theoretical approach

In order to examine how changes in SARs are mechanistically linked to scale-dependent responses of diversity to habitat disturbance, we developed the following algebraic model. The relationship between species number and area is usually described by the equation:

where S is the number of species at a location with area A, and where c (intercept) and z (slope) are constants that depend on the habitat and taxon being sampled (Rosenzweig 1995). Equation 1, a power function, is widely used to model SARs, and hence, we use it in this study, but it is not the only plausible mathematical function. Harte et al. (1999) suggested that this power law arises from a probability rule and self-similarity. If an area of size A₀ containing S₀ species is bisected into two identical self-similar sub-areas A₁, with a fixed (i.e. scale independent and species independent) probability a that a species present in A₀ will be present in a given A₁, then equation 1 will hold with a z value given by a = 2^−z. Thus, arguing inductively, if A_n is an area obtained by n successive bisections of A₀, then the expected number of species in A_n is given by:

This link between z and a, together with the empirical observation that disturbance tends to decrease the value of the slope z (Rosenzweig 1995), can then be further exploited as follows. Consider a large area A₀ of undisturbed habitat, containing a total of S_u species, and which follows a SAR of the form of equation 1 with parameter z_u at spatial scales smaller than A₀. Consider also an ostensibly identical area of disturbed habitat containing S_d species with SAR parameter z_d. We assume that z_u > z_d (so that a_u < a_d in the obvious notation, using the fact that a = 2^−z). Equation 2 implies that at the smallest spatial scale (i.e. with a large number of bisections n), fewer species are observed in the undisturbed habitat compared with the disturbed habitat, because tends towards zero for large n. If S_u ≤ S_d at the largest spatial scale A₀, then the undisturbed habitat will appear less diverse than disturbed habitat at all spatial scales. However, if S_u > S_d, then it is easy to identify the intermediate spatial scale at which undisturbed and disturbed habitats have equal species richness/diversity. The spatial scale at which the two habitats have equal diversity is given by solving (from equation 2), resulting in

(eqn 3)

In equation 3, n* refers to the number of times the original area has been bisected, so that the actual area at which undisturbed and disturbed habitat appear equally diverse is A_e = A₀/2^n*.

This model, whilst appealing in its simplicity, equates diversity with species richness at any given scale. However, this is an incomplete description of diversity because it ignores issues of species abundance within a community. We can address issues of relative abundance within our theoretical context, but only by imposing further assumptions on the way individuals are distributed among species (i.e. by imposing species-abundance relationships on the community). For example, if we assume that when a species is present in both halves of a given bisection it is equally abundant in each half, then we can calculate modified estimates for n* using Margalef's diversity index (D_Mg; Magurran 2004). This index is one of several commonly used indices which combine species richness and evenness into a single measure, and is calculated as:

(eqn 4)

where S is the total number of species recorded and N the total number of individuals recorded. Values for n* are then given by substituting (from equation 2) and N = N_u/2^n* (from the assumption of equal abundances in each bisection) into equation 4, substituting similarly for d subscripts, and equating the two expressions. This results in

(eqn 5)

where N_u and N_d are the total number of individuals in the undisturbed and disturbed habitats, respectively. Alternative formulations of species abundance relationships such as Shannon–Wiener and Simpson's indices are not mathematically tractable (Maddux 2004) and so are not considered here. We used equation 3 (richness) and equation 5 (diversity) to examine how changes in the slope (z-value) of SARs affect the perceived response of diversity to habitat disturbance at different spatial scales. We solved these equations numerically using parameters based on new empirical data (see below).

empirical approach

We compared the findings from our algebraic model with new field data for butterflies from undisturbed and moderately disturbed tropical forest habitats.

theoretical approach

We determined the spatial scale at which estimates of diversity were numerically equal in undisturbed and disturbed habitats (A_e) by solving equation 3 (species richness) and equation 5 (Margalef's diversity index) for decreasing values of z (the slope of SARs; Fig. 1). In order to do this, data on species richness and number of individuals in disturbed and undisturbed habitats were obtained from our empirical data for butterflies (undisturbed habitat; S_u = 56 species, N_u = 1150 individuals; disturbed habitat, S_d = 51, N_d = 1094; Table 1).

For both species richness (equation 3) and diversity (equation 5), the spatial scale at which the measure of diversity was equal between habitats was highly sensitive to small changes in the value of z, and the larger the decrease in z-values following disturbance, the greater the spatial scale at which diversity was equal in the two habitats (Fig. 1). These results show that empirical studies carried out at spatial scales smaller than values for A_e will report increased diversity following habitat disturbance, whereas those carried out at spatial scales larger than A_e will report decreased diversity following disturbance for a given decrease in z.

