a spatially explicit hierarchical approach to...

Ecological Modelling 153 (2002) 7–26

A spatially explicit hierarchical approach to modelingcomplex ecological systems: theory and applications

Jianguo Wu *, John L. DavidDepartment of Plant Biology, Arizona State Uni�ersity, PO Box 871601, Tempe, AZ 85287-1601, USA

Abstract

Ecological systems are generally considered among the most complex because they are characterized by a largenumber of diverse components, nonlinear interactions, scale multiplicity, and spatial heterogeneity. Hierarchy theory,as well as empirical evidence, suggests that complexity often takes the form of modularity in structure andfunctionality. Therefore, a hierarchical perspective can be essential to understanding complex ecological systems. But,how can such hierarchical approach help us with modeling spatially heterogeneous, nonlinear dynamic systems likelandscapes, be they natural or human-dominated? In this paper, we present a spatially explicit hierarchical modelingapproach to studying the patterns and processes of heterogeneous landscapes. We first discuss the theoretical basis forthe modeling approach—the hierarchical patch dynamics (HPD) paradigm and the scaling ladder strategy, and thendescribe the general structure of a hierarchical urban landscape model (HPDM-PHX) which is developed using thismodeling approach. In addition, we introduce a hierarchical patch dynamics modeling platform (HPD-MP), asoftware package that is designed to facilitate the development of spatial hierarchical models. We then illustrate theutility of HPD-MP through two examples: a hierarchical cellular automata model of land use change and a spatialmulti-species population dynamics model. © 2002 Elsevier Science B.V. All rights reserved.

Keywords: Complex systems modeling; Hierarchical patch dynamics; Hierarchical modeling; Scaling; Urban landscape ecology

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1. Introduction

Ecological systems are characterized by diver-sity, heterogeneity and complexity. Complexityoften results from the nonlinear interactionsamong a large number of system componentswhich frequently lead to emergent properties, un-expected dynamics, and characteristics of self-or-ganization (Jørgensen, 1995; Prigogine, 1997;

Levin, 1999). The study of complexity has a his-tory of at least several decades, ranging fromphysical, biological, and to social sciences, and arecent resurgence of interest in complexity issuesis evident as new theories and methods havemushroomed in the past few decades (see Wu andMarceau, this issue and references cited therein).One of the most intriguing and widely-cited theo-ries in the science of complexity is self-organizedcriticality (SOC; Bak et al., 1988; Bak, 1996).According to the theory of SOC, large interactivesystems naturally evolve toward a self-organizedcritical state in which a minor event can lead to a

* Corresponding author. Tel.: +1-480-965-1063; fax: +1-480-965-6899.

E-mail address: [email protected] (J. Wu).

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J. Wu, J.L. Da�id / Ecological Modelling 153 (2002) 7–268

cascading catastrophe. When a system is at theself-organized critical state, the frequency andmagnitude of events follow a power law distribu-tion, and this may be viewed as a statisticallystable, internally controlled state with no charac-teristic scale within the system. At this point,events are correlated across all scales exhibiting astatistical fractal pattern in spatial structure. Bakand Chen (1991) claimed that SOC may explainthe dynamics of a wide range of natural andhuman-related phenomena, including earth-quakes, ecosystems, and social and economic pro-cesses. Furthermore, the title, as well as thecontent, of Bak’s (1996) book even suggested thatSOC was the mechanism of ‘How Nature Works’.

However, while it is extremely intriguing, SOCdoes not seem adequate for explaining the greatdiversity of ecological phenomena (Levin, 1999).Of course, it is not surprising that simple statisti-cal analyses may reveal that some ecological vari-ables in certain ecosystems exhibit power– lawrelationships (e.g. Jørgensen et al., 1998; Sole etal., 1999). However, the existence of a power– lawrelationship alone is not adequate to prove that asystem is at the self-organized critical state be-cause diverse mechanisms may result in such arelationship in both physical and ecological sys-tems (Raup, 1997; Jensen, 1998; Kirchner andWeil, 1998; Levin, 1999). SOC de-emphasizes orcompletely ignores the existence of multiple-scaleconstraints and their significance in influencingsystem dynamics. Bak (1996) asserted that allcomplex self-organizing systems move themselvesto the self-organized critical state just like a sand-pile. To the frantically enthusiastic SOC advo-cates, top-down constraints in controlling systemdynamics do not seem to be important. In thisregard, SOC appears to represent an extremereductionist view. Interestingly, Bak and Chen(1991) claimed that SOC was ‘the only model ormathematical description that has led to a holistictheory for dynamic systems’. It is evident from theabove discussion, however, that the implicationsof the theory of self-organized criticality for eco-logical systems are in sharp contrast with hier-archy theory or any holistic systems theory.

In general, ecological systems are not, and donot behave like, sandpiles. Levin (1999) argued

that heterogeneity, nonlinearity, hierarchical orga-nization, and flows are four key elements of com-plex adaptive systems, like ecosystems, that allowfor self-organization to occur. That is, CAS typi-cally become organized hierarchically into struc-tural arrangements through non-linearbetween-component interactions, and these struc-tural arrangements determine, and are reinforcedby, the flows of energy, materials and informationamong the heterogeneous components. Levin(1999) further suggested that SOC and modularstructure represent the two ends of a continuumalong which most ecosystems are found in themiddle. While we agree with Levin’s postulationin general, we dare to speculate that the majorityof ecosystems, especially once well-developed, arehierarchically structured, so that component di-versity, spatial heterogeneity, process efficiency,and system stability are simultaneously accommo-dated. Simon (1962) convincingly argued that‘complexity frequently takes the form of hier-archy, and that hierarchic systems have somecommon properties that are independent of theirspecific content.’ In other words, hierarchy is acentral structural scheme of the architecture ofcomplexity, and often manifests itself in the formof modularity in nature.

Why are complex systems usually hierarchicallyorganized? For biological and ecological systems,a hierarchical architecture tends to evolve faster,allow for more stability, and thus is favored bynatural selection (Simon, 1962; Whyte et al., 1969;Pattee, 1973; Salthe, 1985; O’Neill et al., 1986).Although not all hierarchical systems are stable,the construction of a complex system using ahierarchical approach is likely to be more success-ful than otherwise as suggested by the watch-maker parable (Simon, 1962; Muller, 1992; Wu,1999). In evolutionary biology it is well docu-mented that complexity is built upon existingcomplexity. This is also frequently the case in thebusiness world, the political arena, and the engi-neering disciplines. For example, to build com-plex, yet stable and efficient software, computersoftware engineers have developed the object-ori-ented paradigm, which is based on the decomposi-tion principle of hierarchy theory (Booch, 1994).In general, successful human problem-solving

J. Wu, J.L. Da�id / Ecological Modelling 153 (2002) 7–26 9

procedures are hierarchical, too. It has been ar-gued that a non-hierarchical complex system can-not be fully described, and even if it could, itwould be incomprehensible (Simon, 1962; Newelland Simon, 1972). In ecology, the hierarchies weconstruct inevitably result from the interactionsbetween the inherent characteristics of the systemunder study and the observer who studies thesystem. While there is no absolute objectivity,how closely a constructed hierarchy correspondsto the structure of the real system significantlyaffects the usefulness and power of using a hierar-chical approach.

In a sense, a hierarchical approach is a way ofbreaking down complexity and a process of dis-covering or rendering order. To do so, a numberof hierarchical modeling methods have been de-veloped in different disciplines (see Wu 1999 for areview). However, the problem of spatial hetero-geneity and the need for spatial explicitnesspresent grand challenges to the application ofhierarchy theory in modeling ecological systems.Based on the hierarchical patch dynamics (HPD)paradigm (Wu and Loucks, 1995; Wu, 1999), wepresent a spatially explicit hierarchical modelingapproach to studying complex ecological systemsand a modeling software platform that was de-signed to facilitate the development of HPD mod-els. Not to be confused with specific modelingmethods such as cellular automata, genetic al-gorithms, and Markov chains, the spatial HPDmodeling approach is a multiple-scale methodol-ogy for studying complex systems that can bringdifferent modeling techniques together in a coher-ent manner.

