Detecting Critical Scales in Fragmented Landscapes

• Abstract
• Introduction
• Methods
• Results
• Discussion
• Responses
• Literature Cited

ABSTRACT

KEY WORDS: connectivity, conservation in fragmented landscapes, dispersal, fragmentation, habitat connectivity vs. dispersal distance, landscape, landscape graphs, metapopulation, percolation, quantifying habitat connectivity at multiple scales, "stepping stone" patch, Strix occidentalis lucida.

INTRODUCTION

Landscape connectivity does not depend on scale alone; the configuration or spatial arrangement of habitats in a landscape is also an important determinant of connectivity (Forman and Baudry 1984, Gardner et al. 1992, Henein and Merriam 1990, Taylor et al. 1993). Heterogeneity in habitat quality, differences in patch shape and size, and variation in isolation among patches lead to spatial variation in landscape connectivity (Gustafson and Gardner 1996). Thus, although indices of landscape connectivity averaged over entire landscapes are useful for comparisons among ecosystems, these indices provide little insight into the local connectivity structure of landscapes. To fully characterize the spatial structure of landscapes, it is necessary to consider connectivity measures associated with local regions or individual habitat patches. Just as overall measures of landscape connectivity are scale dependent, local measures of connectivity are likely to change with scale.

Here, we present a multiscale analysis of landscape connectivity, based on an extension of uniform percolation theory (Stauffer and Aharony 1985, Gardner et al. 1987, Gould and Tobochnik 1988, Creswick et al. 1992) to non-uniform landscape graphs (Cantwell and Forman 1993, Harary 1969). We develop both aggregate measures of landscape connectivity and patch-based measures of individual patch contributions to overall connectivity. We examine the sensitivity of landscape connectivity to changes in landscape configuration, and the relationship between sensitivity and scale. Finally, we consider the implications of our analysis for habitat preservation efforts with regard to threatened species.

METHODS

Landscape data

Our original motivation for choosing ponderosa pine and mixed-conifer forest types was to approximate the distribution of suitable nesting habitat for Mexican Spotted Owls (Strix occidentalis lucida) within the southwestern United States. The data were used in developing a recovery plan for the Mexican Spotted Owl (Keitt et al.1995, USDI Fish and Wildlife Service 1995).

Landscape graphs

We developed an objective means of generating landscape graphs (Fig. 1 ). The input data consisted of a raster habitat map with each grid cell assigned the value one if that cell corresponded to the chosen habitat type, and zero if the cell was considered nonhabitat. A patch was defined as a spatially contiguous set of habitat cells. Two habitat cells were considered to be in the same habitat patch if they were adjacent in the four cardinal directions or diagonally (northeast, northwest, and so on). Once the habitat patches were labeled, we assigned a graph vertex to each patch. Edges were placed between vertices if their corresponding patches were connected according to the criteria to be described. Thus, the landscape graph consisted of a set of vertices, one for each habitat patch, and a set of edges indicating connections among patches. For display purposes, vertices were placed at the geometric center of each patch.

FIG. 1. Construction of a landscape graph. This example landscape contains three separate habitat patches and a single habitat cluster, or ``subgraph.'' The three patches belong to a single cluster because there exists a path along the graph edges (solid lines) that connects all three patches. If either one of the edges shown were removed, there would be two habitat clusters. If all edges were removed, then there would be three habitat clusters, each consisting of a single patch.  

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In classified landscape data, small habitat patches are much more likely to be the product of random error than are large patches. For example, if errors are introduced into an image with probability p << 1, then the probability of creating a patch of errors of size n is approximately p ⁿ. Random errors are unlikely to produce large patches, because as n increases, pⁿ becomes very small. The forest data covered ~ 38% of the map area. Of the patches identified in the forest data, roughly 50% were single-cell patches. We calculated that a random map with 18% cover would also have 50% single-cell patches. Thus, an 18% error rate was sufficient to account for all single-cell patches (i.e., those most likely to be the result of classification error) in the forest data.

