Brad H. McRae

USING CIRCUIT THEORY TO MODEL CONNECTIVITY IN ECOLOGY, EVOLUTION, AND CONSERVATION

Connectivity among populations and habitats is important for a wide range of ecological processes. Understanding, preserving, and restoring connectivity in complex landscapes requires connectivity models and metrics that are reliable, efficient, and process based. We introduce a new class of ecological connectivity models based in electrical circuit theory. Although they have been applied in other disciplines, circuit-theoretic connectivity models are new to ecology. They offer distinct advantages over common analytic connectivity models, including a theoretical basis in random walk theory and an ability to evaluate contributions of multiple dispersal pathways. Resistance, current, and voltage calculated across graphs or raster grids can be related to ecological processes (such as individual movement and gene flow) that occur across large population networks or landscapes. Efficient algorithms can quickly solve networks with millions of nodes, or landscapes with millions of raster cells. Here we review basic circuit theory, discuss relationships between circuit and random walk theories, and describe applications in ecology, evolution, and conservation. We provide examples of how circuit models can be used to predict movement patterns and fates of random walkers in complex landscapes and to identify important habitat patches and movement corridors for conservation planning.

Circuit theory predicts gene flow in plant and animal populations

Maintaining connectivity for broad-scale ecological processes like dispersal and gene flow is essential for conserving endangered species in fragmented landscapes. However, determining which habitats should be set aside to promote connectivity has been difficult because existing models cannot incorporate effects of multiple pathways linking populations. Here, we test an ecological connectivity model that overcomes this obstacle by borrowing from electrical circuit theory. The model vastly improves gene flow predictions because it simultaneously integrates all possible pathways connecting populations. When applied to data from threatened mammal and tree species, the model consistently outperformed conventional gene flow models, revealing that barriers were less important in structuring populations than previously thought. Circuit theory now provides the best-justified method to bridge landscape and genetic data, and holds much promise in ecology, evolution, and conservation planning.

Use of resistance surfaces for landscape genetic studies: considerations for parameterization and analysis

Measures of genetic structure among individuals or populations collected at different spatial locations across a landscape are commonly used as surrogate measures of functional (i.e. demographic or genetic) connectivity. In order to understand how landscape characteristics influence functional connectivity, resistance surfaces are typically created in a raster GIS environment. These resistance surfaces represent hypothesized relationships between landscape features and gene flow, and are based on underlying biological functions such as relative abundance or movement probabilities in different land cover types. The biggest challenge for calculating resistance surfaces is assignment of resistance values to different landscape features. Here, we first identify study objectives that are consistent with the use of resistance surfaces and critically review the various approaches that have been used to parameterize resistance surfaces and select optimal models in landscape genetics. We then discuss the biological assumptions and considerations that influence analyses using resistance surfaces, such as the relationship between gene flow and dispersal, how habitat suitability may influence animal movement, and how resistance surfaces can be translated into estimates of functional landscape connectivity. Finally, we outline novel approaches for creating optimal resistance surfaces using either simulation or computational methods, as well as alternatives to resistance surfaces (e.g. network and buffered paths). These approaches have the potential to improve landscape genetic analyses, but they also create new challenges. We conclude that no single way of using resistance surfaces is appropriate for every situation. We suggest that researchers carefully consider objectives, important biological assumptions and available parameterization and validation techniques when planning landscape genetic studies.

Utility of computer simulations in landscape genetics

Population genetics theory is primarily based on mathematical models in which spatial complexity and temporal variability are largely ignored. In contrast, the field of landscape genetics expressly focuses on how population genetic processes are affected by complex spatial and temporal environmental heterogeneity. It is spatially explicit and relates patterns to processes by combining complex and realistic life histories, behaviours, landscape features and genetic data. Central to landscape genetics is the connection of spatial patterns of genetic variation to the usually highly stochastic space–time processes that create them over both historical and contemporary time periods. The field should benefit from a shift to computer simulation approaches, which enable incorporation of demographic and environmental stochasticity. A key role of simulations is to show how demographic processes such as dispersal or reproduction interact with landscape features to affect probability of site occupancy, population size, and gene flow, which in turn determine spatial genetic structure. Simulations could also be used to compare various statistical methods and determine which have correct type I error or the highest statistical power to correctly identify spatio‐temporal and environmental effects. Simulations may also help in evaluating how specific spatial metrics may be used to project future genetic trends. This article summarizes some of the fundamental aspects of spatial–temporal population genetic processes. It discusses the potential use of simulations to determine how various spatial metrics can be rigorously employed to identify features of interest, including contrasting locus‐specific spatial patterns due to micro‐scale environmental selection.

