Our goals in this chapter are roughly fivefold. First we argue that fractal spatial structure is a useful conceptual and practical null model for species distributions. Second, we introduce a new cross-scale model that allows a range of scaling properties to be described, and for systematic departure from fractal scaling to be quantified. Third, we fit this cross-scale model to bird distribution patterns using novel maximum likelihood methods, allowing us to take an individual species distribution pattern and statistically test for differences between its coarse and fine-scale spatial structure. Fourth, we see how differences found between scaling structure at the species level translate into differences in the distribution and scaling of community (e.g. species richness) patterns. Finally, we discuss whether our results can tell us anything about the existence and properties of the hierarchical spatial processes involved in determining both species distribution and diversity patterns.

Maps of a species' potential range make an important contribution to conservation and invasive species risk analysis. Spatial predictions, however, should be accompanied by an assessment of their uncertainty. Here, we use multimodel inference to generate confidence intervals that incorporate both the uncertainty involved in model selection as well as the error associated with model fitting. In the case of the invasive Argentine ant, we found that it was most likely to occur where the mean daily temperature in mid-winter was 7-14 C and maximum daily temperatures during the hottest month averaged 19-30 C. Uninvaded regions vulnerable to future establishment include: southern China, Taiwan, Zimbabwe, central Madagascar, Morocco, high-elevation Ethiopia, Yemen and a number of oceanic islands. Greatest uncertainty exists over predictions for China, north-east India, Angola, Bolivia, Lord Howe Island and New Caledonia. Quantifying the costs of different errors (false negatives vs. false positives) was considered central for connecting modelling to decision-making and management processes.

1. The distribution patterns of 391 rare and scarce British plants (species recorded in 100 or fewer 10 x 10 km squares) were characterized by their distributional area (area of occupancy at 1-km scale: AOO(1)) and levels of aggregation (as reflected in fractal dimensions measured across two scales: D1-10 and D10-100).

2. Eighteen plant traits were tested for relationships to AOO, and to fractal dimension while controlling for AOO. These included both directly heritable traits (e.g. life-form) and emergent properties that are, at most, indirectly heritable (e.g. typical local density). The latter set included an index of net distributional change and an index of range dynamism.

3. Only two traits, habitat preference and local abundance, were significantly related to AOO(1), but about half were associated with fractal dimension.

4. Relatively aggregated fine-scale distributions (high D1-10) were related to high local abundance, lack of specialized, long-distance dispersal mechanisms, habitat preference and an increasing range size with relatively few local extinctions (i.e. a positive index of change with low dynamism).

5. Relatively aggregated coarse-scale distributions (high D10-100) were related to the use of insect pollinators, obligate outcrossing, habitat preference and relatively stable ranges (low dynamism).

6. Multivariate analyses of subsets of conceptually related variables showed that few variables interacted to affect distributional variables.

7. A highly significant negative relationship between dynamism and fractal dimension appears to be driven primarily by high rates of local extinction, leading to relatively scattered, diffuse range structures. Furthermore, it suggests that recent population trends may be inferred from snapshots of contemporary distribution patterns.

8. The role and interpretation of phylogenetically informed analyses in studies such as this are debatable. However, we found similar relationships in both phylogenetically informed and conventional analyses for all variables except pollination vector (a strongly conserved trait).

9. The spatial pattern of plant species distributions is associated with a range of ecological traits, particularly those describing past changes in distribution. The analysis of distribution patterns therefore has the potential to inform future conservation effort.

Fractals have found widespread application in a range of scientific fields, including ecology. This rapid growth has produced substantial new insights, but has also spawned confusion and a host of methodological problems. In this paper, we review the value of fractal methods, in particular for applications to spatial ecology, and outline potential pitfalls. Methods for measuring fractals in nature and generating fractal patterns for use in modelling are surveyed. We stress the limitations and the strengths of fractal models. Strictly speaking, no ecological pattern can be truly fractal, but fractal methods may nonetheless provide the most efficient tool available for describing and predicting ecological patterns at multiple scales.

