JournalofEnvironmentalManagement(2000)59,265–278 doi:10.1006/jema.2000.0373,availableonlineathttp://www.idealibrary.comon

Landscapeconnectivity:Aconservation applicationofgraphtheory

A.G.Bunn†§*,D.L.Urban† andT.H.Keitt‡

Weusefocal-speciesanalysistoapplyagraph-theoreticapproachtolandscapeconnectivityintheCoastal PlainofNorthCarolina.Indoingsowedemonstratetheutilityofamathematicalgraphasanecological constructwithrespecttohabitatconnectivity.Graphtheoryisawellestablishedmainstayofinformation technologyandisconcernedwithhighlyefficientnetworkflow.Itemploysfastalgorithmsandcompact datastructuresthatareeasilyadaptedtolandscape-levelfocalspeciesanalysis.Americanmink(Mustela vison)andprothonotarywarblers(Protonotariacitrea)sharethesamehabitatbuthavedifferentdispersal capabilities,andthereforeprovideinterestingcomparisonsonconnectionsinthelandscape.Webuiltgraphs usingGIScoveragestodefinehabitatpatchesanddeterminedthefunctionaldistancebetweenthepatches withleast-costpathmodeling.Usinggraphoperationsconcernedwithedgeandnoderemovalwefound thatthelandscapeisfundamentallyconnectedforminkandfundamentallyunconnectedforprothonotary warblers.Theadvantageofagraph-theoreticapproachoverothermodelingtechniquesisthatitisaheuristic frameworkwhichcanbeappliedwithverylittledataandimprovedfromtheinitialresults.Wedemonstratethe useofgraphtheoryinametapopulationcontext,andsuggestthatgraphtheoryasappliedtoconservation biologycanprovideleverageonapplicationsconcernedwithlandscapeconnectivity. 2000AcademicPress

Keywords:landscapeecology,graphtheory,connectivity,modeling,metapopulations,focal ,Americanmink,Mustelavison,prothonotarywarblers,Protonotariacitrea.

Introduction anislandmodel.Agraphrepresentsabinary landscapeofhabitatandnon-,where patchesaredescribedasnodesandthecon- Thecurrenttrendinecologicalresearchand nectionsbetweenthemasedges. landmanagementistofocusonlargebiogeo- Graphtheoryisawidelyappliedframe- graphicareas,whichleavestheresearcher workingeography,informationtechnol- andmanagersearchingforlandscape-scale ogyandcomputerscience.Itisprimarily data(Christensenetal.,1996;Noss,1996). concernedwithmaximallyefficientflow Indeed,theinterpretationoflargespatial orconnectivityinnetworks(Grossand data,conceptuallyandtechnologically,can ŁCorrespondingauthor bethelimitingfactorinmakingconservation Yellen,1999).Tothisend,graph-theoretic approachescanprovidepowerfulleverage †NicholasSchoolofthe biologyandecosystemmanagementatan- Environment,Duke giblegoal.Becausetheinternalheterogene- onecologicalprocessesconcernedwithcon- University,Durham,NC ityoflandscapesmakeshabitat-conservation nectivityasdefinedbydispersal.Inpartic- 27708,USA planningaformidablechallenge,modeling ular,graphtheoryhasgreatpotentialfor ‡NationalCenterfor thespatialaspectsoflandscapesisacrit- useinapplicationsinametapopulationcon- EcologicalAnalysisand icalkeytounderstanding.Untilnow,the text.UrbanandKeitt(2000)haveintroduced Synthesis,SantaBarbara, CA93101,USA variedapproachestobuildingthesemod- -levelgraph-theorytoecologists, elshavefocusedprimarilyontwotypesof andherewebuildonthatworkbyexam- §Presentaddress: spatialdata,coveragesofvectors(polygons) ininghabitatconnectivityfortwospecies MountainResearch Center,MontanaState orrastergrids.Wedemonstratetheutility thatsharethesamehabitatbuthavedif- University,Bozeman,MT ofalessfamiliartypeoflattice,thegraph ferentdispersalcapabilities.Specifically,we 59717,USA (Harary,1969),indetermininglandscape askhowAmericanmink(Mustelavison)and Received31May2000; connectivityusingfocal-speciesanalysisin prothonotarywarblers(Protonotariacitrea) accepted30June2000

0301–4797/00/080265C14$35.00/0 2000AcademicPress 266 A. G. Bunn et al.

