Spatial asynchrony and cross-scale climate interactions in populations of a coldwater stream fish.

Climate change affects populations over broad geographic ranges due to spatially autocorrelated abiotic conditions known as the Moran effect. However, populations do not always respond to broad-scale environmental changes synchronously across a landscape. We combined multiple datasets for a retrospective analysis of time-series count data (5-28 annual samples per segment) at 144 stream segments dispersed over nearly 1,000 linear kilometers of range to characterize the population structure and scale of spatial synchrony across the southern native range of a coldwater stream fish (brook trout, Salvelinus fontinalis), which is sensitive to stream temperature and flow variations. Spatial synchrony differed by life stage and geographic region: it was stronger in the juvenile life stage than in the adult life stage and in the northern sub-region than in the southern sub-region. Spatial synchrony of trout populations extended to 100-200 km but was much weaker than that of climate variables such as temperature, precipitation, and stream flow. Early life stage abundance changed over time due to annual variation in summer temperature and winter and spring stream flow conditions. Climate effects on abundance differed between sub-regions and among local populations within sub-regions, indicating multiple cross-scale interactions where climate interacted with local habitat to generate only a modest pattern of population synchrony over space. Overall, our analysis showed higher degrees of response heterogeneity of local populations to climate variation and consequently population asynchrony than previously shown based on analysis of individual, geographically restricted datasets. This response heterogeneity indicates that certain local segments characterized by population asynchrony and resistance to climate variation could represent unique populations of this iconic native coldwater fish that warrant targeted conservation. Advancing the conservation of this species can include actions that identify such priority populations and incorporate them into landscape-level conservation planning. Our approach is applicable to other widespread aquatic species sensitive to climate change.

Composite dyadic models for spatio-temporal data.

Mechanistic statistical models are commonly used to study the flow of biological processes. For example, in landscape genetics, the aim is to infer spatial mechanisms that govern gene flow in populations. Existing statistical approaches in landscape genetics do not account for temporal dependence in the data and may be computationally prohibitive. We infer mechanisms with a Bayesian hierarchical dyadic model that scales well with large data sets and that accounts for spatial and temporal dependence. We construct a fully connected network comprising spatio-temporal data for the dyadic model and use normalized composite likelihoods to account for the dependence structure in space and time. We develop a dyadic model to account for physical mechanisms commonly found in physical-statistical models and apply our methods to ancient human DNA data to infer the mechanisms that affected human movement in Bronze Age Europe.

Latent trajectory models for spatio-temporal dynamics in Alaskan ecosystems.

The Alaskan landscape has undergone substantial changes in recent decades, most notably the expansion of shrubs and trees across the Arctic. We developed a Bayesian hierarchical model to quantify the impact of climate change on the structural transformation of ecosystems using remotely sensed imagery. We used latent trajectory processes to model dynamic state probabilities that evolve annually, from which we derived transition probabilities between ecotypes. Our latent trajectory model accommodates temporal irregularity in survey intervals and uses spatio-temporally heterogeneous climate drivers to infer rates of land cover transitions. We characterized multi-scale spatial correlation induced by plot and subplot arrangements in our study system. We also developed a Pólya-Gamma sampling strategy to improve computation. Our model facilitates inference on the response of ecosystems to shifts in the climate and can be used to predict future land cover transitions under various climate scenarios.

Melding wildlife surveys to improve conservation inference.

Integrated models are a popular tool for analyzing species of conservation concern. Species of conservation concern are often monitored by multiple entities that generate several datasets. Individually, these datasets may be insufficient for guiding management due to low spatio-temporal resolution, biased sampling, or large observational uncertainty. Integrated models provide an approach for assimilating multiple datasets in a coherent framework that can compensate for these deficiencies. While conventional integrated models have been used to assimilate count data with surveys of survival, fecundity, and harvest, they can also assimilate ecological surveys that have differing spatio-temporal regions and observational uncertainties. Motivated by independent aerial and ground surveys of lesser prairie-chicken, we developed an integrated modeling approach that assimilates density estimates derived from surveys with distinct sources of observational error into a joint framework that provides shared inference on spatio-temporal trends. We model these data using a Bayesian Markov melding approach and apply several data augmentation strategies for efficient sampling. In a simulation study, we show that our integrated model improved predictive performance relative to models for analyzing the surveys independently. We use the integrated model to facilitate prediction of lesser prairie-chicken density at unsampled regions and perform a sensitivity analysis to quantify the inferential cost associated with reduced survey effort.