We then asked, over what patch sizes was the frequency of forest patches similar to the expected frequency of patches created by random noise? We plotted the frequencies of forest patch sizes and of patch sizes observed in 100 random maps with 18% density (Fig. 2 ), and found that the two distributions were similar up to a patch size of 10 km² . Forest patches > 10 km² in size occurred at much higher frequencies than patches of equivalent size in the random maps. Thus, we retained only patches >10 km² in subsequent analyses.

FIG. 2. Relative frequency of patch sizes for Southwest forest data and for 100 random maps with density of 18%. Patch frequencies decay rapidly with increasing patch size for the random maps, but more slowly for the forest data. Forest patches >10 km² in size occurred with much higher frequency than in the random maps, and were thus unlikely to have resulted from classification error.  

Scaling analysis

Connectivity measures

Percolation need not occur on a regular lattice. A more general model involves percolation on graph structures, referred to as "bond percolation" (Stauffer and Aharony 1985). Because regular lattices are a subset of possible graph configurations (Harary 1969), bond percolation naturally incorporates lattice percolation as a special case. In bond percolation, sites are either connected with probability p or are unconnected with probability 1 - p. As in lattice percolation, a large cluster of connected sites occurs at a critical value of p. Our application of percolation theory represents an extension of standard bond percolation. In our model, the probability of a connection between two habitat patches depends on the distance between the patches. Thus, the emergence of a spanning habitat cluster occurs at a critical mean dispersal distance instead of at a particular habitat density, as in traditional lattice percolation. Percolation problems where connection probabilities vary across the landscape are referred to as"gradient" or "non-uniform" percolation (Milne et al. 1996).

In percolation theory, connectivity is associated with the average size of connected clusters. A natural measure of the size of a circular cluster is its radius. However, in general, clusters are not round; they can be irregular, sinuous structures. Therefore, a measure of cluster size must control for irregular shapes. A measure of cluster size used in percolation theory is the "radius of gyration," defined as

where < x > and < y > are the mean x and y coordinates of lattice cells in the cluster, x_i and y_i are the coordinates of the i^th grid cell in the cluster, and n is the total number of cells in the cluster (Creswick et al. 1992). Habitat clusters were defined as sets of patches connected by a subgraph or component of the thresholded landscape graphs. For a cluster comprised of several habitat patches, the sum in Eq. 1 was taken over all habitat cells among all patches in the cluster.

Unlike unitless indices of landscape connectivity, the cluster radius has units of distance and a direct physical interpretation. Imagine a randomly moving particle placed randomly on a habitat cluster. The radius of gyration is the average distance that the particle will move before encountering the cluster edge. Similarly, if a dispersing animal is restricted to moving on a particular habitat cluster (i.e., it has a low probability of traversing any gap separating it from another cluster), its average dispersal range will correspond to the radius of the cluster.

The size-weighted average connectivity of a set of clusters defines the correlation length of a landscape. The correlation length of a set of clusters is given by

where m is the number of clusters and n_s is the number of grid cells in cluster s (Creswick et al. 1992). We used the correlation length as an overall measure of habitat connectivity in a landscape. As with the radius of gyration, correlation length has units of distance: it is the average distance an individual is capable of dispersing before reaching a barrier, if placed randomly on the landscape. Thus, as the correlation length increases, landscape connectivity increases.

Patch removal experiments

is the normalized importance index of patch i. The normalized importance is a relative index of the contribution of each patch to overall landscape connectivity. We chose a normalized, unitless index because it was designed precisely for among-patch comparisons.

We expected that large patches would have larger importance indices, simply owing to their size. Thus, we wanted a measure of per area importance as well. The per area importance index quantified a patch's contribution to overall landscape connectivity per unit area. We defined the per area index as

where n ( i ) is the number of habitat cells in patch i and a_c is the area of a grid cell. The per area index had units of 1/area, e.g., km^-2.