perceive the same landscape. We explore the Focal species sensitivity of landscape connectivity through graph operations concerned with edge defini- Because connectivity occurs at multiple tion. We also examine each habitat patch’s scales and multiple functional levels (Noss, role in maintaining landscape connectiv- 1991), we have chosen two focal species to ity in terms of source strength (Pulliam, apply a graph-theoretic approach to connec- 1988) and long-distance traversability (den tivity. Focal species analysis is an essential Boer, 1968; Levins, 1969) using graph oper- tool for examining connectivity in a real ations concerned with node removal. This landscape, as individual species have differ- type of analysis is done very efficiently ent spatial perceptions (O’Neill et al., 1988). with graph theory. We also present an The American mink and the prothonotary ecologically appealing way to calculate the warbler are appropriate candidates for focal functional distance between habitat patches species analysis as they share very similar using least-cost path modeling. Graph the- habitat but have different ecological require- ory as applied to represents an ments, and fall into different categories as important advance in spatially explicit mod- focal species. Both species are wetland depen- eling techniques because it is an additive dent and indicators of wetland quality and framework: analysis of a simple, preliminary abundance in a landscape. Both are charis- graph can prioritize further data collection to matic. Furthermore, as meso-predators mink improve the graph model. have small but important roles as a keystone species (Miller et al., 1998/1999). American mink are meso-level, semi- Study area and methods aquatic carnivores that occur in riverine, lacustrine and palustrine environments (Gerell, 1970). In chief, they are nocturnal Study area and their behavior largely depends on prey availability. They have a great deal of vari- Our study focuses on the Alligator River ation in their diet according to habitat type, National Wildlife Refuge (NWR) and sur- season and prey availability (Dunstone and rounding counties on the Coastal Plain of Birks, 1987). Muskrats (Odantra zibethicus) North Carolina (35°500N; 75°550W). It is a are a preferred prey item (Hamilton, 1940; riverine and estuarine with an Wilson, 1954), but mink diets in North Car- area of almost 580 000 ha and over 1400 km olina are composed of aquatic and terrestrial of shoreline. The area is rich in wildlife habi- animals, as well as semiaquatic elements tat, dominated by the Alligator River NWR, (e.g. waterfowl; Wilson, 1954). In the south- the Pocosin Lakes NWR, Lake Mattamus- east they have home ranges on the order of keet NWR, Swanquarter NWR, and a variety 1 ha and a dispersal range of roughly 25 km of other federal, state, and private wildlands (Nowak, 1999). (Figure 1). Prothonotary warblers are neotropical mig- The vegetation is characterized by the rants that breed in flooded or swampy mature Southern Mixed Hardwoods forest commu- woodlands. They have two very unusual nity. The area has many diverse vegetation traits in common with wood warblers in that types, including fresh water swamps, pine they are cavity nesters and prefer nest sites woods and coastal vegetation. In the upland over water. They are forest interior birds community, dominant species include many that experience heavy to severe parasitism types of oak (Quercus spp.), American Beech by brown-headed cowbirds (Molothrus ater); (Fagus grandifolia), and evergreen magno- (Petit, 1999). They are primarily insectivo- lia (Magnolia grandiflora). Mature stands rous, occasionally feeding on fruits or seed may have five to nine codominants. The (Curson et al., 1994). Preliminary data indi- wet lowlands are dominated by bald cypress cate that natal dispersal ranges from less (Taxomodium distichum). The pine woods than 1 km to greater than 12 km (Petit, 1999). are dominated by longleaf pine (Pinus palus- Although this is formulated from a small tris), but loblolly pine (P. taeda) and slash sample, it is on the same order as other song pine (P. elliottii) are also important (Vankat, bird dispersal (e.g. Nice, 1933; Sutherland 1979). et al., 2000). Here, we posit warbler dispersal Landscape connectivity 267

Figure 1. Study area in North Carolina with major roads and streams shown along with bottomland hardwood forest (focal species habitat) identified using GIS analysis. to be 5 km and return to the uncertainty of warblers in the Southeast (Price et al., 1995). this statement later. Finer scale spatial information is not readily available. Mink and prothonotary warblers are habi- Geospatial data tat specialists that use the same habitat. To identify habitat patches in the land- To our knowledge there are no current scape we combined data from the US Fish data on the spatial distribution of the focal and Wildlife Service’s National Wetlands species in our study area. The decline in Inventory (http://wetlands.fws.gov) and a trapping of the mink has perversely led to a 1996 land-use coverage from the National decline in good biological information on the Center for Geographic Information and Anal- species. We are unaware of any work done ysis (http://www.negia.ucsb.edu). Both were with mink in the study area since Wilson’s derived from Landsat 7 Thematic Map- (1954) study. The Breeding Bird Survey per imagery with 30-m cells. Cells that indicates that this study area contains one were defined as being bottomland hardwood of the highest concentrations of prothonotary swamp or oak gum cypress swamp, and 268 A. G. Bunn et al.

