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 10^{w}, 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 10^{w}, which represents the ‘first-order’ of the spatial scales at which sampling is being conducted.

A full sampling scheme will extend across *N* orders, with the second-order samples being separated by distances of 10^{2w} 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 *S *=* *3^{N}. 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 (10^{3w}, 10^{4w}, …, 10^{Nw}).

#### 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 (*S*_{min}) scales linearly, rather than exponentially, with the number of spatial scales, with *S*_{min} = 2*N* + 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 (*S*_{com}) between sampling effort and statistical power. In this case *S*_{com} = 5 × 3^{N−2}, where *N *≥* *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.