Landscape Modeling

8/3/2019 Landscape Modeling

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Landscape Ecology vol. 8 no. 4 pp 239 -255 (1993)SPB Academic Publishing bv, The Hague

Landscape graphs: Ecological modeling with graph theory to detectconfigurations common to diverse landscapes

Margot D. Cantwell and Richard T.T. Forman Harvard University, Graduate School of Design, Cambridge, Massachusetts 02138, USA

Abstract

In view of the bewildering diversity of landscapes and possible patterns therein, our objectives w,ere to seeif a useful modeling method for directly comparing land mosaics could be developed based on graph theory,and whether basic spatial patterns could be identified that are common to diverse landscapes. The modelsdeveloped were based on the spatial configuration of and interactions between landscape elements(ecosystems, land uses or ecotopes). Nodes represented landscape elements and linkages represented commonboundaries between elements. Corridors, corridor intersections, and the matrix were successfully incorporat -ed in the models. Twenty -five landscape graphs were constructed from aerial photographs chosen solely torepresent a breadth of climates, land uses and human population densities. Seven distinctive clusters of nodesand linkages were identified and common, three of which, in the forms of a spider, necklace and graphcell, were in > 90% of the graphs. These represented respectively the following configurations of patches,corridors and matrix: (1) a matrix area surrounding or adjoining many patches; (2) a corridor bisecting a het -erogeneous area; and (3) a unit in a network of intersecting corridors. The models also indicated that the con -nectivity or number of linkages for several common elements, such as fields and house clearings, was relative -ly constant across diverse landscapes, and that linear shaped elements such as roads and rivers were the mostconnected. Several additional uses of this graph modeling, including compatibility with systems dynamicsmodels, are pinpointed. Thus the method is useful in allowing simple direct comparisons of any scale andany landscape to help identify patterns and principles. A focus on the common and uncommon configura -tions should enhance our understanding of fluxes across landscapes, and consequently the quality of land

planning and management.

Introduction

The apparent dissimilarity of landscapes with con -trasting climate, land use and human density makesunderstanding the ecology of landscapes a chal -lenge indeed. Yet deciphering patterns or specificconfigurations within land mosaics is critical, be -cause fluxes of species, energy, and materials, and

landscape changes over time are spatially depen-dent (Levin 1976, Dykstra 1981, Forman and Go -dron 1986, Gardner et al. 1989, Merriam 1990).Modeling of these complex systems must be akey component for enhancing ecological under -standing.

The varied approaches to modeling heterogene-ous landscapes (De Angelis et al. 1985, Baker 1989,

*

Present address: Environmental Design and Management, P.O. Box 34110, Halifax, Nova Scotia B3J 3S1, Canada


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Turner and Gardner 1991) include: (a) hexagonalpacking models (Christaller 1933, Haggett et al.1977); (b) general neutral models (Caswell 1976,Gardner el al. 1987, Milne el al. 1989); (c) percola -tion theory (Stauffer 1985, ONeill e l al. 1988,

Gardneret

al. 1989); (d) hierarchy theory (ONeillet al. 1986, 1989); (e) fractal analysis (Burrough1981, Milne 1988); (f) geometric models (Franklinand Forman 1987, Hansen et al. 1992); (g) neigh -borhood models (Shugart and Seagle 1985, Turner1987a); (h) entropy information models (Godron1966, Forman and Godron 1986); (i) varioussimulation models (Sklar el al. 1985, Turner 1988,Wiens and Milne 1989); 6) geographical informa -tion systems models (Burrough 1986, Johnson1990, Burke et al . 1990); (k) cellular automata (Zei -gler 1976, Couclelis 1985); (1) network models(Taaffe and Gauthier 1973, Haggett et al. 1977,Forman and Godron 1986, Forman 1991); (m)patch and corridor simulation models (Fahrig andMerriam 1985, Merriam 1990); and (n) patch dy -namic models (Levin 1976, Pickett and White1985). This array has produced impressive results.Yet no single approach successfully satisfies threekey or minimal ecological criteria: (1) a focus on thespatial configuration of patches, corridors and

matrix as basic elements of the landscape; (2) a fo -cus on interactions of fluxes between elements; and(3) direct comparison of the first two characteristicsat any scale in any landscape. Graph theory mayprovide a useful foundation from which to developmodels that incorporate these three criteria.

Graph theory is an established modeling methodused in a variety of disciplines to describe relation -ships between objects. A graph consists of a finiteset of nodes (points, vertices), a finite set of link -ages (edges, lines), and a rule that defines which

edges join which pairs of vertices (Ore 1963, Harary1972, Rouvray 1973, Wilson 1979). It is customaryto represent a graph by means of a diagram, andrefer to the diagram as the graph. Unlike Euclide -an graphs with horizontal and vertical axes, graphtheory graphs portray topological quantities, pat -terns and relationships. Common applications of graph theory include transportation route maps,molecular graphs in chemistry, and electrical circuitgraphs. In ecology, aspects of graph theory havebeen used to analyze food webs, succession, phyto -

sociological structures, and other patterns (Levins1975, Cohen 1978, 1990, Dale 1985, Roberts 1989,Bruce G. Marcot personal communication, Patrick Kangas personal communication). Only a smallportion of the richness of graph theory will be used