riverine, lacustrine, or palustrine forested as an index to overall traversability of the wetland were selected as habitat. These cells habitat mosaic. were then aggregated into regions using A graph is defined by two data structures: an eight-neighbor rule, and the interven- one that describes its nodes and one that ing matrix was described as non-habitat. We describes its edges. We defined the nodes also used zonal averaging techniques in an (habitat patches) by their spatial centroid attempt to account for functional scaling in and size (x, y, s). We defined the edges by a the habitat and found that the patch defini- distance matrix D whose elements dij are the tion was robust. Transportation and hydrog- functional distances between patches i and j. raphy digital line graphs were obtained For n patches D is n by n but because dij Ddji for the study area from the US Geologi- and dii Ddjj D0, it is sufficient to compute the cal Survey (http://www.usgs.gov) at 1:100 000 lower triangle of the matrix. resolution. Although the spatial array of nodes is sim- ple to produce from a GIS, the other matrices are not as easy to define. Distance between Graph theory patches can be measured in several differ- ent ways: edge to edge, centroid to centroid, Urban and Keitt (2000) give a general centroid to edge, etc. However, measuring description of ecological applications of graph these as Euclidean distance makes little theory and readers should refer to any num- sense when the variance in mortality cost ber of excellent texts on graphs as a primer associated with traversal of the intervening (e.g. Gross and Yellen, 1999). However, this is large, and cost associated with section describes the graph operations and traversal of the intervening habitats is large definitions used in this study. While there are and spatially heterogeneous. Few numerous excellent texts on the formalisms or even ecosystem processes, such as ground- of graph theory (e.g. Gross and Yellen, 1999), water movement or wildfire spread, move in the following largely conforms to Harary’s this way. To differing extents they are all (1969) classic text. A graph G is a set of nodes constrained by the landscape. Good multi- or vertices V.G/ and edges E.G/ such that dimensional models exist to predict some each edge eij Dvivj connects nodes vi and vj. ecosystem processes (e.g. pollution plumes; A path in a graph is a unique sequence of Bear and Verruit, 1987) but not others. Spa- nodes. The distance of a path from vl to vn is tially explicit models that simulate the dis- measured by the length of the unique set of persal of animals have been explored in some edges implicitly defined by the path. A path depth but the process is still poorly under- is closed if nl Dvn. Three or more nodes in a stood (Gaines and Bertness, 1993). Most are closed path is called a cycle. A path with no complex parametric models which are data- cycles is a tree. A tree that includes all the hungry. They require specific information vertices in the graph is a spanning tree. The and are hard to parameterize (Gustafson and minimum spanning tree is the spanning tree Gardner, 1996). in the graph with the shortest total length. For this reason, we have computed D not as The minimum spanning tree in effect repre- Euclidean distances but as a series of least- sents the parsimoniously connected backbone cost paths on a cost surface appropriate to of the graph. the organisms in question. These paths are A graph is connected if a path exists designed to approximate the actual distance between each pair of nodes. An unconnected the focal species (or any other landscape graph may include several connected com- agent, e.g. fire) covers moving from one patch ponents or subgraphs. A graph’s diameter, to the next. For instance, in this riverine d.G/, is the longest path between any two ecosystem, the path a mink might take from nodes in the graph, where the path length one side of a river delta to the other would between those nodes is itself the shortest pos- likely involve traversing the shore for 10 km sible length. If nodes i and j are not adjacent, under cover, rather than a 5-km swim across then the shortest path between them cannot open water. This allows the animal to use be the distance between them but must use stepping-stones of other habitat (low cost) stepping-stones. Here, we use graph diam- along the way rather than set off into an eter (or diameter of the largest component) unknown habitat matrix (high cost). The Landscape connectivity 269 least-cost modeling combines habitat quality possible to predict at 90-m resolution. We are and Euclidean distance in determining dij. not suggesting that the organisms modeled Cost was defined in 90-m cells (aggre- move purely according to least-cost paths. We gated up from 30-m cells to improve pro- use the framework because the distance of the cessing time) by a surface comprised of x, least-cost path is a better approximation of y and z,wherezwas a uniform impedance the actual distance covered than a straight that represented the cost of moving through line between patches. Our goal has been to that cell, i.e. its resistance to dispersal. get a better estimate of distance traveled Weights were approximated, based on per- using least-cost and not to predict corridors. ceived traversability. Cells corresponding to This modeling technique can be applied in areas of habitat were given a weight of 0Ð5, all a GIS, with limited spatial data, making it other forest types were given a weight of one. accessible to land managers and conservation Cells classified as riverine/estuarine herba- practitioners. Despite these advantages, cost- ceous were given a weight of two. Shrubland surface analysis has been only occasionally was given a weight of three. Sparsely vege- used by ecologists (Krist and Brown, 1994; tated cells (cultivated, managed herbaceous) Walker and Craighead, 1997), but widely were given a weight of four. Areas of devel- used in computer science which is concerned opment and large water bodies were given with optimal route planning (e.g. McGeoch, a weight of five. Streams were defined with a 1995; Bander and White, 1998). This type of weight of one. We used grid functions inside analysis is also common in applications of a macro in ArcInfo 7.2.1 (ESRI, 1998) to iter- artificial intelligence (e.g. Xia et al., 1997). atively loop through the array of patches and To focus on scaling between the two focal compute dij for each unique pair of nodes in species we chose to explicitly incorporate only the array. The macro uses area-weighted dis- patches greater than 100 ha in our analyses, tance functions to calculate least-cost paths. as prothonotary warblers are not likely to These functions are similar to Euclidean dis- persist in forest patches less than 100 ha tance functions, but instead of working in (Petit, 1999). Using habitat patches greater geographical units they work in cost units. than 100 ha results in 83 patches, roughly We explored alternative methods for con- 83% of the 53 392 ha of possible habitat. structing D, including Euclidean distance Because all habitat patches, regardless of and resistance-weighted distance between size, are given the lowest value on the cost nodes. We found that the topology of the surface, they are implicitly included in all graph is robust and not sensitive to the analyses in that the species can traverse difference between least-cost path distance them easily as stepping-stones, accruing and Euclidean distance except at the scale of minimal cost. large obstacles in the landscape. For instance, We further defined edges by a dispersal least-cost paths in our model did not cross the probability matrix P that expresses the 5-km mouth of the Alligator river when mov- probability that an individual in patch i will ing from the eastern side of the study area but disperse at least the distance between patch chose a route through habitat instead. In this i and j. We computed the elements of P as case Euclidean distance and least-cost path negative exponential decay: distance were quite different. The least-cost .qÐdij/ path technique is useful to land managers as pij De .1/ the surface can be parameterized based on best available data. Thus, the surface can be where q is an coefficient greater tailored to features in the landscape for which than 0. This way dispersal functions can be the manager has knowledge. The surface can indexed by noting the tail distance corre- be refined as data becomes available, e.g. in sponding to PD0Ð05 is ln.0Ð05/Ðq1. The tail the form of radio tracking. distance for mink and prothonotary warbler Gustafson and Gardner (1996) found that are indexed as 25 km and 5 km, respectively. dispersal routes are difficult to predict in even The tail distance is the distance to a selected slightly heterogeneous landscapes. We have point on the flat tail of the dispersal-distance kept that in mind by building a simple cost function. Other curves are possible and Clark surface that avoids committing the animals et al. (1999) provide a discussion of alternate to movement patterns that are not readily dispersal kernels. 270 A. G. Bunn et al.

