A fractal-based sampling design for ecological surveys quantifying β-diversity


Correspondence author. Email: charles.marsh07@imperial.ac.uk


  1. Species turnover, β-diversity, underpins a number of ecological processes that define patterns of diversity. Estimates of β-diversity are dependent upon the spatial scale investigated, and patterns may vary across spatial scales. This presents us with a logistical problem of how to sample sufficiently at fine, local scales through to broad, landscape scales to provide accurate estimates of β-diversity at all spatial scales.
  2. Here, we present a scalable sampling design based on fractal geometry that is designed to explicitly address questions about β-diversity. Using simulated communities, we assessed the efficacy of the fractal design, along with two further designs representing subsamples of the fractal design and several classical ecological sampling designs (grids and transects) to estimate β-diversity across multiple spatial scales using two measures of β-diversity: community dissimilarity modelled against geographic distance and additive partitioning.
  3. All designs successfully modelled dissimilarity against distance, with the exception of grid sets and transects that were found to be unsuitable. When diversity was partitioned into multiple spatial scales, all sampling designs overestimated large-scale β-diversity.
  4. The accuracy of distance-decay estimates were primarily determined by the spatial configuration of sampling points. By contrast, the accuracy of diversity partitioning estimates was also influenced by sampling effort, with insufficient sampling effort and unsuitable sampling point configuration causing overestimates of β-diversity at larger spatial scales. We recommend that studies investigating β-diversity use a cluster-based configuration of sampling points, such as the fractal-based design presented here, to ensure accurate and comparable estimates at multiple spatial scales. Furthermore, when comparing results between studies, care should be taken to account for differences in sampling grain, sampling effort and the configuration of sampling points.


Identifying patterns of diversity are crucial to our understanding of the maintenance of ecosystems and the mechanisms of the ecological processes that shape them. Many studies estimate diversity at local scales for comparisons between habitats, across natural and non-natural habitat modifications, as well as ecological, latitudinal and elevational gradients. However, fewer studies investigate the spatial component of diversity, β-diversity, although this component may be equally as effected by such processes.

Species turnover underpins numerous ecological processes and rigorous estimates of β-diversity are crucial in our investigation of these fundamental phenomena across a range of spatial scales (Anderson et al. 2011). Examples include assessing the complementarity of species in reserve networks (Su et al. 2004; Wiersma & Urban 2005; Marsh et al. 2010), right through to making predictions of global species richness (Ødegaard 2000), as well as processes that require estimates at multiple spatial scales, such as the distributions of species (Qian, Ricklefs & White 2004) and extinction rates from habitat loss (Hubbell et al. 2008).

Investigations that have partitioned diversity in to α, β and γ components (Lande 1996) have found that the β component may comprise of a significant portion of overall richness, γ-diversity, often much greater than that of local richness (e.g. Gering, Crist & Veech 2003). This important component of diversity may be further partitioned temporally (Tylianakis, Klein & Tscharntke 2005), across multiple spatial scales (Wagner, Wildi & Ewald 2000; Veech et al. 2002; Crist et al. 2003) and ecological gradients (Sobek et al. 2009).

Only rarely, however, are studies explicitly designed to quantify β-diversity, and data are rarely collected in the field with analyses of community turnover as the primary focus. Given the importance of β-diversity to a plethora of ecological and conservation issues, we suggest there is a need to develop a spatial sampling strategy designed explicitly to estimate β-diversity over a range of spatial scales. We approach this problem by designing a range of possible sampling strategies based around fractals, and use simulations of tropical communities to investigate the optimal sampling strategy that compromises between the accuracy of β-diversity estimates and sampling effort.

Fractal mathematics (Mandlebrot 1983) has become increasingly popular to describe a wide range of ecological processes in which Euclidean geometry has proven insufficient. For example, many ecological processes are spatially dependent, and the self-similarity of fractals can infer such patterns by scaling-up or down spatial scales based on observations made at a particular scale. Fractal geometry has been applied to a diverse range of topics in ecology and conservation biology (Brown et al. 2002; Halley et al. 2004). For example, species aggregations are often self-similar across a range of spatial scales (Kunin 1998), so fractals provide a useful model for understanding the spatial distributions of species (Ulrich & Buszko 2003; Storch et al. 2008) and several associated macroecological patterns, such as species–area relationships (Lennon, Kunin & Hartley 2002; Sizling & Storch 2004; Tjørve & Tjørve 2008).

