A self-similarity model for the occupancy frequency distribution

The shapes of interspecific range-size distributions at scales finer than the geographic range are highly variable. However, no numerical model has been developed as a basis for understanding this variation. Using self-similarity conditions, we present an occupancy probability transition (OPT) model to investigate the effect of sampling scale (i.e. sample grain) and species saturation (strongly positively correlated with the fractal dimension) on the shape of occupancy frequency distributions (fine scale expression of range-size distributions). In accordance with empirical observations, the model showed that core-modes are likely to be rare in occupancy frequency distributions. The modal occupancy shifted from core to satellite with an increase in sample grain (from coarse scale to fine scale) at a linear rate after log-transformation of occupancy. Saturation coefficients above a particular threshold generated multimodality. Bimodal distributions arose from a combination of different occupancy probability distributions (OPDs), with species-specific saturation coefficients generating occupancy frequency distributions of the shape commonly observed empirically, i.e. bimodal with a dominant satellite mode. This is a consequence of the statistical properties of the OPD, and is also largely insensitive to species richness. The OPT model thus provides a null model for the shape of occupancy frequency distributions. Furthermore, it demonstrates that the sample grain of a study, sampling adequacy (based on a linear sampling assumption) and the distribution of species saturation coefficients in a community are together largely able to explain the patterns observed in empirical occupancy frequency distributions.     

Hierarchical multi-scale occupancy estimation for monitoring wildlife populations

Occupancy estimation is an effective analytic framework, but requires repeated surveys of a sample unit to estimate the probability of detection. Detection rates can be estimated from spatially replicated rather than temporally replicated surveys, but this may violate the closure assumption and result in biased estimates of occupancy. We present a new application of a multi-scale occupancy model that permits the simultaneous use of presence–absence data collected at 2 spatial scales and uses a removal design to estimate the probability of detection. Occupancy at the small scale corresponds to local territory occupancy, whereas occupancy at the large scale corresponds to regional occupancy of the sample units. Small-scale occupancy also corresponds to a spatial availability or coverage parameter where a species may be unavailable for sampling at a fraction of the survey stations. We applied the multi-scale occupancy model to a hierarchical sample design for 2 bird species in the Black Hills National Forest: brown creeper (Certhia americana) and lark sparrow (Chondestes grammacus). Our application of the multi-scale occupancy model is particularly well suited for hierarchical sample designs, such as spatially replicated survey stations within sample units that are typical of avian monitoring programs. The model appropriately accounts for the non-independence of the spatially replicated survey stations, addresses the closure assumption for the spatially replicated survey stations, and is useful for decomposing the observation process into detection and availability parameters. This analytic approach is likely to be useful for monitoring at local and regional scales, modeling multi-scale habitat relationships, and estimating population state variables for rare species of conservation concern. © 2011 The Wildlife Society.     

A consistent occupancy–climate relationship across birds and mammals of the Americas

At broad spatial scales, species richness is strongly related to climate. Yet, few ecological studies attempt to identify regularities in the individual species distributions that make up this pattern. Models used to describe species distributions typically model very complex responses to climate. Here, we test whether the variability in the distributions of birds and mammals of the Americas relates to mean annual temperature and precipitation in a simple, consistent way. Specifically, we test if simple mathematical models can predict, as a first approximation, the geographical variation in individual species’ probability of occupancy for 3277 non‐migratory bird and 1659 mammal species. We find a Gaussian model, where the probability of occupancy of a 10⁴ km² quadrat decreases symmetrically and gradually around a species ‘optimal’ temperature and precipitation, was generally the best model, explaining an average of 35% of the deviance in probability of occupancy. The inclusion of additional terms had very small and idiosyncratic effects across species. The Gaussian occupancy–climate relationship appears general among species and taxa and explains nearly as much deviance as complex models including many more parameters. Therefore, we propose that hypotheses aiming to explain the broad‐scale distribution of species or species richness must also predict generally Gaussian occupancy–climate relationships. Synthesis Science aims to identify regularities in a complex natural world. General patterns should be identified before one searches for potential mechanisms and contingencies. However, species geographic distributions are often modelled as complex (sometimes black box), species‐specific, functions of their environment. We asked whether a simple model could account for as much of the geographic variation in a species' probability of occupancy, and be widely applicable across thousands of species. As a first approximation, we found that a simple Gaussian occupancy‐climate relationship is very common in Nature, whether it be causal or not.     

