Models of Landscape Structure

Instructor: K. McGarigal Assigned Reading: McGarigal (Lecture notes) Models of Landscape Structure Objective: Provide a basic description of several alternative models of landscape structure, including models based on point, categorical and continuous patterns. Highlight the importance of selecting a meaningful model for the question under consideration given the constraints of data availability and software tools available for analyzing pattern-process relationships. Topics covered: 1. Models of landscape structure 2. Point pattern model 3. Patch mosaic model - island biogeographic and landscape mosaic models 4. Landscape gradient model 5. Graph matrix model

1. Models of Landscape Structure There are many different ways to model or represent landscape structure corresponding to different perspectives on landscape heterogeneity. Here we will review five common alternative models: (1) point pattern model; (2) island biogeographic model based on categorical map patterns; (3) landscape mosaic model based on categorical map patterns; (4) landscape gradient model based on continuous surface patterns; and (5) graph-theoretic model. 7.8

The choice of model in any particular application depends on several criteria: The ecological pattern-process under consideration and the objective of the analysis The spatial character of the landscape with respect to the relevant attributes Available spatial data (type, structure and quality) Available analytical methods (software tools) Available computational resources 7.9

2. Point Pattern Model 2.1. Data Characteristics Point pattern data comprise collections of the locations of entities of interest, wherein the data consists of a list of entities referenced by their (x,y) locations. Familiar examples include: Map of all trees in a forest stand, perhaps by species Map of all occurrences of a focal ecosystem (e.g., seasonal wetland) in a study area Map of all detections of an individual of a species during a season or over a lifetime 7.10

2.2. Data Structure Point pattern data typically are represented using a vector data structure, wherein each point is referenced by its (x,y) location and occupies no real space. Alternatively, it may be convenient in some applications (see below) to represent point patterns using a raster data structure, wherein each point is represented as a cell (or pixel) in a raster grid and thus occupies the space of one cell. Lastly, it may be useful in some applications to address point patterns using a graph matrix data structure, which is described later as a special model of landscape structure. 7.11

2.3. Pattern Elements Point pattern data consist of a single pattern element - points. The points are often indistinguishable from each other (i.e., unweighted), wherein only the (x,y) location of the points is of interest. Alternatively, the points may be distinguished from each other on the basis of one or more attributes (e.g., weights), so that not all points are equal, and this information is then taken into account in the analysis. 7.12

2.4. Pattern Metrics The goal of point pattern analysis is typically to quantify the intensity of points at multiple scales and there are numerous methods for doing so. Here we will review only a few of the more popular approaches. st (1) 1 -order point patterns The primary data for point pattern analysis consist of n points tallied by location within an area of size A (e.g., hundreds of individual trees in a 1-ha stand). The simplest of pattern metrics merely quantify the number and density of points. However, a plethora of techniques have been developed for analyzing more complex aspects of spatial point patterns, some based on sample quadrats or plots and others based on nearest-neighbor distances. A typical distance-based approach is to use the mean point-to-point distance to derive a mean area per point, and then to invert this to get a mean point density (points per unit area), Lambda, from which test statistics about expected point density are derived. There are nearly uncountable variations on this theme, ranging from quite simple to less so (e.g., Clark and Evans 1954). Most of these techniques provide a single global measure of point pattern aimed at distinguishing clumped and uniform distributions from random distributions, but do not help to distinguish the characteristic scale or scales of the point pattern. 7.13

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nd (2) 2 -order point patterns Ripley s K-distribution The most popular means of analyzing (i.e., scaling) point patterns is the use of second-order statistics (statistics based on the co-occurrences of pairs of points). The most common technique is Ripley's K-distribution or K-function (Ripley 1976, 1977), which we discussed previously as a scaling technique for point pattern data. Recall that the K-distribution is the cumulative frequency distribution of observations at a given point-to-point distance (or within a distance class); that is, it is based on the number of points tallied within a given distance or distance class. Because it preserves distances at multiple scales, Ripley's K can quantify the intensity of pattern at multiple scales. In the real-world example shown here for the distribution of vernal pools in a small region in Massachusetts, the K function reveals that pools are more clumped than expected under a spatially random distribution out to a distance of at least 3.5 km, and are perhaps most clumped at a scale of about 400 m. The scale of clumping of pools is interesting given the dependence of many vernal pool amphibians on metapopulation processes such as dispersal of individuals among ponds. For most of these vernal pool-dependent species the pools are highly clumped at scales corresponding to the range of dispersal distances. 7.15

(3) Local pattern intensity Kernel estimators The previous methods provide a numerical summary of the global point pattern; that is, they provide a quantitative description of the average pattern of points across the entire landscape. Often times, however, it is more useful to assess the local point pattern and produce a pattern intensity map. The kernel estimator (Silverman 1986; Worton 1989) is a density estimator, which we discussed previously as a scaling technique for point pattern data. Recall that a kernel estimator involves placing a kernel of any specified shape and width over each point and summing the values to create a cumulative kernel surface that represents a distance-weighted point density estimate. Because we can specify any bandwidth, the kernel estimator can be used to depict the intensity of point pattern at multiple scales. In the example shown here, a bivariate normal kernel was placed over each vernal pool in a small study area from western Massachusetts, and the cumulative kernel surfaces are illustrated here in three dimensions. The kernel surface on the left results from a bivariate normal kernel with a bandwidth (standard deviation) of 200m. The one on the right has a bandwidth of 400m. Clearly, the smaller the bandwidth, the rougher the surface. The peaks represent regions of high vernal pool density, the troughs represent regions of low vernal pool density. 7.16

2.5. Applications (1) Home range estimation and mapping One of the most common applications of point pattern analysis is in home range estimation and mapping, wherein the data consist of a set of point locations for a single animal collected over the course of a fixed time period. While there are a number of methods for estimating home range size and distribution based on such data, the most popular technique is based on kernel estimation. In this approach, a kernel of specified shape and width is centered on each point location. The cumulative kernel surface represents the spatial utilization distribution for the animal. Slicing the cumulative kernel surface at a particular height allows one to depict the area (or areas) used during a specified percentage of the time. Thus, by slicing the kernel surface quite low, for example, one can depict say the 95% utilization distribution; that is, the area containing 95% of the animal s home range use. Conversely, by slicing the kernel surface quite high, one can depict only core high use areas. In the example shown here, the kernel home range based on several utilization thresholds is overlaid on the minimum convex polygon surrounding all locations for a moose in central Massachusetts. It is clear that the kernel approach does a much better job of estimating and depicting the spatial distribution of home range use than the minimum convex polygon. 7.17