Revealing the extent of sea otter impacts on bivalve prey through multi-trophic monitoring and mechanistic models.

Sea otters are apex predators that can exert considerable influence over the nearshore communities they occupy. Since facing near extinction in the early 1900s, sea otters are making a remarkable recovery in Southeast Alaska, particularly in Glacier Bay, the largest protected tidewater glacier fjord in the world. The expansion of sea otters across Glacier Bay offers both a challenge to monitoring and stewardship and an unprecedented opportunity to study the top-down effect of a novel apex predator across a diverse and productive ecosystem. Our goal was to integrate monitoring data across trophic levels, space, and time to quantify and map the predator-prey interaction between sea otters and butter clams Saxidomus gigantea, one of the dominant large bivalves in Glacier Bay and a favoured prey of sea otters. We developed a spatially-referenced mechanistic differential equation model of butter clam dynamics that combined both environmental drivers of local population growth and estimates of otter abundance from aerial survey data. We embedded this model in a Bayesian statistical framework and fit it to clam survey data from 43 intertidal and subtidal sites across Glacier Bay. Prior to substantial sea otter expansion, we found that butter clam density was structured by an environmental gradient driven by distance from glacier (represented by latitude) and a quadratic effect of current speed. Estimates of sea otter attack rate revealed spatial heterogeneity in sea otter impacts and a negative relationship with local shoreline complexity. Sea otter exploitation of productive butter clam habitat substantially reduced the abundance and altered the distribution of butter clams across Glacier Bay, with potential cascading consequences for nearshore community structure and function. Spatial variation in estimated sea otter predation processes further suggests that community context and local environmental conditions mediate the top-down influence of sea otters on a given prey. Overall, our framework provides high-resolution insights about the interaction among components of this food web and could be applied to a variety of other systems involving invasive species, epidemiology or migration.

Scale-dependent influence of the sagebrush community on genetic connectivity of the sagebrush obligate Gunnison sage-grouse.

Habitat fragmentation and degradation impacts an organism's ability to navigate the landscape, ultimately resulting in decreased gene flow and increased extinction risk. Understanding how landscape composition impacts gene flow (i.e., connectivity) and interacts with scale is essential to conservation decision-making. We used a landscape genetics approach implementing a recently developed statistical model based on the generalized Wishart probability distribution to identify the primary landscape features affecting gene flow and estimate the degree to which each component influences connectivity for Gunnison sage-grouse (Centrocercus minimus). We were interested in two spatial scales: among distinct populations rangewide and among leks (i.e., breeding grounds) within the largest population, Gunnison Basin. Populations and leks are nested within a landscape fragmented by rough terrain and anthropogenic features, although requisite sagebrush habitat is more contiguous within populations. Our best fit models for each scale confirm the importance of sagebrush habitat in connectivity, although the important sagebrush characteristics differ. For Gunnison Basin, taller shrubs and higher quality nesting habitat were the primary drivers of connectivity, while more sagebrush cover and less conifer cover facilitated connectivity rangewide. Our findings support previous assumptions that Gunnison sage-grouse range contraction is largely the result of habitat loss and degradation. Importantly, we report direct estimates of resistance for landscape components that can be used to create resistance surfaces for prioritization of specific locations for conservation or management (i.e., habitat preservation, restoration, or development) or as we demonstrated, can be combined with simulation techniques to predict impacts to connectivity from potential management actions.

Multivariate Bayesian clustering using covariate-informed components with application to boreal vegetation sensitivity.

Climate change is impacting both the distribution and abundance of vegetation, especially in far northern latitudes. The effects of climate change are different for every plant assemblage and vary heterogeneously in both space and time. Small changes in climate could result in large vegetation responses in sensitive assemblages but weak responses in robust assemblages. But, patterns and mechanisms of sensitivity and robustness are not yet well understood, largely due to a lack of long-term measurements of climate and vegetation. Fortunately, observations are sometimes available across a broad spatial extent. We develop a novel statistical model for a multivariate response based on unknown cluster-specific effects and covariances, where cluster labels correspond to sensitivity and robustness. Our approach utilizes a prototype model for cluster membership that offers flexibility while enforcing smoothness in cluster probabilities across sites with similar characteristics. We demonstrate our approach with an application to vegetation abundance in Alaska, USA, in which we leverage the broad spatial extent of the study area as a proxy for unrecorded historical observations. In the context of the application, our approach yields interpretable site-level cluster labels associated with assemblage-level sensitivity and robustness without requiring strong a priori assumptions about the drivers of climate sensitivity.

