Using fractal analysis to assess how species perceive landscape structure

To develop a species-centered definition of landscapes, I suggest using a fractal analysis of movement patterns to identify the scales at which organisms are interacting with the patch structure of the landscape. Significant differences in the fractal dimensions of movement patterns of two species indicate that the species may be interacting with the patch structure at different scales. Fractal analysis therefore permits comparisons of landscape perceptions of different species within the same environment.I tested the utility of this fractal application by analyzing the movement patterns of three species of acridid grasshoppers (Orthoptera) in a grassland mosaic. The largest species moved up to 6 times faster than the two smaller species, and species exhibited different responses to microlandscape structure within 25-m² plots. Further, the largest species exhibited different responses to microlandscape structure in two pastures subjected to different intensities of cattle grazing. This species thus is able to integrate information on landscape structure at broad spatial scales. Fractal analysis of movement patterns revealed that the two small species had significantly more tortuous patterns than the larger species, which suggests that these species are interacting with patch structure at a finer scale of resolution than the large species. Fractal analysis can be used to identify the perceptive resolution of a species; that is, the spatial grain and extent at which they are able to perceive and respond to heterogeneity. Analysis of movement patterns across a range of spatial scale may reveal shifts in fractal dimension that reflect transitions in how species respond to the patch structure of the landscape at different scales.     

Sensitivity of landscape resistance estimates based on point selection functions to scale and behavioral state: pumas as a case study

Estimating landscape resistance to animal movement is the foundation for connectivity modeling, and resource selection functions based on point data are commonly used to empirically estimate resistance. In this study, we used GPS data points acquired at 5-min intervals from radiocollared pumas in southern California to model context-dependent point selection functions. We used mixed-effects conditional logistic regression models that incorporate a paired used/available design to examine the sensitivity of point selection functions to the scale of available habitat and to the behavioral state of individual animals. We compared parameter estimates, model performance, and resistance estimates across 37 scales of available habitat, from 250 to 10,000 m, and two behavioral states, resource use and movement. Point selection functions and resistance estimates were sensitive to the chosen scale of the analysis. Multiple characteristic scales were found across our predictor variables, indicating that pumas in the study area are responding at different scales to different landscape features and that multi-scale models may be more appropriate. Additionally, point selection functions and resistance estimates were sensitive to behavioral state; specifically, pumas engaged in resource use behavior had an opposite selection response to some land cover types than pumas engaged in movement behavior. We recommend examining a continuum of scales and behavioral states when using point selection functions to estimate resistance.     

Multiscale sources of variation in ecological variables: modeling spatial dispersion,elaborating sampling designs

Detection of structured spatial variation and identification of spatial scales are important aspects of ecological studies. Spatial structures can correspond to physical features of the environment or to intrinsic characteristics of ecological processes and phenomena. Spatial variability has been approached through several techniques such as classical analysis of variance, or the calculation of fractal dimensions, correlograms or variograms. Under certain assumptions, these techniques are all closely related to one another and represent equivalent tools to characterize spatial structures.Our perception of ecological variables and processes depends on the scale at which variables are measured. We propose simple nested sampling designs enabling the detection of a wide range of spatial structures that show the relationships among nested spatial scales. When it is known that the phenomenon under study is structured as a nested series of spatial scales, this provides useful information to estimate suitable sampling intervals, which are essential to establish the relationships between spatial patterns and ecological phenomena. The use of nested sampling designs helps in choosing the most suitable solutions to reduce the amount of random variation resulting from a survey. These designs are obtained by increasing the sampling intensity to detect a wider spectrum of frequencies, or by revisiting the sampling technique to select more representative sampling units.     