(2) Modeling ecological processes Another way in which kernel point pattern analysis is being applied is in the modeling of ecological processes. Kernels are ideal for modeling point-based ecological processes because the shape and size or width of the kernel can be adjusted to reflect the particular ecological process under consideration. The flexibility in the shape of the kernel allows one to depict nonlinear and even nonparametric ecological processes. In the example shown here, we applied the concept of a dispersal kernel, which describes the scatter of offspring about the parent plant in the form of a probability density function, to a landscape dispersal kernel weighted by percent cover of ponderosa pine in areas adjacent to high-severity patches. Essentially, this involves placing a kernel over every potential seed tree within the burn and surrounding area. We used a Gaussian kernel and varied the smoothing parameter (h) to examine different potential seed dispersal distance functions. Number of seedlings at distances less than 60 m can be high in early post-fire periods, but remotely dispersed individuals may lead to variations at greater distances through time. The Gaussian model approaches zero rapidly with distance, making migration a coherent, stepwise process as compared to fat-tailed models which are expected for higher rates of dispersal. As shown here, an h=150 m had the best fit and shown a strong relationship with ponderosa pine regeneration in terms of both total percent cover and stem density. The resulting kernel map depicts the expected spatial distribution in pine regeneration within the high severity burn patches. 7.18

In this example, Meador et al. (2009) used point pattern analysis to quantify the spatial distribution of ponderosa pine trees in historical stands (pre-european land use) and compare them to the contemporary stand structures (post-european land use) in northern Arizona in an effort to discern the effects of human land use practices on forest structure. Specifically, they surveyed a 2.59 ha stand of ponderosa pine forest characteristic of southwestern ponderosa pine forests in northern Arizona and recorded the location (and size) of all trees existing prior to the first selective harvest in 1894. Historical trees were located based on the presence of old live trees and evidences of historical (pre-1894) trees, based primarily on stumps and logs, which is possible because of the arid conditions and very slow decomposition rates. Based on this survey, they were able to plot the location of every tree (with some confidence) in the pre-1894 stand and every tree in the contemporary stand. 7.19

As shown here, they used Ripley s K distribution to quantify the scale of clumping in the tree distribution. Specifically, they observed a moderate, but significant, clumping of trees at the 0-20 m scale in the pre-harvest (or pre-1894) stand, indicative of trees existing in small clumps and groups of a few to several trees. Post-harvest, the trees were significantly more clumped and at a much coarser scale, or even to some degree at all scales considered. However, in the contemporary stand, the magnitude of clumping decreased dramatically even though it was still significantly clumped at all scales. They interpreted the differences as indicative of contemporary stands having lost the fine-scale clumpy-groupy structure that was characteristic of pre- European land use stands. 7.20

(3) Modeling species-environment relationships Point pattern analysis offers myriad possibilities for modeling species-environment relationships where either the species or the relevant environmental attributes are best represented as point features of the landscape. In the example shown here, we used kernel estimators to depict the spatial distribution of piping plover nests and productivity on Long Island, New York, in relation to several environmental determinants. In this first slide, the distribution of plover nests along a portion of the barrier beach are depicted as point locations. 7.21

In this slide, we have used a Guassian kernel to estimate nest density as a continuous surface. The width of the kernel was selected based on empirical data on territory size. The cumulative kernel surface clearly distinguishes the areas of highest nest density. The following two slides depict the distribution of nest productivity (i.e., number of young fledged per nest)(red) in relation to nest density (blue). Note, the nest productivity distribution is based on weighted point data, where each point location (nest) also contains a weight, in this case, productivity of the nest. In addition, note that high nest density does not always equal high nest productivity why? 7.22

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We also used point-based kernels to depict the distribution of certain environmental variables deemed to be potentially important determinants of plover distribution, abundance and/or productivity. Human activity on beeches is believed to be a major form of disturbance to nesting plovers, and it comes in many forms as shown in this slide. The next two slides depict the distribution of people on the beach recorded during regular aerial surveys and the cumulative kernel surface derived from the point locations. The kernel map only shows the highest values 2 (288-4000 people/km ). 7.24

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This slide shows the plover nest density kernel overlaid on the people on beech kernel and there is some indication of an inverse relationship between nest density and people density (i.e., nest density increases where people density decreases). However, this relationship must be explored statistically before any inferences about the relationship can be made. 7.26

Lastly, we have used the point-based kernel approach to estimate the distribution of several point-based environmental variables. Two others shown here are beach off-road vehicle (ORV) activity (where each ORV is recorded as a separate point location) and gull intensity (where each recorded gull is recorded as a separate point location). Gulls of several species are potential predators of plover chicks. Currently, we are using statistical procedures to examine the relationship between plover nest density and productivity and each of these and several other environmental variables. The key point here is that point pattern analysis underpins most of our analysis into the plover-environment relationship. 7.27

2.6. Pros and Cons Considering the information presented thus far, what are the strengths and limitations of the point pattern model of landscape structure? 7.28

3. Patch Mosaic Model 3.1. Data Characteristics The patch mosaic model represents data in which the system property of interest is represented as a mosaic of discrete patches that can be intuitively defined by the notion of "edge" (a patch is an area with edges adjoining other patches). From an ecological perspective, patches represent relatively discrete areas of relatively homogeneous environmental conditions at a particular scale. The patch boundaries are distinguished by abrupt discontinuities (boundaries) in environmental character states from their surroundings of magnitudes that are relevant to the ecological phenomenon under. The patch mosaic model is the dominant model of landscape structure in use today. Familiar examples include: Map of land cover types Map of ownership parcels 7.29

3.2. Data Structure In the patch mosaic model, the data consists of polygons (vector) or grid cells (raster) classified into discrete classes. There are many methods for deriving a categorical map of patches. Patches may be classified and delineated qualitatively through visual interpretation of the data (e.g., delineating vegetation polygons through interpretation of aerial photographs), as is typically the case with vector maps constructed from digitized lines. Alternatively, with raster grids, information at every location, typically obtained through remote sensing, may be used to classify cells into discrete classes and then to delineate patches by outlining them, and there are a variety of methods for doing this. 7.30

3.3. Pattern Elements In the patch mosaic model three major landscape elements are typically recognized: patches, corridors and matrix, and the extent and configuration of these elements defines the pattern of the landscape. The patch mosaic model is most powerful when meaningful patches can be clearly defined and accurately mapped as discrete patches, and when the variation within a patch is deemed relatively insignificant. The patch mosaic model assumes patches are homogeneous within (or treats them so) and categorically different from one another. For example, breeding habitat for many pond-breeding amphibians can be clearly defined and delineated with relatively little uncertainty, and the variation in habitat quality within ponds is insignificant compared to among-pond differences or pond-upland differences. Similarly, forested woodlots embedded within a contrasting agricultural or urban landscape, fields in a forested landscape, stands of deciduous trees within a coniferous forest, and many other examples can be represented easily and meaningfully in a categorical map. In general, whenever disturbances (natural or anthropogenic) either create discrete patches or leave behind discrete remnant patches, the patch mosaic model is likely to be useful. 7.31