2. Theoretical basis for the spatially explicithierarchical modeling approach

The theoretical basis for the spatially explicithierarchical modeling approach is the hierarchicalpatch dynamics paradigm (HPDP), whichemerges out of the integration between hierarchytheory and patch dynamics (Wu and Loucks,1995; Wu, 1999). The following is a brief discus-sion of the major elements of HPDP and theirecological implications.

2.1. Hierarchy theory

Hierarchy theory emerged from a diversity ofstudies in various disciplines, including manage-ment science, economics, psychology, biology,ecology, and systems science (Simon, 1962, 1973;Koestler, 1967; Whyte et al., 1969; Pattee, 1973;Overton, 1975; McIntire and Colby, 1978). It hasbeen significantly refined and expanded in thecontext of evolutionary biology and ecology by aseries of books published in the past two decades(Allen and Starr, 1982; Salthe, 1985; O’Neill et al.,1986; Ahl and Allen, 1996). Thus, major develop-ments in hierarchy theory are relatively recent,although the concepts of ‘levels’ of organizationand ‘hierarchy’ date back to ancient times(Wilson, 1969). Much of the theory is only perti-nent to nested hierarchies in which lower-levelcomponents are completely contained by the nexthigher level, although some general attributes arefound in both nested and non-nested hierarchicalsystems (Valentine and May, 1996; Wu, 1999).

According to hierarchy theory, complex sys-tems have both a vertical structure that is com-posed of levels and a horizontal structure thatconsists of holons (Fig. 1). Hierarchical levels areseparated, fundamentally, by different characteris-tic rates of processes (e.g. behavioral frequencies,relaxation time, cycle time, or response time).Higher levels are characterized by slower andlarger entities (or low-frequency events) whereaslower levels by faster and smaller entities (orhigh-frequency events). Generally speaking, therelationship between two adjacent levels is asym-metric: the upper level exerts constraints (e.g. asboundary conditions) to the lower level, whereasthe lower provides initiating conditions to theupper. On the other hand, the relationship be-tween subsystems (holons) at each level is rela-tively symmetric in that they interact in bothdirections. The interactions among componentswithin the same holon are more strongly andmore frequently than those between holons.

These characteristics of hierarchical structurecan be explained by virtue of ‘loose vertical cou-pling’, permitting the distinction between levels,and ‘loose horizontal coupling’, allowing the sepa-ration between subsystems (holons) at each level


(Simon, 1973). The existence of vertical and hori-zontal loose couplings is the fundamental reasonfor the decomposability of complex systems (i.e.the feasibility of a system to be disassembled intolevels and holons without a significant loss ofinformation). System decomposition (i.e. the pro-cess of separating and ordering system compo-nents according to their temporal and spatialscales) represent one of the most essential tenetsof hierarchy theory. While the word ‘loose’ sug-gests ‘decomposable’, ‘coupling’ implies interac-tions among components. Completedecomposability only occurs when between-com-ponent interactions do not exist, and thus com-plex systems are usually nearly decomposable(Simon, 1962, 1973). According to the principle ofdecomposition, for a given study that is focusedon a particular level, constraints from higher lev-els are expressed as constants, boundary condi-

tions, or driving functions whereas the rapiddynamics at lower levels are filtered (smoothedout) and only manifest as averages or equilibriumvalues. One of the most important implications ofvertical decomposition is that the short-term dy-namics of subsystems can be effectively and jus-tifiably studied in isolation by ignoring thebetween-subsystem interactions that operate onsignificantly longer time scales. On the otherhand, the long-term dynamics of the entire systemis predominantly determined by slow processes.However, it must be noted that occasional excep-tions to this general rule do exist as certain non-linear effects penetrate through several levelsabove or below (so-called perturbing transitivitiesby Salthe, 1991; also see O’Neill et al., 1991a).

Unfortunately, misinterpretations of the term‘hierarchy’ and hierarchy theory may have been amajor reason for a lot of confusions about, and

Fig. 1. Illustration of the major concepts in hierarchy theory (modified from Wu, 1999 and references cited therein). Much of thetheory is only pertinent to nested hierarchies although some general attributes are found in both nested and non-nested hierarchicalsystems (Valentine and May, 1996; Wu, 1999).


resistance against, hierarchy theory within andoutside the scientific community. Thus, it seemsnecessary to point out that hierarchy, as used inthe scientific context, does not always refer to asystem that is rigidly controlled by overwhelmingtop-down constraints and in which bottom-upeffects generated by local interactions are in-significant. Certainly, hierarchy theory does notsuggest this, either. As discussed earlier, hierarchytheory emphasizes both top-down and bottom-upperspectives. While dominance hierarchies do ex-ist in natural, social, and engineered systems(Whyte et al., 1969), the local dynamics of, andinteractions among, components are fundamentalto the very existence of any functioning hier-archies. Indeed, the relative importance or rela-tionship between top-down constraints andbottom-up forces in determining system dynamicsis a key to understanding most if not all complexsystems. Neither does hierarchy theory imply infl-exibility or a lack of diversity and creativity. Onthe contrary, an appropriate hierarchical, dy-namic structure not only provides opportunitiesfor diversity, flexibility, and creativity, but alsofor higher efficiency and stability that are difficultto obtain in non-hierarchical complex systems.

2.2. Hierarchical structure of landscapes

Landscapes are spatially nested hierarchies andcan be effectively studied as such (Woldenberg,1979; Woodmansee, 1990; Reynolds and Wu,1999; Blaschke, 2001; Hay et al., 2001). For exam-ple, Woodcock and Harward (1992) described aforested landscape as a spatial hierarchy: individ-ual trees form distinctive forest stands that in turnconstitute different forest types. Wu and Levin(1994) modeled a serpentine grassland as a dy-namic spatial hierarchy of patches. Reynolds et al.(1996) demonstrated that the arctic tundra land-scapes in Alaska could be effectively studied asspatially nested hierarchies. The lowest hierarchi-cal level and the smallest landscape spatial unitcorrespond to the individual plant, whose func-tioning is determined by numerous interactionsbetween the plant and its immediate abiotic andbiotic environments. At a coarser spatial scale,plants, soil, and associated local microbial and

faunal communities comprise relatively homoge-neous ‘patch’ ecosystems, which in turn form‘integrated flow systems’—distinctive hydrologi-cal units. Then, the landscape is a mixture ofintegrated flow systems that make up the scale ofinterest.

Reynolds and Wu (1999) argued that complexlandscapes have structural and functional units atdifferent scales on both theoretical and empiricalbases (also see Wu and Levin, 1994, 1997; Wuand Loucks, 1995). Landscapes can be perceivedas near-decomposable, nested spatial hierarchies,in which hierarchical levels correspond to struc-tural and functional units at distinct spatial andtemporal scales. The process of identifying struc-tural and functional units involves finding thecharacteristic scales of ecological processes of in-terest and decomposing landscape systems accord-ingly. The objectives of doing so are twofold: (1)to break down the complexity of landscapes byproviding a hierarchical structure to them; and (2)to identify multiple-scale patterns and processesas well as top-down constraints and bottom–upmechanisms. While simplification is an imperativestep toward understanding, the explicit consider-ation of scale multiplicity, which is closely relatedto hierarchical properties of landscapes, is a keyto successful simplifications of complex systems.