However, fractal geometry has been used surprising rarely to investigate the spatial turnover of species, β-diversity. Early models of species coexistence on fractal landscapes suggested that β-diversity may be a function of the fractal dimension of a landscape (Palmer 1992), and there is some evidence that β-diversity of plants across Britain may be described fractally (Kunin 1997). Only a single study has considered using fractals as a basis to collect data when studying these processes. Kallimanis, Sgardelis & Halley (2002) found that a sampling design based upon a fractal network of rectangles, Cantor grids (Mandlebrot 1983), was more successful in measuring the fractal dimension of a species distribution than classical line transects, but no study has built on this result to investigate optimal sampling strategies to quantify species turnover. Here we explore several possible fractal-based sampling designs for estimating β-diversity.

Criteria for a Sampling Design

Any sampling design for investigating β-diversity should incorporate two main criteria. First, any measure of β-diversity between two points will depend on the distance between points sampled, that is, β-diversity is dependent on the spatial scale at which it is measured (Gaston, Evans & Lennon 2007). Therefore, any sampling design should cover a range of spatial scales, from the fine scales at which individuals interact (Huston 1999), through to broad, landscape scales at which population dynamic processes occur (Gering & Crist 2002). Second, following the principle that the variance can be more important than the mean (Benedetti-Cecchi 2003; Halley et al. 2004) it is vital that any estimate of β-diversity is accompanied by a measure of variance around that estimate.

The need to obtain fine-scale (grain) data over large spatial scales (extent) immediately poses logistical limitations. For studies utilising remotely sensed or pre-existing data, fine-scale data can be seamlessly aggregated up to large spatial scales at almost no additional expense other than processing power (e.g. Kunin 1998). However, for field-based studies, frequently limited by logistical constraints, trade-offs between sampling effort and statistical power are inevitable. Using classical ecological sampling designs, such as regular grids, every order of magnitude increase in spatial scale can require a 10- or 100-fold increase in sampling effort (Kallimanis, Sgardelis & Halley 2002; Halley et al. 2004), which quickly becomes infeasible and forces us to focus our sampling on a subset of spatial scales.

A fractal-based sampling design, based around clusters of sampling points rather than the regular or random placement of sampling points, could allow us to overcome these limitations, scaling up to larger landscape levels with a significantly smaller increase in sampling effort than using a regular grid. As fractal patterns are self-similar across a range of spatial scales, a design such as Cantor grids allows for an efficient method of sampling fine-scale processes while simultaneously sampling large-scale processes with the same design (Kallimanis, Sgardelis & Halley 2002). Furthermore, a fractal-based sampling design ensures that we can generate multiple estimates of the response variable(s) at each spatial scale investigated, as well as a measure of the variance around those estimates.

Cantor grids require four points to define a rectangle. However, only a minimum of three points are necessary to obtain a mean of β-diversity estimates (i.e. three is the minimum number of points required to obtain more than one point-to-point comparison of species identities), although there will be a subsequent decrease in the accuracy of the estimate of the mean and variance. If we wish to increase accuracy, we could increase the number of points to five (pentagons), six (hexagons) or more. However, this gain in precision must be counterbalanced by the increase in sampling effort at a given spatial scale.

This leads to a fractal pattern based on equilateral triangles being an efficient method of sampling across multiple spatial scales while keeping sampling effort to a minimum (Fig. 1a), although with the drawback that it also equates to the fractal-based pattern that is likely to have the highest error (largest confidence intervals) in parameter estimates. To generate the triangle fractal design, we arrange three sampling points on the vertices of an equilateral triangle with sides of length 10w, where w is a scaling factor. The scaling factor will necessarily vary according to the taxa being investigated, with studies on large, vagile taxa requiring a larger scaling factor than smaller, sedentary taxa. This distance defines the value of w, and allows the mean and variance of β-diversity to be estimated over distance 10w, which represents the ‘first-order’ of the spatial scales at which sampling is being conducted.

Figure 1.

Fractal sampling designs based on equilateral triangles and extending across three spatial scales (‘orders’). The sampling points (black dots) are located on the apices of each first-order triangle. Three designs are presented: (a) the full fractal design; (b) a subsampling strategy using the minimum number of points to obtain β-diversity estimates at all spatial scales; and (c) the compromise subsampling strategy that attempts to balance sampling effort and statistical power.