for the modeling of landscapes in this paper.The application of graph theory to landscapeecology (Risser et al. 1984) may prove useful forreducing complex landscapes into an understanda -ble set of spatial configurations, thus uncoveringpatterns of interaction or flow, and possibly crest-ing a framework for the modeling of landscapefluxes. A partial analogy with chemical compoundsmay be helpful. All landscapes can be described ascomposed of patches, corridors and a matrix (e.g.,Forman 1979, Turner 1987b). When these aredescribed by type, one recognizes wooded patches,stream corridors, hedgerows, etc., lets say analo -gous to atoms, such as carbon, hydrogen and oxy -gen. But what are the analogues to benzene ringsand carboxyl groups? Are there widespread repeti -tive combinations of patches and corridors, in -dependent of landscape type? If such structures orbuilding blocks of a landscape can be determined,they should noticeably enhance our ability to ana -lyze landscape fluxes and function.

Several approaches to understanding fluxesthrough a land mosaic relative to graph theory arepossible. For example, connectivity usually focuseson corridors and networks, i.e., channels of move -ment through space (Lowe and Moryadas 1975,Haggett et al. 1977, Forman and Godron 1984,Fahrig and Merriam 1985, Kosova et al. 1986,Henein and Merriam 1990). However, it should bepossible to approach connectivity from the perspec -tive of particular types of landscape elements, e.g.,woods and house clearings. Are there repetitive pat -

terns in the number of connections to surroundinglandscape elements independent of landscape type?Other approaches to understanding fluxes relativeto graph theory briefly considered here includesystems dynamics models (Odum 1983, Robertsonet al. 1991), connectivity matrices (Taaffe andGauthier 1973), and percolation theory (Gardner et al. 1989). Here we will explore the potential appli -cation of these approaches to the spatial configura -tion of patches, corridors and matrix in the land -scape graph.


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City B

CityA

0-q

Fig. 1. Methods of drawing graphs. (a) Transportation graphwith spatially -explicit linkage lengths representing distances (ortime -distances) between cities, and node size indicating city size.(b) to (e) Landscape areas and their corresponding graphs; nodes(circles) represent elements (ecosystems, land uses) in the land -scape, and linkages (lines) represent shared boundaries or pointsbetween elements. Landscape elements recognized are: G-

grassland; B -bean field; W -woods; H -highway; and R -road. (b)Landscape graph of a matrix and two patches, recognizing thematrix as an element. (c) Landscape graph recognizing a cor -ridor (highway) as an element. (d) and (e) Landscape graphswhere a corridor network is represented as comprised of compo -nent elements, or lengths of corridor defined by intersections.(d) Landscape graph recognizing a 3 -way or T-intersection. (e)Landscape graph recognizing a 4 -way or X-intersection, andwhere one corridor axis predominates over the second.

Thus, the objectives of this paper are the fol -

lowing:1 . Develop a method that reduces inherently com -

plex landscapes to a common graph model,based upon the assumption that interactionsbetween adjacent landscape elements are im -portant.

2.1. Identify common and uncommon patterns,if any, in the configuration of patches, cor -ridors and matrix, independent of land -scape type or location.

2.2. Determine the most and least connected ele -ment types in a landscape, and whether thenumber of elements adjacent to a given ele -ment type is independent of landscape con -text.

2.3. Pinpoint links between the graphs and othermodeling approaches.

2. Use this landscape graph to:

1. Development of the graph model

Graphs are comprised of (a) circles called nodes(vertices, points, sites) that represent the spatialelements or parts of an object, and (b) lines knownas linkages (lines, edges, bonds) that connectnodes to represent a relationship between them.

Geography has long used graphs to portray thelandscape. The common transportation route mapdescribes cities as nodes, and cities that may bereached by transportation are connected by link -ages (Taaffe and Gauthier 1973, Rugg 1979). Com -monly the graphs are spatially explicit, with inter -city distances illustrated by linkage length. Cityarea or population may be illustrated by the size of the node (Fig. la). Vector mathematics may be ap -plied to a spatially explicit graph to determine polar

direction, and, if city size is illustrated, a gravity orother interaction model may be used to describe theinfluence of cities on each other. Both the ability toapply vector mathematics and the potential to ap -ply models describing the influence of element sizemay be ecologically useful for landscape graphs.Therefore, in the graphs drawn, nodes are placed inthe approximate center of elements, and node sizereflects relative element size.

Common graph construction methods isolatethe system being described from the backgroundmatrix. Molecular graphs, where atoms are nodes


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and linkages represent atomic bonds, are an exam -ple. The matrix, or surrounding space in this case,is excluded from the description. Graphs used ingeography normally utilize this technique as well(e.g. , the transportation system is isolated from its

landscape setting) (Fig. la). When creating land -scape graphs, this approach results in graphs thatisolate individual systems from the landscape,rather than graphs of patterns and interactions overthe total area. Therefore, in a graph of the land -scape, the matrix must be a key element. An exam -ple of a graph which does not exclude the matrix isa food web graph of species use of resources (Sugi-hara 1983, 1984). In this graph, all resources aremapped as nodes, and linkages represent a relation -ship between nodes (resources) defined by a con -sumer species utilization. In analyzing landscapeswe have taken a similar approach by mapping alllandscape elements present as nodes (Fig. lc) andnot a priori designating one as the matrix to be ex -cluded.