Using species distribution and occupancy modeling to guide survey efforts and assess species status

Habitat loss and fragmentation continue to be major issues affecting the persistence and conservation of species, but identification of critical habitat remains a challenge. Species distribution modeling and occupancy modeling are both approaches that have been used to predict species distributions and can identify critical habitat characteristics associated with species occurrence. Additionally, occupancy sampling can provide measures of detectability, increasing the confidence that a species is truly absent when not detected. While increasingly popular, these methods are infrequently used in synergy, and rarely at fine spatial scales. We provide a case study of using distribution and occupancy modeling in unison to direct survey efforts, provide estimates of species presence/absence, and to identify local and landscape features important for species occurrence. The focal species for our study was Ambystoma jeffersonianum, a threatened salamander in the state of Illinois, U.S.A. We found that fine-scale distribution models accurately discriminated occupied from unoccupied breeding ponds (78–91% accuracy), and surveys could be effectively guided using a well-fit model. We achieved a high detection rate (0.774) through occupancy sampling, and determined that A. jeffersonianum never used ponds inhabited by fish, and the probability of a pond being used for breeding increased as canopy cover increased. When faced with limited resources, combining fine-scale distribution modeling with a robust occupancy sampling design can expedite survey efforts, confidently designate species occupancy status, prioritise habitat for future surveys and/or restoration, and identify critical habitat features. This approach is broadly applicable to other taxa that have specific habitat requirements.     

Spatial scaling of mountain pine beetle infestations

1. The relationship between occupancy and spatial contagion during the spread of eruptive and invasive species demands greater study, as it could lead to improved prediction of ecosystem damage. 2. We applied a recently developed model that links occupancy and its fractal dimension to model the spatial distribution of mountain pine beetle infestations in British Columbia, Canada. We showed that the distribution of infestation was scale-invariant in at least 24 out of 37 years (mostly in epidemic years), and presented some degree of scale-invariance in the rest. There was a general logarithmic relationship between fractal dimension and infestation occupancy. Based on the scale-invariance assumption, we further assessed the interrelationships for several landscape metrics, such as correlation length, maximum cluster size, total edge length and total number of clusters. 3. The scale-invariance assumption allows fitting the above metrics, and provides a framework to establish the scaling relationship between occupancy and spatial contagion. 4. We concluded that scale-invariance dominates the spread of mountain pine beetle. In this context, spatial aggregation can be predicted from occupancy, hence occupancy is the only variable one needs to know in order to predict the spatial distributions of populations. This supports the hypothesis that fractal dispersal kernels may be abundant among outbreaks of pests and invasive species.     

Occupancy frequency distributions: patterns,artefacts and mechanisms

Numerous hypotheses have been proposed to explain the shape of occupancy frequency distributions (distributions of the numbers of species occupying different numbers of areas). Artefactual effects include sampling characteristics, whereas biological mechanisms include organismal, niche-based and meta-population models. To date, there has been little testing of these models. In addition, although empirically derived occupancy distributions encompass an array of taxa and spatial scales, comparisons between them are often not possible because of differences in sampling protocol and method of construction. In this paper, the effects of sampling protocol (grain, sample number, extent, sampling coverage and intensity) on the shape of occupancy distributions are examined, and approaches for minimising artefactual effects recommended. Evidence for proposed biological determinants of the shape of occupancy distributions is then examined. Good support exists for some mechanisms (habitat and environmental heterogeneity), little for others (dispersal ability), while some hypotheses remain untested (landscape productivity, position in geographic range, range size frequency distributions), or are unlikely to be useful explanations for the shape of occupancy distributions 'species specificity and adaptation to habitat, extinction-colonization dynamics). The presence of a core (class containing species with the highest occupancy) mode in occupancy distributions is most likely to be associated with larger sample units, and small homogenous sampling areas positioned well within and towards the range centers of a sufficient proportion of the species in the assemblage. Satellite (class with species with the lowest occupancy) modes are associated with sampling large, heterogeneous areas that incorporate a large proportion of the assemblage range. However, satellite modes commonly also occur in the presence of a core mode, and rare species effects are likely to contribute to the presence of a satellite mode at most sampling scales. In most proposed hypotheses, spatial scale is an important determinant of the shape of the observed occupancy distribution. Because the attributes of the mechanisms associated with these hypotheses change with spatial scale, their predictions for the shape of occupancy distributions also change. To understand occupancy distributions and the mechanisms underlying them, a synthesis of pattern documentation and model testing across scales is thus needed. The development of null models, comparisons of occupancy distributions across spatial scales and taxa, documentation of the movement of individual species between occupancy classes with changes in spatial scale, as well as further testing of biological mechanisms are all necessary for an improved understanding of the distribution of species and assemblages within their geographic ranges.     