A full sampling scheme will extend across N orders, with the second-order samples being separated by distances of 102w and so on. However, the second-order samples are not represented by a single point, but rather by a first-order triangle centred on that point, requiring nine points to simultaneously sample at the first and second-order spatial scales. The number of sampling points (S) in the full sampling design, then, increases exponentially with the number of spatial scales (N) according to the function = 3N. Further increases in spatial scale are added by siting the lower-order sampling pattern on the apices of higher-order triangles along progressively larger scales (103w, 104w, …, 10Nw).

Trade-offs Between Sample Size and Cross-scale Information

Because the number of sample points increases exponentially as we add additional spatial scales, where resources are limited it may become logistically infeasible to employ the full fractal design across numerous spatial scales. However, the self-repeating nature of a fractal design allows for field studies to subsample in ways that reduce the sampling effort required while retaining β-diversity estimates, and variance around those estimates, at all spatial scales.

To reduce sampling effort to the minimum, we can replace two out of every three clusters of points with a single sample point located within the centre of the cluster it replaces (Fig. 1b). This retains the ability to obtain a mean and variance estimate of β-diversity at each spatial scale, although with a reduction to just a single estimate for each scale whereas in the full fractal design the number of estimates increases with decreasing scale. In this situation, the minimum sampling effort required (Smin) scales linearly, rather than exponentially, with the number of spatial scales, with Smin = 2N + 1. This, then, allows for a much larger range of spatial scales to be investigated with relatively small additional investments in sampling effort.

Finally, we present a third sampling design that attempts to balance the number of sampling points with the number of β-diversity estimates at lower orders of spatial scale (Fig. 1c). This design maintains one full triangle of sampling points at the apex of each second-order triangle, which is repeated at each apex of the third-order triangle and so on. While still significantly reducing the number of sampling points, there are now a number of β-diversity estimates at lower-order spatial scales. Therefore, we may consider this as a compromise (Scom) between sampling effort and statistical power. In this case Scom = 5 × 3N−2, where  2. While sampling effort here also scales exponentially with the number of spatial scales, it still approximately halves the number of sampling points needed relative to that of the full fractal design.

Materials and methods

We used simulations to compare the ability of various sampling designs to accurately estimate β-diversity across spatial scales. We generated communities across a ‘landscape’ (Data S1 in Supporting Information) in which we independently tested a range of sampling designs. By controlling species richness and β-diversity, we were able to separate the relative importance of sampling effort and the configuration of sampling points on generating estimates of β-diversity across multiple spatial scales. All simulations were created and analysed in program R version 2.13.1 (R Development Core Team 2011).

Creating a Simulated Community

To provide an ‘ecological space’, we created a hypothetical landscape consisting of a 100 × 100 cell grid, overlaid with a single, linear environmental gradient ranging from −1 to 1 oriented in a random direction. By increasing the range of gradient values, species distribution patterns become more polarised in space and thus β-diversity can be generated in our hypothetical landscape. By randomising the gradient direction, we can investigate the differences in accuracy between sampling designs that are unidirectional (e.g. transects) compared to multidirectional (e.g. regular grids) in situations where we do not know a priori the direction of the gradient(s) driving patterns of β-diversity.

To distribute species across the landscape in patterns that emulate those found in natural ecological communities, we randomly assigned each of i species a value for three parameters: a probability of occurrence (ψi), a degree of density dependency (di), and a response to the environmental gradient (ei). This approach provides the flexibility to alter the probability distributions of each parameter to match those found in real communities. In our study, we generated communities resembling a tropical insect community. Values for ψ were taken from a left skewed beta distribution that corresponds to a community of mostly rare species and few common species. Values of d recorded the strength of density dependence, and were used only if the species was present in a neighbouring cell. Values were selected from between −0·5 and 0·5 sampled from a normal distribution, so that approximately half the species were negatively density dependent with some degree of scattering and half were positively density dependent and exhibited aggregated distributions. Similarly, the values for e were also taken from a normal distribution between −0·5 to 0·5, so that approximately half the species were more abundant on one side of the gradient and the other half were more abundant at the opposite end. Species with a value of = 0 were insensitive to the gradient. Species were populated independently of others, such that multiple species could occupy a cell without influencing the occupation of other species.