Corridors, as important elements or objects ina landscape, are also represented by nodes (Fig.lc). Corridors frequently form interconnected net -works across the landscape, e.g., road systems andhedgerow networks. As linear shaped elements,

different corridor types also intersect each other,e.g., a road crosses a river. The modeling approachrecognizes that, functionally, corridor networks arenot a homogeneous landscape type, but ratherhighly connected individual elements. Where linearelements intersect, the corridor on one side of theintersection often has a different structure than onthe other side (e.g., a road intersecting a highwaymay increase the amount of traffic on the highwayonly in the direction towards the city). This is also

the case where different linear element types inter -sect (e.g., a stream intersecting a strip of woodsmay have a stony bottom upstream and a leaf-littered bottom downstream). For these reasons,each length of corridor, defined by its intersectionwith another length of corridor (whether of thesame element type or not) is represented by a node(Figs. Id and le).

Thus in the landscape graph model developed,nodes represent all landscape elements (local eco -systems, land use units, ecotopes, or biotopes),

whether patch, corridor or matrix. Linkages repre -sent common boundaries between adjacent ele -ments, or points where adjoining elements meet. Itbears emphasis that linkages represent adjacencybetween elements, rather than specific routes or

corridors on maps and photographs. Although em -pirical data are scarce, note that flux rates acrossdifferent boundaries in a mosaic ( i e . , along dif -ferent linkages in a graph) clearly differ. Thus birdsand small mammals in a southern Ontario farm -land moved much more between woods and hedge -row than between woods and field, woods andwoods, or hedgerow and field (Wegner and Mer-riam 1979).

One of the difficult tasks in graph construction[and other modeling methods including geographicinformation systems (GIs)] is deciding how to re -present intersections of linear elements. One com -mon method is to recognize the intersection as aseparate element, and insert a node at that point inthe graph. Alternatively, graph linkages may becrossed over each other, without forming a node. If nodes are inserted to represent intersections, inmany landscapes they permeate the graph and be -come as numerous as element nodes. Such a graphis more a description of juncture distribution than

a portrayal of relationships and flows among land -scape elements. Crossed linkages, on the otherhand, clearly illustrate the network nature of linearelements. In a four -way or X intersection, linkages

join the four corridor nodes together, illustratingall potential interactions. The result is two crossinglinkages in the center of the four nodes, which isresolved by portraying the functionally dominantlinkage (e.g., more animal or traffic movement) asa continuous line and the other linkage as a

broken line (i.e. , with a small half circle) (Fig. le).If the crossed linkages are functionally similar orundetermined, both lines are broken. [A three-way intersection is represented by three nodes (onefor each corridor) that are all connected to oneanother (Fig. Id).]

A final graph construction issue is scale, or deter -mining what to include as elements (represented bynodes). Are home clearings shown individually oras suburban groups, and are individual adjacentfields differentiated or considered as one con-


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tinuous cultivation landscape element? Is a largeclearing with a narrows in the middle (hour -glassshaped) considered one clearing, or two clearingswith a corridor? The scale of resolution of graphsmust be tailored to the question posed; this is notunlike determining what grid cell size to use in GIs.Flexibility in distinguishing elements is considereddesirable so that graphs may be constructed at anyresolution of interest. In the graphs drawn for thispaper, the ability to clearly differentiate relatively -homogeneous elements in an aerial photograph wasthe criterion utilized. For hour -glass shaped ele -ments, if the element narrowed to one -third or lessof its width, it was considered to be two elementsconnected by a corridor. The approach appeareduseful for detecting and comparing patterns, nomatter what the scale ( i . e . , using aerial photographsof areas ranging from approximately 10 to 10,000ha.).

The method developed is an exploratory ap -proach that attempts to model and compare diverselandscapes. It allows the organization of the land -scape as a whole to be modeled by incorporating, orillustrating how to handle, three attributes rarelyencompassed in graph theory applications: the ma -

trix, corridors, and corridor intersections. Foursimple rules emerged from this methodology andare the basis of the graphs mapped and analyzed inthis paper:1. Each landscape element or ecosystem, including

the surrounding matrix if present, is representedby a node placed in the center of the element.Node size corresponds to element size.

2. A common boundary or point where two ad - joining ecosystems meet is represented by a

linkage.3. A graph may be constructed at any scale by iden -

tifying the elements to be included.4. Corridor intersections separate individual cor -

ridor elements. Intersections are represented bylinkages interconnecting the nodes.Using this graph construction method, 25 land -

scape graphs were drawn on clear plastic sheetscovering aerial photographs. Photographs werechosen from a diverse collection on the sole basis of

representing a wide range of regions, climates,vegetation types, human population densities, and

land uses (see Appendix). Predominant land usesincluded forestry, pastureland, cultivation, sub -urbia, and natural landscapes. Diverse areas werepurposely chosen to compare landscapes not other -wise easily Comparable.

2. Uses of landscape graphs

The modeling approach will be used to: (1) identi -fy common configurations within landscapes; (2)examine the connectivity of elements in landscapes;and (3) illustrate potential links with other model -ing approaches. Note that the goal is to detect pat -terns or configurations, not the causes of them suchas geomorphic heterogeneity, natural disturbanceand human activity.