Maintaining or restoring connectivity of modified landscapes: evaluating the least-cost path model with multiple sources of ecological information

Habitat connectivity is an important element of functioning landscapes for mobile organisms. Maintenance or creation of movement corridors is one conservation strategy for reducing the negative effects of habitat fragmentation. Numerous spatial models exist to predict the location of movement corridors. Few studies, however, have investigated the effectiveness of these methods for predicting actual movement paths. We used an expert-based model and a resource selection function (RSF) to predict least-cost paths of woodland caribou. Using independent data for model evaluation, we found that the expert-based model was a poor predictor of long-distance animal movements; in comparison, the RSF model was effective at predicting habitat selection by caribou. We used the Path Deviation Index (PDI), cumulative path cost, and sinuosity to quantitatively compare the spatial differences between inferred caribou movement paths and predicted least-cost paths, and quasi-random null models of directional movement. Predicted movement paths were on average straighter than inferred movement paths for collared caribou. The PDI indicated that the least-cost paths were no better at predicting the inferred paths than either of two null models—straight line paths and randomly generated paths. We found statistically significant differences in cumulative cost scores for the main effects of model and path type; however, post-hoc comparisons were non-significant suggesting no difference among inferred, random, and predicted least cost paths. Paths generated from an expert based cost surface were more sinuous than those premised on the RSF model, but neither differed from the inferred path. Although our results are specific to one species, they highlight the importance of model evaluation when planning for habitat connectivity. We recommend that conservation planners adopt similar techniques when validating the effectiveness of movement corridors for other populations and species.     

Spatial pattern corrections and sample sizes for forest density estimates of historical tree surveys

The U.S. General Land Office land surveys document trees present during European settlement. However, use of these surveys for calculating historical forest density and other derived metrics is limited by uncertainty about the performance of plotless density estimators under a range of conditions. Therefore, we tested two plotless density estimators, developed by Morisita and Pollard, for two, three, and four trees per survey point under simulated ranges of tree densities, non-uniform densities, and different tree spatial distributions. Based on these results, we developed estimator corrections and determined number of survey points needed for reliable density estimates. The Morisita estimator was accurate for densities ranging from 5 to 1,000 trees per unit area, non-uniform densities, random and regular spatial distribution, and outperformed the Pollard estimator. Estimators using points with two or three trees did need a simple correction to account for overestimation. Likewise, for clustered distributions, depending on the number of trees per survey point and the amount of clustering, there should be adjustment for a range of under and overestimation. Sample sizes for survey points with three or four trees should be at least 200 survey points, and 1,000 survey points will have density estimates within ±10% tolerance range of actual density. For survey points with two trees, the minimum sample size should be 600 survey points, and 2,000 survey points should be the target value. These results provide guidelines for researchers to improve density estimates of historical forests.     

Circuit theory emphasizes the importance of edge-crossing decisions in dispersal-scale movements of a forest passerine

Measuring landscape connectivity in ways that reflect an animal’s propensity or reluctance to move across a given landscape is key for planning effective conservation strategies. Resistance distance, based on circuit theory, is one such measure relevant for modeling how broad-scale animal movements over long time periods may lead to gene flow across the landscape. Despite the success of circuit theory in landscape genetic studies, its applicability to model finer-scale processes such as the movement patterns of individual animals within their breeding grounds (e.g., while prospecting for territories) has yet to be tested. Here, we applied both circuit models and least-cost models to understand the relationship between landscape connectivity and return time of Ovenbirds (Seiurus aurocapilla) that had been translocated at least 20 km from their home territory near Québec City, Canada. Using an iterative optimization process, we derived resistance values for three cover types (forest, edge, and open) that resulted in resistance distance values that best explained Ovenbird return times. We also identified the cover-type resistance values that yielded length of least-cost path estimates that best explained return times of the translocated birds. The circuit theory and least-cost path methods were equally supported by the data despite being based on different sets of resistance values. The optimal resistance values for calculating resistance distance indicated that for Ovenbirds, traversing a given distance of edge habitat presented a substantially greater resistance than that of open areas. On the other hand, optimized resistances of edge and open were very similar for calculating length of least-cost path. The circuit theory approach suggested that for an Ovenbird moving through fragmented habitat, the number of forest-open transitions (i.e., edge-crossings) that an individual must make is critical to understanding return times after translocation. The least-cost path approach, on the other hand, suggested that the birds strongly avoid all open areas, regardless of size. Circuit theory offers an important new approach for understanding landscapes from the perspective of individuals moving within their breeding range, at finer spatial scales and shorter time scales than have been previously considered.     