We combined these values to determine the value (Pi,k) that species i will be present in a given cell k according to the equation

display math(eqn 1)

where Di,k represents the presence (= 1) or absence (= 0) of species i in a cell adjacent to cell k, and Ek represents the value of the environmental gradient in cell k. To convert these values into a presence-absence matrix we generated a random number between 0 and 1 from a uniform distribution for each of the k cells (Rk) to mimic the natural stochastic element in species distributions (species may be absent even if the environmental conditions are suitable for it to be present). The presence or absence of species i in cell k was determined using the rule

display math(eqn 2)

where inline image is the presence (1) or absence (0) of species i in cell k.

When the landscape is first populated in this manner, a given cell will not be surrounded by eight cells that have had presence-absence determined and so calculations of density dependency will be inconsistent across the landscape. Therefore, we first populated the landscape in a random order, then repeated the process in a second iteration that produced the final presence-absence matrix for each species. This approach produced a range of realistic species distributions (Fig. 2 provides examples of some possible distributions generated with the procedure). We generated landscapes that had = 100 species.

Figure 2.

Six examples of possible species distributions (presence = black cells) that could be generated using the simulation procedure outlined in the 'Materials and methods'. In the simulated communities created for the analyses approximately half the species will show distributions with some degree of aggregation and half some degree of scattering.

Sampling Designs

We tested seven sampling designs for their ability to estimate β-diversity by comparing them against a control design consisting of 1089 points arranged in a fine-scale regular grid with two cells separating each sample point (Fig. 3). First, we explored the three variations of the triangle fractal design described above (Fig 1): (i) a full fractal design extending across four orders of scale; (ii) the compromise subsampling scheme for a triangle fractal design; and (iii) the minimum number of points necessary to subsample the triangle fractal design. As the second design is taken as a compromise between sampling effort and statistical power, all subsequent designs were designed to have equivalent or nearly equivalent sampling effort, and to have a similar range of distances between sampling points. A fourth sampling design (iv) was composed of five sets of grids that might represent a study investigating, for example, within- and between-plot turnover (e.g. Tylianakis et al. 2006). Two further designs, a regular grid (v) and line transects (vi), are frequently used in ecological investigations and so were also tested in these simulations. Finally, we included a random sampling design (vii), in which sampling points were randomly distributed across the landscape.

Figure 3.

Seven sampling designs used to recover patterns of β-diversity. Black points represent the location of a sampling point. Subtitles indicate the name of the sampling design, the number of sampling points, S, and the range of distances that separate individual sampling points.

The potential robustness of each sampling design to an environmental gradient of unknown direction was visualised by creating a heat map of distances between sampling points and the bearing between those points. A poor design will have large areas of ‘cartesian space’ without sampling points, and so would likely produce inaccurate estimates of β-diversity if the direction of the gradient happened to fall within these bearings. However, if the gradient is the same direction as the bearing between many points (e.g. directly along a transect), highly accurate estimate should be produced.


For each sampling design we distributed 100 species across the landscape. The process was repeated for 100 landscapes for each design and β-diversity estimated within each. To compare the ability of a sampling design to quantify β-diversity, we determined the range of spatial scales over which β-diversity can be estimated for each design and the accuracy of those estimates at different spatial scales. To achieve this, two types of β-diversity were estimated, directional turnover and non-directional variation (Anderson et al. 2011).

Directional turnover was calculated as Bray-Curtis dissimilarity between all pairs of points. Dissimilarity was modelled as a function of distance between points and plotted using the ‘loess’ function in R, a non-parametric technique of fitting a smooth line from a local regression model. The degree of variance among iterations of the sampling simulation represents the likelihood of a sampling design estimating β-diversity accurately. Using the control sampling design, we generated a 90% confidence interval representing uncertainty in the ‘true’ estimate of dissimilarity among points separated by a given distance. Similarly, we generated a 90% confidence interval around dissimilarity estimates for each sampling design which we then compared to the control confidence interval. Sampling designs with confidence intervals that match those of the control were considered to give accurate estimates of β-diversity.

