Ecologists are concerned with the relationships between species composition and environmental factors, and with spatial structure within those relationships. A dissimilaritybased framework incorporating space explicitly is an extremely flexible tool for answering these questions. The R package ecodist brings together methods for working with dissimilarities, including some not available in other R packages. We present some of the features of ecodist, particularly simple and partial Mantel tests, and make recommendations for their effective use. Although the partial Mantel test is often used to account for the effects of space, the assumption of linearity greatly reduces its effectiveness for complex spatial patterns. We introduce a modification of the Mantel correlogram designed to overcome this restriction and allow consideration of complex nonlinear structures. This extension of the method allows the use of partial multivariate correlograms and tests of relationship between variables at different spatial scales. Some of the possibilities are demonstrated using both artificial data and data from an ongoing study of plant community composition in grazinglands of the northeastern United States.

Ecologists are familiar with two data structures commonly used to represent landscapes. Vector‐based maps delineate land cover types as polygons, while raster lattices represent the landscape as a grid. Here we adopt a third lattice data structure, the graph. A graph represents a landscape as a set of nodes (e.g., habitat patches) connected to some degree by edges that join pairs of nodes functionally (e.g., via dispersal). Graph theory is well developed in other fields, including geography (transportation networks, routing applications, siting problems) and computer science (circuitry and network optimization). We present an overview of basic elements of graph theory as it might be applied to issues of connectivity in heterogeneous landscapes, focusing especially on applications of metapopulation theory in conservation biology. We develop a general set of analyses using a hypothetical landscape mosaic of habitat patches in a nonhabitat matrix. Our results suggest that a simple graph construct, the minimum spanning tree, can serve as a powerful guide to decisions about the relative importance of individual patches to overall landscape connectivity. We then apply this approach to an actual conservation scenario involving the threatened Mexican Spotted Owl (Strix occidentalis lucida). Simulations with an incidence‐function metapopulation model suggest that population persistence can be maintained despite substantial losses of habitat area, so long as the minimum spanning tree is protected. We believe that graph theory has considerable promise for applications concerned with connectivity and ecological flows in general. Because the theory is already well developed in other disciplines, it might be brought to bear immediately on pressing ecological applications in conservation biology and landscape ecology.

Identifying adaptive loci can provide insight into the mechanisms underlying local adaptation. Genotype-environment association (GEA) methods, which identify these loci based on correlations between genetic and environmental data, are particularly promising. Univariate methods have dominated GEA, despite the high dimensional nature of genotype and environment. Multivariate methods, which analyse many loci simultaneously, may be better suited to these data as they consider how sets of markers covary in response to environment. These methods may also be more effective at detecting adaptive processes that result in weak, multilocus signatures. Here, we evaluate four multivariate methods and five univariate and differentiation-based approaches, using published simulations of multilocus selection. We found that Random Forest performed poorly for GEA. Univariate GEAs performed better, but had low detection rates for loci under weak selection. Constrained ordinations, particularly redundancy analysis (RDA), showed a superior combination of low false-positive and high true-positive rates across all levels of selection. These results were robust across the demographic histories, sampling designs, sample sizes and weak population structure tested here. The value of combining detections from different methods was variable and depended on the study goals and knowledge of the drivers of selection. Re-analysis of genomic data from grey wolves highlighted the unique, covarying sets of adaptive loci that could be identified using RDA. Although additional testing is needed, this study indicates that RDA is an effective means of detecting adaptation, including signatures of weak, multilocus selection, providing a powerful tool for investigating the genetic basis of local adaptation.

Graph theory is a body of mathematics dealing with problems of connectivity, flow, and routing in networks ranging from social groups to computer networks. Recently, network applications have erupted in many fields, and graph models are now being applied in landscape ecology and conservation biology, particularly for applications couched in metapopulation theory. In these applications, graph nodes represent habitat patches or local populations and links indicate functional connections among populations (i.e. via dispersal). Graphs are models of more complicated real systems, and so it is appropriate to review these applications from the perspective of modelling in general. Here we review recent applications of network theory to habitat patches in landscape mosaics. We consider (1) the conceptual model underlying these applications; (2) formalization and implementation of the graph model; (3) model parameterization; (4) model testing, insights, and predictions available through graph analyses; and (5) potential implications for conservation biology and related applications. In general, and for a variety of ecological systems, we find the graph model a remarkably robust framework for applications concerned with habitat connectivity. We close with suggestions for further work on the parameterization and validation of graph models, and point to some promising analytic insights. Graphs are models of landscapes -that is, simplifications of a more complicated reality -and so it is appropriate to consider the application of these models in the same way we might evaluate other models used in ecology. This invites a series of very pragmatic questions: What is the underlying conceptual model of the system? How might we formalize and implement (codify) this conceptual model? How will the model be parameterized? Which parameters are most sensitive, most uncertain? What insights might be garnered from a formal analysis of the model? Can the model be extended to applications beyond those used to build it initially, and how might these extensions be validated with independent data? Importantly, can graph models provide predictions about landscapes that are not available from other models we already use?Here we review recent applications of graph theory to habitat mosaics, focusing on applications in landscape

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