The graphs are described most succinctly 100 repetitions), by minimum area, and by by an adjacency matrix A in which aij D1if endnodes with the smallest area (Urban and nodes i and j are connected and 0 if not. We Keitt, 2000). An endnode in a graph is a leaf set aij D1ifdij Ä25 km for mink and dij Ä5km in the spanning tree (here based on area- for prothonotary warbler. weighted flux) that is adjacent to only one We can also define the graph’s edges in other node. All edges incident to the removed terms of dispersal fluxes. Combining P and s node were also removed. At each iteration of allows us to compute dispersal flux from i to j: the removal process the graph was analyzed to determine the importance of the patch s i 0 to the graph’s area-weighted dispersal flux fij D Ðpij .2/ stot (F), and traversability (T). Area-weighted dispersal flux was indexed as: where si is relativized as the proportion of 0 the total habitat area stot in i and pij is pij Xn Xn normalized by the row sum of i in P.Because FD pijsi .4/ dispersal flux is asymmetrical .fij 6Dfji/ when i j,i6Dj si 6D sj we average the directions between nodes to give area-weighted dispersal flux where si is the size of node i and pij is from Equation (1) above. wij:   f Cf Traversability was indexed as the diameter w Dw D1 ij ji .3/ ij ji 2 of the largest component in the graph formed by the removal of the node: Subtracting from 1 allows the flux value 0 to have the smaller fluxes at greater dis- TDd.G /.5/

tances. The area-weighted dispersal flux 0 matrix allows us to compute a version of a where G is the largest component of G.We minimum spanning tree with more dispersal use F as an index of a patch’s source strength, biology incorporated. after Pulliam (1988). We use T in the sense of spreading-of-risk or rescue from catastrophe, after den Boer (1968) and Levins (1969). Graph operations Finally, we determined the importance of individual nodes to the entire landscape by With graph construction complete we per- assessing their individual contribution to formed two types of graph operations relat- area-weighted dispersal and traversability ing to connectivity: edge thresholding and in the graph by computing F and T for node removal. Edge thresholding allows us the entire landscape, and then recomputing to determine connectivity for mink and each with a single node removed from the prothonotary warblers based on their tail graph. That node’s impact is the difference dispersal distances. It also allows us to between the intact metric and the metric gauge the importance of variation of the that its removal elicited. Furthermore, we tail distance. We removed edges from the sought to determine the landscape’s overall graph iteratively with a edge distance sensitivity to scale by repeating this process thresholded at 100 m to 50 000 m in 100 m with edge definition thresholds from 2Ð5km increments. At each iteration the num- to 25 km, increasing in 2Ð5-km increments. ber of graph components, the number of We assessed the robustness of the patches’ nodes in the largest component and the sensitivity rankings on F and T with Spear- diameter of the largest component were man’s rank correlation, using the middle recorded. edge distance of 12Ð5 km to as the reference Node removal is a way to examine the case. relative importance of habitat area and connectivity in the landscape. We used node Results removal to tell us about the dynamics of the entire landscape under different habitat- loss scenarios. Nodes were removed from the The mean distance between patches in the graph iteratively. We began with the entire 83ð83 matrix is 62Ð7 km. The study area and graph and removed nodes randomly (with habitat patches are illustrated in Figure 1. Landscape connectivity 271