2.3. Hierarchical patch dynamic and the scalingladder approach

Spatial patchiness is ubiquitous in ecologicalsystems. The theory of patch dynamics, assumingthat ecological systems are dynamic patch mosa-ics, studies the structure, function and dynamicsof patchy systems with an emphasis on theiremergent properties that arise from interactions atthe patch level (Levin and Paine, 1974; Pickettand White, 1985; Wu and Levin, 1994, 1997;Pickett et al., 1999). On the one hand, hierarchytheory provides useful guidelines for ‘decompos-ing’ complex systems and focuses on a ‘vertical’perspective. On the other hand, patch dynamicsdeals explicitly with the spatial heterogeneity andits change, an apparent ‘horizontal’ or landscapeperspective (Wu, 1999, 2000). The hierarchicalpatch dynamics (HPD) paradigm integrates hier-


Table 1Main tenets of hierarchical patch dynamics paradigm(modified from Wu and Loucks, 1995 and Wu, 1999).

Ecological systems are spatially nested patch hierarchies, inwhich larger patches are made of smaller patches.Dynamics of an ecological system can be studied as the

composite dynamics of individual patches and theirinteractions at adjacent hierarchical levels.

Pattern and process are scale dependent, and they areinteractive when operating in the same domain of scalein space and time.

Non-equilibrium and stochastic processes are not onlycommon, but also essential for the structure andfunctioning of ecological systems.

Ecological stability frequently takes the form ofmeta-stability that is achieved through structural andfunctional redundancy and incorporation in space andtime.

change on geological time scales; thus, landformssubstantially constrain, but are not significantlyaffected by, ecological processes such as commu-nity and ecosystem dynamics (Rowe, 1988; Swan-son et al., 1988). On the other hand, landformsinteract with regional climatic regimes (Rowe,1988), and the leaf-level photosynthetic processesaffect and are affected by the spatial pattern ofmicrometeorological conditions surrounding indi-vidual leaves (Baldocchi, 1993; Wu et al., 2000a).Regional climate patterns surely affect the latentheat fluxes over a landscape, but contribute littleto the understanding of the photosynthetic pro-cess of individual leaves. Some biochemists maywish that their precise understanding of rubisco’scarbon-fixing mechanisms could somehow be di-rectly extrapolated to the global scale, so that thebiospheric responses to elevated CO2 could beequally well predicted in the same way. Unfortu-nately, this is absurdly unrealistic because of thescale separation of several orders of magnitude inspace and time between the enzyme molecule andthe planet. In other words, non-linearity, emer-gent properties, and spatiotemporal heterogeneityin the real world suggest that such a scalingstrategy is theoretically flawed and practicallyformidable.

In an attempt to develop a methodology forstudying the relationship among pattern, processand scale and for extrapolating informationacross heterogeneous landscapes, Wu (1999) pro-posed an HPD multiple-scale modeling and scal-ing strategy-the scaling ladder approach. TheHPD scaling ladder approach is composed ofthree steps: (1) identifying appropriate patch hier-archies. Ecological processes always interact withspatial patterns, but not all spatial patterns matterto ecological processes. In complex ecological sys-tems, reliable spatial scaling must be based on anadequate account for the spatial heterogeneity ofthe landscape (e.g. spatially explicit or statisticalrepresentations). Thus, it is immensely helpful tobe able to identify the spatial patterns— the patchhierarchies— that are relevant to the ecologicalprocesses of interest. The identified patch hier-archies can serve as ‘scaling ladders’ that facilitatemulti-scale modeling and spatial scaling. To iden-tify patch hierarchies is to decompose complex

archy theory and patch dynamics, and emphasizesthe dynamic relationship among pattern, process,and scale in a landscape context (Table 1). As aresult of the integration of the two perspectives,HPD unites structural and functional componentsof a spatially extended system, like a landscape,into a coherent hierarchical framework.

The relationship between pattern and process isscale dependent. In view of hierarchical patchdynamics, pattern and process are only interactivewhen both of them operate on the same or similarspatiotemporal scales. When a spatial pattern ismore or less static relative to the process understudy, only the effect of pattern on process, notprocess on pattern, needs to be considered. Whena spatial pattern changes much faster than theprocess under study, only the spatially filteredaverage property is relevant to the pattern andprocess relationship. In neither of these two casesdoes a reciprocal relationship exist between pat-tern and process. However, when a spatial patternand an ecological process operate at similar ratesin the same spatial domain, their relationship may(but not necessarily) become interactive—by defi-nition, reciprocal. For example, few would imag-ine that centimeter-scale grass clumping patternscould directly affect the behavior of eagles, eventhough these fine-grained patterns certainly influ-ence the movement of beetles. On one hand,landforms cover large geographic areas and


spatial systems. In general, decomposing a com-plex system may invoke a top-down (partitioning)or bottom-up (aggregation) scheme or both (Fig.2). A top-down approach identifies levels andholons by progressively partitioning the entiresystem downscale, whereas a bottom-up schemeinvolves successively aggregating or grouping sim-

ilar entities upscale. A number of quantitativemethods in spatial pattern analysis exist for iden-tifying patch hierarchies. For example, variabilityusually changes abruptly when pattern and pro-cess shift their characteristic domains of scaleacross a heterogeneous landscape. These conspic-uous changes reveal scale breaks that may be

Fig. 2. Illustration of the process of decomposing a complex system to find an appropriate patch hierarchy and building hierarchicalmodels.


indicative of hierarchical levels (e.g. O’Neill et al.,1991b; Cullinan et al., 1997; Wu et al., 2000b).

(2) Making observations and developing modelsat focal levels. Once an appropriate patch hier-archy is established, ecological processes can bestudied at focal levels (corresponding to charac-teristic domains of scale), by properly choosinggrain size (sampling interval or spatial resolution)and extent (study duration or area). There arealways many factors affecting a given ecologicalprocess, but usually only a few are dominant for agiven spatiotemporal domain of scale (Holling,1992). Thus, a process-relevant patch hierarchyeffectively groups these factors into relatively sep-arate regions according to their characteristicscales in space and time. It is crucial to under-stand the role of scale in making observations.The phenomena of interest are only observable atthe appropriate scale of observation. Simon(1973) explained well how temporal scale shouldbe chosen by dividing system behaviors into high,medium, and low ranges of characteristic frequen-cies. The medium range corresponds to the focallevel. If the total time span for a study is T, andif the temporal resolution of the observation (ortime interval between measurements) is �, thebehavior of the system that is much faster than1/� (high frequency events) appears to be noise,and its meaning (signal) to the focal level isrevealed by its statistical averages. On the otherhand, system dynamics that are much slower than1/T (low frequency events) will not be observedand can be treated as constants at the focal level.This principle remains equally valid for the rela-tionship between system dynamics and character-istic spatial scales where T and � are the spatialextent and grain size of the observation, respec-tively. The above argument provides the essentialtheoretical basis for adopting the so-called triadicstructure of hierarchy in research. That is, whenone studies a phenomenon at a particular hierar-chical level (level x), the mechanistic understand-ing comes from the next lower level (level x−1),whereas the significance of that phenomenon isrevealed at the next higher level (level x+1).

(3) Extrapolating information across the do-mains of scale hierarchically. Scaling or extrapo-lating information across scales (or levels) over

spatially heterogeneous landscapes has proven tobe a formidable task because of complex pattern–process interactions. The most salient aspect ofthis complexity is the non-linearity in time andspace that is the fundamental source of emergentproperties. Thus, a major role of a patch hier-archy identified in step one is to serve as a scalingladder that is composed of the domains of scalerelevant to a particular study. Scaling can beaccomplished by changing the grain size and ex-tent of models along the patch hierarchy (Fig. 3).While a variety of specific scaling techniques canbe applied here (e.g. Iwasa et al., 1987, 1989;Ehleringer and Field, 1993; van Gardingen et al.,1997; Jarvis, 1995; see Wu, 1999 for a review), ageneral approach is to link models along thescaling ladder that are built individually arounddistinctive focal levels. One of the most sensibleways of doing so is to use the output of lower-level models as the input to upper-level models.Sometimes, the input may take the form of re-sponse curves or surfaces that are generated usingstatistical methods based on the output from alower-level model (e.g. Reynolds et al., 1993).Similarly, such hierarchical scaling can be imple-mented from top down—using the output ofhigher-level models to constrain or drive lower-level models. This top-down approach has be-come increasingly appealing and feasible asremote sensing data, with high temporal and spa-tial resolutions, are readily available over largegeographic areas.