Quantifying scale-dependent effects of animal movement with simple percolation models

A simple model of animal movement on random and patterned landscapes was used to explore the problems of extrapolating information across a range of spatial scales. Simulation results indicate that simple relation- ships between pattern and process will produce a variety of scale-dependent effects. These theoretical studies can be used to design experiments for determining the nature of scale-dependent processes and to estimate parameters for extrapolating information across scales.     

Scaling properties in landscape patterns: New Zealand experience

In this paper we present a case study of spatial structure in landscape patterns for the North and South Islands of New Zealand. The aim was to characterise quantitatively landscape heterogeneity and investigate its possible scaling properties. The study examines spatial heterogeneity, in particular patchiness, at a range of spatial scales, to help build understanding on the effects of landscape heterogeneity on water movement in particular, and landscape ecology in general.We used spatial information on various landscape properties (soils, hydrogeology, vegetation, topography) generated from the New Zealand Land Resource Inventory. To analyse this data set we applied various methods of fractal analyses following the hypothesis that patchiness in selected landscape properties demonstrates fractal scaling behaviour at two structural levels: (1) individual patches; and (2) mosaics (sets) of patches.Individual patches revealed scaling behaviour for both patch shape and boundary. We found self-affinity in patch shape with Hurst exponent H from 0.75 to 0.95. We also showed that patch boundaries in most cases were self-similar and in a few cases of large patches were self-affine. The degree of self-affinity was lower for finer patches. Similarly, when patch scale decreases the orientation of patches tends to be uniformly distributed, though patch orientation on average is clearly correlated with broad scale geological structures. These results reflect a tendency to isotropic behaviour of individual patches from broad to finer scales. Mosaics of patches also revealed fractal scaling in the total patch boundaries, patch centers of mass, and in patch area distribution. All these reflect a special organisation in patchiness represented in fractal patch clustering. General relationships which interconnect fractal scaling exponents were derived and tested. These relationships show how scaling properties of individual patches affect those for mosaics of patches and vice-versa. To explain similarity in scaling behaviour in patchiness of different types we suggest that the Self-Organised Criticality concept should be used. Also, potential applications of our results in landscape ecology are discussed, especially in relation to improved neutral landscape models.     

Spatial versus temporal variation in precipitation in a semiarid ecosystem

Spatial and temporal variations in precipitation are central features of semiarid ecosystems, influencing patterns of plant productivity and the distribution of native fauna. Although temporal variation in precipitation has been studied extensively, far less is known about the spatial scale and pattern of precipitation variability in semiarid regions. I used long-term precipitation records to examine spatial variation across the 63 km² Central Plains Experimental Range in northeastern Colorado, and across the 117,000 km² region of shortgrass steppe in eastern Colorado. Relative to temporal variation, spatial variation was low at scales <10 km, increased linearly across scales of 40–120 km, and was nearly equal in magnitude to temporal variation across distances of 120–160 km. Although I hypothesized that most spatial variation would be generated by early-summer convective thunderstorms in June, I found that the magnitude and spatial pattern of variation was similar for precipitation received in June compared to cumulative precipitation received during the full growing season. The degree of spatial autocorrelation in precipitation across all distances that I evaluated was similar for drought, dry, above-average and wet years. Across distances of 10–120 km, spatial variation within a single growing season was approximately two times greater than spatial variation in long-term mean growing-season precipitation, indicating spatial shifting in the locations of patches of high and low precipitation over multiple years. Overall, these findings suggest spatial variation at scales of 10–160 km may have been an important factor influencing vegetation patterns and migratory fauna of the shortgrass steppe, and have implications for livestock producers and future assessments of climate change.     