(1) Patch In the patch mosaic model, landscapes are composed of a mosaic of patches (Urban et al. 1987). Landscape ecologists have used a variety of terms to refer to the basic elements or units that make up a patch mosaic, including ecotope, biotope, landscape component, landscape element, landscape unit, landscape cell, geotope, facies, habitat, and site (Forman and Godron 1986). Any of these terms, when defined, are satisfactory according to the preference of the investigator. Like the landscape, patches comprising the landscape are not self-evident; patches must be defined relative to the phenomenon under consideration. For example, from a timber management perspective a patch may correspond to the forest stand. However, the stand may not function as a patch from a particular organism's perspective. From an ecological perspective, patches represent relatively discrete areas (spatial domain) or periods (temporal domain) of relatively homogeneous environmental conditions where the patch boundaries are distinguished by discontinuities in environmental character states from their surroundings of magnitudes that are perceived by or relevant to the organism or ecological phenomenon under consideration (Wiens 1976). From a strictly organism-centered view, patches may be defined as environmental units between which fitness prospects, or "quality", differ; although, in practice, patches may be more appropriately defined by nonrandom distribution of activity or resource utilization among environmental units, as recognized in the concept of "Grain Response". 7.32

Patches are dynamic and occur on a variety of spatial and temporal scales that, from an organismcentered perspective, vary as a function of each animal's perceptions (Wiens 1976 and 1989, Wiens and Milne 1989). A patch at any given scale has an internal structure that is a reflection of patchiness at finer scales, and the mosaic containing that patch has a structure that is determined by patchiness at broader scales (Kotliar and Wiens 1990). Thus, regardless of the basis for defining patches, a landscape does not contain a single patch mosaic, but contains a hierarchy of patch mosaics across a range of scales. 7.33

Patch boundaries are artificially imposed and are in fact meaningful only when referenced to a particular scale (i.e., grain size and extent). For example, even a relatively discrete patch boundary between an aquatic surface (e.g., lake) and terrestrial surface becomes more and more like a continuous gradient as one progresses to a finer and finer resolution. However, most environmental dimensions possess one or more "domains of scale" (Wiens 1989) at which the individual spatial or temporal patches can be treated as functionally homogeneous; at intermediate scales the environmental dimensions appear more as gradients of continuous variation in character states. Thus, as one moves from a finer resolution to coarser resolution, patches may be distinct at some scales (i.e., domains of scale) but not at others. 7.34

KEY POINT: It is not my intent to argue for a particular definition of patch. Rather, I wish to point out the following: (1) that patch must be defined relative to the phenomenon under investigation or management; (2) that, regardless of the phenomenon under consideration (e.g., a species, geomorphological disturbances, etc), patches are dynamic and occur at multiple scales; and (3) that patch boundaries are only meaningful when referenced to a particular scale. It is incumbent upon the investigator or manager to establish the basis for delineating among patches and at a scale appropriate to the phenomenon under consideration. 7.35

(2) Corridor Corridors are linear landscape elements that can be defined on the basis of structure or function. Forman and Godron (1986) define corridors as narrow strips of land which differ from the matrix on either side. Corridors may be isolated strips, but are usually attached to a patch of somewhat similar vegetation. These authors focus on the structural aspects of the linear landscape element and recognize three different types of structural corridors: (1) line corridors, in which the width of the corridor is too narrow to allow for interior environmental conditions to develop; (2) strip corridors, in which the width of the corridor is wide enough to allow for interior conditions to develop; and (3) stream corridors, which are a special category. Alternatively, these authors also classify corridors based on the agent of formation, including: (1) disturbance corridors, in which the corridor is established by a disturbance event, usually anthropogenic (e.g., roads, utility lines, fences); (2) remnant corridors, in which the corridor is the result of disturbance around the corridor, leaving the corridor as an intact remnant of the formally widespread cover type; and (3) environmental corridors, in which the corridor is the result of a strong linear environmental gradient, such as a riparian corridor created by the landwater interface along a stream. 7.36

As a consequence of their form and context, structural corridors may function as habitat, dispersal conduits, barriers, or as a source of abiotic and biotic effects on the surrounding matrix: Habitat Corridor.--Linear landscape element that provides for survivorship, natality, and movement (i.e., habitat), and may provide either temporary or permanent habitat. Habitat corridors passively increase landscape connectivity for the focal organism(s). Facilitated Movement Corridor. Linear landscape element that provides for survivorship and movement, but not necessarily natality, between other habitat patches. Facilitated movement corridors actively increase landscape connectivity for the focal organism(s). Barrier or Filter Corridor. Linear landscape element that prohibits (i.e., barrier) or differentially impedes (i.e., filter) the flow of energy, mineral nutrients, and/or species across (i.e., flows perpendicular to the length of the corridor). Barrier or filter corridors actively decrease matrix connectivity for the focal process. Source of Abiotic and Biotic Effects on the Surrounding Matrix. Linear landscape element that modifies the inputs of energy, mineral nutrients, and/or species to the surrounding matrix and thereby effects the functioning of the surrounding matrix. 7.37

Most of the past attention and debate has focused on facilitated movement corridors. It has been argued that this corridor function can only be demonstrated when the immigration rate to the target patch is increased over what it would be if the linear element was not present (Rosenberg et al. 1997). Unfortunately, as Rosenberg et al. point out, there have been few attempts to experimentally demonstrate this. In addition, just because a corridor can be distinguished on the basis of structure, it does not mean that it assumes any of the above functions. Moreover, the function of the corridor will vary among organisms due to the differences in how organisms perceive and scale the environment. More recently, the attention has shifted to the role of corridors as barriers or impediments to ecological flows. KEY POINT: Corridors are distinguished from patches by their linear nature and can be defined on the basis of either structure or function or both. If a corridor is specified, it is incumbent upon the investigator or manager to define the structure and implied function relative to the phenomena (e.g., species) under consideration. 7.38

(3) Matrix In the patch mosaic model, the matrix is the most extensive and most connected element, and therefore plays the dominant role in the functioning of the landscape. For example, in a large contiguous area of mature forest embedded with numerous small disturbance patches, the mature forest constitutes the matrix because it is greatest in extent, is mostly connected, and exerts a dominant influence on the biota and ecological processes. In most landscapes the matrix is obvious to the investigator or manager. However, in some landscapes, or at a certain point in time, the matrix will not be obvious, and it may not be appropriate to consider any element as the matrix. The designation of a matrix is largely dependent upon the phenomenon under consideration. For example, in the study of geomorphological processes, the geological substrate may serve to define the matrix and patches; whereas, in the study of vertebrate populations, vegetation structure may serve to define the matrix and patches. In addition, what constitutes the matrix is dependent on the scale of investigation or management. For example, at a particular scale, mature forest may be the matrix with disturbance patches embedded within; whereas, at a coarser scale, agricultural land may be the matrix with mature forest patches embedded within. KEY POINT: A matrix element is not inherent to the landscape. It is incumbent upon the investigator or manager to determine whether a matrix element exists and should be designated given the scale and phenomenon under consideration. 7.39