2.1. Common configurations

Graphs of the 25 diverse landscapes were examinedfor the presence of repeated and/or distinctive pat -terns or building blocks within the landscapes. Suchgraph patterns would represent configurations of

patches, corridors and matrix elements indepen -dent of landscape type or location.

Each landscape graph as a whole had a uniquearrangement of nodes and linkages. Within thegraphs seven distinctive patterns of nodes and link -ages were identified as common, or present in morethan three graphs. Three of these patterns (so-called necklace, spider, and graph cell patternsdescribed below) were detected in > 90% of thegraphs. Three additional patterns were each detect -

ed in a single landscape graph. The ten patterns of nodes and linkages identified from the graphs arenamed for familiar forms that they mimic.

2.1.1. Description of graph patternsThe necklace pattern (Fig. 2) is typical of linearelements: roads, hedgerows, powerline corridors,road verges (margins), and rivers (Figs. 3 and 4).Necklace nodes are all the same landscape elementtype linked together to form a pattern resembling

separated beads on a necklace. Crossing linkagesare common in necklaces where corridors intersect


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Necklace

Graph Cell

Satellite

Candelabra

Cross

Rigid Polygon

Fig. 2. Common and uncommon graph patterns identified fromtwenty -five diverse landscapes. The first seven patterns (left toright, top to bottom) were common. Three patterns were un -common: rigid polygon, variegated necklace, and spider-satel-lite (see text).

(e.g., road intersections). Necklaces typically ex -hibit many linkages when penetrating heterogene -ous space and few linkages in homogeneous land -scapes. The variegated necklace pattern looks

identical to the necklace except that nodes repre -senting different landscape elements are strungtogether, rather than nodes of the same type. Thispattern, representing sequential zonation, wasfound in a graph of the Grand Canyon (USA)where elevation defined bands of different rock formations. These formations were represented bynodes strung together from the canyon top to bot -tom in a variegated necklace pattern.

A graph figure with a central node and > 4linkages to other landscape element nodes is

labelled a spider (Fig. 2). The central, or spidernode is often the matrix, and a large spider nodeelement tends to have more linkages than a smallerone. Woods, grassland and fields are commonspider node elements (Figs. 3 and 4). A variation on

the spider pattern, the spider -satellite, was foundwhere almost all of the nodes attached to a spidernode had no other linkages. This uncommon pat -tern was found in a landscape where isolated wood -ed patches were surrounded by pastureland.

A corridor network in a landscape is representedby interconnecting necklace patterns in a graph.The component sections or units formed by thisnecklace network are referred to here as graphcells (Fig. 2). In the graphs constructed, roads were

the primary necklace elements delineating graphcells. Spider patterns were typically found withingraph cells (Fig. 3).

The cross pattern is an 8 -linkage pattern, withfour almost polar linkages off a central cross nodeto four surrounding nodes that are interconnected(Fig. 2). Several types of cross patterns were presentin the landscape graphs. They differ in the numberof element types surrounding the cross node, andwhether the cross node represents a patch or cor -

ridor. For example, a field bounded by four hedg -erows, or a suburban housing block surrounded byfour roads would be patch -centered cross with onesurrounding element type. One, two, three andfour -element type patch -centered crosses werefound in the landscape graphs. In contrast, cor -ridor -centered crosses have a corridor cross node.Two, three, and four -element corridor -centeredcrosses were found, although two -element typecrosses were most common. An example is thenarrows in an hour -glass shaped field that is linkedto the two large field areas and to the forest on eachside.

The satellite pattern represents an isolatedpatch surrounded by a single element type. The lonebog in a forest, oasis in a desert, or remnant wood -lot in a field are common satellites (Figs. 3 and 4).This pattern is recognized by a node with only onelinkage (Fig. 2). The satellite node, in our analysis,was usually a small element in a landscape dominat -ed by one large patch or area of matrix. In one case


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one side, and to one element type on the other side(Figs. 2 and 4). Floodplains were the most commoncentral or candelabra nodes, but road verges (roadsides), dunes, and salt water marshes also exhibitedthis pattern.

One uncommon pattern, in addition t o the varie -gated necklace and spider -satellite, was detected.The rigid polygon pattern occurs when four ormore nodes join together at one point. For exam -ple, four fields with different crops that all touchedat one point would form a rigid polygon represent -ed by four completely connected nodes in a square.In this analysis the pattern was found only in a par -tially logged forest landscape of old growth, clear -ing, and regeneration stands.

Thus landscape graphs contain a limited numberof repetitive graph patterns, of which spiders, neck -laces, and graph cells predominate. In actual land -scapes these represent, respectively, the followingconfigurations: (1) a matrix or large landscapepatch surrounding or adjacent to many patches; (2)a corridor bisecting the landscape; and, (3) the unitformed by a network of intersecting corridors. Afocus on these and other landscape building blocks,including uncommon configurations, should en -hance our understanding of the structure and func -tioning of landscapes, and consequently the qualityof land planning and management.

2.1.2 Ecological and management implications.Many of these identified patterns, representing con -figurations in a landscape, are useful for ecologicalunderstanding as well as landscape planning andmanagement. Several examples for each pattern aregiven. While some examples are speculative, thepurpose is to illustrate the richness of uses and im -plications of repeated configurations or patterns.

