1205 CONCEPTS & SYNTHESIS EMPHASIZING NEW IDEAS TO STIMULATE RESEARCH IN ECOLOGY Ecology, 82(5), 2001, pp. 1205���1218 q 2001 by the Ecological Society of America LANDSCAPE CONNECTIVITY: A GRAPH-THEORETIC PERSPECTIVE DEAN URBAN1,3 AND TIMOTHY KEITT2,4 1Nicholas School of the Environment, Duke University, Durham, North Carolina 27708 USA 2National Center for Ecological Analysis and Synthesis, Santa Barbara, California 93101 USA Abstract. Ecologists are familiar with two data structures commonly used to represent landscapes. Vector-based maps delineate land cover types as polygons, while raster lattices represent the landscape as a grid. Here we adopt a third lattice data structure, the graph. A graph represents a landscape as a set of nodes (e.g., habitat patches) connected to some degree by edges that join pairs of nodes functionally (e.g., via dispersal). Graph theory is well developed in other fields, including geography (transportation networks, routing ap- plications, siting problems) and computer science (circuitry and network optimization). We present an overview of basic elements of graph theory as it might be applied to issues of connectivity in heterogeneous landscapes, focusing especially on applications of metapo- pulation theory in conservation biology. We develop a general set of analyses using a hypothetical landscape mosaic of habitat patches in a nonhabitat matrix. Our results suggest that a simple graph construct, the minimum spanning tree, can serve as a powerful guide to decisions about the relative importance of individual patches to overall landscape con- nectivity. We then apply this approach to an actual conservation scenario involving the threatened Mexican Spotted Owl (Strix occidentalis lucida). Simulations with an incidence- function metapopulation model suggest that population persistence can be maintained de- spite substantial losses of habitat area, so long as the minimum spanning tree is protected. We believe that graph theory has considerable promise for applications concerned with connectivity and ecological flows in general. Because the theory is already well developed in other disciplines, it might be brought to bear immediately on pressing ecological ap- plications in conservation biology and landscape ecology. Key words: connectivity conservation biology dispersal graph theory habitat fragmentation habitat patches and landscape connectivity habitat pattern landscape ecology metapopulation theory minimum spanning tree Strix occidentalis lucidus. INTRODUCTION Ecological work is being done at increasingly larger scales. For example, conservation biology is necessar- ily concerned with large biogeographic areas (Noss 1991), and ecosystem management is inherently large scale (Christensen et al. 1996). This has lead us to work with new sorts of data sets summarizing the spatial attributes of landscapes. Indeed, construction and anal- ysis of spatial landscape data is the first step in virtually all habitat conservation planning. In the development of landscape-scale conservation plans, we typically encounter one of three classes of spatial data (Cressie 1993): (1) spatial point patterns Manuscript received 12 July 1999 revised 24 March 2000 accepted 29 May 2000 final version received 7 July 2000. 3 E-mail: deanu@duke.edu 4 Present address: Ecology and Evolution, State University of New York, Stony Brook, New York 11794 USA. that comprise a set of locations of entities of interest (e.g., locations or distributional records of species of concern) (2) geostatistical data that represent mea- surements at locations separated by some distance and (3) lattices that assign a measurement or value to re- gions within the landscape. In geographic information systems (GIS) these data classes provide for two al- ternative conceptual models of landscapes (Goodchild 1994). In the field view, a landscape is a continuous surface defined by some variable(s) that can be mea- sured at any point on the surface. Examples of fields would include elevation, surface temperature, or veg- etation biomass. Fields that are represented exhaus- tively (i.e., the entire surface) are lattices, while in- completely sampled representations are geostatistical data. Alternatively, one might view a landscape as fea- tures or objects, discrete entities that occupy positions in an otherwise undifferentiated space. Elevation benchmarks (points), temperature isopleths (lines), and lakes (polygons) are examples of features.