3.4. Perspectives on Categorical Landscapes There are at least two different perspectives on the patch mosaic mode and categorical map patterns that have profoundly influenced the development of landscape ecology and have important implications for the analysis of landscape patterns. (1) Island biogeographic model In the island biogeographic model, the emphasis is on a single patch type; disjunct patches (e.g., habitat fragments) are viewed as analogues of oceanic islands embedded in an inhospitable or ecologically neutral background (matrix). This perspective emerged from the theory of island biogeography (MacArthur and Wilson 1967) and subsequent interest in habitat fragmentation (Saunders et al. 1991). Under this perspective, there is a binary patch structure in which the focal patches (fragments) are embedded in a neutral matrix. Here, the emphasis is on the extent, spatial character, and distribution of the focal patch type without explicitly considering the role of the matrix. Under this perspective, for example, connectivity may be assessed by the spatial aggregation of the focal patch type without consideration of how intervening patches affect the functional connectedness among patches of the focal class. The island biogeography perspective has been the dominant perspective since inception of the theory. The major advantage of the island model is its simplicity. Given a focal patch type, it is quite simple to represent the structure 7.40

of the landscape in terms of focal patches contrasted sharply against a uniform matrix, and it is relatively simple to devise metrics that quantify this structure. Moreover, by considering the matrix as ecologically neutral, it invites ecologists to focus on those patch attributes, such as size and isolation, that have the strongest effect on species persistence at the patch level. The major disadvantage of the strict island model is that it assumes a uniform and neutral matrix, which in most real-world cases is a drastic over-simplification of how organisms interact with landscape patterns. (2) Landscape mosaic model In the landscape mosaic model, landscapes are viewed as spatially complex, heterogeneous assemblages of patch types, which can not be simply categorized into discrete elements such as patches, matrix, and corridors (With 2000). Rather, the landscape is viewed from the perspective of the organism or process of interest. Patches are bounded by patches of other patch types that may be more or less similar to the focal patch type, as opposed to highly contrasting and often hostile habitats, as in the case of the island model. Connectivity, for example, may be assessed by the extent to which movement is facilitated or impeded through different patch types across the landscape. The landscape mosaic perspective derives from landscape ecology (Forman 1995) and has only recently emerged as a viable alternative to the island biogeographic model. The major advantage of the landscape mosaic model is its more realistic representation of how organisms perceive and interact with landscape patterns. Few organisms, for example, exhibit a binary (all or none) response to habitats (patch types), but rather use habitats proportionate to the fitness they confer to the organism. Moreover, movement among suitable habitat patches usually is a function of the character of the intervening habitats. The major disadvantage of the landscape mosaic model is that it requires detailed understanding of how organisms interact with landscape pattern, and this has delayed the development of additional quantitative methods that adopt this perspective. 7.41

3.5. Pattern Metrics Regardless of data format (raster or vector) and method of classifying and delineating patches, the goal of categorical map pattern analysis with such data is to characterize the composition and spatial configuration of the patch mosaic, and a plethora of metrics has been developed for this purpose. We will explore these pattern metrics in detail in a subsequent lecture (and lab), so for now, suffice it to say that there are many different metrics available for quantifying the composition and configuration of the patch mosaic. It is worth noting that in contrast to the emphasis on identifying the scale of pattern with point data and continuous data, so-called scaling techniques for categorical map data are less commonly employed in landscape ecology. This may seem somewhat surprising given the predominant use of categorical data in landscape ecological investigations after all, the predominant patch mosaic model of landscape structure is based on a categorical data format. However, there has been a plethora of landscape metrics developed for quantifying various aspects of categorical map patterns and these have largely taken the place of the more conventional scaling techniques. In addition, in applications involving categorical map patterns, the relevant scale of the mosaic is often defined a priori based on the phenomenon under consideration. In such cases, it is usually assumed that it would be meaningless to determine the so-called characteristic scale of the mosaic after its construction. 7.42

3.6. Applications (1) Landscape ecological assessment Landscape ecological assessment based on the patch mosaic model of landscape structure is increasingly common. Given the inherent complexity of ecological systems and the daunting challenges confronting managers seeking to sustain ecosystems in the face of increasing human pressures, it is not too surprising that managers are increasingly seeking effective ecological indicators. Landscape metrics that quantify the composition and configuration of the patch mosaic are increasingly being used as course-scale ecological indicators of change. In the example shown here, several different pattern metrics were evaluated for their sensitivity to land use change between 1974 and 1999 at Fort Benning, Georgia. In the realm of wetland ecological assessment there has been a similar explosion in the use of landscape pattern metrics to evaluate the condition of wetlands as part of a regional wetlands monitoring and assessment program. While the current methods almost exclusively adopt pattern metrics based on the patch mosaic model of landscape structure, it is important to note that this is only by convention and that metrics derived from other conceptual models of landscape structure apply equally well. 7.43

(2) Landscape disturbance-succession modeling One of the more common applications of the patch mosaic model in land management has been in landscape disturbance-succession modeling (LDSM). In most LDSMs the landscape is represented as patch mosaic, where the patches represent discrete land cover types (e.g., vegetation communities) and disturbance and succession processes operate in various ways to alter the structure of the mosaic over time, such as shown in the animation on this slide. Pattern metrics based on the patch mosaic are used to quantify the structure of the landscape at each point in time during the simulation. Statistical summaries of the resulting trajectory of change can then be used to quantify the range of variation in landscape structure under the particular scenario simulated. This is a popular basis for characterizing the historic range of variability (HRV) in landscapes and in comparing alternative future land management scenarios, two subjects that we will explore in greater depth in subsequent lectures and labs. 7.44

(3) Modeling species-environment relationships Perhaps the most common application of pattern analysis based on the patch mosaic model is in modeling species-environment relationships, and there are myriad examples of such applications. An increasingly common example involves quantifying the landscape structure around sample locations for the purpose of building statistical models to explain and/or predict the distribution, abundance or performance of a species. In the example shown here, we adopted the patch mosaic model to map the distribution of several major cover types deemed potentially relevant to a variety of moth species in the pine barrens of southeastern Massachusetts. Conventional landscape pattern analysis involves quantifying the structure of the entire landscape mosaic and returning a single computed value for each metric. 7.45

However, in this study we were more interested in quantifying the local neighborhood structure around each point location (represented as a cell in the raster grid). To do this, we passed a moving window of specified shape and size across the landscape one cell at time. Within each window, we compute the desired landscape metric and returned the value to the focal (center) cell. By doing this for every cell in the input landscape, the result was a new grid depicting a continuous surface representing the local landscape structure as measured by the particular metric. As shown in the next two slides, we repeated this process for several different landscape metrics at several different scales (i.e., window sizes). These local landscape structure surfaces represented independent variables in a logistic regression, where the dependent variable was presence/absence of a particular moth species. Using statistical procedures, we identified the combination of metrics and scales that best predicted each species presence/absence. By overlaying the predicted distribution of each species, we were able to identify areas of high moth richness, which might serve to inform conservation decisions regarding where to focus land conservation efforts. 7.46

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3.7. Pros and Cons Considering the information presented thus far, what are the strengths and limitations of the patch mosaic model of landscape structure? 7.48

4. Landscape Gradient Model 4.1. Data Characteristics An alternative approach to the conventional patch mosaic model involves representing heterogeneity continuously as a gradient. In this model, heterogeneity does not exist in discrete patches, but rather exists as a continuously varying property of the local environment and landscape. Here, the data can be conceptualized as representing a three-dimensional surface, where the measured value at each geographic location is represented by the height of the surface. For example, instead of representing habitat as discrete patches, it is represented as a suitability or capability index, where the value at each location represents the quality of habitat and can take on continuous values. In practice, habitat suitability or capability is often classified into categories representing, say, high-, moderate-, and low-quality habitat, but this is more often done for convenience to facilitate further analysis with conventional categorical-based procedures or to simplify the presentation of results. Other familiar examples of inherently continuous gradients include: Map of elevation Map of burn severity Map of leaf area index 7.49