We explored possible values of A_e in the field by incorporating observed z-values of SARs from our empirical butterfly data. In order to do this, we produced plots of SARs and modelled them to a power function using linear regression (Fig. 2). We subdivided data from the two study grids using a rectangular bisection method (Harte et al. 1999) by first dividing the grid into two non-equal rectangles and then continuing to bisect areas until the smallest spatial scale was reached (a single trap), resulting in nine different spatial scales. SARs in the two habitats (Fig. 2) were described by the following equations: (i) undisturbed forest, log S = 0·420 (SE = 0·024) × log A + log 0·988 (SE = 0·031), (linear regression, F_1,7 = 315·55, P < 0·001, R² = 0·98); (ii) disturbed forest, log S = 0·400 (SE = 0·027) × log A + log 0·975 (SE = 0·035), (linear regression, F_1,7 = 230·43, P < 0·001, R² = 0·97).

Thus, our empirical data indicate similar z-values between habitats, with only a small decrease in z-value reported following disturbance (decrease in z = 0·020). These z-values were very similar regardless of how study grids were bisected and regardless of whether or not species richness values were rarefied (to account for the effects of abundance on species richness) prior to analysis (range of z-values, undisturbed forest z = 0·321–0·424; disturbed forest z = 0·301–0·410; change in z = 0·014–0·020). Incorporating these z-values into equation 3 (species richness) gave values of n* = 6·75 which shows species richness to be numerically equal between habitats at a spatial scale (A_e) of 0·73 ha. Numerical solutions to equation 5 (Margalef's index) revealed similar values of n* to those predicted by species richness (n* = 5·69; A_e = 1·53 ha).

empirical approach

There were significant positive linear relationships between spatial scale and both species richness and α diversity (Shannon–Wiener, Margalef and Simpson's indices) in undisturbed and disturbed forests (F_1,23 ≥ 107·60, P < 0·001, R² ≥ 0·82 for all regressions). However, measures of species richness and diversity increased at a significantly faster rate in undisturbed forest than in disturbed forest (ancova of species richness or diversity in undisturbed and disturbed forests with spatial scale as a covariate; habitat by spatial scale interaction; species richness, F_1,46 = 8995·79, P < 0·001; Margalef index, F_1,46 = 29·58, P < 0·001; Simpson's index, F_1,46 = 19·01, P < 0·001; but there was no significant interaction effect with Shannon–Wiener's index, P = 0·95). As a consequence, butterfly diversity generally decreased following disturbance when data were analysed at a large spatial scale by combining data from all traps on the study grid, whereas little difference in diversity was observed when data were analysed at a small spatial scale per trap (Table 1). These differences in α diversity patterns between habitats were supported by analyses of β diversity which showed that Morisita–Horn's index was significantly positively correlated with geographical distance in undisturbed forest but not in disturbed forest (Fig. 3; Mantel test, using 10 000 randomizations; undisturbed habitat, r = 0·23, P = 0·001; disturbed habitat, r = 0·04, P = 0·3).

Thus, higher z-values from SARs in undisturbed forest were associated with higher species turnover and greater increases in α diversity with increasing spatial scale in undisturbed forest, compared with disturbed habitat. These findings were supported by semivariogram analyses which showed that measures of Simpson's index were positively spatially autocorrelated in undisturbed forest but not in disturbed forest (Fig. 4). The distribution of semivariogram values fitted a Gaussian model best (IGF = 0·0007) and produced the following values: nugget = 2·49 (22% of the variance attributed to error); sill = 8·93 (78% of the variance attributed to spatial autocorrelation); and range = 345·4 m (distance at which samples become spatially independent). The high proportion of the variance explained by the sill (78%) suggests that Simpson's index in undisturbed forest was strongly positively spatially autocorrelated between pairs of sampling points separated by distances smaller than 345·4 m. Semivariogram values for the other diversity indices (Margalef and Shannon-Weiner) were all shown to exhibit a pure nugget effect in both undisturbed and disturbed forest, and thus, there was no detectable spatial autocorrelation in these data.