3. A hierarchical patch dynamics model of thePhoenix urban landscape (HPDM-PHX)

In this section, we demonstrate how to imple-ment the hierarchical patch dynamics paradigmand the scaling ladder approach in modeling com-plex ecological systems through an example, thehierarchical patch dynamics model for thePhoenix urban landscape (HPDM-PHX). This ex-ample is a part of our on-going modeling effortsassociated with the Central Arizona-PhoenixLong-Term Ecological Research (CAP-LTER)and related research projects. Although the sys-tem under study is an urban landscape, the spa-


Fig. 3. Illustration of hierarchical scaling or extrapolating information along a hierarchical scaling ladder. Scaling up or down canbe implemented by changing model grain size and extent successively across domains of scale.

tially explicit hierarchical modeling approach isgeneral, and can be used for other complex eco-logical systems.

3.1. Background

Urbanization has drastically transformed natu-ral landscapes everywhere throughout the world,inevitably exerting profound effects on the struc-ture and function of ecosystems. In particular, theconversion of natural and agricultural areas tohighly artificially modified urban land uses hasbeen taking place at an astonishing rate. Accord-ing to the United Nations, the world urban popu-lation was only a few percent of the globalpopulation in the 1800s, but increased to nearly30% in 1950 and reached 50% in 2000. It has beenprojected that 60% of the world population willlive in urban areas by 2025.

Land use and land cover changes associatedwith urbanization significantly affect the composi-tion of plant communities by fragmenting thelandscape, removing and introducing species, andaltering water, carbon, and nutrient pathways.Although urban areas represent arguably themost important habitats for humans, they are

among the least understood ecosystems of all, andurban ecology has not been considered part of themainstream ecology worldwide (Collins et al.,2000). It is true that ecological studies in urbanareas have a long history that dates back to theearly 1900s or even earlier (Breuste et al., 1998).Also, much research has been done to understandspatial pattern and urban dynamics by geogra-phers and social scientists with little or only su-perficial consideration of ecology in and aroundcities. However, a full understanding of how ur-ban ecosystems work does not come from iso-lated, disciplinary studies, be they ecological,sociological, or geographic. The urban whole islarger than the sum of its biotic and abiotic parts.The ecology of urban systems as integratedwholes needs new and integrative perspectives(Pickett et al., 1997, Zipperer et al., 2000).

In the southwest US, the Phoenix metropolitanarea in particular, urbanization has profoundlychanged the desert landscape. In fact, Phoenix hasbecome the sixth largest city with the highestpopulation growth rate in the United States. Tounderstand the interactions between urbanizationand ecological conditions, we have been develop-ing models based on the hierarchical patch dy-


namics paradigm to simulate the pattern and pro-cess of urban growth and its ecological conse-quences. This section describes the generalstructure of the hierarchical patch dynamicsmodel for the Phoenix metropolitan landscape(HPDM-PHX). The main goal of the currentversion of HPDM-PHX is to develop an under-standing of how urbanization affects ecosystemproductivity and biogeochemical cycles at localand regional scales.

3.2. General model structure of the HPDM-PHX

A spatially nested patch hierarchy is used forHPDM-PHX, which consists of local ecosystems,local landscapes, and the regional landscape (Figs.4 and 5). In the case of modeling ecosystemprocesses, this patch hierarchy is essentially ahierarchical implementation of the ecosystemfunctional type (EFT) concept (Reynolds et al.,1997; Reynolds and Wu, 1999). Local ecosystemscorrespond to land cover types that have a rela-tively homogeneous vegetation–soil complexwithin (e.g. cotton fields, urban centers, residen-

tial areas, parks, creosote bush-dominated desertcommunities). The land cover EFTs are readilydetectable from air photos and remote sensingdata (e.g. Landsat TM images), and largely corre-spond to the categories of the Anderson et al.’s(1976) level II classes. A local landscape is a patchmosaic of local ecosystems, in which spatial pat-terns emerge. Local landscapes are characterizedby dominant land cover types, and several differ-ent types can thus be recognized (e.g. urban,rural, agricultural, and natural landscapes). Thus,the structure and function of a landscape EFT isa function of its (non-spatial) composition and(spatial) configuration. Finally, the regional EFTis a mixture of local landscapes, and characterizedby climate, geomorphology, hydrology, soils, andvegetation at the regional scale. Because the EFTconcept emphasizes ecosystem attributes and pro-cesses such as primary productivity, biogeochem-istry, and hydrology, it gives concrete meanings topatches and thus reinforces the less tangible func-tional aspect of the hierarchical patch dynamicsparadigm.

Fig. 4. Hierarchical ecosystem functional types (EFTs) for the Phoenix metropolitan area. The EFT hierarchy consists of localecosystems, local landscapes, and the region. Each of these hierarchical levels is characterized by a set of distinct structural andfunctional features.


Fig. 5. Diagrammatic representation of the basic structure of the hierarchical patch dynamics model of the Phoenix urban landscape(HPD-PHX).

At the local ecosystem level, we use modifiedversions of two ecosystem process models: CEN-TURY, a general model of terrestrial biogeo-chemistry originally developed for the GreatPlains grassland ecosystem by Parton et al. (1987,1988) and PALS, a patch-level arid ecosystemsimulator developed by Reynolds and his associ-ates for the Jornada basin, New Mexico(Reynolds et al., 1993, 1997). CENTURY simu-lates the long-term dynamics of carbon, nitrogen,phosphorus, sulfur, and plant production and hasbeen tested for a number of grassland ecosystemsworldwide (Parton et al., 1993). PALS simulatescarbon, water, nitrogen, and phosphorus cycles,and takes into account variations in patch type,

plant characteristics, soil resources, and climaticfactors. The abiotic components of PALS includemicrometeorological conditions (e.g. temperatureand moisture within and above the canopy) andsoil properties (e.g. water flux, nutrients, and tem-perature). PALS is well-suited to explore ques-tions related to nutrient cycling and has beenparameterized for the Jornada LTER site, theCalifornia chaparral, and a grassland in Kansas(Reynolds et al., 1997; Reynolds and Wu, 1999).We use these two ecosystem models in parallel forthe following reasons. CENTURY and PALSrepresent different levels of mechanistic details insimulating ecosystem processes, and thus compar-ing them can help us understand what details can


be ignored in the process of scaling up from thelocal ecosystem to the region. Model comparisonsalso provide a means for increasing our confi-dence in estimating ecological variables especiallywhen data are rarely available (Schimel et al.1997). Moreover, ecosystem models that are tai-lored for different land cover types found in thePhoenix metropolitan area can be more effectivelydeveloped based on CENTURY and PALS.

While our land cover change (sub)model inHPDM-PHX shares some of the similarities ofthe Markov-cellular automata approach (e.g. Liand Reynolds, 1997), it is integrated directly withthe ecosystem model. The regional model is theintegration of various component landscapes withexplicit consideration of their horizontal interac-tions (Fig. 5). The land cover change model isdriven by local rules and top-down constraintswhich are in turn influenced by socioeconomicprocesses in the region. Changes in landscapepattern then result in changes in ecosystem pro-cesses at both local and regional scales. Althoughthe effects of land use and land cover change onecological processes are often more obvious anddominant than the feedback of changed ecologicalconditions to land use decisions, the latter doesexist and will become more important as urban-ization continues to progress. While still on-going,our model evaluation process involves severalsteps: (1) to assess the reasonableness of themodel structure and the interpretability of func-tional relationships within HPDM-PHX; (2) tosimulate ecosystem processes across a gradient ofland cover types; (3) to evaluate the correspon-dence between model behavior and empiricallyobserved patterns at local ecosystem, landscape,and regional scales; and (4) to conduct a series ofsensitivity and uncertainty analysis to HPDM-PHX.