Rapid proliferation of the parasitic copepod, Salmincola californiensis (Dana), on kokanee salmon, Oncorhynchus nerka (Walbaum), in a large Colorado reservoir.

Ecologically and economically valuable Pacific salmon and trout (Oncorhynchus spp.) are widespread and susceptible to the ectoparasite Salmincola californiensis (Dana). The range of this freshwater copepod has expanded, and in 2015, S. californiensis was observed in Blue Mesa Reservoir, Colorado, USA, an important kokanee salmon (O. nerka, Walbaum) egg source for sustaining fisheries. Few S. californiensis were detected on kokanee salmon in 2016 (<10% prevalence; 2 adult S. californiensis maximum). By 2020, age-3 kokanee salmon had 100% S. californiensis prevalence and mean intensity exceeding 50 adult copepods. Year and kokanee salmon age/maturity (older/mature) were consistently identified as significant predictors of S. californiensis prevalence/intensity. There was evidence that S. californiensis spread rapidly, but their population growth was maximized at the initiation (the first 2-3 years) of the invasion. Gills and heads of kokanee salmon carried the highest S. californiensis loads. S. californiensis population growth appears to be slowing, but S. californiensis expansion occurred concomitant with myriad environmental/biological factors. These factors and inherent variance in S. californiensis count data may have obscured patterns that continued monitoring of parasite-host dynamics, when S. californiensis abundance is more stable, might reveal. The rapid proliferation of S. californiensis indicates that in 5 years a system can go from a light infestation to supporting hosts carrying hundreds of parasites, and concern remains about the sustainability of this kokanee salmon population.

Improving inferences about private land conservation by accounting for incomplete reporting.

Private lands provide key habitat for imperiled species and are core components of function protectected area networks; yet, their incorporation into national and regional conservation planning has been challenging. Identifying locations where private landowners are likely to participate in conservation initiatives can help avoid conflict and clarify trade-offs between ecological benefits and sociopolitical costs. Empirical, spatially explicit assessment of the factors associated with conservation on private land is an emerging tool for identifying future conservation opportunities. However, most data on private land conservation are voluntarily reported and incomplete, which complicates these assessments. We used a novel application of occupancy models to analyze the occurrence of conservation easements on private land. We compared multiple formulations of occupancy models with a logistic regression model to predict the locations of conservation easements based on a spatially explicit social-ecological systems framework. We combined a simulation experiment with a case study of easement data in Idaho and Montana (United States) to illustrate the utility of the occupancy framework for modeling conservation on private land. Occupancy models that explicitly accounted for variation in reporting produced estimates of predictors that were substantially less biased than estimates produced by logistic regression under all simulated conditions. Occupancy models produced estimates for the 6 predictors we evaluated in our case study that were larger in magnitude, but less certain than those produced by logistic regression. These results suggest that occupancy models result in qualitatively different inferences regarding the effects of predictors on conservation easement occurrence than logistic regression and highlight the importance of integrating variable and incomplete reporting of participation in empirical analysis of conservation initiatives. Failure to do so can lead to emphasizing the wrong social, institutional, and environmental factors that enable conservation and underestimating conservation opportunities in landscapes where social norms or institutional constraints inhibit reporting.

