We take a spatial pattern-matching approach, by first generating spatial patterns of species richness and turnover at different scales, and then statistically matching these with each other. We have five main methodological objectives: (i) to fit two types of SAR model to the species distribution data, and to map spatial variation in the SAR model parameters; (ii) to map species richness patterns at different spatial scales; (iii) to define and discuss the various measures of species turnover we employ; (iv) to map species turnover patterns, using these measures, again across a series of spatial scales; (v) to statistically test for relationships between these maps of species richness, species turnover and SAR parameters. As mentioned in the Introduction, the latter are measures of turnover based on species accumulation with area; we go on to show that the two SAR models used imply two types of scale invariance for species richness.

#### species richness and the species–area relationship

We generated maps of species richness at a series of successively larger quadrat sizes (coarser scales), using a moving-window algorithm: a larger quadrat was positioned centrally over each base 10-km quadrat (Fig. 1) and all species present counted. This introduces additional spatial autocorrelation into the species richness patterns, which is dealt with using appropriate statistical methods.

The process of generating these maps involves fitting SAR models, but we are also interested in these in their own right. At coarse scales there is clearly a problem caused by the encroachment of sea into the larger quadrats. We compensated for this by using a SAR model to correct upwards the species richness of these coastal quadrats to that expected if there was 100% land. We calculated a separate SAR for each of the 2406 10-km quadrats (Fig. 1). For each such quadrat, we gathered richness and area data points by expanding a ‘window’ around it (from 10 km × 10 km to 290 km × 290 km in steps of 20 km, giving 15 points). We then fitted two alternative SAR models to each quadrat’s set of 15 points. These were the familiar log(species)–log(area) relationship logR =*c* +*z* log A (the power SAR –Arhennius 1921), and the species–log(area) relationship *R* = *k* +*m* log A (the logarithmic SAR –Gleason 1922), where R and A are species richness and area, respectively. The richness of each coastal quadrat was then corrected (using the prediction for 100% land area) by applying whichever of the SAR models (power or logarithmic) that gave the best fit (i.e. highest *r*^{2}) to its local data. We were able to map spatial variation in *m* and *z* by assigning the parameter estimates to their corresponding 10-km quadrat.

Applying this land-area correction raises some interesting problems. First, the species–area relationship is not spatially homogenous, i.e. the coefficients of the SAR vary spatially. So each fitted curve is an integration of different relationships. Secondly, the correction applied in each quadrat may be biased by the geometry of the surrounding landmass. For a given area, long strips of quadrats contain more species than a more compact arrangement (e.g. Goodall 1952; Kunin 1997). In the former circumstance, the correction applied may be too large. Thirdly, the richness data used to calculate each SAR are not independent. This affects all nested-design (rather than inter-island) species–area curves. The magnitude of these three problems is difficult to quantify precisely, but it is highly likely that the corrected patterns reflect true species richness at different scales more accurately. Unless otherwise mentioned, we use the corrected patterns in subsequent analyses.

#### turnover measured using indices

In contrast with the components of turnover that have units of numbers of species, turnover indices (essentially dissimilarity indices) implicitly correct for the number of pairwise comparisons (species) – they measure turnover on a ‘per-species’ basis. Two such indices that do not adjust for local species richness gradients are those of Whittaker (1960) and Sorenson. The first of these is the most widely used index of turnover:

where *R*_{9} is the species richness of a large quadrat encompassing the focal quadrat and its eight neighbours (when all are available), and *R*_{1} is the species richness of the focal quadrat alone. It has a lower limit of zero but no upper limit. Clearly, areas with a stronger SAR will have higher β_{w} values.

The second index is a composite Sorenson index, defined as the average dissimilarity of the focal quadrat with each of its neighbours:s

where *n* is the number of pairwise comparisons (*n* = 8 when all neighbours are available). We use this index to show that the average pairwise dissimilarity (uncorrected for local species richness gradients) measures essentially the same property as β_{w}. β_{sor} has an upper limit of one (when the focal quadrat has no species in common with any neighbouring quadrat) and a lower limit of zero (if the focal quadrat and all neighbouring quadrats have identical species lists).

If there is a large difference in richness between quadrats, then β_{sor} and β_{w} will always be large. To focus more precisely on compositional differences, we introduce another turnover index:

*S*_{2} resembles Simpson’s asymmetric index (Simpson 1943), but our form is symmetric. Any difference in species richness inflates either *b*′ or *c*′, so choosing the value of the smaller of these decreases the influence that any local species richness gradient has on dissimilarity. β_{sim} has an upper limit of one (focal quadrat has no species in common with any neighbour) and a lower limit of zero (all have identical species lists).

We can empirically verify that β_{sim} adjusts for local richness gradients (between adjacent quadrats), i.e. it is not influenced by *local* differences in richness, while β_{sor} does not, i.e. is inflated by these differences. This can be done by comparing β_{g} (below) with the difference between β_{sim} and β_{sor}. First, we subtract the maps of β_{sor} from β_{sim} on a quadrat by quadrat basis. Then we compare the resulting map of differences with a map of local species richness gradients. There should be a good match (strong correlation) if β_{sim} removes the local species richness gradient component from β_{sor}. As we expect β_{w} to behave like β_{sor}, this test can be applied to it, too. We define the local richness gradients for a focal quadrat as β_{g}, the average proportional difference in richness between the focal quadrat and its neighbours:

where α_{f} and α_{n} are species richness of the focal and neighbouring quadrats. β_{g} has a lower limit of zero (identical richness throughout) and an upper limit of two (when the focal or all neighbouring quadrats have no species at all, so *a* and either *b* or *c* are zero).

As with species richness, all measures of turnover were calculated at different spatial scales: 10, 30, 50, 70 and 90 km. At each scale, we calculated turnover on the 3 × 3 neighbourhood basis (Fig. 1), and drew maps of each measure. Again, at coarser scales more quadrats have less than 100% land. In this case, we attempted no correction, because this is a much more difficult problem. The presence or absence of particular species would have to be predicted accurately. A related issue is that differences in land area and peninsularity will tend to produce differences in species richness between the quadrats and so inflate turnover for all measures except β_{sim}. The magnitude of this effect will depend on the strength of the species–area relationship. It is not clear whether this effect, if real, will be evident in the spatial pattern of turnover, because variation in turnover caused by other factors may be large relative to variation produced by these area differences alone. For example, because values of the exponent *z* in *R* = cA^{z} are relatively small (mean *z* = 0·12) in the present study, differences in species richness caused by differences in land area may be substantially less than those attributable to other causes of turnover. However, finding such a coastal effect is not necessarily evidence of a methodological artefact: there may be sound biological reasons for increased turnover at the coast. We check for coastal effects on the turnover maps.

##### Turnover as SAR model parameters

The strength of the SAR (magnitude of *z* or *m*) can reasonably be considered to be a measure of turnover based on species gain. Clearly *z* and *m* are affected by richness gradients. We estimated *z* and *m* using a nested set of 15 quadrats around each 10-km square, as described above.