4.2. Data Structure Continuous surface pattern data typically are represented using a raster data structure, wherein each cell (or pixel) takes on a continuous value. Alternatively, it may be convenient in some applications to represent continuous surface data using a vector data structure, wherein the surface is represented as a series of contours (lines) as in the familiar topographic map. 7.50

4.3. Pattern Elements Interestingly, there are no pattern elements in the landscape gradient model, making it the most parsimonious of the landscape structure models. 7.51

4.4. Pattern Metrics A wide variety of methods have been developed for quantifying the intensity and scale of pattern in regionalized quantitative variables. Recall that a regionalized variable is one that takes on values based on its spatial location and that the analysis of the spatial dependencies (or autocorrelation the ability to predict values of a variable from known values at other locations, as described by a structure function ) in the measured characteristic is the purview of geostatistics. Recall that a variety of techniques exist for measuring the intensity and scale of this spatial autocorrelation, and while the location of the data points (or quadrats) is known and of interest, it is the values of the measurement taken at each point that are of primary concern. (1) Autocorrelation Structure Functions The most basic and common measures of pattern in regionalized quantitative variables (i.e., landscape gradients) are based on autocorrelation structure functions, including for example Moran s I autocorrelation coefficient and semivariance, which we described previously as scaling techniques for continuous gradient data. Both measures are typically used to describe the magnitude of autocorrelation as a function of distance between locations, as expressed by the correlogram and variogram (or semi-variogram), respectively, and illustrated here for a gradient in the topographic moisture index along a transect at the Coweeta Experiment Station. 7.52

(2) Other Structure Functions There are many other methods for analyzing the intensity and scale of pattern with continuous data, especially when the data is collected along continuous transects or two-dimensional surfaces, which we described previously as scaling techniques for continuous gradient data. Like the autocorrelation structure functions, these techniques are typically used to quantify spatial dependencies in a quantitative variable in relation to scale (distance). Recall that no one technique has been found to be superior for all applications. 7.53

(3) Surface metrology metrics The autocorrelation and other related structure functions described above can provide useful indices to quantitatively compare the intensity and extent of autocorrelation in quantitative variables among landscapes. However, while they can provide information on the distance at which the measured variable becomes statistically independent, and reveal the scales of repeated patterns in the variable, if they exist, they do little to describe other interesting aspects of the surface. For example, the degree of relief, density of troughs or ridges, and steepness of slopes are not measured. Fortunately, a number of gradient-based metrics that summarize these and other interesting properties of continuous surfaces have been developed in the physical sciences for analyzing three-dimensional surface structures (Barbato et al. 1996, Sout et al. 1994, Villarrubia 1997). In the past ten years, researchers involved in microscopy and molecular physics have made tremendous progress in this area, creating the field of surface metrology (Barbato et al. 1996). 7.54

In surface metrology, several families of surface pattern metrics have become widely utilized. One so-called family of metrics quantify intuitive measures of surface amplitude in terms of its overall roughness, skewness and kurtosis, and total and relative amplitude. Another family records attributes of surfaces that combine amplitude and spatial characteristics such as the curvature of local peaks. Together these metrics quantify important aspects of the texture and complexity of a surface. A third family measures certain spatial attributes of the surface associated with the orientation of the dominant texture. A final family of metrics is based on the surface bearing area ratio curve (or Abbott curve). The Abbott curve is computed by inversion of the cumulative height distribution histogram. The curve describes the distribution of mass in the surface across the height profile. A number of indices have been developed from the proportions of this cumulative height-volume curve which describe structural attributes of the surface. Many of the patch-based metrics for analyzing categorical landscapes have analogs in surface metrology. For example, compositional metrics such as patch density, percent of landscape and largest patch index are matched with peak density, surface volume, and maximum peak height. Configuration metrics such as edge density, nearest neighbor index and fractal dimension index are matched with mean slope, mean nearest maximum index and surface fractal dimension. Many of the surface metrology metrics, however, measure attributes that are conceptually quite foreign to conventional landscape pattern analysis. Landscape ecologists have not yet explored the behavior and meaning of these new metrics; it remains for them to demonstrate the utility of these metrics, or develop new surface metrics better suited for landscape ecological questions. 7.55

McGarigal et al. (2009) examined landscapes in Turkey defined using both the landscape gradient and patch mosaic models according to a variety of landscape definition schemes and conducted multivariate statistical analyses to identify the universal, consistent and important components of surface patterns and their relationship to patch-based metrics. They observed four relatively distinct components of landscape structure based on empirical relationships among 17 surface metrics across 18 landscape gradient models: 1. Surface roughness The dominant structural component of the surfaces was actually a combination of two distinct sub-components: (1) the overall variability in surface height and (2) the local variability in slope. The first sub-component refers to the nonspatial (composition) aspect of the vertical height profile; that is, the overall variation in the height of the surface without reference to the horizontal variability in the surface, and is represented by three surface amplitude metrics: average roughness (Sa), root mean square roughness (Sq), and ten-point height (S10z)(Appendix). These metrics are analogous to the patch type diversity measures (e.g., Simpson's diversity index) in the patch mosaic paradigm, whereby greater variation in surface height equates to greater landscape diversity. Importantly, while these metrics reflect overall variability in surface height, they say nothing about the spatial heterogeneity in the surface. 7.56

The second sub-component refers to the spatial (configuration) aspect of surface roughness with respect to local variability in height (or steepness of slope), and includes two surface metrics: surface area ratio (Sdr) and root mean square slope (Sdq)(Appendix). These metrics are analogous to the edge density and contrast metrics (e.g., contrast-weighted edge density, total edge contrast index) in the patch mosaic paradigm, whereby greater local slope variation equates to greater density and contrast of edges. Interestingly, while these surface metrics reflect something akin to edge contrast, they do so without the need to supply edge contrast weights because they are structural metrics. These two metrics appear to have the greatest overall analogy to the patch-based measures of spatial heterogeneity and overall patchiness. A fine-grained patch mosaic (as represented by any number of common patch metrics, such as mean patch size or density) is conceptually equivalent to a rough surface with high local variability. On conceptual and theoretical grounds, these spatial and nonspatial aspects of surface roughness are independent components of landscape structure; however, in the landscape gradients we examined these two aspects were highly correlated empirically. This distinction between conceptually and/or theoretically related metrics and groupings based on their empirical behavior has also been demonstrated for patch metrics. 7.57

