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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Sea-level rise (SLR) will greatly alter littoral ecosystems, causing habitat change and loss for coastal species. Habitat loss is widely used as a measurement of the risk of extinction, but because many coastal species are migratory, the impact of habitat loss will depend not only on its extent, but also on where it occurs. Here, we develop a novel graph-theoretic approach to measure the vulnerability of a migratory network to the impact of habitat loss from SLR based on population flow through the network. We show that reductions in population flow far exceed the proportion of habitat lost for 10 long-distance migrant shorebirds using the East Asian-Australasian Flyway. We estimate that SLR will inundate 23-40% of intertidal habitat area along their migration routes, but cause a reduction in population flow of up to 72 per cent across the taxa. This magnifying effect was particularly strong for taxa whose migration routes contain bottlenecks-sites through which a large fraction of the population travels. We develop the bottleneck index, a new network metric that positively correlates with the predicted impacts of habitat loss on overall population flow. Our results indicate that migratory species are at greater risk than previously realized.

Connectivity is a vital component of metapopulation and landscape ecology, influencing fundamental processes such as population dynamics, evolution, and community responses to climate change. Here, we review ongoing developments in connectivity science, providing perspectives on recent advances in identifying, quantifying, modelling and analysing connectivity, and highlight new applications for conservation. We also address ongoing challenges for connectivity research, explore opportunities for addressing them and highlight potential linkages with other fields of research. Continued development of connectivity science will provide insights into key aspects of ecology and the evolution of species, and will also contribute significantly towards achieving more effective conservation outcomes.

Connectivity among marine populations is critical for persistence of metapopulations, coping with climate change, and determining the geographic distribution of species. The influence of pelagic larval duration (PLD) on connectivity has been studied extensively, but relatively little is known about the influence of other biological parameters, such as the survival and behavior of larvae, and the fecundity of adults, on population connectivity. Furthermore, the interaction between the seascape (habitat structure and currents) and these biological parameters is unclear. We explore these interactions using a biophysical model of larval dispersal across the Indo-Pacific. We describe an approach that quantifies geographic patterns of connectivity from demographically relevant to evolutionarily significant levels across a range of species. We predict that at least 95% of larval settlement occurs within 155 km of the source population and within 13 days irrespective of the species' life history, yet long-distant connections remain likely. Self-recruitment is primarily driven by the local oceanography, larval mortality, and the larval precompetency period, whereas broad-scale connectivity is strongly influenced by reproductive output (abundance and fecundity of adults) and the length of PLD. The networks we have created are geographically explicit models of marine connectivity that define dispersal corridors, barriers, and the emergent structure of marine populations. These models provide hypotheses for empirical testing.

Can genetic adaptation in reef-building corals keep pace with the current rate of sea surface warming? Here we combine population genomics, biophysical modeling, and evolutionary simulations to predict future adaptation of the common coral Acropora millepora on the Great Barrier Reef (GBR). Genomics-derived migration rates were high (0.1–1% of immigrants per generation across half the latitudinal range of the GBR) and closely matched the biophysical model of larval dispersal. Both genetic and biophysical models indicated the prevalence of southward migration along the GBR that would facilitate the spread of heat-tolerant alleles to higher latitudes as the climate warms. We developed an individual-based metapopulation model of polygenic adaptation and parameterized it with population sizes and migration rates derived from the genomic analysis. We find that high migration rates do not disrupt local thermal adaptation, and that the resulting standing genetic variation should be sufficient to fuel rapid region-wide adaptation of A. millepora populations to gradual warming over the next 20–50 coral generations (100–250 years). Further adaptation based on novel mutations might also be possible, but this depends on the currently unknown genetic parameters underlying coral thermal tolerance and the rate of warming realized. Despite this capacity for adaptation, our model predicts that coral populations would become increasingly sensitive to random thermal fluctuations such as ENSO cycles or heat waves, which corresponds well with the recent increase in frequency of catastrophic coral bleaching events.

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Dispersal of planktonic larvae can create connections between geographically separated adult populations of benthic marine animals. How geographic context and life history traits affect these connections is largely unresolved. We use data from genetic studies (species level FST) of benthic teleost fishes combined with linear models to evaluate the importance of transitions between biogeographic regions, geographic distance, egg type (benthic or pelagic eggs), pelagic larval duration (PLD), and type of genetic marker as factors affecting differentiation within species. We find that transitions between biogeographic regions and egg type are significant and consistent contributors to population genetic structure, whereas PLD does not significantly explain population structure. Total study distance frequently contributes to significant interaction terms, particularly in association with genetic markers, whereby FST increases with study distance for studies employing mtDNA sequences, but allozyme and microsatellite studies show no increase in FST with study distance. These results highlight the importance of spatial context (biogeography and geographic distance) in affecting genetic differentiation and imply that there are inherent differences in dispersal ability associated with egg type. We also find that the geographic distance over which the maximum pairwise FST between populations occurs (relative to total study distance) is highly variable and can be observed at any scale. This result is consistent with stochastic processes inflating genetic differentiation and/or insufficient consideration of geographic and biological factors relevant to connectivity.