Spider and spider -satellite. The former repre -sents a large patch or matrix surrounding or adja -cent to many landscape patches, and the latterrepresents a patch entirely embedded in a matrix.Examples are a forest reserve in a suburban land -scape and bogs in a boreal forest. These patternssuggest: (a) strong exchanges across patch -matrixboundaries; (b) that the matrix exerts a strong con -trol over patches and patch species, which in turn

are at risk and could easily disappear; (c) areas of

low habitat diversity, if the patches are of the sametype; (d) that the matrix is subject to many in -fluences, if the patches are of different types; (e) ab -sence of coverts or convergency points [where threeor more habitats converge, a useful spot for certain

wildlife (Forman and Godron 1986)]; (f) metapopu-lation dynamics [frequent extinctions and somerecolonizations in patches (Merriam 1990, Harmsand Opdam 1990)l; and (g) genetic inbreeding inspider -satellite patches, and genetic heterogeneityfor many species that are in more than one patch.Spider and spider -satellite configurations are areasin a landscape requiring careful monitoring andmanagement.

Necklace. This represents a corridor penetrating

the landscape. Examples are roads, hedgerowsand streams. Necklaces penetrating heterogeneousspace have many linkages as compared with thosethrough a homogeneous matrix. Necklaces withmany linkages exhibit: (a) more landscape resis -tance or a less efficient conduit for animal move -ment, due to exerting caution as adjacent habitatschange; (b) more microhabitat heterogeneity andmore edge species; and (c) less of a barrier effect foran animal, i.e., greater ease in finding a suitablecrossing spot, such as natural vegetation on bothsides of a road. Necklaces with few linkages ex -hibit: (a) less landscape resistance; (b) microha-bitat homogeneity (c) and either extreme land -scape resistance for an animal seeking an appro -priate crossing point, or rapid exchange if thehabitat is appropriate, e.g., corn field to corn fieldfor pest spread. Management should often favornecklaces with many linkages to minimize speciesisolation in a landscape, enhance populations of two -plus species (animals requiring or using twoor more habitats), and provide coverts (contingencypoints) for certain wildlife using three adjacenthabitats.

Graph Cell. This represents a section or unit in anetwork of necklaces. Graph cells may be formedby one necklace type, e.g., a road network, or twoor more necklace types, e.g., roads and powerlines(as in Fig. 3). Landscape grain is typically measuredby the mean area or diameter of the patches andcorridors in a landscape (Forman and Godron

1986). One -necklace -type graph cells can also be a


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measure of grain at a somewhat coarser scale, andtwo -necklace -type graph cells can be a measure of grain at a still coarser scale. It would be interestingto interpret ecologically the means and variances of grain at these nested scales. For example, decreas -ing graph cell size (or landscape grain), appears tocorrelate with increasing human population densi -ty as road networks, in particular, become moredense. Graph cells, just as for any circuit, offer op -tional routes for animals and people between twopoints in a corridor system. Graph cells are goodcandidates for individual or local management, be -cause of the clear boundaries and the normallygood access from all sides.

Candelabra. This represents an elongated land -scape element separating one element on one sidefrom many elements on the other side. Examplesare a wetland floodplain separating a river frommany adjacent upland land uses, and a green beltseparating a town from diverse adjoining land usesbeyond. The elongated landscape element, or cen -tral node, functions as a narrows or bottleneck orfilter. In the floodplain case, nutrients from the up -land must pass through the wetland to enter theriver, or an animal from the river area must crossthe wetland to reach the upland resources. Thecandelabra pattern gradually disappears as thefloodplain loses its connectivity and becomes sub -divided into small land uses. The central node of acandelabra indicates a landscape element requiringspecial management focus.

Mesh. This represents at least three landscapeelement types in which patches are of approximate -ly equal size. Crop fields of three types in farmland,and forest patch cuts of different regeneration ages

are examples. A mesh indicates a: (a) landscape un -der intensive management where no one patch typepredominates; (b) abundance of coverts; (c) paucityof corridors (and no graph cells) dividing up thelandscape; and (d) broad scale homogeneity of fine-scale patch clusters repeated over the landscape. Amesh may indicate an area of a landscape wheresustainability is unlikely and a management changeis warranted.

Satellite. This represents a lone patch surrounded

by matrix or a large patch. The satellite may indi -cate: (a) a patch or patch species in danger of disap -

pearing; (b) genetic inbreeding in the lone patch,and (c) the necessity of an animal to be compatiblewith the matrix in order to enter or leave the patch.Again, monitoring and possibly active managementmay be indicated by a satellite.

Rigid Polygon. This represents four or morepatches joining at a point. In such a configuration:(a) the point is a covert; (b) a funnel effect of animalmovement (slowly due to behavioral caution)through the point is likely; and (c) water and windfluxes (accelerated due to the Bernoulli effect)through the point are likely.

Variegated Necklace. This represents banding orzonation of landscape elements, such as from laketo upland, or along a mountain slope. The configu -ration: (a) is usually along an environmental gra -dient; and (b) has the end node embedded in a ringor donut shaped patch (the penultimate node). Asin several preceding cases, special monitoring, plan -ning and management typically is appropriate forthe distal nodes of a variegated necklace.