To estimate non-directional variation, we hierarchically partitioned diversity into multiple additive components (Lande 1996). Diversity was estimated at four spatial scales, calculated using moving windows of sizes (i) 1 × 1, (ii) 5 × 5, (iii) 15 × 15, and (iv) 29 × 29 cells. For a given spatial scale, N, α-diversity at that scale, αN, was the mean number of species recorded in every moving window. For the regular grid, line transects and random designs which are not a priori designed for partitioning diversity, sampling points could be included in more than one moving window. There is, however, no such pseudoreplication in the fractal designs. By taking total diversity, γ, as the total number of species recorded from all sampling points, for our largest spatial scale (= 4 and a window of 29 × 29 cells), we calculated β-diversity as β4 = γ−α3, and for decreasing spatial scales as βN = γ−αN−1. Repeated on the 100 simulated landscapes, this returned mean (and variance) estimates of α, β1 through to β4, and γ for each sampling design. To quantify the ability of each design to recover known patterns of diversity, we plotted all diversity estimates against those generated from the control design. Best fit lines were generated from linear models with polynomial functions. For each sampling design we used Tukey's honest significant difference tests to determine if the mean estimate for diversity at each spatial scale was significantly different from that estimated from the control design.


Heat maps of the distances between sampling points highlighted large differences in the configurations between sampling designs (Fig. 4). Random and regular designs had the most consistent coverage although the latter had no estimates of β-diversity at very local scales. The fractal designs had consistent coverage across all distances, although the minimal subsampling design was unsurprisingly sparse given the extremely small number of sampling points. Transects had very little coverage at small scales and grid sets produced an uneven distribution.

Figure 4.

Heat maps showing the distances between points against the bearings between points for each sampling design. Values are shown on a colour scale extending from white (indicating no pairwise comparisons) to dark red (the most pairwise comparisons). Designs that minimise the quantity of white areas will be more accurate at estimating β-diversity in situations where we do not know a priori the direction of environmental gradient(s) driving spatial turnover.

All sampling designs that we tested produced similar dissimilarity-distance slopes when averaged over the 100 runs (Fig. 5). However, there were considerable differences in the range of distances over which each sampling design was able to estimate dissimilarity, and the between-run variance in those estimates. Unsurprisingly, given the very limited sampling effort, the minimal subsampling design had the highest variance and therefore the least accurate β-diversity estimates (Fig. 6). The other two triangle fractal designs tended to slightly underestimate β-diversity at all distances, but produced slopes similar to that of the control. Of the classical designs, the regular grid provided the most accurate estimates across all distances, but was unable to generate estimates at very small scales. The random design consistently overestimated β-diversity. The final two designs, grid sets and transects provided variable estimates, with dissimilarity-distance slopes that varied depending on the range of distances being examined. Grid sets underestimated or overestimated β-diversity at different spatial scales, whereas transects were accurate at small scales but produced highly variable estimates at medium or large distances.

Figure 5.

Bray-Curtis dissimilarity against distance between sampling points. Light grey lines represent loess functions fitted to the data generated from 100 simulation runs. Here, we present as polygons the region delimited by the inner 90 lines for each sampling design (90% confidence interval; green polygons), and compare them to the 90% confidence interval generated by the control sampling design (red polygons).

Figure 6.

The proportion of the confidence interval for each sampling design that falls outside the confidence interval of the control design. The control subplot (top left) indicates the width of the 90% confidence interval surrounding β-diversity estimates for the control (from Fig. 5), with lower values representing more accurate estimates. For all other sampling designs, polygons represent the proportion of the 90% confidence interval surrounding β-diversity estimates at a given distance that were greater than (positive values, green polygons) or less than (negative values, blue polygons) the confidence interval determined in the control. Polygons that fall within the red shaded area indicate distances at which < 5% of the confidence interval was outside the confidence interval of the control.

Estimates of γ-diversity were predictably associated with sampling effort. Sampling designs with 45 sampling points recorded around 95% of all species present (Fig. 7a). The full fractal design, with almost twice the sampling effort, recorded 98% of species, whereas the seven points of the minimal subsampling design recorded on average only 55% of all species. The proportion of diversity attributable to local richness (α-diversity) was even across designs (15–17%), except for the minimal subsampling design where α composed 28% of diversity. The classical designs of regular grids, transects or random sampling were unsuitable for partitioning diversity and were not able to partition diversity at the smallest spatial scales (Table 1). The fractal and grid sets designs produced β estimates more similar to the control than the other designs, although only the fractal designs were able to partition β-diversity across all four spatial scales. However, all sampling designs significantly overestimated the proportion of β-diversity at the largest spatial scales (Fig. 7b).

Figure 7.