slightly at greater thresholding distances 80 (Figure 3). The edges are drawn as straight lines between patch centroids with 5, 10, 15 70 and 20 km thresholding distances in Figure 4, 60 even though the actual paths are computed 50 by least-cost and are circuitous. The distinct threshold at a 19-km func- 40 tional edge distance (Figures 2–4) implies 30 that the landscape as it stands now is per- ceived as being connected for species with 20

10 200

Number of graph components/Graph order 0 5 101520253035404550 175 Threshold distance (km) 150

Figure 2. Number of graph components ( ) 125 and graph order ( ) as a function of effective edge distance. 100 75 Diameter (km) Edge thresholding 50

The graph begins to disconnect and frag- 25 ment into subgraphs at a 19 km edge dis- tance, and quickly fragments into numerous 0 5 10 15 20 25 30 35 40 45 50 components containing only a few nodes Threshold distance (km) (Figure 2). The diameter of the largest Figure 4. Diameter of the largest component component increases quickly with threshold remaining in the graph with increasing thresholding distance, peaking at 20 km and declining distance.

Figure 3. All graphs edges with increasing thresholded distances from 5 to 20 km. 272 A. G. Bunn et al.

Figure 5. Minimum spanning tree for mink and prothonotary warbler based on distance.

a dispersal range of at least 20 km, and For mink, the minimum spanning tree unconnected for species with a dispersal on distance (Figure 5) is an excellent first range of less than 20 km. Using this edge- look at habitat-specific connectivity in the thresholding scenario, and the language of landscape. The minimum spanning tree rep- percolation theory, the landscape percolates resents the backbone of the habitat in the for mink but not for prothonotary warblers matrix. The minimum spanning tree based (Gardner et al., 1987, 1992). Another way to on area-weighted dispersal flux (Figure 6) envision this landscape is that prothonotary is very different. Couched in the mainland- warblers may have a tendency to act as many island model of Harrison (1994), the tree discrete populations, while the robust con- is now weighted by larger patches which nectivity of the landscape indicates that mink are expected to produce a larger number of will act as one patchy population (Harrison, propagules. The largest patch now radiates 1994). spokes which illustrates the spatial effect on For organisms with a 5-km dispersal dis- dispersal under these kernels. tance, like the prothonotary warbler, the landscape graph divides into subgraphs. The implications from edge-thresholding opera- Node removal tions are that some portions of the landscape have natural units to partition for manage- Node removal is habitat removal. We mea- ment. Edge thresholding also indicates nodes sured the effects of node removal in two ways that are easily isolated. This can serve as an which can indicate a landscape’s potential early blueprint for decisions regarding habi- to provide conditions that foster metapopula- tat acquisition or enhancement. For instance, tions. Flux (F), as governed by area and dis- this analysis indicates useful areas for patch persal potential, measures a node’s influence creation via wetland restoration. to a landscape-level metapopulation. Flux This preliminary exploration of edge thres- can measure the patch’s potential to act as a holding can provide some idea of landscape source in a source-sink metapopulation model connectivity relative to the dispersal capa- (Pulliam, 1988). Traversability (T) is a func- bilities (however uncertain) of mink and tion of the graph’s diameter. In this light it warblers. Using this framework it is easy to can be thought of as a proxy for spreading-of- highlight important nodes and edges under risk or long distance rescue (den Boer, 1968; different dispersal distances. Levins, 1969). T has the possibility to point Landscape connectivity 273

Figure 6. Area-weighted minimum spanning tree for mink with 25-km tail dispersal.

25 000 8 20 000

6 15 000

10 000

4 Diameter (m)

Total flux Total 5000

2 0 20 40 60 80 Number of nodes removed

0 20 40 60 80 Figure 8. Traversability (T) of the largest graph Number of nodes removed component as a function of three different node prunning scenarios. ( ), random; ( ), minnode; Figure 7. Area-weighted dispersal flux (F)asa ( ), endnode. Graph defined with 5 km adjancency function of three different node pruning scenarios. threshold. ( ), random; ( ), minnode; ( ), endnode. Graph defined with 25 km adjancency threshold. in the graph (Figure 8). Traversability of the graph is maintained with a majority of the out important stepping-stone patches in the graph nodes removed. The effect of endnode landscape. Source strength and long-distance pruning on this landscape may indicate that rescue are well established in conservation this riverine ecosystem has a high degree of biology. F and T are codified versions of those natural connectivity that an ecosystem not that fit into the graph context. comprised of linearly connected features may The different node removal scenarios give not posses. different pictures of the landscape. The better Area-weighted dispersal flux relies on P performance of endnode pruning over random and s and is functionally similar for mink or minimum area pruning for F indicates the and prothonotary warblers under random, tendency for endnodes to be less connected endnode, and minimum area pruning. The to the landscape (Figure 7). The advantage three thinning procedures produce similar of endnode pruning is clear in its effect on T results, although endnode pruning resulted 274 A. G. Bunn et al.