Least-cost path length versus accumulated-cost as connectivity measures

Least-cost modelling has become a popular method for measuring connectivity. By representing the landscape as a cost-surface, least-cost paths can be calculated that represent the route of maximum efficiency between two locations as a function of the distance travelled and the costs traversed. Both the length and the accumulated-cost of a least-cost path have been used as measures of connectivity between pairs of locations. However, we are concerned that in some situations the length of a least-cost path may provide a misleading measure of connectivity as it only accounts for the distance travelled while ignoring the costs traversed, and results in a measure that may be little better than Euclidean distance. Through simulations using fractal landscapes we demonstrate that least-cost path length is often highly correlated with Euclidean distance. This indicates that least-cost path length provides a poor measure of connectivity in many situations, as it does not capture sufficient information about the ecological costs to movement represented by the cost-surface. We recommend that in most situations the accumulated-cost of a least-cost path provides a more appropriate measure of connectivity between locations as it accounts for both the distance travelled and costs traversed, and that the generation of vector least-cost paths should be reserved for visualisation purposes.     

How key habitat features influence large terrestrial carnivore movements: waterholes and African lions in a semi-arid savanna of north-western Zimbabwe

Within a landscape where prey has an aggregated distribution, predators can take advantage of the spatial autocorrelation of prey density and intensify their search effort in areas of high prey density by using area-restricted search behaviour. In African arid and semi-arid savannas, large herbivores tend to aggregate around scarce water sources. We tested the hypothesis that water sources are a key determinant of habitat selection and movement patterns of large free-ranging predators in such savannas, using the example of the African lion. We used data from 19 GPS radio-collared lions in Hwange National Park, Zimbabwe. Maps of lions’ trajectories showed that waterholes are key loci on the lions’ route-maps. Compositional analyses revealed that lions significantly selected for areas located within 2 km of a waterhole. In addition, analysis of lions’ night paths showed that when lions are close to a waterhole (<2 km), they move at lower speed, cover shorter distances per night (both path length and net displacement) and follow a more tortuous path (higher turning angle, lower straightness index and higher fractal dimension) than when they are further from a waterhole. Hence, our results strongly suggest that lions adopt area-restricted searching in the vicinity of waterholes, and reduce their search effort to minimize the time spent far from a waterhole. They provide an illustration of how key habitat features that determine the dispersion of prey (e.g. waterholes in this study) have an influence on the spatial ecology and movement patterns of terrestrial predators.     

Animal movements and population dynamics in heterogeneous landscapes

Organisms respond to environmental heterogeneity at different scales and in different ways. These differences are consequences of how the movement characteristics of animals—their movement rates, directionality, turning frequencies, and turning angles—interact with patch and boundary features in landscape mosaics. The interactions of movement patterns with landscape features in turn produce spatial patterns in individual space-use, population dynamics and dispersion, gene flow, and the redistribution of nutrients and other materials. We describe several theoretical approaches for modeling the diffusion, foraging behavior, and population dynamics of animals in heterogeneous landscapes, including: (1) scaling relationships derived from percolation theory and fractal geometry, (2) extensions of traditional patch-based metapopulation models, and (3) individual-based, spatially explicit models governed by local rules. We conclude by emphasizing the need to couple theoretical models with empirical studies and the usefulness of ‘microlandscape’ investigations.     

Scales of movement by elk (<Emphasis Type="Italic">Cervus elaphus</Emphasis>) in response to heterogeneity in forage resources and predation risk

Animals may respond to spatial and temporal heterogeneity by altering their movement patterns. The time an animal spends in an area of a given size is termed ȁ8first-passage timeȁ9 and can be used to identify the scales at which different movement processes occur. Using first-passage time and 2-h observations, we identified nested spatial scales representing three movement behaviours for elk (Cervus elaphus) – inactive/resting (moves < 50 m), active/foraging (x̄ = 276.7 m, SD = 56.6), and active/relocating (x̄ = 1628.3 m, SD = 436.6). Our ability to identify inactive behaviour was limited by GPS accuracy. The scale separating relocating and foraging behaviour ranged 550–1650 m across individuals and varied quadratically with the mean patch size of cutover forest in an animal’s home range. We classified path segments into the 3 movement behaviours and related behaviours to local environmental conditions. Elk were likely to be inactive in areas having a low predicted use by wolves (Canis lupus), farther than 50 m from anthropogenic linear clearings, and where microclimatic conditions were cool (high shrub cover and north to east-facing slopes). In contrast, elk were most likely to forage in areas having intermediate levels of herbaceous biomass and low movement costs. Elk were most likely to be relocating when in areas of high wolf use, when close to linear clearings, and in energetically costly situations such as moving upslope. We discuss how elk use of potential foraging habitats may be restricted in this landscape by risks imposed by predators, humans, or both.     