In short, the ecological implications pinpointedsuggest a wide range of ecological insights possiblefrom identifying configurations in a landscape. Themanagement implications noted suggest areas wi -thin a landscape warranting special care or focus,or even change in management, independent of landscape type.

2.2. Connectivity

Connectivity is considered a measure of the strengthof the relationships described by the graph (Ore1963, Taaffe and Gauthier 1973, Wilson 1979) and

is one of the most common measurements. Ecologi -cally, high connectivity implies much interactionor movement of animals, plants, heat energy,water, and materials among elements (Forman andGodron 1986, Turner 1987b, Forman 1991).

A common measure of connectivity is the degreeof a node in mathematics, or beta index in trans -portation geography, which is simply the numberof linkages connected to a given node. In landscapegraphs, where linkages represent adjacencies, the

degree of a node of any given landscape element isan indication of how accessible that element is, as


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well as its overall relative influence in the landscape(Bruce G. Marcot, personal communication). Thedegree of a node was determined for each node inthe 25 landscape graphs, and the mean number of

linkages calculated for each element type in eachgraph (see Appendix and Fig. 5). Although a fullstatistical analysis of this data set is beyond thescope of this paper, several results illustratingpromising uses of graph models are noted.

As might be expected, larger elements (relative toothers in the graph) had more linkages than smallerones. However, small elements were well connectedif they had a linear shape or were in a finely dissect -ed landscape (e.g., a pond surrounded by manyland uses). Roads and rivers were the most connect -ed elements (grand means of 9.1 and 10.1 respec -tively, versus 2.8 to 6.5 for other element types)(Figs. 3 and 5). Rivers with intact floodplains (notdissected or fragmented) and roads with continuousverges exhibited low connectivity, whereas the adja -cent floodplain or verge was well connected (e.g.,Fig. 4). Linear shaped hedgerows were also wellconnected, especially when they formed an inter -connected network.

Woods elements (Fig. 5) were the most variablein relative size and connectivity. Isolated patchesexhibited one linkage, and small patches adjacentto stream corridors or in hedgerow networks hadmore. Large wooded patches that were not thematrix were well connected to other landscape ele -ment types, but rarely to each other. Such patcheshad more connections than a forest matrix of logged clearing and forest stands. The highestdegree of the node for woods was found at the tran -sition mosaic zone between two forest types, where

each of the two large intersecting patches werespider -satellites.

Two element types, fields and house clearings,exhibited a relatively constant number of linkages.Fields almost always had four linkages (Fig. 5). Ex -ceptions on the high side indicated intensive cultiva -tion areas without hedgerows, but with remnantpatches (e.g., Fig. 4). Pastureland had many link -ages due to small clusters of remaining trees. Houseclearings (the openings around buildings) typically

had only two linkages (Fig. 5 ) , to a road and t o thesurrounding element type (e.g., Fig. 3). Only in cer -

tain agricultural locations where the farmer sur -rounded the home with diverse resources, and inthe suburban/rural fringe where house clearingswere at the juncture of two landscape types, were

house clearings better connected.The degree of a node is a simple measure andonly quantifies adjacency (e.g., a patch two -stepsaway is not considered). Other measures of connec -tivity may be applied to graphs (Taaffe and Gau-thier 1973, Haggett et al. 1977, Rugg 1979, Wilson1979, Forman and Godron 1984, 1986). Measuresof connection in food webs are also available (e.g.,Sugihara 1984, Cohen 1990). Elements may be exa -mined for how frequently they are connected to aparticular type. Also, the robustness of the graphmay be evaluated by examining linkage and nodeconnectivity (e.g., the number of nodes or linkagesthat must be removed in order to disconnect thegraph). This measure allows the connectivity of theentire landscape to be assessed.

For example, in the two graphs illustrated,woods are the most connected landscape elementtype in Fig. 3, while fields are in Fig. 4. In Fig. 3 ,home lots are the least connected, and 100% of thelots are connected to woods and to roads. Woodsare connected to every landscape element type. I ncontrast, in Fig. 4, shrubland/woods are the leastconnected element type, and 100% of them are con -nected to fields. The graph in Fig. 4 is not veryrobust. Removal of the floodplain (FP), large field(F), or large shrubland (S) node would disconnectthe graph into two graphs. The landscape pattern of Fig. 3 is more robust as removal of any node doesnot severely affect the graph structure.

This brief assessment emphasizes that some land -

scape elements are inherently better connected thanothers. Of special interest is the high connectivityexhibited by linear elements, suggesting substan -tially more influence on the landscape than a meas -ure of relative area would indicate. The connectivi -ty analysis also emphasizes that some landscape ele -ment types have a relatively constant number of connections, irrespective of context. This rein -forces the conclusion in Section 2.1. on commonconfigurations, that a landscape, like other objects,

is composed of distinct spatial patterns or buildingblocks that may be combined in different ways.


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2.3. Additional uses

In addition to the diverse measures and applica -tions provided by the field of graph theory, a graphtheoretic landscape graph is a structure to which

other modeling approaches may be applied. For ex -ample, to better understand fluxes, systems dynam -ics models, connectivity matrices and percolationtheory may be applied. Hierarchy theory may beuseful for studying the internal structure of nodes,and insights into landscape management andchange may be possible through further patternanalysis. These additional uses are briefly describedto suggest the richness of the landscape graph ap -proach.