The spatial distribution of a species can be characterized at many different spatial scales, from fine-scale measures of local population density to coarse-scale geographical-range structure. Previous studies have shown a degree of correlation in species' distribution patterns across narrow ranges of scales, making it possible to predict fine-scale properties from coarser-scale distributions. To test the limits of such extrapolation, we have compiled distributional information on 16 species of British plants, at scales ranging across six orders of magnitude in linear resolution (1 in to 100 km). As expected, the correlation between patterns at different spatial scales tends to degrade as the scales become more widely separated. There is, however, an abrupt breakdown in cross-scale correlations across intermediate (ca. 0.5 km) scales, suggesting that local and regional patterns are influenced by essentially non-overlapping sets of processes. The scaling discontinuity may also reflect characteristic scales of human land use in Britain, suggesting a novel method for analysing the 'footprint' of humanity on a landscape.

In developing red data books of threatened species, the World Conservation Union (IUCN) uses measures of rarity, rates of decline, and population fragmentation to categorize species according to their risk of extinction. However, most quantitative measures of these three concepts are sensitive to the scale at which they are made. In particular, definitions of rarity based on an area-of-occupancy threshold can nearly always be met if area of occupancy is calculated from a sufficiently fine-scale (high-resolution) grid. Recommendations for dealing with scale dependency include (1) choosing a standard scale of measurement, (2) using multiple scales of measurement, and (3) developing indices that combine information from multiple scales. As an example of the second and third approach, the construction of a species' scale-area curve represents a unifying method for quantifying all three indicators of extinction risk-rarity, rate of decline, and population fragmentation-as functions of area of occupancy and measurement scale. A multiscale analysis is also of practical importance because measurements made at different scales are relevant to different extinction processes. Coarse-scale measures of rarity are most appropriate when threat is assessed on the basis of spatially autocorrelated events of a large extent, such as global climate change, whereas fine-scale measures may best predict extinction risk due to local processes such as demographic stochasticity. We illustrate our arguments with a case study of the British distributions of two related plant species that show a 200-fold reversal in their relative rarity when measured at different scales.

1. The aggregation model of coexistence has been used widely to explain the coexistence of competing species that utilize patchy and ephemeral resources. Over the years, it has been reformulated in many different ways, using different assumptions, indices and analyses, leading sometimes to contradictory conclusions. We present a general framework, from which many of the alternative approaches are derived as special cases.

2. A generalized distribution, composed of the distribution of visits across patches and the distribution of eggs per visit, is used to model changes in the mean individual-level experience of density that occur at different population-level densities.

3. New and more general criteria for coexistence are derived, based upon standard invasability analysis of Lotka-Volterra competition equations applied to a patchy system.

4. An important parameter in the new coexistence criteria is the mean per capita density of individuals in a single clutch (c*). Until now this measure has been relatively ignored, experimentally and theoretically.

5. We confirm earlier findings that the random distribution of clutches may be a sufficient cause of aggregated egg distributions to allow coexistence between species of unequal competitive ability, but only if the product of competition coefficients is less than one.

We derive the species-area relationship (SAR) expected from an assemblage of fractally distributed species. If species have truly fractal spatial distributions with different fractal dimensions, we show that the expected SAR is not the classical power-law function, as suggested recently in the literature. This analytically derived SAR has a distinctive shape that is not commonly observed in nature: upward-accelerating richness with increasing area (when plotted on log-log axes). This suggests that, in reality, most species depart from true fractal spatial structure. We demonstrate the fitting of a fractal SAR using two plant assemblages (Alaskan trees and British grasses). We show that in both cases, when modelled as fractal patterns, the modelled SAR departs from the observed SAR in the same way, in accord with the theory developed here. The challenge is to identify how species depart from fractality, either individually or within assemblages, and more importantly to suggest reasons why species distributions are not self-similar and what, if anything, this can tell us about the spatial processes involved in their generation.