(a) Additive partitioning of diversity into five components of increasing spatial scale: α-diversity (very dark grey); β1 (dark grey); β2 (grey); β3 (light grey); and β4 (very light grey). Estimates of γ-diversity are presented above each sampling design. Error bars are 95% confidence intervals for the diversity estimate at each spatial scale. (b) The estimates for the seven sampling designs were plotted against the partitioned diversity estimates from the control sampling design. Lines of best fit were obtained from a polynomial linear model. The dotted black line represents the 1:1, which would reflect a sampling design that partitioned diversity exactly in accordance with the control. Fitted lines above the 1:1 line underestimate β-diversity, while those below overestimate β-diversity. Sampling designs are C = control; TF = triangle fractal; CS = compromise subsample; MS = minimal subsample; GS = grid sets; Reg = regular; Tr = transects; and Ran = random.

Table 1. Relative performances to the control of each sampling design in estimating dissimilarity against distance and partitioning diversity across multiple spatial scales
Sampling design(a) Dissimilarity-distance(b) Diversity partitioning
  1. (a) Comparison of directional dissimilarity estimates between sampling designs and the control design at small (distance = 5), medium (distance = 50) and large (distance = 95) scales. Values represent scales at which > 5% of the dissimilarity-distance estimates fell above (+) or below (−) the confidence limits of dissimilarity estimates generated by the control design. Tick marks (✓) indicate scales at which > 95% of estimates fell within the control estimates, and NA represents scales at which dissimilarities could not be calculated by a sampling design. (b) Significant differences between the additive diversity partitions estimated for each sampling design and those estimated from the control design across four spatial scales. Symbols indicate if the estimate of diversity was greater than (+), less than (−), or did not differ significantly (✓) from the estimate from the control. NA represents diversity partitions that could not be calculated by a sampling design.

Triangle fractal+
Compromise subsample++
Minimal subsample+/−+/−NA++
Grid sets+/−NA+


Due to limited time and resources there will necessarily be a trade-off between collecting data able to accurately detect spatial patterns at fine vs. broad-scales. An appropriate sampling design should attempt to allow researchers to estimate β-diversity at a range of spatial scales, and with a similar level of accuracy at all scales using a variety of complementary methods (Gaston, Evans & Lennon 2007). No sampling design produced estimates with the same accuracy as the control but there were clear differences between the performances of each design (Table 1), highlighting that when sampling effort is consistent the configuration of sampling points may have large impacts on our estimates of β-diversity at different spatial scales.

Towards an Optimal Sampling Design to Investigate Community Turnover

When estimating directional turnover, there was a clear distinction between designs that produced consistent slopes of the dissimilarity-distance curve across all spatial scales in a manner equivalent to the control, and designs such as grid sets and transect designs that produced variable slope estimates (Fig. 5). The minimal subsample design also falls in to this category, although this is unsurprising given the extremely small number of sampling points. The explanation for the relatively high variability returned by these designs is that they have the greatest amount of ‘Cartesian space’ unoccupied by pairwise comparisons of points (Fig. 4). Consequently, these designs have very high uncertainty in β-diversity estimates when the direction of an environmental gradient does not correspond with the ‘direction’ of the sampling design. Of the other designs, the regular grid provided the most consistent estimates (Fig. 6) but has the limitation that if the same range of spatial scales is to be covered as other designs it becomes impossible to also sample at the smallest spatial scales. The random design and the fractal designs also performed well, although they overestimated and underestimated β-diversity respectively.

All designs were inferior at partitioning diversity than at estimating dissimilarity-distance patterns. Grid sets produced the most accurate diversity partitioning estimates, although the fractal designs were the only designs that were able to partition diversity across all four spatial scales (Fig. 7). All designs produced diversity estimates at large spatial scales that were significantly lower than the control (Table 1). Sampling effort is important in any estimate of an ecological process (Kallimanis, Sgardelis & Halley 2002) and it is apparent from the relative performances of the compromise subsample, full fractal and control designs that a reduction in the quantity of sampling points results in an increase in large-scale β-diversity estimates. There is, however, an important role played by the spatial configuration of sampling points when sampling effort is kept constant. In particular, sampling designs that are not cluster based appear unsuitable for partitioning diversity across multiple spatial scales, and are also likely to attribute a greater proportion of diversity to large scales than is actually the case.