Habitat barriers limit gene flow and illuminate historical events in a wide-ranging carnivore, the American puma.

We examined the effects of habitat discontinuities on gene flow among puma (Puma concolor) populations across the southwestern USA. Using 16 microsatellite loci, we genotyped 540 pumas sampled throughout the states of Utah, Colorado, Arizona, and New Mexico, where a high degree of habitat heterogeneity provides for a wide range of connective habitat configurations between subpopulations. We investigated genetic structuring using complementary individual‐ and population‐based analyses, the latter employing a novel technique to geographically cluster individuals without introducing investigator bias. The analyses revealed genetic structuring at two distinct scales. First, strikingly strong differentiation between northern and southern regions within the study area suggests little migration between them. Second, within each region, gene flow appears to be strongly limited by distance, particularly in the presence of habitat barriers such as open desert and grasslands. Northern pumas showed both reduced genetic diversity and greater divergence from a hypothetical ancestral population based on Bayesian clustering analyses, possibly reflecting a post‐Pleistocene range expansion. Bayesian clustering results were sensitive to sampling density, which may complicate inference of numbers of populations when using this method. The results presented here build on those of previous studies, and begin to complete a picture of how different habitat types facilitate or impede gene flow among puma populations.

Toward Best Practices for Developing Regional Connectivity Maps

To conserve ecological connectivity (the ability to support animal movement, gene flow, range shifts, and other ecological and evolutionary processes that require large areas), conservation professionals need coarse-grained maps to serve as decision-support tools or vision statements and fine-grained maps to prescribe site-specific interventions. To date, research has focused primarily on fine-grained maps (linkage designs) covering small areas. In contrast, we devised 7 steps to coarsely map dozens to hundreds of linkages over a large area, such as a nation, province, or ecoregion. We provide recommendations on how to perform each step on the basis of our experiences with 6 projects: California Missing Linkages (2001), Arizona Wildlife Linkage Assessment (2006), California Essential Habitat Connectivity (2010), Two Countries, One Forest (northeastern United States and southeastern Canada) (2010), Washington State Connected Landscapes (2010), and the Bhutan Biological Corridor Complex (2010). The 2 most difficult steps are mapping natural landscape blocks (areas whose conservation value derives from the species and ecological processes within them) and determining which pairs of blocks can feasibly be connected in a way that promotes conservation. Decision rules for mapping natural landscape blocks and determining which pairs of blocks to connect must reflect not only technical criteria, but also the values and priorities of stakeholders. We recommend blocks be mapped on the basis of a combination of naturalness, protection status, linear barriers, and habitat quality for selected species. We describe manual and automated procedures to identify currently functioning or restorable linkages. Once pairs of blocks have been identified, linkage polygons can be mapped by least-cost modeling, other approaches from graph theory, or individual-based movement models. The approaches we outline make assumptions explicit, have outputs that can be improved as underlying data are improved, and help implementers focus strictly on ecological connectivity.

Comparison of Bayesian clustering and edge detection methods for inferring boundaries in landscape genetics

Recently, techniques available for identifying clusters of individuals or boundaries between clusters using genetic data from natural populations have expanded rapidly. Consequently, there is a need to evaluate these different techniques. We used spatially-explicit simulation models to compare three spatial Bayesian clustering programs and two edge detection methods. Spatially-structured populations were simulated where a continuous population was subdivided by barriers. We evaluated the ability of each method to correctly identify boundary locations while varying: (i) time after divergence, (ii) strength of isolation by distance, (iii) level of genetic diversity, and (iv) amount of gene flow across barriers. To further evaluate the methods’ effectiveness to detect genetic clusters in natural populations, we used previously published data on North American pumas and a European shrub. Our results show that with simulated and empirical data, the Bayesian spatial clustering algorithms outperformed direct edge detection methods. All methods incorrectly detected boundaries in the presence of strong patterns of isolation by distance. Based on this finding, we support the application of Bayesian spatial clustering algorithms for boundary detection in empirical datasets, with necessary tests for the influence of isolation by distance.