Self-similar mitochondrial DNA

We show that repeated sequences, like palindromes (local repetitions) and homologies between two different nucleotide sequences (motifs along the genome), compose a self-similar (fractal) pattern in mitochondrial DNA. This self-similarity comes from the looplike structures distributed along the genome. The looplike structures generate scaling laws in a pseudorandom DNA walk constructed from the sequence, called a Lévy flight. We measure the scaling laws from the generalized fractal dimension and singularity spectrum for mitochondrial DNA walks for 35 different species. In particular, we report characteristic loop distributions for mammal mitochondrial genomes.     

Analytical evidence for scale-invariance in the shape of species abundance distributions

The distribution of species abundances within an ecological community provides a window into ecological processes and has important applications in conservation biology as an indicator of disturbance. Previous work indicates that species abundance distributions might be independent of the scales at which they are measured which has implications for data interpretation. Here we formulate an analytically tractable model for the species abundance distribution at different scales and discuss the biological relevance of its assumptions. Our model shows that as scale increases, the shape of the species abundance distribution converges to a particular shape given uniquely by the Jaccard index of spatial species turnover and by a parameter for the spatial correlation of abundances. Our model indicates that the shape of the species abundance distribution is taxon specific but does not depend on sample area, provided this area is large. We conclude that the species abundance distribution may indeed serve as an indicator of disturbances affecting species spatial turnover and that the assumption of conservation of energy in ecosystems, which is part of the Maximum Entropy approach, should be re-evaluated.     

Occupancy versus colonization–extinction models for projecting population trends at different spatial scales

Understanding spatiotemporal population trends and their drivers is a key aim in population ecology. We further need to be able to predict how the dynamics and sizes of populations are affected in the long term by changing landscapes and climate. However, predictions of future population trends are sensitive to a range of modeling assumptions. Deadwood‐dependent fungi are an excellent system for testing the performance of different predictive models of sessile species as these species have different rarity and spatial population dynamics, the populations are structured at different spatial scales, and they utilize distinct substrates. We tested how the projected large‐scale occupancies of species with differing landscape‐scale occupancies are affected over the coming century by different modeling assumptions. We compared projections based on occupancy models against colonization–extinction models, conducting the modeling at alternative spatial scales and using fine‐ or coarse‐resolution deadwood data. We also tested effects of key explanatory variables on species occurrence and colonization–extinction dynamics. The hierarchical Bayesian models applied were fitted to an extensive repeated survey of deadwood and fungi at 174 patches. We projected higher occurrence probabilities and more positive trends using the occupancy models compared to the colonization–extinction models, with greater difference for the species with lower occupancy, colonization rate, and colonization:extinction ratio than for the species with higher estimates of these statistics. The magnitude of future increase in occupancy depended strongly on the spatial modeling scale and resource resolution. We encourage using colonization–extinction models over occupancy models, modeling the process at the finest resource‐unit resolution that is utilizable by the species, and conducting projections for the same spatial scale and resource resolution at which the model fitting is conducted. Further, the models applied should include key variables driving the metapopulation dynamics, such as the availability of suitable resource units, habitat quality, and spatial connectivity.     