La incorporación de las tierras privadas a la planeación de la conservación regional y nacional ha sido un reto a pesar de su importancia como hábitat para especies en peligro y como componentes nucleares de las redes funcionales de áreas protegidas. La identificación de las localidades en donde sea probable que los propietarios privados participen en las iniciativas de conservación puede ayudar a evitar conflictos costosos y a aclarar las compensaciones entre los beneficios ecológicos y los costos sociopolíticos. La evaluación empírica y espacialmente explícita de los factores asociados con la conservación en tierras privadas es una herramienta emergente usada para la identificación de oportunidades de conservación en el futuro. Sin embargo, la mayoría de los datos sobre la conservación en tierras privadas es reportada voluntariamente y está incompleta, lo cual complica realizar estas evaluaciones. Usamos una aplicación novedosa de los modelos de ocupación para analizar la presencia de la mitigación por conservación en tierras privadas. Comparamos diferentes formulaciones de los modelos de ocupación con un modelo de regresión logística para predecir las localidades de la mitigación por conservación con base en un marco de trabajo de un sistema socioecológico espacialmente explícito. Combinamos un experimento de simulación con un estudio de caso sobre datos de mitigación en Idaho y Montana (Estados Unidos) para ilustrar la utilidad del marco de trabajo de ocupación para el modelado de la conservación en tierras privadas. Los modelos de ocupación que consideraron explícitamente la variación en los reportes produjeron estimados de los predictores que estuvieron sustancialmente menos sesgados que los estimados producidos por la regresión logística bajo todas las condiciones simuladas. Los modelos de ocupación produjeron estimaciones para seis predictores que evaluamos en nuestro estudio de caso, los cuales fueron mayores en magnitud pero menos certeros que aquellos producidos por la regresión logística. Estos resultados sugieren que los modelos de ocupación tienen como resultado inferencias cualitativamente diferentes a la regresión logística con respecto a los efectos de los predictores sobre la presencia de mitigación por conservación y resaltan la importancia de la integración de los reportes variables e incompletos sobre la participación dentro del análisis empírico de las iniciativas de conservación. Si se falla en lo anterior se puede terminar enfatizando el factor social, institucional y ambiental equivocado que permite la conservación, además de subestimar las oportunidades de conservación en paisajes en donde las normas sociales o las restricciones institucionales inhiben el reporte de datos.

Iterative near-term ecological forecasting: Needs, opportunities, and challenges.

Two foundational questions about sustainability are "How are ecosystems and the services they provide going to change in the future?" and "How do human decisions affect these trajectories?" Answering these questions requires an ability to forecast ecological processes. Unfortunately, most ecological forecasts focus on centennial-scale climate responses, therefore neither meeting the needs of near-term (daily to decadal) environmental decision-making nor allowing comparison of specific, quantitative predictions to new observational data, one of the strongest tests of scientific theory. Near-term forecasts provide the opportunity to iteratively cycle between performing analyses and updating predictions in light of new evidence. This iterative process of gaining feedback, building experience, and correcting models and methods is critical for improving forecasts. Iterative, near-term forecasting will accelerate ecological research, make it more relevant to society, and inform sustainable decision-making under high uncertainty and adaptive management. Here, we identify the immediate scientific and societal needs, opportunities, and challenges for iterative near-term ecological forecasting. Over the past decade, data volume, variety, and accessibility have greatly increased, but challenges remain in interoperability, latency, and uncertainty quantification. Similarly, ecologists have made considerable advances in applying computational, informatic, and statistical methods, but opportunities exist for improving forecast-specific theory, methods, and cyberinfrastructure. Effective forecasting will also require changes in scientific training, culture, and institutions. The need to start forecasting is now; the time for making ecology more predictive is here, and learning by doing is the fastest route to drive the science forward.

What processes must we understand to forecast regional-scale population dynamics?

An urgent challenge facing biologists is predicting the regional-scale population dynamics of species facing environmental change. Biologists suggest that we must move beyond predictions based on phenomenological models and instead base predictions on underlying processes. For example, population biologists, evolutionary biologists, community ecologists and ecophysiologists all argue that the respective processes they study are essential. Must our models include processes from all of these fields? We argue that answering this critical question is ultimately an empirical exercise requiring a substantial amount of data that have not been integrated for any system to date. To motivate and facilitate the necessary data collection and integration, we first review the potential importance of each mechanism for skilful prediction. We then develop a conceptual framework based on reaction norms, and propose a hierarchical Bayesian statistical framework to integrate processes affecting reaction norms at different scales. The ambitious research programme we advocate is rapidly becoming feasible due to novel collaborations, datasets and analytical tools.

Running on empty: recharge dynamics from animal movement data.

Vital rates such as survival and recruitment have always been important in the study of population and community ecology. At the individual level, physiological processes such as energetics are critical in understanding biomechanics and movement ecology and also scale up to influence food webs and trophic cascades. Although vital rates and population-level characteristics are tied with individual-level animal movement, most statistical models for telemetry data are not equipped to provide inference about these relationships because they lack the explicit, mechanistic connection to physiological dynamics. We present a framework for modelling telemetry data that explicitly includes an aggregated physiological process associated with decision making and movement in heterogeneous environments. Our framework accommodates a wide range of movement and physiological process specifications. We illustrate a specific model formulation in continuous-time to provide direct inference about gains and losses associated with physiological processes based on movement. Our approach can also be extended to accommodate auxiliary data when available. We demonstrate our model to infer mountain lion (Puma concolor; in Colorado, USA) and African buffalo (Syncerus caffer; in Kruger National Park, South Africa) recharge dynamics.