2. Shape of the surface height distribution Another important nonspatial (composition) component of the surfaces we examined was the shape of the surface height distribution. This component was comprised of five metrics: skewness (Ssk), kurtosis (Sku), surface bearing index (Sbi), valley fluid retention index (Svi), and core fluid retention index (Sci). All of these metrics measure departure from a Gaussian distribution of surface heights, but emphasize different aspects of departure from normality (Appendix). Ssk and Sku measure the familiar skewness and kurtosis of the surface height distribution, while the surface bearing metrics, Sbi, Sci and Svi, measure different aspects of the surface height distribution in its cumulative form. This component was universally present across landscape models, but the composition of metrics varied somewhat among models reflecting the complexities inherent in measuring non-parametric shape distributions. There were no strong patch mosaic analogs to these surface metrics; however, departure from a Gaussian distribution of surface heights was weakly correlated with, and conceptually most closely related to, patch-based measures of landscape dominance (or its compliment, evenness) such as Simpson's evenness index (SIEI) and largest patch index (LPI). Importantly, these five surface metrics measure the 'shape' of the surface height distribution and are not affected by the surface roughness (as defined above) per se. 7.58

3. Angular texture A third prominent component of the surfaces we examined was the angular orientation (direction) of the surface texture and its magnitude. This component is inherently spatial, since the arrangement of surface peaks and valleys determines whether the surface has a particular orientation or not, and is represented by four spatial metrics: dominant texture direction (Std), texture direction index (Stdi), and two texture aspect ratios (Str20 and Str37). The computational methods behind these metrics are too complex to describe here (but see Appendix), but are based on common geostatistical methods (Fourier spectral analysis and autocorrelation functions) that determine the degree of anisotropy (orientation) in the surface. Not surprisingly given our knowledge of the study landscape, we did not observe sample landscapes with a strong texture orientation. We did observe mild levels of texture orientation in some landscapes, but many were without apparent orientation. Importantly, the measurement of texture direction has no obvious analog in the patch mosaic paradigm; indeed, we observed no pairwise correlation greater than ±0.22 between any of these four surface metrics and any of the 28 patch metrics. 7.59

4. Radial texture The fourth prominent component of the surfaces we examined was the radial texture of the surface and its magnitude. Radial texture refers to repeated patterns of variation in surface height radiating outward in concentric circles from any location. Like angular texture, this component is inherently spatial, since the arrangement of surface peaks and valleys determines whether the surface has any radial texture or not, and is represent by three spatial metrics: dominant radial wavelength (Srw), radial wave index (Srwi), and fractal dimension (Sfd). Again, the computational methods behind these metrics are based on common geostatistical methods. A limitation of these and other metrics based on Fourier spectral analysis and autocorrelation functions is that they are only sensitive to repeated, regular patterns. We observed that in the absence of a prominent radial texture, the dominant radial wavelength (Srw) ends up being equal to the diameter of the sample landscape. As a result, in some of our landscape gradient models we observed too little variation in this metric and were forced to drop it from the final analyses. Despite these limitations, we observed sample landscapes with varying degrees of radial texture based on the other two metrics. In contrast to angular texture, the measurement of radial texture has at least one conceptual analog in the patch mosaic paradigm - mean and variability in nearest neighbor distance. On conceptual grounds, Srw should equate to mean nearest neighbor distance, and Srwi and Sfd should equate to the coefficient of variation in nearest neighbor distance. However, in our study the corresponding pairwise correlations did not exceed ±0.22, nor were there any pairwise correlations greater than ±0.40 between either of these surface metrics and any of the 28 patch metrics. 7.60

4.5. Applications (1) Scaling gradient patterns By far the most common application of surface pattern metrics in landscape ecology has been to identify the characteristic scale or scales of the ecological phenomena or to elucidate the nature of the scaling relationship. One thing that is true of pattern, as we have already seen, is that its expression varies with scale. Thus, a careful evaluation of the relationship between scale and pattern is often considered an important first step in the characterization of pattern. In addition, we noted previously, quantifying the scaling relationship of the ecological phenomenon of interest can provide unique insights into potential pattern-process relationships. For example, multi-scaled patterns (i.e., those with two or more distinctive characteristic scales) revealed through scaling techniques can indicate the existence of multiple agents of pattern formation operating at different scales. Conversely, scaling techniques can be used to assess a priori hypotheses concerning agents of pattern formation; e.g., is the expected scale or scales of pattern expressed in the data? 7.61

In the example shown here, semivariance was used to examine the scale of pattern in the abundance of coastal cutthroat trout in several Pacific Northwest streams. In this case, the regionalized quantitative variable of interest is fish abundance and the spatial lag is stream distance; i.e., the distance between two points on the stream along the flow path. The semivariograms reveal different patterns of variation among streams. For example, the pattern of variation for Glenn Creek in the Coast Range shows a distinct sill with a range of approximately 1100 m. In contrast, the pattern for Miller Creek in the Cascade Range reveals a nested patchy pattern with a gradient; that is, a distinct scale of patchiness at roughly 300-400 m superimposed on a coarse-scale gradient of increasing dissimilarity in fish abundance with increasing stream distance. These patterns may reveal something about the differences between these stream systems in the underlying distribution of habitat (e.g., the distribution of channel unit types). 7.62

In the example shown here (from Meador et al. 2009), continuing with the example shown earlier in the point pattern analysis section, correlograms were used to discern the pattern of spatial autocorrelation in tree diameter (dbh) as a proxy fo establishment age in historical forest stands (pre-european land use) versus contemporary forest stands (post-european land use) in ponderosa pine forests of northern Arizona. In all cases, there was strong autocorrelation out to15-30 m in the pre-1894 and contemporary stands, which the authors interpreted as evidence of the clumpy-groupy nature of tree establishment, whereby trees establish in small groups largely in the openings between the patches of canopy. 7.63

They also estimated tree establishment age for all the historical trees and contemporary trees and fit a semivariogram to the data (as shown) and then used the process of krigging to create an interpolated surface of tree establishment age. In the figure shown here, the colored areas represent contours of tree establishment age, with the immediate post-harvest 1909 residual trees shown as points (with the symbol sized proportionate to dbh). The pattern reveals that trees largely established in cohorts up until the last regeneration event in 1953 and established first in the openings in clumps/groups and then gradually established closer to the existing canopies and lastly underneath the existing canopy. 7.64

(2) Landscape disturbance-succession modeling Another application of landscape gradient model in land management has been in landscape disturbance-succession modeling (LDSM). As noted previously, most LDSMs adopt the patch mosaic model of landscape structure; however, some models utilize a gradient-based approach or a hybrid approach in which some aspects of the landscape are represented as a continuous surface. For example, the LANDIS model represents vegetation as a continuous surface where each cell can take on continuous values for the abundance of each tree species and age cohort. Disturbance and succession processes operate in various ways to alter the distribution of individual species and age cohorts over time, essentially modifying the continuous surface structure in species abundance and age. Simple landscape composition metrics that quantify the total area occupied by each species and age class can then be computed for each timestep of the model and summarized for the simulation as a whole. And while surface metrology metrics have not yet been employed to describe the changing landscape structure in connection with this simulation model, the potential exists to do so. 7.65