On the scaling patterns of species spatial distribution and association

Understanding how species distribution (occupancy and spatial autocorrelation) and association (that is, multi-species co-distribution) change across scales is fundamental to unlocking the pattern formation in population ecology and macroecology. Based on the Bayesian rule and join-count statistics, I present here a mathematical model that can demonstrate the effect of spatial scale on the observation of species distribution and association. Results showed that the intensity of spatial autocorrelation and species association declines when the grain in the spatial analysis increases, although the category of species distribution (aggregated or segregated) and association (positive or negative) remains the same. Random distribution and species independence were proved to be scale-free. Regardless of the possible patterns of species distribution and association, species tend to be randomly distributed and independent from each other when scaling-up (an increasing grain), reflecting a percolation process. This model, thus, grasps the statistical essence of species scaling pattern and presents a step forward for unveiling mechanisms behind species distributional and macroecological patterns.     

Universal scaling of species‐abundance distributions across multiple scales

The scale‐dependent species abundance distribution (SAD) is fundamental in ecology, but few spatially explicit models of this pattern have thus far been studied. Here we show spatially explicit neutral model predictions for SADs over a wide range of spatial scales, which appear to match empirical patterns qualitatively. We find that the assumption of a log‐series SAD in the metacommunity made by spatially implicit neutral models can be justified with a spatially explicit model in the large area limit. Furthermore, our model predicts that SADs on multiple scales are characterized by a single, compound parameter that represents the ratio of the survey area to the species’ average biogeographic range (which is in turn set by the speciation rate and the dispersal distance). This intriguing prediction is in line with recent empirical evidence for a universal scaling of the species‐area curve. Hence we hypothesize that empirical SAD patterns will show a similar universal scaling for many different taxa and across multiple spatial scales.     

Self-similar patterns of nature: insect diversity at local to global scales

The insects are probably the most hyperdiverse and economically important metazoans on the planet, but there is no consensus on the best way to model the dimensions of their diversity at multiple spatial scales, and the huge amount of information involved hinders data synthesis and the revelation of 'patterns of nature'. Using a sample of more than 600k insect species in the size range 1-100mm, we analysed insect body sizes and revealed self-similar patterns persisting across spatial scales from several hectares to the World. The same patterns were found in both Northern and Southern Hemispheres. The patterns include: parallel rank-abundance distributions; flatter species-area curves in smaller insects-indicating their wider geographical distribution; the recurrence of the same species-rich family in the same body-size class at all spatial scales-which generates self-similar size-frequency distributions (SFDs)-and the discovery that with decreasing mean body size, local species richness represents an increasing fraction of global species richness. We describe how these 'rationalizing' patterns can be translated into methods for monitoring and predicting species diversity and community structure at all spatial scales.     

A dynamic multi‐scale occupancy model to estimate temporal dynamics and hierarchical habitat use for nomadic species

Distribution models are increasingly being used to understand how landscape and climatic changes are affecting the processes driving spatial and temporal distributions of plants and animals. However, many modeling efforts ignore the dynamic processes that drive distributional patterns at different scales, which may result in misleading inference about the factors influencing species distributions. Current occupancy models allow estimation of occupancy at different scales and, separately, estimation of immigration and emigration. However, joint estimation of local extinction, colonization, and occupancy within a multi‐scale model is currently unpublished. We extended multi‐scale models to account for the dynamic processes governing species distributions, while concurrently modeling local‐scale availability. We fit the model to data for lark buntings and chestnut‐collared longspurs in the Great Plains, USA, collected under the Integrated Monitoring in Bird Conservation Regions program. We investigate how the amount of grassland and shrubland and annual vegetation conditions affect bird occupancy dynamics and local vegetation structure affects fine‐scale occupancy. Buntings were prevalent and longspurs rare in our study area, but both species were locally prevalent when present. Buntings colonized sites with preferred habitat configurations, longspurs colonized a wider range of landscape conditions, and site persistence of both was higher at sites with greener vegetation. Turnover rates were high for both species, quantifying the nomadic behavior of the species. Our model allows researchers to jointly investigate temporal dynamics of species distributions and hierarchical habitat use. Our results indicate that grassland birds respond to different covariates at landscape and local scales suggesting different conservation goals at each scale. High turnover rates of these species highlight the need to account for the dynamics of nomadic species, and our model can help inform how to coordinate management efforts to provide appropriate habitat configurations at the landscape scale and provide habitat targets for local managers.     