4. Developing a hierarchical patch dynamicsmodeling platform (HPD-MP)

Constructing and evaluating hierarchical patchdynamics models like HPDM-PHX can be techni-cally complex in terms of programming, datahandling, and model linkage and interface. To

facilitate the development of such models, there-fore, we have been building an HPD-based mod-eling platform (HPD-MP). In this section wedescribe the general structure of HPD-MP, andillustrate how it is being used in our effort todevelop HPDM-PHX.

4.1. Description of HPD-MP

The hierarchical patch dynamics modeling plat-form is designed to facilitate spatially explicithierarchical modeling by taking a ‘fine-grained’approach to program interoperability. Practicallyspeaking, this means that we begin by approach-ing modeling problems from a programmer’s viewpoint: first developing necessary objects (or mod-ules) and application programming interfaces(APIs), and then using them to further developtools and utilities that allow users to developmodels with a minimal amount of programming.The fine-grained approach differs from ‘coarse-grained’ methods that adopt existing modelingand data management tools (e.g. simulation andGIS packages) via common interchange linkagesand languages (e.g. COM, DCOM, CORBA,XML). While these two approaches represent dif-ferent modeling perspectives, they are not mutu-ally exclusive. By taking a fine-grained approach,HPD-MP provides a high degree of flexibility thatallows for modeling a variety of complex systemsand a high level of user-friendliness that eases itsapplications.

The hierarchical patch dynamics modeling plat-form consists of software libraries, algorithms,data converters, and a series of tools and utilitiesfor model development and integration. All thesecomponents are organized into two levels withinHPD-MP (Fig. 6). The API/Data Format levelgives users the flexibility to develop new objects,tools and utilities in C++ and to build modelsfrom ground up. The Tools/Utilities level allowsusers to develop models using tools and utilitiesprovided by HPD-MP, and to link them withother models and data management tools externalto HPD-MP. An application programming inter-face (API) is simply a set of routines, protocols,and tools for building software applications. Itserves as a software interface for other programs


Fig. 6. Illustration of the structure of the hierarchical patch dynamics modeling platform (HPD-MP), showing the majorcomponents and their relationship.


such as image manipulation routines, and makesit easier to develop applications by providingnecessary building blocks.

Spatially explicit models often have to deal withquestions similar to those in computational ge-ometry (CG). For example, is a given point inside,on, or outside a polygon? What is the surface areaof a polygon or higher dimensional object? Dotwo geometric objects intersect? If so, what arethe intersection and union of these objects? Whatare the nearest neighbors of a given object? Todeal with these computational problems, HPD-MP incorporates a number of CG algorithms andmethods (O’Rourke, 1998; de Berg et al., 2000).

When contemplating a modeling project we areoften confronted with the choice between rasterand vector data. While tradeoffs of choosing oneover the other are well documented and under-stood, specific data formats (e.g. DEM, GIS cov-erages, digitized aerial photos) have no commonapplication programming interfaces (APIs) thatdeal with both raster and vector data interchange-ably, much less with some mechanisms throughwhich spatial queries can be readily made. Toovercome this problem, HPD-MP contains utili-ties to read and translate between a variety ofdata formats. The Hierarchical Data Format(HDF; Brown et al., 1993; Schmidt, 2000) is usedas a base for information interchange because itcan handle large (greater than 2 gigabytes) andmultidimensional data sets, permits the use ofboth vector and rater data layers within the samefile, and allows user-definable data types. In addi-tion, we also use the ImageMagick Studio imageprocessing libraries (http://magick.imagemagick.org) to provide a transpar-ent interface to read and write over 60 differentcommon image formats. This capability of HPD-MP greatly facilitates I/O and visualization pro-cesses. For example, the data and imageprocessing facilities are not only convenient fordeveloping utilities like converting ARC GRIDand SHAPE files (yet to be implemented), butalso allow for the automatic generation of outputinto common image formats like GIF and JPEG.

There are three basic ways to use HPD-MP: (1)to use the existing tools and utilities, as well asprebuilt models provided by the platform to run

simulations by simply specifying model parame-ters and input data; (2) to use high-level modelingtools and utilities provided by the platform todevelop models for user-defined problems; and (3)to develop new objects or modules using a pro-gramming language (e.g. C++) and, at the sametime, take advantage of the capabilities of theexisting tools and utilities. Currently, HPD-MPincludes several high-level utilities, includingMINT—a model interpreter that reads in andruns an existing model (e.g. STELLA equations;HPS, 1996), MODEL2XX—a utility to translatevarious modeling languages (e.g. STELLA) intostand-alone programs in C++ (and JAVA in fu-ture), HPD–STATS—a collection of statisticalroutines and tools for spatial analysis, andDATA–CONV—a data converter that supportsa number of data formats, including ImageMag-ick Studio, HDF, and ArcInfo/ArcView.

4.2. Examples of using HPD-MP

As a demonstration of HPD-MP, we presenttwo examples here: (1) a land use change modelfor the Phoenix area; and (2) a spatial multi-spe-cies population dynamics model. These exampleshere are used only for the purpose of illustratingthe use of HPD-MP. A thorough evaluation ofthe structure and behavior of these models is notintended, therefore.

4.2.1. A hierarchical stochastic cellular automatamodel of land use change

The land use change model is a hierarchicalstochastic cellular automata model. Stochastic cel-lular automata (CA) typically model the statetransition probability as a function of only localor neighborhood rules. These local-interactionsare assumed to be the only driving forces togenerate the global pattern of system dynamics.However, in real-world phenomena, local-interac-tions are frequently modified by patterns andfactors at broader scales that act as top-downconstraints or driving functions. These may eitherbe spatially fixed (as in the case of propertyownership boundaries and zoning ordinance re-strictions) or variable (such as domains of influ-ence or the land use change due to the proximity

http://magick.imagemagick.org




to roads). They are important in simulating ob-served land use change patterns. HPD-MP wasused to facilitate such multiple-scale modelingwhereby local processes may be modified by top-down constraints or driving functions and, at thesame time, bottom-up propagation of informationis allowed to hierarchically link interactions be-tween scales.

Specifically, we modeled the land use change inthe Phoenix area by explicitly considering localurban growth factors, domains of urbanizationinfluence, and effects of ownership (Fig. 7). Thetransition probability between different land usetypes (urban, agriculture, and desert) was treated

as a function of the three groups of factors atdifferent spatial scales, i.e. Pchange= f (local rules,domain of influence, ownership). The domains ofinfluence were intended to reflect heterogeneousurbanization situations between the scale corre-sponding to the pixel size and the entire region. Inreality, they may modify land use transition prob-abilities through legal and zoning restrictions. Asa first approximation we assumed that these do-mains of influence operated independently of thelocal-scale processes. The data used to parameter-ize the model were historic land-use maps fromCAP-LTER (Knowles-Yanez et al., 1999). In anearlier study, Jenerette and Wu (2001) had used

Fig. 7. Illustration of the hierarchical structure of the stochastic CA model of land use change in Phoenix (A), and a comparisonbetween observed and simulated land use patterns (B).


this data set (for years 1975 and 1995) to developa cellular automata urban growth model using agenetic algorithm (GA) optimization approach. Inthis study, we adopted the optimized parameterset from Jenerette and Wu (2001) for the initialvalues of land use change transition probabilitiesat the pixel size of 250 by 250 m (same as inJenerette and Wu, 2001). Ownership informationwas obtained from a 1988 data set provided bythe Arizona Land Resource Information System(ALRIS).