Across all the designs we considered, we suggest that the fractal design represents the most parsimonious design to capture relatively accurate estimates for both dissimilarity-distance and diversity partitions across all spatial scales. Furthermore, the similarity in performance of the optimal subsampling scheme to the full fractal design suggests that sampling effort can be reduced to a minimum level with only negligible decreases in the accuracy of β-diversity estimates, representing a considerable advantage for data collection in the field. Our results do, however, illustrate the difficulty in defining a single sampling design that can be confidently relied upon to accurately estimate different forms of β-diversity. For example, a design based upon grid sets appears accurate in partitioning diversity but not for estimating directional turnover. By contrast, the regular grid and random designs estimate turnover accurately but are unsuitable for diversity partitioning as well as being unable to produce fine-scale β-diversity estimates and being impractical to implement in the field respectively. The most commonly employed classical design, transects, performed poorly in recovering both measures of β-diversity.

Limitations of Quantifying β-Diversity

A feature common across all designs was that variance in β-diversity estimates increased with spatial scale. This could be a result of a decrease of pair-wise point comparisons at large distances and thus a greater influence of outliers, or due to the randomisation of the environmental gradient with respect to the orientation of the sampling designs between runs. If the former is true then studies should remain vigilant of results at the largest spatial scales, and consider, if possible, sampling at scales slightly larger than required to ensure accurate estimates at the scales relevant to ecological analyses and data interpretation. If the latter is true, it becomes more difficult to design field studies, with the recommendation being that studies should attempt to ascertain the direction(s) of any environmental gradient(s) before commencing surveys. However, a multi-directional design, such as those based on fractal patterns, will be more likely to provide accurate β-diversity estimates than a unidirectional sampling design such as transects in situations where we do not know a priori the direction of the gradient(s) that are driving species turnover.

Our results from dissimilarity modelling appeared relatively robust to sampling effort, with the spatial orientation of sampling points being a more critical determinant of the observed patterns of β-diversity than sample size. The partitioning of diversity, however, was greatly affected by both configuration and sampling effort. Contributions of α- and total β-diversity (i.e. the sum of all β components) were similar across designs, except in cases of severe undersampling. This can be easily explained as the sampling grain, here one cell, determined the values of α-diversity regardless of sampling effort, which only provided a more accurate estimate of the mean α-diversity, whereas sampling effort determined the values for γ-diversity (Crist & Veech 2006). Thus, undersampling will lead to a greater relative contribution of α- than β- to γ-diversity.

In our simulation > 40 points appeared sufficient to adequately estimate γ-diversity. However, where β-diversity was further partitioned in to multiple spatial scales, the sampling effort and spatial orientation of sampling points became important. All sampling designs overestimated the contribution of β-diversity at larger spatial scales. In all likelihood, with every additional partition of β-diversity at a larger spatial scale we would need to increase the sampling effort to provide an accurate estimate. If this is the case, then we must be careful of confounding the relative importance of large-scale β-diversity in determining community richness and composition patterns with a reduced number of replicates at those scales. Furthermore, in comparisons between studies we should account for differences in sampling grain (the size of the basic unit sampled), sampling effort and sampling point configuration.

Recommendations for Sampling Community Turnover

The accuracy of any estimate of a scale dependent ecological process will depend upon the total area covered and the sampling effort committed (Kallimanis, Sgardelis & Halley 2002). However, the differences found between sampling designs highlight that the spatial pattern by which data are collected may also affect our estimates of β-diversity. Sampling designs that naturally partition across multiple spatial scales are able to generate a greater resolution and accuracy of diversity patterns than more systematic designs (Fortin, Drapeau & Legendre 1989). Ideally, we would bespoke our sampling designs according to the taxa and the processes we wish to study, but in reality this will rarely be possible because of insufficient pre-existing information about the scales at which those taxa and processes operate. Given that many ecological processes exhibit fractal or multi-fractal patterns (Brown et al. 2002), a sampling design based upon fractals would seem to be an appropriate approximation to the ideal sampling design. We found that a sampling design of triangle fractals was more successful at estimating our two measures of β-diversity than classical ecological sampling designs, covering a wider range of spatial scales and at higher accuracy. Furthermore, we have shown that the sampling design could be subsampled reducing sampling effort with little reduction in accuracy, while maximising the range of spatial scales that can be covered. Therefore, we recommend that future field studies investigating β-diversity utilise a fractal-based sampling design to increase the accuracy of results.


C.J.M. was funded through a Natural Environment Research Council (NERC) studentship. This work further benefitted from funding to R.M.E. from the Sime Darby Foundation as part of the Stability of Altered Forest Ecosystems (SAFE) project. We thank James Rosindell for insightful assistance, and Tim Coulson, Simon Leather and two anonymous reviewers who provided valuable comments on earlier versions of this manuscript.