forest structure and habitat heterogeneity

Habitat structure and heterogeneity differed between habitats. Undisturbed forest had significantly taller, larger trees and higher overstorey, ground and canopy cover compared with disturbed forest (Table 2; t-test assuming unequal variance; tree height, t_36·73 = 6·92, P < 0·001; tree girth, t_38·81 = 2·67, P = 0·01; overstorey cover, t_34·61 = −2·35, P = 0·024; ground cover, t_47·60 = 2·74, P = 0·01; canopy cover t_26·36 = 4·99, P < 0·001). Undisturbed forest also had a significantly higher proportion of Dipterocarp trees and saplings and a significantly lower proportion of trees and saplings of the pioneer genus Macaranga (Table 2; Dipterocarp trees, t_46·44 = 9·87, P < 0·001; Dipterocarp saplings, t_45·65 = 5·38, P < 0·001; Macaranga spp. trees, t_24·00 = −7·39, P < 0·001; Macaranga spp. saplings, t_24·00 = −3·35, P < 0·001). Measures of tree height, tree girth and canopy cover had significantly greater variance in undisturbed forest than in disturbed forest, indicating more homogeneous vegetation following disturbance (Table 2; Levene's test for equality of variances, tree height, F_1,48 = 7·51, P = 0·009; tree girth, F_1,48 = 9·80, P = 0·003; canopy cover, F_1,48 = 41·42, P < 0·001).

diversity and spatial scale

Our theoretical approach produced a novel mathematical explanation for how a decrease in the slope of SARs can produce a scale-dependent response of diversity to habitat disturbance. Our theoretical model predicted numerically equal species richness and diversity in undisturbed and disturbed habitats at spatial scales of approximately 0·73–1·55 ha, with increased diversity following disturbance at smaller spatial scales and decreased diversity following disturbance at larger spatial scales. These findings were supported by our empirical data for butterflies showing significantly lower diversity in the disturbed habitat at large (78·5 ha) spatial scales but little difference between habitats at smaller (3·14 ha) spatial scales. However, by contrast with the model, our empirical data did not provide any evidence of increased diversity following logging at very small spatial scales. This is probably because even the smallest spatial scale we sampled (3·14 ha) was relatively large compared with our predicted A_e; other published studies carried out at much smaller spatial scales using walk-and-count transect methods (0·1–0·9 ha) have shown increased diversity of Lepidoptera following disturbance (Hill & Hamer 2004).

Ideally, we should have tested our theoretical findings with empirical data spanning a wider range of spatial scales than those we considered here, and including spatial scales well above and below the value at which diversity is numerically equal (Fig. 1). However, no single field study has yet been carried out over a sufficiently wide range of spatial scales to test this; most single-taxon studies generally use a single sampling method (e.g. walk-and-count transects or baited traps for butterflies), resulting in studies generally being either at small (i.e. transects) or large (i.e. traps) spatial scales (Hamer & Hill 2000). In addition, a thorough testing of our model not only requires data to be collected over a range of spatial scales from different habitats, but samples need to be contiguous and follow a spatially explicit design that allows species–area relationships to be examined. The empirical data presented in this study are from a comparison of only two sites, and thus, further data are required to test the robustness of our findings. Nonetheless, our data indicate that not only may the sampling method chosen by researchers have an impact on whether or not diversity is reported to increase or decrease following disturbance, but the way in which data are subsequently analysed may affect whether these diversity changes are significant. Other published studies (Tittensor et al. 2007), support our theoretical explanation for how a decrease in the slope of SARs may be associated with a scale-dependent response of diversity to disturbance.

methodological issues

There are some caveats to our theoretical explanation. Importantly, Harte et al.'s (1999) bisection algorithm, which forms the basis of our algebraic model, is not universally accepted. For example, Maddux (2004) demonstrated that Harte et al.'s (1999) probability rule for producing power law SARs only works when certain types of ‘well-shaped’ rectangles are bisected. However, arguments used for equation 3 (species richness), although derived using the bisection method, do not rely on this method and the same result can be derived by considering the intersection of two continuous species area curves in log–log space. Thus, potential problems associated with the bisection method only affect equation 5 (diversity index). Nonetheless, Harte et al.'s (1999) bisection algorithm is one of very few robust methods with which to produce power function SARs that can be used to test further hypotheses. In addition, SARs derived from empirical data in this study were fitted to a power function and results were qualitatively the same regardless of whether sub-divisions of the study grids followed a rectangular bisection method or not (see Maddux 2004). Perhaps more importantly, changes in z-values following disturbance were similar regardless of how data were combined across study grids to construct SARs. Thus, we are confident that using Harte et al.'s (1999) bisection algorithm to produce power function SARs provides a robust method on which to test further hypotheses.