in slightly higher flux values (Figure 7). The In the more commonly used Pulliam model, effect of different types of patch removal F is analogous to source-sink strength and is on traversability is markedly different for not sensitive to scale because it is influenced mink and prothonotary warblers. For mink, most by close patches (short distance). with a 25-km functional adjacency thresh- We have chosen two focal species defined old, the three removal methods produce very by extremes in dispersal. In our model, mink similar results. For prothonotary warblers, can disperse five times farther than warblers with 5 km functional adjacency threshold, through the landscape. Our results indicate the random and minimum area pruning pro- that the high degree of connectivity for the duce similar linear results but the effect mink and low connectivity for the warbler do of endnode pruning is substantially dif- not cause meaningful interpretation of node ferent. Traversability of the graph is not sensitivity at that scale. However, the great effected until ¾75% of the nodes are removed flexibility of the graph approach is the ability (Figure 8). to instantly posit other degrees of dispersal based on edge distance. Figures 2–4 illus- trate that the landscape begins to fragment Node sensitivity seriously with a functional distance thresh- old between 10 and 15 km. These distances The spatial arrangement of habitat patches become important if we are concerned with in a landscape in combination with scale issues of connectivity, as this is the scale that can influence measures of connectivity (Keitt the landscape begins to meaningfully con- et al., 1997). Our two main metrics for con- nect. Figure 10 shows a false-color composite nectivity, F and T, show differing responses of patch sensitivity at 12Ð5-km effective edge- to scale. Traversability, T, is indexed inde- distance that displays each patch’s sensitivity pendently of patch area and is quite scale- to flux and traversability. We separated the dependent, showing little to no rank correla- metric F used above into recruitment poten- tion between scales (Figure 9). Conversely, tial (R) and dispersal flux .F0/. Here, F0 is F is calculated explicitly with patch area a dispersal flux coefficient not influencedP by and is very robust across scales. This is area and computed only with P .FD pij/ likely to be a function of patch area, and so as to separate it from area. R is a neu- illuminates interesting management and eco- tral model of connectivity that is computedP logical aspects of the landscape. In a Levins as a function of patch size alone .RD sij/. metapopulation model, T is analogous to Each patch in the landscape was tested for spreading-of-risk and is sensitive to scale. sensitivity, and scored for the three met- rics. This allows us to send R, F0, and T to the red, green and blue color guns respec- 1.5 tively. When the patches are displayed in 1.25 a false-color composite (Figure 10), some 1 interesting patterns emerge. In this image, patches that register high on metrics R, F0, 0.75 and T, saturate on all the colors and show 0.5 up as white. Conversely, patches that show up as a dark color have registered low on 0.25 every metric. Various other shades are read- Rank correlation 0 ily interpretable for each patch. Thus, node –0.25 sensitivity analysis can illuminate nodes that have contextual importance. For instance, –0.5 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 the blue patch indicated by the arrow in Effective edge distance (km) Figure 10 contributes to T but could be easily dismissed by a land manager as being unim- Figure 9. Correlation length of the vectors for portant because it is small and somewhat area-weighted dispersal (F;-- --) and traversability (T; ) at varying scales. Reference variables are isolated. This type of view on the landscape respective vectors at 12Ð5 km edge distance. Filled can indicate crucial linkages or bottlenecks symbols are significant at P<0Ð005. to connectivity. Landscape connectivity 275

Figure 10. False-color composite of node sensitivity with 12Ð5 km functional edge distance. Red, recruitment potential (R); green, dispersal flux .F0/; and blue, Traversability (T). For instance, the white arrow points to a patch with a high score for T and low scores for R and F.