2.3.1. Systems dynamicsSystems dynamics is a modeling paradigm to whichthe landscape graph is particularly well suited. Sys -tems models are conceptualized as a set of compart -ments, with flows between them. The landscapegraph may be used as the skeletal structure for sucha model, where nodes correspond to compartmentsand linkages to flows. Because the landscape graphincludes all element types whether patch, matrix orcorridor, it is a complete description of the entirelandscape as a series of interconnected compart -ments, where flows occur as a result of element ad -

j acencies .The usefulness of the landscape graph as a sys -

tems dynamics framework may be illustrated by asimple model created using STELLA I P software.Consider the graph of a landscape where a Woodsis adjacent to a Field which in turn contains aWoods Patch. In the graph, linkage length is pro -portional to boundary length, where the Woods/

Fields boundary is twice as long as the Field/WoodsPatch boundary.

WoodsWoods Field Patch

Systems dynamics flows occur according to anequation expressing the rate of flow between com -partments in terms of the values of compartments,parameters and other variables in the model. As-

100

80-!I

SystemEquilibrium

planted plantedUCmpA lCmpB

1o 5 i o (5 20 25 30 35 w 45 50 55 60 65 ro 75

Time

Fig. 6. Systems dynamics model results showing quantity of material in each node per unit time. Solid lines show quantity inWoods, Field and Woods Patch if Crop A is planted; dashedlines show quantity in these elements if Crop B is planted. Seetext for initial values, exchange rates and questions.

sume material moves between the nodes in thegraph above according to a diffusion type dispersalequation (modified from Hastings 1990): Dispersalat time t = (Rate of exchange between nodes a and6 ) x (Amount of material in b - Amount of material in a). The dynamics of each node (or com -partment) is governed by a differential equationwhich is formed from the sum of the inflows intothe compartment minus the outflows from it. Thus:Amount of Material in Node at time t = (Amountof Material in Node at time t - At) + [(At) x (Flowinto the Node - Flow out of the Node)], where Atis the time interval for calculations used to approxi -mate the differential equation curve.

In the example, assume that a material (species,pollen, etc.) exists only in the Woods initially(Woods initial value = 100; Field and Woods Patch

initial values=

0). A farmer wishes to compare theeffect that planting different crops in the Field hason the movement of the material. If Crop A offershalf the landscape resistance that Crop B does,and exchange rates are proportional to boundarylength, then the following exchange rates may beassumed: Woods and Crop A = .40; Crop A andWoods Patch = .20; Woods and Crop B = .20;Crop B and Woods Patch = .lo). The results of running of this systems dynamics model show thatwith both planting options the quantity of material


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in the Field will initially overshoot (more and soon -er with Crop A than B) before reaching equilibrium(Fig. 6). With Crop A equilibrium of the system willbe reached in 34 days, whereas with Crop B, 74 daysare required.

This simple systems dynamics model may be en -hanced by including other parameters important tounderstanding the system such as birth and deathrates (Andow et al. 1990, Henein and Merriam1990, Hastings 1990), and by modeling more com -plex landscapes. As graph complexity increases andflow parameters become less well understood, thelink with systems dynamics models becomes in -creasingly useful because of the limited requirementto define quantitative terms. It is indeed possible tospecify complex and realistic models without enter -ing a single equation, simply by defining relation -ships (Robertson et al. 1991).

2.3.2. Connectivity and accessibility matricesMatrices may be constructed from landscapegraphs to more fully understand the connectivity oraccessibility of elements in the graph (Alan R.Johnson, personal communication, Taaffe andGauthier 1973). A connectivity matrix (C), where

nodes are numbered and listed in both the columnsand rows of the matrix, may be used to determinedirect and indirect connectivity. For example, as -suming six nodes, a 6 by 6 matrix is constructed,where 0 indicates no direct linkage and 1 a directlinkage between landscape elements (nodes). Whenthe matrix is squared, 1s appear where there aretwo -step paths between a given pair of nodes orlandscape elements. Cubed matrices identify ele -ments that are linked by three steps, etc. The con -

nectivity matrix may be useful to determine link -age or interaction between non -adjacent elements,such as a bird feeding in two woods separated by afield.

The accessibility matrix (A) is the sum of the con -nectivity matrix (C) and all matrices that enumerateindirect paths between nodes of the graph. Thus, A= C + C2 + C3 + . . . C, where n is equal tothe maximum number of steps to traverse thegraph. Accessibility matrices may be useful for un -derstanding landscape resistance or animal andplant dispersion across a landscape (Forman and

Godron 1986). There are many other possiblematrix applications.