Modeling stream fish distributions using interval‐censored detection times

Controlling for imperfect detection is important for developing species distribution models (SDMs). Occupancy‐detection models based on the time needed to detect a species can be used to address this problem, but this is hindered when times to detection are not known precisely. Here, we extend the time‐to‐detection model to deal with detections recorded in time intervals and illustrate the method using a case study on stream fish distribution modeling. We collected electrofishing samples of six fish species across a Mediterranean watershed in Northeast Portugal. Based on a Bayesian hierarchical framework, we modeled the probability of water presence in stream channels, and the probability of species occupancy conditional on water presence, in relation to environmental and spatial variables. We also modeled time‐to‐first detection conditional on occupancy in relation to local factors, using modified interval‐censored exponential survival models. Posterior distributions of occupancy probabilities derived from the models were used to produce species distribution maps. Simulations indicated that the modified time‐to‐detection model provided unbiased parameter estimates despite interval‐censoring. There was a tendency for spatial variation in detection rates to be primarily influenced by depth and, to a lesser extent, stream width. Species occupancies were consistently affected by stream order, elevation, and annual precipitation. Bayesian P‐values and AUCs indicated that all models had adequate fit and high discrimination ability, respectively. Mapping of predicted occupancy probabilities showed widespread distribution by most species, but uncertainty was generally higher in tributaries and upper reaches. The interval‐censored time‐to‐detection model provides a practical solution to model occupancy‐detection when detections are recorded in time intervals. This modeling framework is useful for developing SDMs while controlling for variation in detection rates, as it uses simple data that can be readily collected by field ecologists.     

Modeling bird species distribution change in fire prone Mediterranean landscapes: incorporating species dispersal and landscape dynamics

Species distribution models (SDMs) have traditionally been founded on the assumption that species distributions are in equilibrium with environmental conditions and that these species–environment relationships can be used to estimate species responses to environmental changes. Insight into the validity of this assumption can be obtained from comparing the performance of correlative species distribution models with more complex hybrid approaches, i.e. correlative and process‐based models that explicitly include ecological processes, thereby accounting for mismatches between habitat suitability and species occupancy patterns. Here we compared the ability of correlative SDMs and hybrid models, which can accommodate non‐equilibrium situations arising from dispersal constraints, to reproduce the distribution dynamics of the ortolan bunting Emberiza hortulana in highly dynamic, early successional, fire driven Mediterranean landscapes. Whereas, habitat availability was derived from a correlative statistical SDM, occupancy was modeled using a hybrid approach combining a grid‐based, spatially‐explicit population model that explicitly included bird dispersal with the correlative model. We compared species occupancy patterns under the equilibrium assumption and different scenarios of species dispersal capabilities. To evaluate the predictive capability of the different models, we used independent species data collected in areas affected to different degree by fires. In accordance with the view that disturbance leads to a disparity between the suitable habitat and the occupancy patterns of the ortolan bunting, our results indicated that hybrid modeling approaches were superior to correlative models in predicting species spatial dynamics. Furthermore, hybrid models that incorporated short dispersal distances were more likely to reproduce the observed changes in ortolan bunting distribution patterns, suggesting that dispersal plays a key role in limiting the colonization of recently burnt areas. We conclude that SDMs used in a dynamic context can be significantly improved by using combined hybrid modeling approaches that explicitly account for interactions between key ecological constraints such as dispersal and habitat suitability that drive species response to environmental changes.     

城市景观空间自相关与自相似的尺度特征研究

在RS与GIS技术支持下,以上海市外环以内城市中心区为区域背景,研究城市景观格局空间自相关与分形结构在不同粒度下的变化特征,揭示其对粒度的响应特征和敏感程度。结果表明,城市景观格局的空间自相关性和自相似性依赖于粒度的选择,MoranⅠ与各种景观类型的分维数均存在尺度效应;城市景观格局空间自相关性与自相似性在一定尺度范围内具有尺度依赖性,但二者对尺度的依赖程度不同。空间自相关性具有较高的尺度依赖性;而不同景观类型的自相似性随尺度的变化没有统一规律;城市景观格局的空间自相关性和自相似性对尺度变化的敏感点不同。居住景观的MoranⅠ对尺度变化的敏感点为50m,其他景观类型无明显的敏感点;分维数对尺度的敏感点因景观类型不同而异。     