The model was initialized with the1975 land-usemap and run for 55 years with a time step of 1year to year 2030. This period of simulation waschosen to conform to the 50-year plan initiated bythe local government in 1980. The simulated landuse map for 1995 was compared with the empiri-cal map for the same year, and the two mapsmatched each other well (Fig. 7). The observedurbanized area was approximately 1975 km2 whilethe predicted value was 1973 km2. This high accu-racy in the simulated urbanized area, however,should be of no surprise given that the originalparameterization of the model, as in all otherMarkov chain or transition probability models oflandscape change, was based on the 1975 and1995 land use maps (Jenerette and Wu, 2001).Nevertheless, the high accuracy in spatial patternof urban growth, attributable to the incorporationof hierarchical constraints, was noteworthy. Wecontinued the simulation up to 2030 when a totalof 3659 km2 or roughly 69% of the total privatelyowned land in the study area was urbanized (Fig.7). This example not only demonstrates the utilityof HPD-MP, but also the increased accuracy andinterpretability of the CA approach due to theaddition of a hierarchical structure.

4.2.2. A spatial multi-species population dynamicsmodel

While the Phoenix land use change model in-volves only the simulation of landscape pattern,our second example focuses on the linkage be-tween spatial pattern and process-based models.This is a pilot model we developed to test thetools and utilities of the HPD modeling platform.Through this example we intend to illustrate someof the basic procedures for implementing process-

based models, such as the ecosystem process mod-els encapsulated in the HPDM-PHX which iscurrently under construction.

In the arid environment of the southwesternUnited States, water resources are often over-allo-cated, and this has resulted in devastating modifi-cation to the natural flood regimes. These naturalprocesses are necessary to provide suitable germi-nation sites and conditions for native plants suchas willow (Salix gooddingii ) and cottonwood(Populus fremontii ). The modification to the floodregimes in turn has profoundly impacted nativevertebrates such as birds. The modification to thenatural wetland habitats has allowed a number ofinvasive exotics, such as salt cedar (Tamarixramosissima), to become established by out-com-peting the native plant species. We hypothesizedthat one possible method to restore the naturalriparian communities is to mimic the naturalflood regimes with managed dam releases.

As an example of how HPD-MP can be used todevelop spatial process-based models and in orderto test the above hypothesis on riparian habitatrestoration, we developed a plant competitionmodel for the drought-tolerant salt cedar and theinundation-tolerant willow and cottonwood, anda suitable habitat-based bird population dynamicmodel. Both of these models were developed usingSTELLA (as are several components of HPDM-PHX under development). The plant competitionmodel was of the Lotka–Volterra type, but ex-plicitly incorporated the spatial variations in to-pography and the water table. For the purposesof demonstration, the topography was generatedusing Rosenbrock’s (1960) multimodal mathemat-ical function: f(x,y)= (1−x)2+100(y−x2)2,where −1.5�x�1.5, −0.5�y�1.5. Thisfunction is a well-established mathematical sur-face and can be used to portray a generalizedriverbed. The maximum difference in elevationbetween high and low points for the generatedtopographic surface was 5 m. The water table wasin turn varied according to three hypotheticalscenarios of water management: (1) static waterlevel— the water level remains at a fixed height of0.5 m (as measured from the bottom of the riverchannel); (2) random water level— the water levelis uniformly random between 0 and 1.0 m; and (3)


pulsed water level— the water level remains low inbetween periodic dam releases. The results of theplant competition model then produced the habi-tat suitability map for the bird population model.To run the STELLA models spatially, we firstexported the STELLA models as finite differenceequations (FDEs), and then these FDEs togetherwith spatially-gridded information on topographyand hydrology were input into HPD-MP’s modelinterpreter (MINT) and run within each cell onthe landscape. Although this was a model basedon contrived data, it was interesting to notice thatthe periodic flood scenario produced the greatestnative plant recruitment, which was in agreementwith observed riparian vegetation dynamics (Mid-dleton, 1999).

5. Discussion and conclusions

A distinctive feature of the prevailing theme inthe science of complexity is that local interactionsamong components are essential for the organiza-tion and global dynamics of complex systems. AsMitchell et al. (1994) pointed out, ‘a central goalof the sciences of complex systems is to under-stand the laws and mechanisms by which compli-cated, coherent global behavior can emerge fromthe collective activities of relatively simple, locallyinteracting components.’ This view has been rein-forced by the wide-spread use of such approachesas cellular automata, genetic algorithms, andagent-based modeling, all of which rely heavily ona bottom-up, rather than a top-down, perspective.While admitting that local interactions and bot-tom-up forces are essential, we argue that top-down constraints and hierarchical linkages arealso crucial for understanding and predicting thedynamics of many, if not most, complex systems.In general, ecological systems are not sandpiles,but hierarchical patch dynamic systems withevolving structures and changing components.

Therefore, to deal with the complexity of eco-logical systems we advocate the hierarchical patchdynamics paradigm (Wu and Loucks, 1995; Pick-ett et al., 1999; Reynolds and Wu, 1999; Wu,1999). The HPD paradigm integrates hierarchytheory and patch dynamics, and represents a spa-

tially explicit theory of pattern, process, scale andhierarchy. Because complexity always involvesmultiple scales, HPD provides a sensible andpowerful approach to modeling complex ecologi-cal systems and spatial scaling over heterogeneouslandscapes. Scaling in ecology is inevitable for atleast two important reasons (Wu, 1999; Wu andQi, 2000). First, most environmental and resourcemanagement issues can only be dealt with effec-tively at broad scales whereas much of the empir-ical information has been collected at local scales.Second and more profoundly, to understand howecological systems work we must be able to relatebroad-scale patterns to fine-scale processes andvice versa. In both cases, transferring informationbetween scales is indispensable. The HPD scaling-ladder strategy (Wu, 1999) provides a hierarchicalway of dealing with spatial heterogeneity andmodularizing nonlinearity so as to facilitate theextrapolation and translation of informationacross scales.

Based on the hierarchical patch dynamicsparadigm and the scaling ladder concept, in thispaper we have articulated a spatially hierarchicalmodeling approach to studying complex systems.Then, we described how this approach has beenused in developing the hierarchical patch dynam-ics model of the Phoenix urban landscape(HPDM-PHX) that simulates the land use changeand related ecosystem processes. In addition, wepresented the hierarchical patch dynamics model-ing platform (HPD-MP)—a software package-from which multi-scale ecological models can bedeveloped and integrated in an efficient and co-herent manner. To illustrate the utility of HPD-MP, we discussed two examples: a hierarchicalstochastic CA model of land use change and aspatial population dynamics model. This model-ing platform is still being developed and will becontinuously refined through the development ofthe hierarchical patch dynamics model of thePhoenix urban landscape and related modelingprojects at the Landscape Ecology and ModelingLaboratory (LEML) at Arizona State University.

The spatially explicit hierarchical modeling ap-proach we have presented here is but one ap-proach to modeling and understanding complexecological systems. At a time when complexity


and diversity are recognized and emphasized, plu-ralism in ecology is not only appropriate but alsonecessary. However, neither extremely reduction-ist nor metaphysically holistic approaches seem tobe productive when dealing with such phenomenaas self-organization and emergent properties. Toeffectively deal with complexity, we need morethan simply using both approaches in parallel; weneed to integrate them and produce new, moreeffective approaches. We believe that most suchapproaches are hierarchical in one way or another(Wu, 1999). In this regard, it is always refreshingand enlightening to cite Morrison (1966): ‘Theworld is both richly strange and deeply simple.That is the truth spelled out in the graininess ofreality; that is the consequence of modularity.Neither gods nor men mold clay freely; ratherthey form bricks.’

Acknowledgements

We thank Darrel Jenerette, Habin Li, MattLuck and Thomas Meyer for their comments onthis paper. Darrin Thome and Hoski Schaasfmahelped with the development of the spatial multi-species population dynamics model. JW wouldlike to acknowledge the support for his researchin hierarchical patch dynamics from US Environ-mental Protection Agency grant R827676-01-0and US National Science Foundation grant DEB97-14833 (CAP-LTER). Although the researchdescribed in this paper has been funded in part bythe above mentioned agencies, it has not beensubjected to the Agencies’ required peer and pol-icy review and therefore does not necessarilyreflect the views of the agencies and no officialendorsement should be inferred.

