oriented model if the underlying hypothesisare consciously derived from the biological informationand no data dredging (Burnham and Anderson 1998)was done. Clearly, the challenges and problems forpopulation models in biological conservation are notmuch different from that in other fields of science wheremodels are used for inference, and our modeling strategywill be similar to that in other fields. Our main objective,however, is to point out to problems related to theapplication of models in conservation and to provide apractical strategy to overcome them, rather to invent a‘‘new wheel’’ for science in general. The pattern-orientedmodeling strategy provides a framework for consciouslyderiving hypotheses on processes to be included in amodel and for testing them against each other by using(multiple) independent data (i.e. the patterns). Therebyit facilitates the selection of a ‘‘best approximationmodel’’ regarding a given set of independent patterndata and the detection of deficiencies in the modelstructure when the model does not match one or morepatterns. Thus, the pattern-oriented modeling strategy isin some respects an analogue to statistical approaches ofmodel selection and inference for mathematical models(Hilborn and Mangel 1997, Burnham and Anderson1998, Wood 2001), which are not necessarily based onmathematical equations but may be based on simula-tions.

Lessons from the examples

The three examples presented in this article highlightdifferent features of the pattern-oriented modeling strat-egy. In the first example, a simple non-spatial individual-based model was closely adjusted to the time-series dataof the observed number of female brown bears withcubs. While the model itself is quite simple, the proce-

dure of parameter estimation was time-consuming andrequired non-standard methods. However, as compensa-tion for this effort the additional data provided by theobserved pattern helped adjust the formerly unknownmodel parameters and prevented error propagation.These are important considerations that solved two ofthe most serious problems in the application of modelsin conservation biology: the inherent paucity and uncer-tainty of ‘‘conventionally’’ collected data, and sensitivityof model predictions to parameter values. The majortake-home message emerging from this example is thatfield research should more consciously search and docu-ment patterns that can potentially deliver informationthat is impossible to obtain with traditional methods.Determination of unknown or uncertain demographicparameters (e.g. mortality rates) to a similar precisionwould be impossible with traditional methods, becausethat would require extensive telemetric studies and trap-ping of a high percentage of the entire population, anundertaking that is impossible to perform in practice (J.Naves, pers. comm.). Because data in the pattern canonly be used via methods of indirect parameter estima-tion, a close interaction between field research andmodeling is required.

The striking feature of the savanna model is the closeinteraction between field investigation and modelingduring several cycles of pattern-oriented modeling forspecifying initially unknown processes. Early results ofthe model pointed to critical processes which werenecessary to obtain tree distributions typically for theKalahari savanna. Hypotheses on these processesderived from the initial modeling exercise guided furtherfield investigations that specified the processes. Includ-ing the new biological information into the modelproduced new insight on the status of the savanna thatcould hardly have been obtained with traditional meth-ods. The possibility to more effectively plan field inves-tigations with the aid of pattern-oriented models is animportant advantage, especially in times of scarce re-sources and pressing environmental problems. However,the savanna model showed not only that the pattern-ori-ented strategy can be effectively used at different stagesof model development, but also that it can serve toimprove different parts of the model. While the ‘‘big’’pattern was the spatial distribution of the savanna trees,data on the mean densities of savanna trees was used asan additional pattern for adjusting the unknown valueof the model parameter ‘‘rate of annual seed patchformation’’. Finally, the savanna model is a good exam-ple of how a model can facilitate a deeper understandingof spatial pattern formation in ecological systems, aquestion that has attracted much interest recently (Levin1992). A key element in the savanna example was thepossibility to scan the internal model performance witha virtual observer. Such an analysis goes beyond theusual approach to sensitivity and error analysis (Grimmet al. 1999).

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The incidence function model is an excellent exampleof how a general minimalistic model can be applied tospecific situations; it may represent a signpost of how toreconcile ecological theory with ecological applicationand conservation. This simple stochastic metapopula-tion model ignores systematically all biological infor-mation (that might to some extent be available) belowthe scales and the level of detail dictated by the ob-served pattern. Such conscious sacrifice of detail makesthe model general. The model itself is an ‘‘empty gen-eral structure’’, which summarizes very generally howextinction and colonization processes depend on mea-surable landscape variables and general life-historytraits of the species. However, when combined with thespatially explicit information contained in the observedpattern – the locations and areas of suitable habitatpatches and the observed incidence of the species inthese patches – the model can be closely tied to specificsituations.

Advantages and shortcomings of thepattern-oriented modeling strategy

Although the pattern-oriented modeling strategy pre-sented in this article has a number of advantages, wehave to discuss also some potential problems of thisstrategy.