Accounting for phenology in the analysis of animal movement.

The analysis of animal tracking data provides important scientific understanding and discovery in ecology. Observations of animal trajectories using telemetry devices provide researchers with information about the way animals interact with their environment and each other. For many species, specific geographical features in the landscape can have a strong effect on behavior. Such features may correspond to a single point (eg, dens or kill sites), or to higher dimensional subspaces (eg, rivers or lakes). Features may be relatively static in time (eg, coastlines or home-range centers), or may be dynamic (eg, sea ice extent or areas of high-quality forage for herbivores). We introduce a novel model for animal movement that incorporates active selection for dynamic features in a landscape. Our approach is motivated by the study of polar bear (Ursus maritimus) movement. During the sea ice melt season, polar bears spend much of their time on sea ice above shallow, biologically productive water where they hunt seals. The changing distribution and characteristics of sea ice throughout the year mean that the location of valuable habitat is constantly shifting. We develop a model for the movement of polar bears that accounts for the effect of this important landscape feature. We introduce a two-stage procedure for approximate Bayesian inference that allows us to analyze over 300 000 observed locations of 186 polar bears from 2012 to 2016. We use our model to estimate a spatial boundary of interest to wildlife managers that separates two subpopulations of polar bears from the Beaufort and Chukchi seas.

Monitoring dynamic spatio-temporal ecological processes optimally.

Population dynamics vary in space and time. Survey designs that ignore these dynamics may be inefficient and fail to capture essential spatio-temporal variability of a process. Alternatively, dynamic survey designs explicitly incorporate knowledge of ecological processes, the associated uncertainty in those processes, and can be optimized with respect to monitoring objectives. We describe a cohesive framework for monitoring a spreading population that explicitly links animal movement models with survey design and monitoring objectives. We apply the framework to develop an optimal survey design for sea otters in Glacier Bay. Sea otters were first detected in Glacier Bay in 1988 and have since increased in both abundance and distribution; abundance estimates increased from 5 otters to >5,000 otters, and they have spread faster than 2.7 km/yr. By explicitly linking animal movement models and survey design, we are able to reduce uncertainty associated with forecasting occupancy, abundance, and distribution compared to other potential random designs. The framework we describe is general, and we outline steps to applying it to novel systems and taxa.

Estimating abundance of an open population with an N-mixture model using auxiliary data on animal movements.

Accurate assessment of abundance forms a central challenge in population ecology and wildlife management. Many statistical techniques have been developed to estimate population sizes because populations change over time and space and to correct for the bias resulting from animals that are present in a study area but not observed. The mobility of individuals makes it difficult to design sampling procedures that account for movement into and out of areas with fixed jurisdictional boundaries. Aerial surveys are the gold standard used to obtain data of large mobile species in geographic regions with harsh terrain, but these surveys can be prohibitively expensive and dangerous. Estimating abundance with ground-based census methods have practical advantages, but it can be difficult to simultaneously account for temporary emigration and observer error to avoid biased results. Contemporary research in population ecology increasingly relies on telemetry observations of the states and locations of individuals to gain insight on vital rates, animal movements, and population abundance. Analytical models that use observations of movements to improve estimates of abundance have not been developed. Here we build upon existing multi-state mark-recapture methods using a hierarchical N-mixture model with multiple sources of data, including telemetry data on locations of individuals, to improve estimates of population sizes. We used a state-space approach to model animal movements to approximate the number of marked animals present within the study area at any observation period, thereby accounting for a frequently changing number of marked individuals. We illustrate the approach using data on a population of elk (Cervus elaphus nelsoni) in Northern Colorado, USA. We demonstrate substantial improvement compared to existing abundance estimation methods and corroborate our results from the ground based surveys with estimates from aerial surveys during the same seasons. We develop a hierarchical Bayesian N-mixture model using multiple sources of data on abundance, movement and survival to estimate the population size of a mobile species that uses remote conservation areas. The model improves accuracy of inference relative to previous methods for estimating abundance of open populations.