References

Ahl, V., Allen, T.F.H., 1996. Hierarchy Theory: A Vision,Vocabulary, and Epistemology. Columbia UniversityPress, New York, 206 pp.

Allen, T.F.H., Starr, T.B., 1982. Hierarchy: Perspectives forEcological Complexity. University of Chicago Press,Chicago, 310 pp.

Anderson, J.R., Hardy, E.E., Roach, J.T., Witmer, R.E., 1976.A Land Use and Land Cover Classification System for Usewith Remote Sensor Data. United States Goverment Print-ing Office, Washington, D.C., 28 pp.

Bak, P., 1996. How Nature Works: The Science of Self-Orga-nized Criticality. Copernicus (an imprint of Springer-Ver-lag New York, Inc.), New York, 212 pp.

Bak, P., Chen, K., 1991. Self-organized criticality. Sci. Am.264, 46–53.

Bak, P., Tang, C., Wiesenfeld, K., 1988. Self-organized criti-cality. Phys. Rev. A 38, 364–374.

Baldocchi, D.D., 1993. Scaling water vapor and carbon diox-ide exchange from leaves to a canopy: Rules and tools. In:Ehleringer, J.R., Field, C.B. (Eds.), Scaling PhysiologicalProcesses: Leaf to Globe. Academic Press, San Diego, pp.77–114.

Blaschke, T., 2002. Continuity, complexity and change: Ahierarchical geoinformation-based approach to explorepatterns of change in a cultural landscpae in Germany. In:U. Mander, H. Palang (Eds.), Multifunctional Landscapes:Continuity and Change. WIT Press, in press.

Booch, G., 1994. Object-Oriented Analysis and Design withApplications. Addison-Wesley, Reading, 589 pp.

Breuste, J., Feldmann, H., Uhlmann, O. (Eds.), 1998. UrbanEcology. Springer, Berlin, 714 pp.

Brown, S.A., Folk, M., Goucher, G., Rew, R., 1993. Softwarefor portable scientific data management. Comput. Phys. 7,304–308.

Collins, J.P., Kinzig, A., Grimm, N.B., Fagan, W.F., Hope,D., Wu, J., Borer, E.T., 2000. A new urban ecology.American Scientist 88, 416–425.

Cullinan, V.I., Simmons, M.A., Thomas, J.M., 1997. ABayesian test of hierarchy theory: scaling up variability inplant cover from field to remotely sensed data. LandscapeEcol. 12, 273–285.

de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf,O., 2000. Computational Geometry: Algorithms and Ap-plications. Springer, New York, 367 pp.

Ehleringer, J.R., Field, C.B. (Eds.), 1993. Scaling PhysiologicalProcesses: Leaf to Globe. Academic Press, San Diego, 388pp.

Hay, G., Marceau, D.J., Dube, P., Bouchard, A., 2001. Amultiscale framework for landscape analysis: object-spe-cific analysis and upscaling. Landscape Ecol. 16, 471–490.

Holling, C.S., 1992. Cross-scale morphology, geometry, anddynamics of ecosystems. Ecol. Monogr. 62, 447–502.

HPS, 1996. An Introduction to Systems Thinking. High Per-formance Systems Inc., Hanover, 172 pp.

Iwasa, Y., Andreasen, V., Levin, S.A., 1987. Aggregation inmodel ecosystems: I. Perfect aggregation. Ecol. Model. 37,287–302.

Iwasa, Y., Levin, S.A., Andreasen, V., 1989. Aggregation inmodel ecosystems: II. Approximate aggregation. IMA J.Math. Appl. Med. Biol. 6, 1–23.

Jarvis, P.G., 1995. Scaling processes and problems. Plant CellEnviron. 18, 1079–1089.


Jenerette, G.D., Wu, J., 2001. Analysis and simulation of landuse change in the central Arizona–Phoenix region. Land-scape Ecol. 16, 611–626.

Jensen, H.J., 1998. Self-Organized Criticality: Emergent Com-plex Behavior in Physical and Biological Systems. Cam-bridge University Press, New York, 153 pp.

Jørgensen, S.E., 1995. Complex ecology in the 21st century. In:B.C. Patten, S.E. Jørgensen, S.I. Auerbach (Eds.), Com-plex Ecology: The Part–Whole Relation in Ecosystems.Prentice-Hall, Englewood Cliffs, NJ, pp. xvii–xix.

Jørgensen, S.E., Mejer, H., Nielsne, S.N., 1998. Ecosystem asself-organizing critical systems. Ecol. Model. 111, 261–268.

Kirchner, J.W., Weil, A., 1998. No fractals in fossil extinctionstatistics. Nature 395, 337–338.

Knowles-Yanez, K., Moritz, C., Fry, J., Redman, C.L.,Bucchin, M., McCartney, P.H., 1999. Historic Land Use:Phase I Report on Generalized Land Use. Central Ari-zona–Phoenix Long-Term Ecological Research(CAPLTER), Phoenix, 21 pp.

Koestler, A., 1967. The Ghost in the Machine. RandomHouse, New York, 384 pp.

Levin, S.A., 1999. Fragile Dominions: Complexity and theCommons. Perseus Books, Reading, 250 pp.

Levin, S.A., Paine, R.T., 1974. Disturbance, patch formationand community structure. Proc. Natl. Acad. Sci. USA 71,2744–2747.

Li, H., Reynolds, J.F., 1997. Modeling effects of spatial pat-tern, drought, and grazing on rates of rangeland degrada-tion: a combined Markov and cellular automata approach.In: Quattrochi, D.A., Goodchild, M.F. (Eds.), Scales inRemote Sensing and GIS. Lewis Publishers, Boca Raton,FL, pp. 211–230.

McIntire, C.D., Colby, J.A., 1978. A hierarchical model oflotic ecosystems. Ecol. Monogr. 48, 167–190.

Middleton, B., 1999. Wetland Restoration, Flood Pulsing, andDisturbance Dynamics. Wiley, New York, 388 pp.

Mitchell, M., Crutchfield, J.P., Haraber, P.T., 1994. Dynam-ics, computation, and the ‘edge of chaos’: a re-examina-tion. In: Cowan, G.A., Pines, D., Meltzer, D. (Eds.),Complexity: Metaphors, Models, and Reality. PerseusBooks, Reading, pp. 497–513.

Morrison, P., 1966. The modularity of knowing. In: Kepes, G.(Ed.), Module, Proportion, Symmetry, Rhythm. Braziller,New York, pp. 1–19.

Muller, F., 1992. Hierarchical approaches to ecosystem theory.Ecol. Model. 63, 215–242.

Newell, A., Simon, H.A., 1972. Human Problem Solving.Prentice-Hall, Englewood Cliffs, NJ, 920 pp.

O’Neill, E.G., O’Neill, R.V., Norby, R.J., 1991a. Hierarchytheory as a guide to mycorrhizal research on large-scaleproblems. Environ. Pollut. 73, 271–284.

O’Neill, R.V., DeAngelis, D.L., Waide, J.B., Allen, T.F.H.,1986. A Hierarchical Concept of Ecosystems. PrincetonUniversity Press, Princeton, 253 pp.

O’Neill, R.V., Gardner, R.H., Milne, B.T., Turner, M.G.,Jackson, B., 1991b. Heterogeneity and spatial hierarchies.In: Kolasa, J., Pickett, S.T.A. (Eds.), Ecological Hetero-geneity. Springer-Verlag, New York, pp. 85–96.

O’Rourke, J., 1998. Computational Geometry in C. Cam-bridge University Press, Cambridge, 376 pp.

Overton, W.S., 1975. The ecosystem modeling approach in theconiferous forest biome. In: Patten, B.C. (Ed.), SystemsAnalysis and Simulation in Ecology. Academic, NewYork, pp. 117–138.