In this study, our theoretical model assumes that habitat disturbance reduces the slope of SARs (z-values). This assumption is supported empirically across a range of disturbance types and taxa (Cannon, Peart & Leighton 1998; Reilly, Wimberly & Newell 2006). Nonetheless, increases in z-values following disturbance may occur (Reilly et al. 2006). For example, habitat disturbance could increase z-values if the frequency and intensity of disturbance was such that it produced a mosaic of habitat types and thus increased habitat heterogeneity. In our study, incorporation into our algebraic model of increased z-values following habitat disturbance would produce results opposite to those presented. Thus, we would have reported decreased diversity following disturbance when measured at small spatial scales, but increased diversity when measured at larger spatial scales. Although this was not observed in our study, such effects may occur in other taxa (Hill & Hamer 2004) and deserve more study.

impacts of spatial scale and disturbance

Butterfly species richness and α diversity were positively related with spatial scale in both undisturbed and disturbed habitats, although these relationships increased at a significantly faster rate in undisturbed forest compared with disturbed forest. β diversity was also positively correlated with increasing distance between samples in undisturbed forest but not in disturbed forest. In addition, measures of α diversity (Simpson's index) were spatially autocorrelated in undisturbed forest, but not in disturbed forest. As a consequence of these relationships between diversity and spatial scale, there was a significant decrease in butterfly α diversity following disturbance when measured at a large spatial scale but no significant change in diversity at a small spatial scale (Table 1). Thus, higher z-values in undisturbed forest were associated with higher β diversity and greater increases in α diversity with increasing distance in undisturbed forest, compared with disturbed forest.

These patterns of butterfly diversity were associated with changes in habitat structure following disturbance. Analysis of vegetation data showed that the variance of some variables was higher in undisturbed forest compared with disturbed forest, indicating a reduction in the structural heterogeneity of forests following disturbance. For example, undisturbed forest had significantly greater heterogeneity in canopy cover and tree size, which in turn is likely to affect the amount of light penetrating the forest, and thus the distribution of butterflies throughout the forest (Hill et al. 2001). In this study, the reduction in habitat heterogeneity following disturbance was associated with a reduction in the rate at which α and β diversity increased with distance between samples in disturbed habitats, and affected patterns of spatial autocorrelation. These diversity changes resulted in a reduction in the slope (z-value) of SARs in disturbed habitats (Drakare, Lennon & Hillebrand 2006), which our model showed was sufficient to result in scale-dependent estimates of diversity changes following disturbance. Thus, for butterflies, changes in habitat heterogeneity were associated with changes in α and β diversity, consistent with the observed reduction in z-value following disturbance, which in turn resulted in a scale-dependent response of diversity to disturbance in both our theoretical and empirical studies. Thus, understanding habitat heterogeneity and how it affects SAR z-values appears crucial to understanding responses of species to disturbance.

differences among taxa in scale-dependence

This study suggests that responses of butterflies to habitat disturbance are scale-dependent, but the role of spatial scale may differ among taxa. For example, the responses of birds to moderate tropical forest disturbance are scale-dependent, but bird studies at large spatial scales report increased diversity following disturbance and decreased diversity at small spatial scales, the opposite response to that for Lepidoptera (Hill & Hamer 2004). Part of the reason for this difference may be that birds are usually sampled over much larger spatial scales than Lepidoptera, and the higher dispersal ability of birds may result in them experiencing habitat structural characteristics differently from Lepidoptera, even when the spatial scale of studies are apparently similar (Loreau 2000; Hill & Hamer 2004). Overall, conclusions from bird and Lepidoptera studies indicate that we might expect increased diversity following disturbance at both very small and very large spatial scales and decreased diversity following disturbance at intermediate spatial scales, although the precise values for spatial scale are likely to vary among taxa. The challenge now is to determine the conditions under which our algebraic model might support these conclusions; non-linear relationships between spatial scale and diversity might arise from non-linear relationships between z-values and spatial scale (Crawley & Harral 2001; but see Smith et al. 2005), and more empirical and theoretical studies clearly are needed.

sensitivity of response to small changes in sars

This study suggests that only very small changes in z-values from SARs following disturbance were sufficient to result in qualitatively different responses of diversity to disturbance at large and small spatial scales. This apparent sensitivity to very small changes in SARs and diversity-area relationships (Figs 2 and 3) highlights that no matter how minor the habitat disturbance, quantifying the effect of that disturbance on diversity is likely to be dependent on the spatial scale of analyses. A far broader ecological implication is that when comparing the diversity of any two communities, if there is even a minor difference in SARs between communities, then the estimated difference in diversity may be dependent on the spatial scale of sampling. Techniques that assess differences in community composition, rather than diversity, overcome these problems, but this does not mean that measures of diversity should be ignored, and such measures are likely to continue to be useful in highly diverse areas such as tropical forests.