(3) Modeling pattern-process relationships There are myriad possibilities for using the landscape gradient model to explore pattern-process relationships, which we will illustrate here with a couple of examples. In the first example, Moody et al. (2007) used the gradient model to examine the relationship between burn severity and runoff after a large wildfire. Extreme floods often follow wildfire in mountainous watersheds. However, a quantitative relation between the runoff response and burn severity at the watershed scale has not been established. Runoff response was measured as the runoff coefficient C, which is equal to the peak discharge per unit drainage area divided by the average maximum 30 min rainfall intensity during each rain storm. The magnitude of the burn severity was expressed as the change in the normalized burn ratio, a continuous surface derived from LandsatTM imagery. A new burn severity variable, hydraulic functional connectivity Ç i was developed and incorporates both the magnitude of the burn severity and the spatial sequence of the burn severity along hillslope flow paths. The runoff response and the burn severity were measured in seven subwatersheds in the upper part of Rendija Canyon burned by the 2000 Cerro Grande Fire near Los Alamos, New Mexico, USA. The runoff coefficient was a linear function of the mean hydraulic functional connectivity of the subwatersheds. Moreover, the variability of the mean hydraulic functional connectivity was related to the variability of the mean runoff coefficient, and this relation provides physical insight into why the runoff response from the same subwatershed can vary for different rainstorms with the same rainfall intensity. 7.66

In the second example, Cushman et al. (2006) used the gradient model to examine the relationship between gene flow and landscape structure in black bears in northern Idaho. They tested several hypotheses about environmental factors affecting gene flow, including the hypothesis that observed gene flow was best explained by landscape resistance represented as a continuous surface based on a factorial combination of elevation, slope, vegetative cover, and roads. Here, landscape resistance derived from multiple environmental attributes was portrayed as a continuous surface using the landscape gradient model of landscape structure. As will be explained in more detail in a subsequent lecture, they used statistical procedures to compare the genetic distance among the individual bears sampled across the landscape to the ecological distance among sample locations based on least cost distances derived from the resistant landscape surface. 7.67

4.6. Pros and Cons Considering the information presented thus far, what are the strengths and limitations of the landscape gradient model of landscape structure? 7.68

5. Graph matrix model 5.1. Data Characteristics An altogether different approach for representing landscape structure that in many ways adopts aspects of all the preceding models is based on graph theory. In the graph-theoretic model, the data consists of a collection of nodes (patches) and linkages (connections) represented as data matrices. In this model, the landscape is simplified into a graph depicting the focal points or patches as the nodes and the linkages between nodes as connections. In the example shown here, the grass openings, representing habitat patches, are the nodes and the shortest pathway between patches are the links. Graph theory is widely used in computer science and engineering, and became popular in food-web theory and landscape ecology in the early 90's. In landscape ecology, graph theory has been described as a tool that bridges the gap between structural measures and functional measures of a landscape, where it has been proven useful in examining connectivity and ecological flows. 7.69

5.2. Data Structure The graph-matrix model is distinctive in representing the landscape structure as a set of data matrices, wherein the spatial heterogeneity is summarized in two non-spatial data matrices: 1. Node data matrix contains the x,y coordinates of the nodes and may also contain additional information about the nodes such as the size and/or quality of the patch. 2. Link data matrix square matrix containing information pertaining to the links between every pair of nodes. There are at least three different sorts of edge matrices possible: a. Distance matrix containing ecological distances between every pair of nodes, based on simple Euclidean distance or functional distance, for example using least cost paths derived from resistance surfaces. In addition, distances can be from nearest patch edge to patch edge or from patch centroid to patch centroid. b. Flux/Dispersal probability matrix containing rates of ecological flow (e.g., dispersal) between every pair of nodes, derived from the distance matrix by applying some function to the distance, from some attribute of the nodes (e.g., patch size, population size), or from empirical data collected in the field. c. Adjacency matrix binary indicator matrix containing 0's for disconnected links and 1's for connected links. The adjacency matrix could be created from either of above matrices using a threshold distance to define connectivity. 7.70

5.3. Pattern Elements Graph matrices contain two basic elements: nodes and links. 1. Nodes nodes (also referred to as vertices) can represent any point or patch features of the landscape relevant to the phenomenon under consideration. It is important to recognize that in the graph nodes are depicted as points, but in reality they may represent patches of widely varying sizes. In most cases, the patch centroid is used to represent the patch location in the graph. 2. Links links (also referred to as edges in the graph theoretic framework) represent the connections between nodes. Again, it is important to recognize that in the graph links are depicted as straight lines between nodes, but in reality they may represent nonlinear pathways between nodes or flux rates between nodes that don t have an explicit spatial representation. Moreover, the links can have directionality; that is, the distance or flux rate between two nodes may not be the same in both directions. For example, if flux rate is influenced by gravitational fields, then the distance between node A downslope and node B upslope may depend dramatically on whether the link as from A to B or the reverse. 7.71

In addition to the two basic elements of a graph (nodes and links), there are a number of other attributes of the graph that warrant brief description because that are often utilized in the pattern metrics: Path (or Walk) = any series of nodes in a graph. Cycle = closed Path of at least 3 nodes. Tree = Path that has no Cycles; note, there can be many different trees in a graph. Spanning Tree = Tree that connects all nodes in a graph. Minimum Spanning Tree = shortest Path of all Spanning Trees in a graph; in other words, the ree that connects all nodes in a graph using the shortest possible total linkage distance. Component (or Cluster) = Nodes that are connected to each other. 7.72

Cut-node/Cut-link = a Node/Link that singularly disconnects a connected graph. 5.4. Pattern Metrics Not surprisingly, there are many ways to measure the differences between graphs. Here, we review some of the more common ones: Number = number of components (disjunct clusters) in a graph. Order = number of nodes in the largest component. Diameter = diameter of the largest component, where diameter is defined as the distance between the two nodes furthest apart and the path taken is always the shortest (also called the Maximum Eccentricity). Expected cluster size = area-weighted mean component (cluster) size, where the component size is defined as the total area contained in the corresponding nodes. Correlation length = area-weighted mean component (cluster) radius of gyration, where the radius of gyration is defined as the mean distance between each point or cell in a cluster and the centroid of the cluster. Note, this measure is conceptually very similar to expected cluster size, but uses the average distance traversed within a cluster as opposed to the absolute area of the cluster. Thus, the unit of measurement is a distance instead of an area. 7.73

Connected graph = logical or binary indicator of whether all nodes are connected to all other nodes or not. Node-connectivity = number of nodes that need to be removed to disconnect a connected graph. Edge-connectivity = number of links that need to be removed to disconnect a connected graph. 7.74

It is important to remember that graph links can be defined on the basis of any measure of ecological distance, univariate or multivariate, and it is this feature that provides great flexibility to the graph matrix model. 7.75

5.5. Applications (1) Examining connectivity thresholds The vast majority of applications have involved examining connectivity thresholds using a mixture of both theoretical and empirical approaches. Let s begin with a brief theoretical example fro Urban and Keitt (2001). One way to explore connectivity thresholds is to sequentially remove links by progressively decreasing the threshold distance for determining whether nodes are connected or not the so called edge thinning approach. In the hypothetical example shown here, the graph is nearly connected at a threshold of 1500 m there are but two components (clusters). As the threshold distance is progressively decreased, the graph is increasingly fragmented into a larger number of smaller components. 7.76