Distributional dot maps are one of the most ubiquitous and useful outputs to emerge from a wide range of recording schemes. As well as suggesting interesting relationships between a species and its habitat requirements, comparisons between maps have been used to assess the relative rarity or commoness of different species. However, the number of occupied squares is not the only property of a distribution map, the relative positions of the occupied squares can also be used to make useful inferences regarding the local frequency of a species. It is a common experience that species with a restricted but relatively compact distribution of occupied squares, are likely to be much more abundant "on the ground", than those species occupying the same number of grid squares, but in a diffuse and scattered distribution. One way of formalising this intuition is through the use of fractal geometry. In this paper we demonstrate that the distribution of D. submontana is very close to a perfect statistical fractal. Thus, using information about the one kilometre distribution of this species we were able to make accurate predictions of its coverage at a one metre resolution. However, this was not the case for the other species we examined, Dianthus armeria. The reasons for these different scaling behaviours are not clear, but further work is being undertaken to examine the relationship between this and species' traits such as dispersal ability and habitat preference.

He & Gaston (this issue) present a novel method for estimating both the fine-scale grid-occupancy and the number of individuals in a species' distribution, given that one only has presence/absence data collected over a set of contiguous quadrats. Their method assumes a negative binomial distribution (NBD) of individuals with a scale-independent k. In an analysis of a 50-ha plot of tropical trees it compares favourably with an alternative extrapolation used by Kunin (1998) which assumes a fractal distribution. In this note, we explore some of the strengths and weaknesses of the two approaches. In particular, we find that when analysing data of species distributions collected over much larger extents at coarser scales (e.g. national data at 20 kilometre resolution and above), the fractal method tends to provide better predictions of finer-scale occupancy than the NBD method. However, one definite advantage of the NBD model, is its ability to predict population numbers: something the fractal model cannot achieve.

At the landscape scale, temporal and spatial studies of semi-natural vegetation change, successional pathways and pattern, frequently use both up-to-date and archival aerial photography as a data source. However, the relatively straightforward task of mapping vegetation patch boundaries from aerial photography can be affected by multiple sources of error associated with the processes of georefrencing, digitising and subjective photointerpretation. Different measurements of landscape pattern and change will vary in their sensitivity to the different sources of error. To date, relatively little attention has been focused on developing an integrated approach to the identification and estimation of error in vegetation mapping tasks where the data subsequently form the basis for quantitative measurement of landscape change. In this hapter a simple empirical method is proposed for estimating positional boundary error due to the separate and combined effects of georeferencing, digitising and subjective photointerpretation. The findings of this study, which reveal that up to one-third of the total area may be in error, lead to a strong recommendation that researchers make use of such empirical methods to assess the constraints inherent in their data, prior to undertaking complex or time-consuming analyses.

1. Recent attempts to examine the role of different mechanisms in generating a positive abundance-occupancy relationship failed to properly distinguish between Brown's (1984) sampling artefact, and the form of relationship to be expected from a random distribution of individuals.

2. Because random distributions generate a positive relationship, one can never predict that removing the influence of some or all of the mechanisms will lead to 'no relationship'.

3. In considering how the spatial aggregation of individuals might influence the form of the abundance-occupancy relationship it is demonstrated that curvilinear and triangular relationships are expected, and that correlation coefficients and linear regression statistics are unlikely to be sensitive to the addition and removal of mechanisms.

4. Examining distributional data with alternative indices of spatial structure may lead to a more intuitive understanding of how different mechanisms influence the form of abundance-occupancy relationships.

A rapid assessment of woodland flora was made at sixty-seven plots distributed across fifty-five woodlands in North and West Yorkshire. Half of the sites were located in semi-natural broadleaf woodland, one-quarter were mixed, and the remainder were either conifer plantation, broadleaf plantation or coppice. The study area was broadly divided into three regions: 1) lower Wharfedale, rolling farmland with an underlying geology of Millstone grit, 2) a more heavily wooded band of Magnesian limestone and 3) the Vale of York, an intensively farmed plain of sandstone and alluvial deposits. Details of the regional frequency and within-plot abundance of the commoner species are presented. In particular, sycamore, ash and then oak, were found to be the most common canopy-forming tree species. Hawthorn and elder were the most frequent under-storey shrubs; and brambles and bluebells were the most frequent constituents of the ground cover. The most striking regional pattern was shown by dog's mercury: of twenty-two occurences, seventeen were on the Magnesian limestone, four in lower Wharfedale and only one in the Vale of York. Bluebells were under-represented on the limestone, probably due to their requirement for deeper, moister soils. Himilayan balsalm only occurred in ten of the plots, but it appears to be more widespread in the Vale of York.