2.3.3. Percolation theoryPercolation theory may be useful to landscapegraphs. Through a simple application of the perco -lation backbone, graphs may be used to determinecircuitry, or the number of optional routes in alandscape (Alan R. Johnson, personal communica -tion, Stanley 1977, Haggett et al. 1977, Staurffer1985, Forman and Godron 1986, Gardner et al.1989). For example, by isolating from the graph theelement types that a species is likely to traverse, andestablishing initial and terminal travel nodes, allnodes in between may be classified as either: (a)nodes through which the species must pass, (b)nodes through which the species may pass, or (c)dead ends, which do not lead toward the terminalnode. From this analysis, the most direct route (orbackbone), and number of optional circuits(paths), available to travel across the landscape bygoing through only compatible landscape elementsis revealed. Once outlined, paths may be visuallyexamined for other factors, such as the number of element types adjacent to the path.

2.3.4. DirectionalityA graph may be used to model movements of spe -cies, or disturbances such as fire (Turner 1987b). Byassigning directionality to the linkages (essentiallythinking of them as vectors) and resistance factorsto the nodes (Forman and Godron, 1986, Harmsand Opdam 1990), net movement may be calculat -ed. It may also be possible to make assumptionsabout linkage directionality and node resistance

from the relative suitability of adjacent elements,or their spatial configuration.

2.3.5. Hierarchy theoryHierarchy theory may be applied to the graphs tomodel the internal structure of nodes (Allen et al.1984, ONeill et al. 1986, 1989). For example, anode may represent an entire valley bottom at onescale; however, nested within that node is a sub-setof channels, stream banks, wetlands, and well -drained soils. This application may be useful wherethe spatial configuration at the finer scale is impor-


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tant to interactions at the broader scale, or viceversa.

2.3.6. Management optionsLandscape graphs may also be manipulated to iden -

tify and resolve what if type management options(Gross and Dykstra 1989). By removing or chang -ing elements, a new graph is created. This graphmay be compared with the original for pattern, con -nectivity, circuitry, movement, or other ecologicalmeasures.

2.3.7. Landscape changeFinally, landscape graphs should be useful in iden -tifying and comparing the patterns produced bydifferent processes of landscape change. Thus thespatial changes produced by deforestation, deser-tification and suburbanization are readily com -pared, as well as a particular process in differentparts of the world. The connectivity and interactiondimension of landscape graph modeling wouldcomplement dynamic models of spatial structure,such as for the 50 years of landscape change dueto many processes in Georgia (USA) (Turner 1988,Odum and Turner 1990), or the geometric modelsof landscape and ecological changes produced by

different logging regimes (Franklin and Forman1987, Hansen et a l . 1992).

Conclusion

The graph construction method outlined is a way of describing landscape structure. It is also possible tomodel landscape element size, connectivity, anddirection of flow. This makes the method potential -ly

useful for ecological modeling of landscape func-

tioning and future changes that result from interac -tions between adjacent ecosystems. As a manage -ment tool and disturbance evaluator, its value isbased on the ready comparison of diverse areas byreducing the landscape to a common structure.Thus land managers can compare the pattern andexpected ecological effect of proposed landscapemodifications.

One of the most important characteristics pin -pointed in drawing graphs is the importance of

linear elements. The graphs drawn depict highlyconnected linear elements which divide the land -scape into component cells or units. The impor -tance of linear elements in the graph contrasts shar -ply with ecologists traditional focus on characteris -tics within a patch or the matrix. Similarly, theattention given to linear elements in landscapegraphs differs from the results of commonly -usedraster GIS systems, where each cell expresses onlythe most dominant element type present, andtherefore smaller corridors (country roads, hedge -rows, etc.) easily disappear (Burrough 1986). Li-near elements, such as roads, are major movementconduits and sources of pollution and energy con -sumption, and many animals tend to avoid crossingeven narrow roads (Lyon 1983; Forman 1991).

The flexibility of the basic landscape graphmodeling approach permits many additional objec -tives or variables, including mathematical descrip -tions, to be incorporated. In a patchy landscapewith few corridor intersections, each corridor seg -ment could be determined by adjacent land use.Thus in Fig. lc, two H nodes would be present,highlighting the ecological difference of being nextto woods versus beanfield. Other examples include:adjusting node size to accurately reflect the area co -

vered by the element; using the length of the linkageto describe the length of the common boundary be -tween elements; using line thickness or differentline types to reflect the quality of linkages (e.g.,quantity of fluxes between elements); varying thelinkage to reflect the fractal dimension of a patchedge; and a look -up table, like the periodic tableof elements, to describe other ecological and land-use characteristics of the landscape elements. In thefuture one might envision graphs with databasesbehind each node and linkage, describing landscaperelations based upon size, flow, and other measura -ble characteristics.

Finally, drawing landscape graphs requires oneto carefully examine spatial juxtaposition and rela -tionships. This in itself is an important step, if therelationship between the spatial configuration of patches, corridors and matrix in a land mosaic, andthe movement and change of organisms, energyand materials, is the central paradigm for under -standing the ecology of landscapes.


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Acknowledgments

We are pleased to thank Alan R. Johnson, MichaelW. Binford, and Dennis P. Dykstra for helpful in -formation and comments on the manuscript, andGeorge Sugihara, Patrick Kangas, and Bruce Mar-cot for suggesting the promise of applying graphtheory. This research was partially sponsored bythe Canada Mortgage and Housing Corporation,The OBrien Foundation, and the Urban LandInstitute.

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APPENDIX: The Landscapes Modeled withGraph Theory

Landscapes are listed in general order of increased human popu -

lation density. A linkage indicates that two elements share a


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