What can one do when no pattern exists? The mostobvious shortcoming of our modeling strategy is that agenuine pattern is needed. So what can one do whenapparently no pattern exists? The most important pointto note is that the pattern does not need to be thatdistinct and spectacular as e.g. the wave-like spread ofrabies (Jeltsch et al. 1997). The pattern can be anycharacteristic non-random structure in observations ordata (Grimm et al. 1996). The term ‘‘non-random’’implies that there were certain processes or mechanismsthat created this pattern, and in turn that the patterncontains information on these processes or mechanisms.While searching for helpful patterns one may ask anumber of questions: (1) Are there spatial structuresthat are related to my question? (2) Are there temporalstructures or time-series data that are related to myquestion? (3) Are there comparable systems that behavedifferently from my system and are these differencesperhaps connected to my question? (4) Are there rele-vant data or information available on my system that Icannot include directly into the rule-set of my model?(5) Does the system behave markedly differently frommy expectations?

Vague and genuine patterns. Different patterns maycontain a different amount of information that can beused with the pattern-oriented modeling strategy.‘‘Vague’’ patterns that contain relatively little informa-tion may easily be overlooked, but the use of severalvague patterns in combination with the Pareto Optimal

Model Assessment Cycle POMAC (Reynolds and Ford1999, Ford 2000) may provide just as much informationas one strong pattern (Wiegand et al., in press). PO-MAC uses binary error measures as vector assessmentcriteria and by revealing which combination of assess-ment criteria the model can satisfy simultaneously itguides the researcher to locate deficiencies in both themodel structure and the assessment criteria.

If the number of parameters to be estimated is largeor if the pattern is vague, several parameter combina-tions might produce equally good results and it mightnot be possible to determine even with several patternsan optimal model parameterization. In such cases wehave to accept that the current knowledge is insuffi-cient; i.e. the model rules and the filter procedure areonly stated at a certain level of detail and may containinevitable errors and subjective interpretations, and thepattern(s) may not carry sufficient information to re-duce the degree of freedom of the system. It is impor-tant to note that the pattern-oriented modeling strategyis only a strategy that can help to construct a bettermodel rather than a recipe that can solve all problems.

One may argue that one can always find a model thatis able to reproduce an observed pattern, provided thatone adds enough parameters and processes in themodel. Such a critique may hold true for ‘‘blind’’statistical fits and descriptive regression models that donot rely on information on the underlying biologicalprocesses (an example is conventional statistical timeseries analysis). In contrast, the pattern-oriented ap-proach is a process-oriented approach that uses biolog-ical information on patterns and processes rather thanarbitrary functions for fitting. Also, the criterion ofminimalism prevents arbitrary inclusion of functionalrelationships into the model, and the critical assessmentof whether or not the pattern is genuine (see third stepof the protocol for pattern-oriented modeling) ensuresthat predicted patterns which match the observed pat-terns are neither a singular output of the model nor canthey be reproduced with arbitrary combinations ofparameters and/or processes.

Significance for ecological applications andconservation

We argue that a successful application of the pattern-oriented modeling strategy to its full potential mayrequire a major paradigm shift in ecological modelingas well as in data collection.

Paradigm shift in modeling. Modelers should con-sciously employ observed patterns to construct modelsuseful for practical applications as well as to developnew methods for indirect parameter estimation that relyon observed patterns. A systematic classification ofpatterns that can be used for pattern-oriented modelingis needed, and modelers should investigate and state

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which type of pattern contains most information fordetermining parameters and processes. Such guidelineswould be extremely helpful in conservation practice forenhancing the value of the biological information thathas to be collected.

Paradigm shift in data collection. Ecologists need todevelop a new consciousness about data by recognizingthe potential of observed patterns as a rich source ofbiological information. Therefore they should look ac-tively for patterns at a high hierarchic level that mayprovide independent data and document and measuresuch patterns. In general, some demographic data andsome patterns at higher hierarchical levels will be rela-tively easy to obtain. Both will have some uncertainty,and it is very unlikely that all the demographic parame-ters needed in a model can easily be obtained withsmall uncertainties. That is why our approach is impor-tant in spatial modeling. On the other hand, it may behard to say a priori whether information of one partic-ular type or another (patterns versus single demo-graphic parameters) will be more economical to obtainor have less associated uncertainty, and more theoreti-cal and empirical studies are needed to resolve thisissue.

Reconciling ecological theory and conser�ation prac-tice. Pattern-oriented modeling provides means for rec-onciling theory and practice (Grimm 1994, Grimm etal., unpubl.). At present, most models are either tootheoretical and general to be useful for practical appli-cations, or they are too detailed for more generalapplications to other species and situations. Pattern-ori-ented models with the ‘‘right’’ scales and the ‘‘right’’level of detail capture the essence of a system and cantherefore often be easily refined for applied questions aswell as generalized for a certain class of species orcircumstances. A good example of such ‘‘intermediategeneral’’ models is the incidence function model whichcaptures the essence of population-level spatial dynam-ics for a wide range of metapopulations. Our hope isthat such intermediate general models, which arise outof specific case studies but which are tied to a generalmodeling strategy, may provide further opportunitiesfor advancing our understanding of more general ques-tions in conservation biology.

Acknowledgements – We thank W. Batista, F. Davis, F.Knauer, B. Kendall, K. Moloney, J. Naves, H. Possingham,K. Wiegand, A. Tyre, S. Schadt, C. Wissel, and especially D.DeAngelis and E. Revilla for assistance during the develop-ment of ideas and for comments on drafts of this manuscript.

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