Measuring boundary convexity at multiple spatial scales using a linear “moving window” analysis: an application to coastal river otter habitat selection

Landscape metrics have been used to quantify ecological patterns and to evaluate relationships between animal presence/abundance and habitat at multiple spatial scales. However, many ecological flows occur in linear systems such as streams, or across patch/landscape boundaries (ecotones). Some organisms and flows may depend on the boundary shape, but metrics for defining linear boundary characteristics are scarce. While sinuosity and fractal dimension address some elements of shape, they fail to specify the dominate shape direction (convexity/concavity). We propose a method for measuring boundary convexity and assess its utility, along with sinuosity and fractal dimension, for predicting site selection by coastal river otters. First, we evaluate the characteristics of boundary convexity using a hypothetical boundary. Second, to compare convexity with other linear metrics boundary convexity, sinuosity and fractal dimension were calculated for the coastline of a set of islands in Prince William Sound, AK. Finally, we use logistic regression in an information-theoretic framework to assess site selection of river otters as a function of these linear metrics. Boundary convexity, fractal dimension and sinuosity are relatively uncorrelated at all scales. Otter latrine sites occurred at significantly more convex locations on the coastline than random sites. Using logistic regression and convexity values at the 100 m window-size, 69.5% of the latrine sites were correctly classified. Coastal terrestrial convexity appears to be a promising landscape-scale metric for predicting otter latrine sites. We suggest that boundary convexity may be an important landscape metric for describing species use or ecological flows at ecotones.     

Movement rules for herbivores in spatially heterogeneous environments: responses to small scale pattern

Foraging herbivores respond to the spatial pattern of resources at a variety of scales. At small scales of space and time, existing models capture the essence of the feeding process and successfully predict intake rates. Models that operate over larger scales have not exhibited a similar success, in part because we have a limited understanding of the rules used by animals to make decisions in spatially complex environments, or of the consequences of departing from these rules. To evaluate the rules that large herbivores use when navigating between forages, we examined movements of bighorn sheep foraging on apparent prey (alfalfa plants) in hand-constructed patches of plants. Observations of movements and path lengths were compared to simulations that used a variety of different rules-of-thumb to determine a search path. Rules used in simulations ranged from a random walk with various detection distances, to more complicated rules that solved a variant of the travelling salesman problem. Simulations of a random walk yielded movement lengths that exceeded observations by a factor of 3 for long detection distances, and by 30-fold for short detection distances. Observed move distances were most closely approximated by simulations based on a nearest-neighbor ruleover 75 % of all moves by bighorn sheep were to the closest available plant. Movement rules based on random walks are clearly inappropriate for many herbivores that typically consume visually apparent plants, and we suggest the use of a nearest-neighbor rule for modelling foraging by large herbivores.     

Fractal dimension estimates of a fragmented landscape: sources of variability

Although often seen as a scale-independent measure, we show that the fractal dimension of the forest cover of the Cazaville Region changes with spatial scale. Sources of variability in the estimation of fractal dimensions are multiple. First, the measured phenomenon does not always show the properties of a pure fractal for all scales, but rather exhibits local self-similarity within certain scale ranges. Moreover, some sampling components such as area of sampling unit, the use of a transect in the estimation of the variability of a plane, the location, and the orientation of a transect all affect, to different degrees, the estimation of the fractal dimension. This paper assesses the relative importance of these components in the estimation of the fractal dimension of the spatial distribution of woodlots in a fragmented landscape. Results show that different sources of variability should be considered when comparing fractal dimensions from different studies or regions.     