Concepts & Synthesis 1206 DEAN URBAN AND TIMOTHY KEITT Ecology, Vol. 82, No. 5 FIG. 1. An example of a graph defined by the sets of p 5 6 nodes {a,b,d,c,e,f} and q 5 8 edges {ab, bc, be, bd, cd, de, df, ef}. A lattice often is represented in one of two familiar data structures (other less common structures are used for certain applications, see Goodchild 1994). In vector maps the patches are represented by vectors of coor- dinates outlining each patch as a polygon. Patches are treated as internally homogeneous. In the class of me- tapopulation models called ������island models,������ the patch- es are islands of suitable habitat in a sea of inhospitable matrix (nonhabitat). This modeling approach converts the lattice from a field into a set of features (habitat patches). Raster lattices or mosaics are grids in which each cell is assigned to a discrete state or assumes some value. The size of the grid cells defines the minimum resolution or grain of the mosaic. A raster mosaic is a field model. The island and mosaic data structures have complementary strengths and weaknesses: vector files are compact but sacrifice fine-grain information mo- saics retain this information at the expense of data vol- ume. Ecologists generally are comfortable choosing or converting between these two forms to match the data structure to a particular application. For example, hab- itat patches might be defined as regions (clusters) of more-or-less similar cells, converting a mosaic into polygon features. Here we adopt a less familar lattice data structure, the graph. A graph represents a landscape of habitat patches as a set of nodes (points) connected to some extent by edges between nodes (these are not the ������edg- es������ of field���forest ecotones, although the term might indeed connote a similar sense of adjacency between patches). An edge between two nodes implies there is some ecological flux between the nodes, such as via propagule dispersal or material flow. Graph theory is widely applied in various disciplines (computer sci- ence, operations research) for a wide variety of appli- cations concerned with maximally efficient flow or routing in networks or circuits (Harary 1969, Thula- siraman and Swamy 1992, Gross and Yellen 1999). The theory is currently being stretched to even greater al- gorithmic efficiency through its extension to applica- tions on huge networks such as the worldwide web (Hayes 2000a, b). While long used as a framework for food-web theory in ecology (e.g., Pimm 1982), the for- malisms of graph theory are not widely appreciated in landscape ecology (Cantwell and Forman 1993). A graph-theoretic perspective would seem to provide powerful leverage on ecological applications con- cerned with connectivity or ecological fluxes (van Langevelde et al. 1998). In particular, graphs are ame- nable to applications concerned with metapopulations and conservation biology (Fahrig and Merriam 1985, Verboom and Lankester 1991, Taylor et al. 1993, Schip- pers et al. 1996). Here we illustrate the power and potential utility of applying graph theory to landscape analysis. We begin with an overview of graphs, and then illustrate this approach with an empirical appli- cation to habitat pattern for the Mexican Spotted Owl (Strix occidentalis lucida). Our purpose is not to argue against other approaches already used in ecology, but rather to present an additional option that has much to offer to large-scale ecological applications. GRAPHS AND GRAPH THEORY Definitions Graph theory has rich vocabulary and some defini- tions are necessary to the following discussion. The following largely follows Harary���s classic (1969) text. A graph G is a set of nodes or vertices V(G) and edges E(G) such that each edge e 5 vivj connects nodes vi and vj. In this, nodes vi and vj are adjacent and each is incident to their shared edge. A graph of m nodes and n edges is G(m,n) and has order m and value n. The graph G is defined by its sets {vi} and {ei}, but it is common to represent these sets as a diagram (Fig. 1). A path in this graph is a sequence of nodes���a walk from v0 to vn, such that each node is unique (i.e., no node is visited more than once). This implies that the edges of a path are also unique. The length of a walk is the sum of the lengths of its edges. A walk is closed if v0 5 vn (i.e., the node first is revisited), and a closed path of three or more nodes is a cycle. A path that includes no cycles is a tree. A tree that includes every node in the graph is a spanning tree. There might be several of these for any given graph. The spanning tree with the shortest length is the graph���s minimum span- ning tree. The path defined by the edges ab, bc, cd, df, de is the minimum spanning tree in Fig. 1. A graph is connected if there exists a path between each pair of nodes, that is, if every node is reachable from some other node. An unconnected graph may con- sist of several subgraphs. A graph component is a con- nected subgraph, that is, a subgraph in which every node is adjacent to at least one other node in the sub- graph. A connected graph that can be disconnected by the removal of a key node has a cut-node at that point. The minimum number of nodes that must be removed from a connected graph before it disconnects is its node-connectivity k(G). Equivalently, the minimum number of edges that must be removed to disconnect a graph is its edge- or line-connectivity l(G), and an edge whose removal disconnects a graph is a cut-edge or bridge. (Ecologists use a variety of terms to connote connectivity. The connectivity terms above have a spe- cial meaning in graph theory that does not correspond