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Appendix S2 Analyses of potential data biases. We first investigate the influence of sampling method on density. The estimation method varies and includes mark-recapture studies (37% of studies, 40% islands and 35% mainlands), continuous observation of marked individuals in life-history studies (18% of studies, 6% islands and 25% mainlands), censusing of quadrats (27% of studies, 39% islands and 20% mainlands), and strip transects (5%, of studies 0% islands and 9% mainlands). When considering only mark-recapture studies and controlling for energetic constraints, island densities remain an order of magnitude higher than those on mainlands (N island º 10 1.00±0.14 N mainland ). There is no interaction between the island or mainland factor and sampling method when controlling for energy use and availability. Sampling method has a relatively small, but significant effect on densities when included as a factor in the regression analysis along with energy use and availability and the island/mainland factor (F=11.6, P<1x10-8 for sampling method in an ANOVA). The island effect is somewhat reduced, but still large, when including sampling method as a factor (N island º 10 0.91±0.15 N mainland ; F=186.4, P<1x10-15 for island in an ANOVA; F [7, 649]=120.3, r 2 =0.56, P<1x10-15 ). In these analyses, transect sampling, which accounts for just 5% of studies, is the only method that significantly influences (depresses) densities (P<0.001). We next investigate the influence of sampling area on density. Sample areas approximately ranged from 100m linear transects to 500,000m 2 (which has the potential to bias abundance estimates, Gaston et al. 1999). A subset of studies (25%) were ranked for sampling area. There was a tendency toward smaller sampling areas on islands (>5ha: 3% of studies on islands and 16% mainlands; 1-5ha: 9% of studies on islands and 10% mainlands; 0.1-0.99ha: 39% of studies on islands and 53% mainlands; <0.1ha: 48% of studies on islands and 21% mainlands). While 74% of studies were smaller than 1 hectare on mainlands, the percentage was 87% on islands. Sampling area had a significant effect on densities when included as a covariate in the regression analysis along with energy use and availability and the island/mainland factor (F=53.8, P<1x10-15 for sampling area in an ANOVA). The island effect remains an order of magnitude when including sampling method as a covariate (N island º 10 1.19±0.84 N mainland ; F=92.6, P<1x10-15 for island in an ANOVA; F [9, 155] =54.9, r 2 =0.76, P<1x10-15 ). We included the interaction between sampling area and island to avoid confounding these effects. While the largest sampling areas (>5 ha) depressed densities, the other sample area classes were little differentiation in their influence on density. A subset of population density records (40%) were ranked by Rodda on at least one other element of sampling quality including representativeness of habitat, sample size, temporal sampling, isolation of sampling area, and method validation. There were no systematic differences in these quality measures between islands and mainlands.

Table S1 Ecological models of lizard population density (lizards/ha) across islands and mainlands and for each group independently accounting for phylogenetic autocorrelation. Models include energy use (B, kj/year) and supply (P, annual minimum net primary productivity) and competitor (COMP), predator (PRED), or combined (BOTH) species richness (SR). Smaller AIC values indicate better models. The AIC values can be compared to those of the non-phylogenetic null model (N B+P) for each group (combined: 1001.5; mainlands: 574.1; islands: 347.6). Pagel s λ indicates phylogenetic autocorrelation and ranges between 0 (phylogenetic independence) and 1 (species traits covary in direct proportion to their shared evolutionary history). Phylogenetic autocorrelation is not significant for mainland populations. For the combined mainland and island analysis, phylogenetic autocorrelation was not significant when considering grouped predators or competitors. However, phylogenetic autocorrelation was significant for some particular predator and competitor taxa.

Table S2 Coefficients for predator and competitive release models accounting for phylogenetic autocorrelation. Coefficients for energy use (B, kj/year) and supply (P, annual minimum net primary productivity) in lizard density (lizards/ha) models from Table S1. Values in brackets indicate standard errors.

Table S3 Models examining the influence of the richness of different taxa on combined mainland and island lizard densities. Lizard density models (lizards/ha) across the island and mainland groups. All models include energy use (B, kj/year) and supply (P, annual minimum net primary productivity) and competitor or predator species richness (see Table S4 for coefficients). Smaller AIC values indicate better models. Spatial autocorrelation is accounted for using lag models with an 800km neighborhood. Higher absolute values of Moran s I indicate stronger spatial autocorrelation.

Table S4 Coefficients for models examining the influence of taxa richness on combined mainland and island lizard densities. Coefficients for energy use (B, kj/year) and supply (P, annual minimum net primary productivity) in lizard density (lizards/ha) models from Table S3. Values in brackets indicate standard errors.

Table S5 Coefficients for predator and competitive release models explaining lizard densities. Coefficients for energy use (B, kj/year) and supply (P, annual minimum net primary productivity) in lizard density (lizards/ha) models from Table 1. Values in brackets indicate standard errors.