Parton, W.J., Schimel, D.S., Cole, C.V., Ojima, D.S., 1987.Analysis of factors controlling soil organic matter levels inGreat Plains grasslands. Soil Sci. Soc. Am. J. 51, 1173–1179.

Parton, W.J., Stewart, J.W.B., Cole, C.V., 1988. Dynamics ofC, N, P and S. in grassland soils: a model. Biogeochemistry5, 109–131.

Parton, W.J., Scurlock, J.M.O., Ojima, D.S., Gilmanov, T.G.,Scholes, R.J., Schimel, D.S., Kirchner, T., Menaut, J.-C.,Seastedt, T., Moya, E.G., Kamnalrut, A., Kinyanmario,J.I., 1993. Observations and modeling of biomass and soilorganic matter dynamics for grassland biome worldwide.Global Biogeochem. Cycles 7, 785–809.

Pattee, H.H. (Ed.), 1973. Hierarchy Theory: The Challenge ofComplex Systems. George Braziller, New York, 156 pp.

Pickett, S.T.A., White, P.S., 1985. The Ecology of NaturalDisturbance and Patch Dynamics. Academic Press, Or-lando, 472 pp.

Pickett, S.T.A., Burch, J.W.R., Dalton, S.E., Foresman, T.W.,Grove, J.M., Rowntree, R., 1997. A conceptual frameworkfor the study of human ecosystems in urban areas. UrbanEcosyst. 1, 185–199.

Pickett, S.T.A., Wu, J., Cadenasso, M.L., 1999. Patch dynam-ics and the ecology of disturbed ground. In: Walker, L.R.(Ed.), Ecosystems of Disturbed Ground. Ecosystems of theWorld 16. Elsevier, Amsterdam, pp. 707–722.

Prigogine, I., 1997. The End of Certainty: Time, Chaos, andthe New Laws of Nature. Free Press, New York, 240 pp.

Raup, D.M., 1997. A breakthrough book? Complexity 2,30–32.

Reynolds, J.F., Wu, J., 1999. Do landscape structural andfunctional units exist? In: Tenhunen, J.D., Kabat, P.(Eds.), Integrating Hydrology, Ecosystem Dynamics, andBiogeochemistry in Complex Landscapes. Wiley,Chichester, pp. 273–296.

Reynolds, J.F., Hilbert, D.W., Kemp, P.R., 1993. Scalingecophysiology from the plant to the ecosystem: a concep-tual framework. In: Ehleringer, J.R., Field, C.B. (Eds.),Scaling Physiological Processes: Leaf to Globe. AcademicPress, San Diego, pp. 127–140.

Reynolds, J.F., Tenhunen, J.D., Leadley, P.W., 1996. Patchand landscape models of Arctic Tundra: potentials andlimitations. In: Reynolds, J.F., Tenhunen, J.D. (Eds.),Ecological Studies. Springer-Verlag, Berlin, pp. 293–324.

Reynolds, J.F., Virginia, R.A., Schlesinger, W.H., 1997. Defin-ing functional types for models of desertification. In:Smith, T.M., Shugart, H.H., Woodward, F.I. (Eds.), PlantFunctional Types. Cambridge University Press, Cam-bridge, pp. 195–216.

Rosenbrock, H.H., 1960). Automatic method for finding thegreatest or least value of a function. Comput. J. 3, 175–184.


Rowe, J.S., 1988. Landscape Ecology: The Study of TerrainEcosystems. In: Moss, M.R. (Ed.), Landscape Ecology andManagement. Polyscience Publ. Inc., Montreal, pp. 35–42.

Salthe, S.N., 1985. Evolving Hierarchical Systems: TheirStructure and Representation. Columbia University Press,New York, 343 pp.

Salthe, S.N., 1991. Two forms of hierarchy theory in westerndiscourses. Int. J. Gen. Syst. 18, 251–264.

Schimel, D.S., Participants, V., B.H., B., 1997. Continentalscale variability in ecosystem processes: models, data, andthe role of disturbance. Ecological Monographs, 67: 251-271.

Schmidt, L.J., 2000. The Universal Language of HDF-EOS.The NASA Distributed Active Archive Centers (http://earthobservatory.nasa.gov/Study/HDFEOS).

Simon, H.A., 1962. The architecture of complexity. Proc. Am.Philos. Soc. 106, 467–482.

Simon, H.A., 1973. The organization of complex systems. In:Pattee, H.H. (Ed.), Hierarchy Theory: The Challenge ofComplex Systems. George Braziller, New York, pp. 1–27.

Sole, R.V., Manrubia, S.C., Benton, M., Kauffman, S., Bak,P., 1999. Criticality and scaling in evolutionary ecology.Trends Ecol. Evol. 14, 156–160.

Swanson, F.J., Kratz, T.K., Caine, N., Woodmansee, R.G.,1988. Landform effects on ecosystem patterns and pro-cesses. BioScience 38, 92–98.

Valentine, J.W., May, C.L., 1996. Hierarchies in biology andpaleontology. Paleobiology 22, 23–33.

van Gardingen, P.R., Foody, G.M., Curran, P.J. (Eds.), 1997.Scaling-Up: From Cell to Landscape. Cambridge Univer-sity Press, Cambridge, 386 pp.

Whyte, L.L., Wilson, A.G., Wilson, D. (Eds.), 1969. Hierar-chical Structures. American Elsevier, New York, 322 pp.

Wilson, D., 1969. Forms of hierarchy: A selected bibliography.In: Whyte, L.L., Wilson, A.G., Wilson, D. (Eds.), Hierar-chical Structures. American Elsevier, New York, pp. 287–314.

Woldenberg, M.J., 1979. A periodic table of spatial hier-archies. In: Gale, S., Olsson, G. (Eds.), Philosophy inGeography. D. Reidel Publishing Company, Dordrecht,pp. 429–456.

Woodcock, C., Harward, V.J., 1992. Nested-hierarchical scenemodels and image segmentation. Int. J. Remote Sens. 13,3167–3187.

Woodmansee, R.G., 1990. Biogeochemical cycles and ecologi-cal hierarchies. In: Zonneveld, I.S., Forman, R.T.T. (Eds.),Chaning Landscapes: An Ecological Perspective. Springer-Verlag, New York, pp. 57–71.

Wu, J., 1999. Hierarchy and scaling: extrapolating informationalong a scaling ladder. Can. J. Remote Sens. 25, 367–380.

Wu, J., 2000. Landscape Ecology: Pattern, Process, Scale andHierarchy. Higher Education Press, Beijing, 258 pp.

Wu, J., Levin, S.A., 1994. A spatial patch dynamic modelingapproach to pattern and process in an annual grassland.Ecol. Monogr. 64 (4), 447–464.

Wu, J., Levin, S.A., 1997. A patch-based spatial modelingapproach: conceptual framework and simulation scheme.Ecol. Model. 101, 325–346.

Wu, J., Loucks, O.L., 1995. From balance-of-nature to hierar-chical patch dynamics: a paradigm shift in ecology. Q.Rev. Biol. 70, 439–466.

Wu, J., Qi, Y., 2000. Dealing with scale in landscape analysis:an overview. Geographic Inf. Sci. 6, 1–5.

Wu, J., Liu, Y., Jelinski, D.E., 2000a. Effects of leaf areaprofiles and canopy stratification on simulated energyfluxes: the problem of vertical spatial scale. Ecol. Model.134, 283–297.

Wu, J., Jelinski, D.E., Luck, M., Tueller, P.T., 2000b. Multi-scale analysis of landscape heterogeneity: Scale varianceand pattern metrics. Geographic Inf. Sci. 6, 6–19.

Zipperer, W.C., Wu, J., Pouyat, R.V., Pickett, S.T.A., 2000.The application of ecological principles to urban and ur-banizing landscapes. Ecol. Appl. 10, 685–688.

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