Animal response to nested self-similar patches: a test with woolly bears

To gain insight into how animals respond to resource patchiness at different spatial scales, we envision their responses in environments comprised of nested, self-similar patches. In these environments, all resources reside within the smallest patches, and resource density declines as a constant exponent of patch size. Accordingly, we use simple mathematical formulations to describe a self-similar environment and a null model of how animals should respond to this environment if they do not perceive resource distribution. We then argue that animals that can perceive resource distribution should partition space by reducing the relative time searching between patches as patch size increases. On an experimental landscape, we found that woolly bear caterpillars Grammia geneura could partition space in this manner, but the range of patch sizes over which they did so tended to increase with resource aggregation. Nevertheless, scaling efficiency (i.e. the scaling of search time versus the scaling or resource density) was similar in all distributions when averaged over all patch sizes. These disparate patterns with similar outcomes resulted from differences in caterpillars' abilities to discriminate spatially among patches of different sizes via their movement pathways, and differences in their use of speed to detect resource items. Our work is relevant to the characterization of resource availability from an animal's perspective, and to the linking of optimal foraging theory to the modeling of search behavior.     

Spatial patterns and scaling behaviors of Steller sea lion (Eumetopias jubatus) distributions and their environment

Fractal geometry and other multi-scale analyses have become popular tools for investigating spatial patterns of animal distributions in heterogeneous environments. In theory, changes in patterns of animal distributions with changes in scale reflect transitions between the controlling influences of one environmental factor or process over another. In an effort to find linkages between Steller sea lions (Eumetopias jubatus) and their environment, the objective of this study was to determine if the spatial distribution of Steller sea lions at sea displayed similar scaling properties to the variation of two environmental features, including bathymetry and sea surface temperature (SST). Additionally, distributions of Steller sea lion point patterns were examined with respect to measurements of bathymetric complexity. From February 2000 to May 2004, satellite transmitters were deployed on 10 groups of juvenile Steller sea lions (n=52) at eight different locations within the Aleutian Islands and Gulf of Alaska. Indices of fractal dimension were calculated for each group of sea lions using a unit square box-counting method, whereas indices of bathymetry and SST patchiness were derived by conducting a variance ratio analysis over the same scales. Distributions of Steller sea lions at sea displayed self-similar fractal patterns, suggesting that individuals were distributed in a continuous hierarchical set of clumps within clumps across scales, and foraging behavior was likely influenced by a scale invariant mechanism. Patterns of bathymetric variability also were self-similar, whereas patterns of SST variability were scale dependent and failed to retain self-similar spatial structure at larger scales. These results indicate that the distributions of Steller sea lions at sea were more influenced by bathymetry than SST at the scales examined, but scale-dependent patterns in the distribution of Steller sea lions at sea or linkages with SST may have been apparent if analyses were conducted at finer spatial scales.     

The importance of local and landscape-scale processes to the occupancy of wetlands by pond-breeding amphibians

Variation in the distribution and abundance of species across landscapes has traditionally been attributed to processes operating at fine spatial scales (i.e., environmental conditions at the scale of the sampling unit), but processes that operate across larger spatial scales such as seasonal migration or dispersal are also important. To determine the relative importance of these processes, we evaluated hypothesized relationships between the probability of occupancy in wetlands by two amphibians [wood frogs (Lithobates sylvaticus) and boreal chorus frogs (Pseudacris maculata)] and attributes of the landscape measured at three spatial scales in Rocky Mountain National Park, Colorado. We used cost-based buffers and least-cost distances to derive estimates of landscape attributes that may affect occupancy patterns from the broader spatial scales. The most highly ranked models provide strong support for a positive relationship between occupancy by breeding wood frogs and the amount of streamside habitat adjacent to a wetland. The model selection results for boreal chorus frogs are highly uncertain, though several of the most highly ranked models indicate a positive association between occupancy and the number of neighboring, occupied wetlands. We found little evidence that occupancy of either species was correlated with local-scale attributes measured at the scale of individual wetlands, suggesting that processes operating at broader scales may be more important in influencing occupancy patterns in amphibian populations.