The structural changes in the graph occurring during this edge thinning experiment are shown here. Specifically, the figure shown here depicts the changes in the number of components (number), number of nodes in the largest remaining component (order), and diameter of the largest remaining component (diameter). As edges are progressively loss, moving from the right to the left on the graph, at first there is no change in the metrics. This is because the edgeconnectivity of the graph is greater than one, which means that each node is connected by multiple links to other nodes so that the loss of any one link does not effect the overall connectivity of the graph. However, at some point the diameter of the largest component goes up before eventually going down, which seems counter-intuitive at first. This happens because the direct paths between nodes are lost at first- then stepping stones are lost. Thus, the cluster order remains the same (i.e., no nodes are lost), but the shortest pathway between the two furthest apart nodes (diameter) increases. As links continue to be lost, at some point there is a rather abrupt change in all of the metrics. This threshold-like behavior reflects sudden and dramatic changes in the connectivity of the graph as the connected graph begins to get fragmented into many smaller components. Ecologically, the implications are obvious there may be critical ecological distances (e.g., dispersal distances) that once crossed dramatically reduce connectivity of the landscape. 7.77

An alternative way to explore connectivity thresholds is to sequentially remove nodes the so called node thinning approach. In the hypothetical example shown here, nodes were progressively removed using three different approaches: (1) randomly, (2) the smallest nodes (in terms of area), and (3) the end-node with the smallest area. The right-hand figure shows the relationship between the number of nodes removed under the three removal methods and the graph diameter. There were a couple of important findings from this study. First, the exact shape of the resulting curves was landscape dependent, preventing generally applicable conclusions. Second, in this particular landscape, end-node removal showed an advantage over the other removal methods and exhibited threshold behavior, suggesting that node position in the landscape can be important and that connectivity may change abruptly with the loss of critical nodes. 7.78

Now let s consider an empirical example. In the example shown here, the graph matrix model was used to examine connectivity thresholds in winter habitat of woodland Caribou in Manitoba, Canada (O Brien et al. 2006). Caribou, a threatened species in Canada, require lichen-rich, mature conifer habitat, especially in winter. Past forestry practices have favored white-tailed deer and moose over caribou, though caribou can adapt to some habitat fragmentation as long as enough connected habitat is provided. 7.79

In this study, the graph matrix model was used to depict the distribution of high-quality winter habitat patches and the potential linkages between those patches using data from 20 radiocollared animals. More specifically, nodes were defined as high-quality (lichen-rich jack pine and sparsely treed rock) winter habitat patches >5 ha. It is worth noting that this approach essentially treats all other land cover as non-habitat that is only traversed to get between habitat patches. This may or may not be a reasonable assumption. Links were defined as the least-cost pathways between habitat patches, where cost was based on a resource selection function derived using logistic regression. 7.80

The figure shown here depicts the relationship between expected cluster size (defined earlier), interpreted as a measure of overall landscape connectivity, and the cost distance based on least cost pathways. Several things are evident in this figure. First, habitat patches connect gradually over small scales, with large increases in ECS occurring at cost distances of 800, 1600 and 2250. These correspond to Euclidean distances of approximately 650 m, 1250 m, and 1750 m, respectively. These thresholds represent scales where the graph includes sub-clusters connected by only a few links into one much larger cluster. Second, at a cost distance of roughly 2500 m (approximately 2000 m Euclidean distance), the graph is completely connected (i.e., all habitat patches are connected into a single large cluster). Third, the curve for the late winter habitat use is greater than the curve for all winter and very close to the curve for the maximum cluster size, indicating that late winter animals utilize the largest available habitat clusters. 7.81

The maps shown here depict the graph structure for thresholds corresponding to the important transitions in ECS. Generally, clusters of well-connected high-quality habitat exist in the north, central and southeast regions of the winter home range. For cost distances of up to 800, clusters of high-quality habitat in these areas form gradually where patches are aggregated. Above a cost distance of 800, two large clusters in the central region of the winter home range link together, corresponding to the large increase in ECS at this threshold, while clusters in the north and south remain isolated. Including links up to a cost distance of 1600 joins the northern cluster with the central cluster resulting in another large and rapid increase in ECS. At a cost distance of 2250 the large southern cluster coalesces with the main central cluster. At this threshold the ECS approaches the maximum, but small clusters in the northwest and central portions of the winter home range remain isolated. 7.82

Lastly, O Brien et al. (2006) compared the graph structure for the current winter range with the historical winter range. In both the Owl Lake and Kississing ranges, expected cluster size values increased more rapidly at short cost distances (0 2000 units) in the winter home range than in the historical home range. High-quality habitat patches within the winter home range are spaced more closely together and coalesce into clusters more rapidly than patches in the historical home range. 7.83

(2) Identifying critical linkages A closely related application of graph theory involves not only examining connectivity thresholds but identifying the critical linkages and/or nodes associated with threshold changes in connectivity. In the classic study by Keitt et al. (1997), the authors used the graph matrix model to represent habitat patches of the Mexican spotted owl in Colorado, Utah, Arizona and New Mexico. The links were simply Euclidean distances between patches. 7.84

In the following series of figures, habitat patches are depicted in yellow and the links are in purple. In the first figure, all links < 10 km are highlighted. The graph is highly disconnected. The second figure depicts all links <30 km. The graph is more connected, but still largely comprised of several separate clusters. The next three figures depict the graph structure for threshold distances of 50, 70 and 90 km. By definition, the correlation length of the graph (defined above) increases monotonically with increasing threshold distance. However, the graph shows a distinct threshold in the vicinity of 40 km, where correlation length almost doubles as the threshold distance increases slightly. Interestingly, 40 km corresponds roughly to the dispersal distance of juvenile spotted owls. 7.85

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The figure shown here depicts the graph sensitivity at different threshold distances. Patches are ordered by size. Sensitivity is the change in correlation length when a patch was removed. Not surprisingly, at pretty much all threshold distances the largest patches are the most sensitive. That is because they constitute a disproportionately large area and their removal results in a significant change in correlation length. However, at intermediate threshold distances (40-50 km), several smaller patches showed significant sensitivity, indicating that their position in the landscape makes them particular important as stepping stones between large clusters and that their removal has a disproportionate impact on graph connectivity as measured by correlation length. 7.87

This figure portrays the results of the sensitivity analysis in map form. The left-hand figure depicts the importance index of each patch averaged over all threshold distances between 0-100 km. Not surprisingly, the few largest patches have the highest overall absolute importance; that is, they are essential to the connectivity of the graph. However, the right-hand figure depicts the importance index on a per unit area basis, revealing that a few relatively small patches have a very high importance due largely to their position in the landscape. By connecting two large clusters to the north and south, they serve a critical role as stepping stones at threshold distances of 40-50 km. These particular habitat patches might be the logical focus of conservation efforts aimed at maintaining habitat connectivity for spotted owls. 7.88

5.6. Pros and Cons Considering the information presented thus far, what are the strengths and limitations of the graph matrix model of landscape structure? 7.89