When mechanism matters: Bayesian forecasting using models of ecological diffusion.

Ecological diffusion is a theory that can be used to understand and forecast spatio-temporal processes such as dispersal, invasion, and the spread of disease. Hierarchical Bayesian modelling provides a framework to make statistical inference and probabilistic forecasts, using mechanistic ecological models. To illustrate, we show how hierarchical Bayesian models of ecological diffusion can be implemented for large data sets that are distributed densely across space and time. The hierarchical Bayesian approach is used to understand and forecast the growth and geographic spread in the prevalence of chronic wasting disease in white-tailed deer (Odocoileus virginianus). We compare statistical inference and forecasts from our hierarchical Bayesian model to phenomenological regression-based methods that are commonly used to analyse spatial occurrence data. The mechanistic statistical model based on ecological diffusion led to important ecological insights, obviated a commonly ignored type of collinearity, and was the most accurate method for forecasting.

Inferring infection hazard in wildlife populations by linking data across individual and population scales.

Our ability to infer unobservable disease-dynamic processes such as force of infection (infection hazard for susceptible hosts) has transformed our understanding of disease transmission mechanisms and capacity to predict disease dynamics. Conventional methods for inferring FOI estimate a time-averaged value and are based on population-level processes. Because many pathogens exhibit epidemic cycling and FOI is the result of processes acting across the scales of individuals and populations, a flexible framework that extends to epidemic dynamics and links within-host processes to FOI is needed. Specifically, within-host antibody kinetics in wildlife hosts can be short-lived and produce patterns that are repeatable across individuals, suggesting individual-level antibody concentrations could be used to infer time since infection and hence FOI. Using simulations and case studies (influenza A in lesser snow geese and Yersinia pestis in coyotes), we argue that with careful experimental and surveillance design, the population-level FOI signal can be recovered from individual-level antibody kinetics, despite substantial individual-level variation. In addition to improving inference, the cross-scale quantitative antibody approach we describe can reveal insights into drivers of individual-based variation in disease response, and the role of poorly understood processes such as secondary infections, in population-level dynamics of disease.

Leveraging constraints and biotelemetry data to pinpoint repetitively used spatial features.

Satellite telemetry devices collect valuable information concerning the sites visited by animals, including the location of central places like dens, nests, rookeries, or haul-outs. Existing methods for estimating the location of central places from telemetry data require user-specified thresholds and ignore common nuances like measurement error. We present a fully model-based approach for locating central places from telemetry data that accounts for multiple sources of uncertainty and uses all of the available locational data. Our general framework consists of an observation model to account for large telemetry measurement error and animal movement, and a highly flexible mixture model specified using a Dirichlet process to identify the location of central places. We also quantify temporal patterns in central place use by incorporating ancillary behavioral data into the model; however, our framework is also suitable when no such behavioral data exist. We apply the model to a simulated data set as proof of concept. We then illustrate our framework by analyzing an Argos satellite telemetry data set on harbor seals (Phoca vitulina) in the Gulf of Alaska, a species that exhibits fidelity to terrestrial haul-out sites.

An integrated data model to estimate spatiotemporal occupancy, abundance, and colonization dynamics.

Ecological invasions and colonizations occur dynamically through space and time. Estimating the distribution and abundance of colonizing species is critical for efficient management or conservation. We describe a statistical framework for simultaneously estimating spatiotemporal occupancy and abundance dynamics of a colonizing species. Our method accounts for several issues that are common when modeling spatiotemporal ecological data including multiple levels of detection probability, multiple data sources, and computational limitations that occur when making fine-scale inference over a large spatiotemporal domain. We apply the model to estimate the colonization dynamics of sea otters (Enhydra lutris) in Glacier Bay, in southeastern Alaska.

The basis function approach for modeling autocorrelation in ecological data.

Analyzing ecological data often requires modeling the autocorrelation created by spatial and temporal processes. Many seemingly disparate statistical methods used to account for autocorrelation can be expressed as regression models that include basis functions. Basis functions also enable ecologists to modify a wide range of existing ecological models in order to account for autocorrelation, which can improve inference and predictive accuracy. Furthermore, understanding the properties of basis functions is essential for evaluating the fit of spatial or time-series models, detecting a hidden form of collinearity, and analyzing large data sets. We present important concepts and properties related to basis functions and illustrate several tools and techniques ecologists can use when modeling autocorrelation in ecological data.