Discussion spatial perceptions, the various habitat- removal scenarios allow us to determine patch function in reference to both species. We found that mink and warblers perceived Specifically, they allow us to envision, and this landscape differently, as a function of then prioritize, habitat loss in the landscape. their dispersal capabilities. For instance, and loss is one of the Figures 2–4 show that the landscape has a greatest threats to and a great great variability in connectivity depending deal of management decisions focus around on dispersal distance. According to the dis- minimizing the impact of habitat reduction persal estimates used, we found that mink (Burgess and Sharp, 1981; Harris, 1984). perceive this landscape as connected, while In this landscape, like many others, habi- prothonotary warblers do not. Given a sin- tat is managed by many different agencies gle patchy environment for mink we are and private landowners. Conflicting manage- able to exercise the graph and highlight the ment paradigms virtually guarantee habitat minimum spanning trees based on distance alteration and loss. Given this, node removal and area-weighted distance (Figures 5 and also allows us to quickly gauge the tendency 6). These represent the parsimoniously con- of a species to act like a metapopulation. For nected backbone of the landscape. For the instance, we found that individual patches warbler, which experiences this landscape as have different functions based on their size fragmented components, these graph struc- and position in the landscape. In the node tures are not as meaningful, but minimum removal graph perturbations, we found a ten- spanning trees based on connected subgraphs dency for endnodes to be poorly connected can provide utility in examining connectiv- and therefore contribute weakly to dispersal ity on a finer scale (not shown). The edge- flux and traversability. These results held for thresholding operations serve as a continuous mink (Figure 7) and prothonotary warblers picture of connectivity in the landscape, and (Figure 8). can be applied to other focal species. The node-sensitivity results show that In contrast to edge-thresholding proce- patches also have contextual importance. dures, which highlight mink and warbler Figure 10 is a powerful depiction of each 276 A. G. Bunn et al.

patch’s contribution to landscape connectiv- to landscapes is that it allows managers and ity, describing the landscape based on an edge researchers to take an initial, but thorough, threshold of 12Ð5 km. Given our dispersal look at the spatial configuration of a land- estimates, this is an intermediate dispersal scape. It is applicable at any scale. threshold not of direct importance to mink Another benefit of a graph-theoretic appro- or warblers. However, this distance serves ach is that dispersal biology does not need two pertinent functions in this landscape. to be fully understood for the graphs to be First, it is approximately the distance at interpretable. To give context to the graph which the landscape begins to meaningfully framework, we have postulated that disper- connect. Second, it highlights the important sal for mink is 25 km and for prothonotary versatility of the graph-theoretic approach warblers, 5 km. We used a conservative neg- and lets us instantly posit a gradient of ative exponential decay curve for dispersal dispersal thresholds. Given the overwhelm- probability, matrix P. An advantage of the ing complexity of dispersal biology, the node graph-theoretic approach is that gaming with sensitivity analysis provides an initial esti- alternative kernels is easy, and will affect dis- mate of the relative importance of individual persal in the landscape based on the spatial patches in the landscape. These preliminary arrangement of patches. Dispersal biology is analyses can also marshall further study by incredibly complex, and precise distances are identifying those patches where field studies virtually always unknown. Here our results should be concentrated. For example, the blue and their interpretation are largely inter- patch highlighted in Figure 10, and surround- pretable despite this uncertainty and can ing green patches, offer themselves as likely be immediately tailored to different disper- candidates to determine the effectiveness of sal estimates. Edge thresholding and node T and F. removal, as well as node sensitivity, are Challenges persist in developing macro- graph descriptors that are useful macroscopic scopic landscape models. Although focal metrics when dispersal can only be estimated species analysis can enrich macroscopic (see Keitt et al., 1997 for an additional exam- approaches by producing a species-specific ple). From a management perspective, the perspective to the analyses (O’Neill et al., graph can provide a powerful visualization of 1988; Pearson et al., 1996), reliable habi- connectivity when used in conjunction with tat definition from relatively coarse spatial dispersal estimates such as those based on data (e.g. 30-m cells) is challenging for many allometric relationship to body mass (see species, and limited to habitat specialists. Sutherland et al., 2000). The use of the intervening non-habitat matrix is especially important, as this affects the functional scale at which patches are defined. Prospectus Edge definition in a graph calls for dispersal biology that is often difficult to parameter- Land and conservation management is incre- ize. However, well chosen focal species in a asingly concerned with regional-scale habitat landscape can provide ecological and political analyses. The development of graph theory in effectiveness in issues of connectivity. an ecological framework represents a promis- When appropriate species such as mink ing step forward in that regard. Graph the- and warblers are available in a land- ory rests on a foundation of intensive study scape, then focal species analysis is partic- for computer networks which must be effi- ularly well-suited to graphic representation, cient. Therefore, the theory and algorithms because ecological flux is a primary concern. are well developed; many are computation- The graph-theoretic approach differs from ally optimal. Like metapopulation theory, most focal species analyses as it allows one the graph can merge landscape configuration to use surrogates as a rapid assessment tool and focal species biology to arrive at process- without long-term population data, although based measures of connection (Hanski, 1998; population data can (and should) be incorpo- Urban and Keitt, 2000). The advantage of rated as knowledge of the system improves. graph-theoretic approaches to conservation It is a heuristic framework which is a robust planners and researchers is that, while rea- way to represent connectivity in the land- sonable quality spatial data are required, scape. The utility of applying graph theory long-term population data are not. Landscape connectivity 277