Stochastic landscape graphs

where d was the minimum distance between patches. An important property of the threshold function was that a given threshold distance resulted in only a single landscape graph structure. However, the percolation problem can be generalized to include any probability function. Beyond the threshold function, the simplest probability function is the cumulative negative exponential

where p ( d )  is the probability of dispersing at least distance d, and k is the dispersal coefficient. The dispersal coefficient has units of 1/distance; its inverse, 1/k, is the average dispersal distance under the distribution. Distributions requiring more parameters, such as the gamma distribution, could be used as well.

Unlike the threshold function, the negative exponential function resulted in a large number of landscape graphs for any given dispersal coefficient. Following standard techniques in percolation theory, we used a Monte Carlo procedure to generate multiple landscape configurations. We varied the dispersal coefficient, k , from 1.0 (average dispersal distance = 1 km) to 0.01 (average dispersal distance = 100 km). For each value of the dispersal coefficient, 100 randomly connected landscape configurations were generated and their correlation length was recorded. Connections between patches in each configuration were determined by comparing a uniform random number between 0.0. and 1.0 to the probability of dispersal occurring between patches, according to Eq. 6. If the random number was less than the the dispersal probability, the patches were joined into the same subgraph or cluster. Over many random configurations, the expected frequency of two patches being joined was equal to the probability of successful dispersal between them. Patches far apart were rarely connected, whereas patches whose borders were in close proximity were often connected, the frequency being determined by the dispersal coefficient.

RESULTS

Thresholded landscape graphs

FIG. 4. Correlation length of the habitat distribution vs. threshold distance.  

Landscape sensitivity

FIG. 5. Graph sensitivity at different threshold distances. Patches are ordered by size. Sensitivity is the change in correlation length when a patch was removed.  

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The pattern of landscape sensitivity changed abruptly at 45 and 50 km. Although the largest patch still had a relatively high importance index, graph sensitivity (Fig. 5 ) exhibited several large spikes generated by much smaller patches. The large sensitivity associated with these patches was not due to their size; rather it reflected their role as "stepping stone" patches connecting large areas of habitat. These stepping stone patches corresponded to articulation points in the graph.

At distances above the transition (60 and 80 km), graph sensitivity returned to a pattern in which the largest patches were most important. However, the overall sensitivity was much lower, particularly at 80 km, because as the threshold distance was increased, the number of alternate pathways between any two patches increased, and the removal of any one patch had little effect on connectivity.

Patch importance

Connectivity in stochastic landscape graphs

When the dispersal coefficient was varied between k =0.01 (i.e., long-range dispersal) and k =1.0 (i.e., short-range dispersal), the relationship between dispersal behavior and connectivity of the landscape was profoundly nonlinear, exhibiting a strong inflection at intermediate values (Fig. 8 ). Furthermore, the distribution of correlation lengths was clearly divided into two distinct phases, higher values indicating the connected phase and lower values the disjoint phase. At intermediate values, both phases existed for the same value of the dispersal coefficient. Thus, the phase transition observed previously for maximum dispersal distances was, in the Monte Carlo analysis, directly related to a particular dispersal coefficient. For forest habitats in the Southwest, the critical dispersal coefficient ( k _c ) was ~ 0.06-0.07, i.e., an average dispersal distance of 14-17 km.

FIG. 8. Landscape connectivity for different dispersal coefficients. The coefficient is k  in p(d) =e^-kd where p(d) is the probability of an individual dispersing at least distance d . Each circle represents a single random landscape configuration.  

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DISCUSSION

An important finding of our analysis was that habitat loss has a highly scale-dependent effect on landscape connectivity. For organisms that perceive the landscape at fine scales, landscape configuration and stepping stone patches are of little consequence, because populations are restricted to local habitat patches. Similarly, movements of species capable of long-range dispersal will not be strongly influenced by the configuration of individual patches. For species near the percolation transition, landscape configuration may play a significant role in determining landscape connectivity. Near the percolation transition, individual patches can act as corridors or stepping stones, bridging gaps in the habitat distribution. Thus, we expect that landscape configuration will be important to species whose dispersal behavior places them near the landscape percolation threshold.