Use of Linkage Mapping and Centrality Analysis Across Habitat Gradients to Conserve Connectivity of Gray Wolf Populations in Western North America

Centrality metrics evaluate paths between all possible pairwise combinations of sites on a landscape to rank the contribution of each site to facilitating ecological flows across the network of sites. Computational advances now allow application of centrality metrics to landscapes represented as continuous gradients of habitat quality. This avoids the binary classification of landscapes into patch and matrix required by patch-based graph analyses of connectivity. It also avoids the focus on delineating paths between individual pairs of core areas characteristic of most corridor- or linkage-mapping methods of connectivity analysis. Conservation of regional habitat connectivity has the potential to facilitate recovery of the gray wolf (Canis lupus), a species currently recolonizing portions of its historic range in the western United States. We applied 3 contrasting linkage-mapping methods (shortest path, current flow, and minimum-cost-maximum-flow) to spatial data representing wolf habitat to analyze connectivity between wolf populations in central Idaho and Yellowstone National Park (Wyoming). We then applied 3 analogous betweenness centrality metrics to analyze connectivity of wolf habitat throughout the northwestern United States and southwestern Canada to determine where it might be possible to facilitate range expansion and interpopulation dispersal. We developed software to facilitate application of centrality metrics. Shortest-path betweenness centrality identified a minimal network of linkages analogous to those identified by least-cost-path corridor mapping. Current flow and minimum-cost-maximum-flow betweenness centrality identified diffuse networks that included alternative linkages, which will allow greater flexibility in planning. Minimum-cost-maximum-flow betweenness centrality, by integrating both land cost and habitat capacity, allows connectivity to be considered within planning processes that seek to maximize species protection at minimum cost. Centrality analysis is relevant to conservation and landscape genetics at a range of spatial extents, but it may be most broadly applicable within single- and multispecies planning efforts to conserve regional habitat connectivity.

Where to Restore Ecological Connectivity? Detecting Barriers and Quantifying Restoration Benefits

Landscape connectivity is crucial for many ecological processes, including dispersal, gene flow, demographic rescue, and movement in response to climate change. As a result, governmental and non-governmental organizations are focusing efforts to map and conserve areas that facilitate movement to maintain population connectivity and promote climate adaptation. In contrast, little focus has been placed on identifying barriers—landscape features which impede movement between ecologically important areas—where restoration could most improve connectivity. Yet knowing where barriers most strongly reduce connectivity can complement traditional analyses aimed at mapping best movement routes. We introduce a novel method to detect important barriers and provide example applications. Our method uses GIS neighborhood analyses in conjunction with effective distance analyses to detect barriers that, if removed, would significantly improve connectivity. Applicable in least-cost, circuit-theoretic, and simulation modeling frameworks, the method detects both complete (impermeable) barriers and those that impede but do not completely block movement. Barrier mapping complements corridor mapping by broadening the range of connectivity conservation alternatives available to practitioners. The method can help practitioners move beyond maintaining currently important areas to restoring and enhancing connectivity through active barrier removal. It can inform decisions on trade-offs between restoration and protection; for example, purchasing an intact corridor may be substantially more costly than restoring a barrier that blocks an alternative corridor. And it extends the concept of centrality to barriers, highlighting areas that most diminish connectivity across broad networks. Identifying which modeled barriers have the greatest impact can also help prioritize error checking of land cover data and collection of field data to improve connectivity maps. Barrier detection provides a different way to view the landscape, broadening thinking about connectivity and fragmentation while increasing conservation options.

Connectivity Planning to Address Climate Change

As the climate changes, human land use may impede species from tracking areas with suitable climates. Maintaining connectivity between areas of different temperatures could allow organisms to move along temperature gradients and allow species to continue to occupy the same temperature space as the climate warms. We used a coarse-filter approach to identify broad corridors for movement between areas where human influence is low while simultaneously routing the corridors along present-day spatial gradients of temperature. We modified a cost-distance algorithm to model these corridors and tested the model with data on current land-use and climate patterns in the Pacific Northwest of the United States. The resulting maps identified a network of patches and corridors across which species may move as climates change. The corridors are likely to be robust to uncertainty in the magnitude and direction of future climate change because they are derived from gradients and land-use patterns. The assumptions we applied in our model simplified the stability of temperature gradients and species responses to climate change and land use, but the model is flexible enough to be tailored to specific regions by incorporating other climate variables or movement costs. When used at appropriate resolutions, our approach may be of value to local, regional, and continental conservation initiatives seeking to promote species movements in a changing climate. Planificación de Conectividad para Atender el Cambio Climático.