The conservation potential of graph theory Cantwell, M. D. and Forman, R. T. T. (1993). is far from realized. The existing body of eco- Landscape graphs: ecological modeling with logical work that considers landscape graphs graph-theory to detect configurations common to diverse landscapes. Landscape 8, is slim (Cantwell and Forman, 1993; Keitt 239–255. et al., 1997; Urban and Keitt, 2000). The Christensen, N. L., Bartuska, A. M., Brown, J. H., most appealing feature of graph theory as Carpenter, S., D’Antonio, C., Francis, R. et al. applied to ecology is that it is a heuristic (1996). The report of the ecological society framework for management which is neces- of America committee on the scientific basis for ecosystem management. Ecological Applica- sarily perpetual. With very little data, one tions 6, 665–691. can construct a graph of loosely-defined habi- Clark, J. S., Silman, M., Kern, R., Macklin, E. tat patches and then explore the structure of and HillRisLambers, J. (1999). Dispersal near the graph by considering a range of thresh- and far: patterns across temperate and tropical old distances to define edges. It is important forests. Ecology 80, 1475–1494. Curson, J., Quinn, D. and Beadle, D. (1994). that as more ecological information is col- Warblers of the Americas: An Identification lected it can be infused into the graph and Guide. Massachusetts: Houghton Mifflin. consequently add more precision and con- den Boer, P. J. (1968). Spreading of risk and stabi- fidence to the analyses. Graph theory can lization of animal numbers. Acta Biotheoretica provide initial processing of landscape data 18, 165–194. Dunstone, N. and Birks, J. D. S. (1987). Feeding and can serve as a guide to help develop and ecology of mink (Mustela vison) in coastal marshall landscape-scale plans, including the habitat. Journal of Zoology 212, 69–84. identification of sensitive areas across scales. ESRI (Environmental Services Research Incorpo- This does not mean that graph theory should rated) (1998). ArcInfo v. 7.2.1. Redlands, CA. displace alternative approaches. We suggest Gaines, S. D. and Bertness, M. (1993). The dynam- ics of juvenile dispersal: why field ecologists graph theory as a computationally power- must integrate. Ecology 74, 2430–2435. ful adjunct to these other approaches. The Gardner, R. H., Milne, B. T., Turner, M. G. simplicity and flexibility of graph-theoretic and O’Neill, R. V. (1987). Neutral models for approaches to landscape connectivity offers the analysis of broad-scale landscape pattern. much to land practitioners and can increase 1, 19–28. Gardner, R. H., Turner, M. G., Dale, V. H. the scope and effectiveness of resource man- and O’Neill, R. V. (1992). A percolation model agement. of ecological flows. In Landscape Boundaries: Consequences for Ecological Flows (A. Hancen and F. di Castri, eds), pp. 259–269. New York: Springer-Verlag. Acknowledgements Gerell, R. (1970). Home ranges and movements of the mink Mustela vison in southern Sweden. Oikos 21, 160–173. The authors thank Patrick N. Halpin, Robert Gross, J. and Yellen, J. (1999). Graph Theory and S. Schick, and three anonymous reviewers whose Its Applications. Florida: CRC Press. comments greatly improved earlier drafts of this Gustafson, E. J. and Gardner, R. H. (1996). manuscript. The Landscape Ecology Lab at Duke The effect of landscape heterogeneity on the University provided much useful technical sup- probability of patch colonization. Ecology 77, port. AGB was supported by the Stanback Foun- 94–107. dation for the duration of this project. Hamilton, W. J. (1940). The summer food of minks and raccoons on the Montezuma Marsh, New York. Journal of Wildlife Management 4, 80–84. References Hanski, I. (1998). Metapopulation dynamics. Nature 396, 41–49. Harary, F. (1969). Graph Theory. Massachusetts: Bander, J. L. and White, C. C. (1998). A heuristic Addison-Wesley. search algorithm for path determination with Harris, L. D. (1984). The Fragmented Forest: learning. IEEE Transactions On Systems Man Island Biogeography Theory and the Preserva- and Cybernetics Part A-Systems and Humans tion of Biotic Diversity. Illinois: University of 28, 131–134. Chicago Press. Bear, J. and Verruit, A. (1987). Modeling Ground- Harrison, S. (1994). Metapopulations and conser- water Flow and Pollution: With Computer Pro- vation. In Large-scale Ecology and Conserva- grams for Sample Cases. MA: Kluwer Academic tion Biology (P. J. Edwards, N. R. Webb and Publishers. R. M. May, eds), pp. 111–128. Oxford: Blackwell. Burgess, R. L. and Sharpe, D. M. (eds) (1981). Keitt, T. H., Urban, D. L. and Milne, B. T. Forest Island Dynamics in Man-dominated (1997). Detecting critical scales in fragmented Landscapes. New York: Springer-Verlag. landscapes. Conservation Ecology 1, 4, (online) 278 A. G. Bunn et al.

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