How applicable these results are to different species will depend to some extent on the mode of dispersal used. We designed our analyses to capture the essential features of owls dispersing between the "sky-island" habitats characteristic of the Southwest. Species using other modes of dispersal will interact with landscape patterns in a different manor. An organism that must walk or run over a landscape may encounter barriers not represented in our habitat maps. However, we see no reason why our approach could not be modified to incorporate other modes of dispersal and more detailed spatial information.

In general, we have too few data available to precisely predict dispersal patterns of Mexican Spotted Owls. However, there are several reports (reviewed in Keitt et al. 1995) of juvenile owls dispersing > 45 km, and it is typical, at least in some parts of the owls' range, for juveniles to establish new territories 10-20 km from their natal territory (Keitt et al. 1995). Thus, we believe that the Mexican Spotted Owls probably exist as a metapopulation and that landscape connectivity is an important consideration for their survival.

Management perspectives

The importance of regional- and landscape-scale analyses is even greater in situations in which multiple agencies manage habitats throughout a species range. For example, market forces may induce forest managers to harvest timber from a small region of habitat such as Mount Taylor, thinking that the patch is of little ecological importance. However, a high per area index (Fig. 6 ) for the same patch might lead managers to recognize the role of the patch as a crucial link within a broader region composed of several management jurisdictions. Alteration of the patch based on nonspatial, economic perspectives could jeopardize conservation goals among many agencies. Within a jurisdiction, such as a national forest, managers might reallocate harvests to patches that have small per area indices in order to preserve the connectivity value of the entire habitat network. Alternatively, one or more agencies may trade land or extracted resources to accomplish management goals while maintaining network fidelity.

We also wish to emphasize that our results are not restricted to single-species studies. Habitat patches that have high importance over a wide range of scales (Fig. 7 ) may represent important dispersal habitat for many species at once. A multipecies, ecosystem-level analysis could also be constructed by overlaying the results of distance-percolation studies from many habitat types and combinations of habitat types. The resulting map of habitat importance could then be used, along with appropriate supporting data and studies, as a basis for ecosystem management.

Dispersal, biogeography, and climate change

Critical dispersal also has implications for the origin and maintenance of biological diversity. For example, in Wright 's (1970) shifting-balance theory, intermittent gene flow among isolated populations is an important mechanism allowing individual populations to explore many genetic possibilities. Such a mechanism, combined with critically connected populations, might explain the high diversity of mammals associated with montane environments in the western United States (Kerr and Packer 1997).

Climate change may also play a role, both in the origin of diversity and in the isolation and extinction of species. Typically, tree distributions are expected to move upward in elevation as climate warms, thereby eliminating summit habitats and reducing the area of mid- to high- elevation vegetation types. Shifts that reduce habitat area may produce species extinctions Lomolino et al. (1989). In addition, elevational shifts should alter network geometry and require dispersal over greater distances to effectively link patches. Additional local extinctions or population fragmentation may occur if the resultant critical distance exceeds a particular species' average dispersal ability. Thus, climate changes or land use practices that alter patch network geometries may interact with dispersal distances, thereby hampering some species while facilitating others. We see an open and interesting area of research in the study of how landscape graphs evolve over time with changes in climate and topography, and how these changes relate to extinction and species diversity.

RESPONSES TO THIS ARTICLE

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Acknowledgments

We wish to thank A. Johnson for suggestions and comments on early drafts of this manuscript. We thank members of the Mexican spotted owl recovery team--B. Block, G. White, A. Franklin, P. Ward, S. Rinkevitch, S. Spangle, and J. Ganey--and three anonymous reviewers for critically reviewing this work. We greatfully acknowledge the support of the U.S. Fish and Wildlife Service (USFWS and UNM cooperative agreement no. 1448-00002-94-0810).

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