Beyond the least-cost path: evaluating corridor redundancy using a graph-theoretic approach

The impact of the landscape matrix on patterns of animal movement and population dynamics has been widely recognized by ecologists. However, few tools are available to model the matrix’s influence on the length, relative quality, and redundancy of dispersal routes connecting habitat patches. Many GIS software packages can use land use/land cover maps to identify the route of least resistance between two points—the least-cost path. The limitation of this type of analysis is that only a single path is identified, even though alternative paths with comparable costs might exist. In this paper, we implemented two graph theory methods that extend the least-cost path approach: the Conditional Minimum Transit Cost (CMTC) tool and the Multiple Shortest Paths (MSPs) tool. Both methods enable the visualization of multiple dispersal routes that, together, are assumed to form a corridor. We show that corridors containing alternative dispersal routes emerge when favorable habitat is randomly distributed in space. As clusters of favorable habitat start forming, corridors become less redundant and dispersal bottlenecks become visible. Our approach is illustrated using data from a real landscape in the Brazilian Atlantic forest. We explored the effect of small, localized disturbance on dispersal routes linking conservation units. Simulated habitat destruction caused the appearance of alternative dispersal routes, or caused existing corridors to become narrower. These changes were observed even in the absence of significant differences in the length or cost of least-cost paths. Last, we discuss applications to animal movement studies and conservation initiatives.     

Spatial scale influence on longterm temporal patterns of a semi-arid grassland

Longterm (45 years) temporal data were used to assess the influence of spatial scale on temporal patterns of a semi-arid west Texas grassland. Temporal basal area dynamics of common curlymesquite (Hilaria belangeri (Steud.) Nash) collected from permanent plots within two areas that were released from disturbance (longterm overgrazing and drought), were evaluated at two spatial scales (quadrat, site). Wiens (1989) proposed hypotheses to characterize the influence of scale on variability, predictability, and equilibrium. These hypotheses were tested for this grassland and temporal patterns observed were different for each spatial scale. The large scale (site) was characterized by low variation between units, high variation within units, high potential predictability, and possible movement toward a fluctuating but relatively stable or equilibrial state. At the small scale (quadrat), variation between units was high, predictability low, and there was no indication of movement toward a stable state; chaotic behavior may be expressed at this scale although the length of the temporal record may not be sufficient to evaluate this phenomenon.     

Identifying structural self-similarity in mountainous landscapes

Digital elevation model data were used to partition a mountainous landscape (northwestern Montana, USA) into watershed/hillslope terrain units at several different spatial scales. Fractal analysis of the perimeter to area relationships of the resulting partition polygons identified statistical self-similarity across a range of spatial scales (approximately four orders of magnitude in partition area). The fractal dimension was higher for a relatively complex fluvially-dominated terrain than for a structurally simpler glacially-dominated terrain (1.23 vs. 1.02, respectively). The structural self-similarity exhibited by this landscape has direct implications in scaling up ecosystem process models for landscape to regional simulations.     

Evidence of Hierarchical Patch Dynamics in an East African savanna?

The Hierarchical Patch Dynamics Paradigm provides a conceptual framework for linking pattern, process and scale in ecosystems, but there have been few attempts to test this theory because most ecological studies focus on only one spatial scale, or are limited in their temporal scope. Here I use palaeoecological techniques (analysis of fossil pollen and stable carbon isotopes) to compare vegetation heterogeneity in an east African savanna at three spatial scales, over hundreds of years. The data show that patterns of vegetation change are different at the three spatial scales of observation, and suggest that different ecological processes dominate tree abundance at micro, local and landscape scales. Interactions between plants, disturbance (e.g., by fire and herbivores), climate and soil type may influence tree density at differing spatial and temporal scales. This hierarchical explanation of savanna vegetation dynamics could inform future biodiversity conservation and management in savannas. This revised version was published online in July 2006 with corrections to the Cover Date.