Supplementary Figures and Tables Supplementary Figure 1: Niche overlap, niche equivalency and similarity tests according to the PCA-env approach of Broennimann et al. (2012). Upper panels show the kernel density estimates of the native and invasive populations in PCA space. Darker shading indicate higher density of occurrence records, wherein the available climate space of native and invasive populations is indicated by solid (100%) and dashed lines (50%). The histograms show the results of null-models of niche equivalency and similarity according to Warren et al. (2008). The red diamonds indicate the observed niche overlap. For abbreviations of the climate variables see Methods. Supplementary Figure 2: N-dimensional hypervolumes of native and invasive populations of the wasp spider. Solid points indicated actual species records wherein light dots indicate the density of species records in environmental space. The correlations of PCs with climatic variables are provided in Suppl. Table 1. Supplementary Figure 3: Population genetic parameters, estimated from our modelling approach for 100 datasets of each 400 ancestry informative SNPs in invasive Northern European populations. a(f) represents the estimated founding allele frequency for European alleles. a(m) shows the estimated frequency of European migrant alleles. M shows the migration rate (4Nm). t represents the time scaled by 2N generations. Supplementary Table 1 Bioclimatic variables used for SDM development and niche quantification. Variable bio1 = Annual Mean Temperature bio2 = Mean Diurnal Range bio3 = Isothermality bio4 = Temperature Seasonality bio5 = Max Temperature of Warmest Month bio6 = Min Temperature of Coldest Month bio7 = Temperature Annual Range bio8 = Mean Temperature of Wettest Quarter bio9 = Mean Temperature of Driest Quarter bio10 = Mean Temperature of Warmest Quarter bio11 = Mean Temperature of Coldest Quarter bio12 = Annual Precipitation bio13 = Precipitation of Wettest Month bio14 = Precipitation of Driest Month bio15 = Precipitation Seasonality bio16 = Precipitation of Wettest Quarter bio17 = Precipitation of Driest Quarter bio18 = Precipitation of Warmest Quarter bio19 = Precipitation of Coldest Quarter PET_HE_bio1 = Annual Mean Potential Evapotranspiration PET_HE_bio4 = Potential Evapotranspiration Seasonality PET_HE_bio5 = Max Monthly Potential Evapotranspiration PET_HE_bio6 = Min Monthly Potential Evapotranspiration PET_HE_bio7 = Annual Range of Potential Evapotranspiration PET_HE_bio10 = Mean Potential Evapotranspiration of Highest Quarter PET_HE_bio11 = Mean Potential Evapotranspiration of Lowest Quarter Supplementary Table 2: Sampling sites for all populations used for genome and transcriptome sequencing. ID Japan Baltic Sweden Portugal Italy Reference Genome Reference transcriptome Region East Asia Invasive European Invasive European Native European Native European Extra European Invasive European Native European Thermal stress RNA seq Invasive European Country Region City GPS N GPS E Japan Ibaraki 36 140,1 Japan Kagoshima 31,6 130,56 Japan Nagano Tsukuba Shimofukmotochö Azusagawa-Azusa 36,23 137,97 Japan Kagoshima Amami-Oshima 28,28 129,43 Japan Shiga Ozigaoka 35,12 136,07 Latvia Dobele Annenieki Parish 56,7 23,13 Latvia Limbazi Skulte 57,35 24,45 Estonia Pärnu Pärnu 58,3 24,62 Latvia Limbazi Ainazi 57,87 24,37 Sweden Kalmar county Soderakra 56,45 16,67 Sweden Kalmar county Kalmar 56,63 16,22 Sweden Skane county Kristianstad 55,95 14, 10 Portugal Alentejo Corte Pequena 37,7 -7,85 Portugal Alentejo Santo Jao 37,67 -7,83 Portugal Alentejo Mertola 37,68 -7,6 Portugal Alentejo Mertola 37,65 -7,6 Portugal Alentejo Beja 37,8 -7,85 Italy Campania Giugliano 40,93 14,05 Italy Calabria Crosia 39,57 16,72 Italy Puglia Taranto 40,47 17,3 Portugal Madeira Camacha 32,7 -16,8 Germany Schleswig Holstein Ascheberg 54,13 10,32 Portugal Alentejo Corte Pequena 37,7 -7,85 Portugal Alentejo Mertola 37,68 -7,6 Portugal Alentejo Mertola 37,65 -7,6 Lithuania Klaipeda Klaipeda 55,8 21,15 Latvia Liepaja Niedasciems 56,08 21, 12 Latvia Kurzemes Viduskrogs 56,68 22,13 Latvia Dobele Annenieki Parish 56,7 23,13 Estonia Pärnu Pärnu 58,3 24,62 Estonia Pärnu Pärnu 58,31 24,6 Latvia Limbazi Ainazi 57,87 24,37 Sweden Kalmar county Soderakra 56,45 16,67 Sweden Skane county Kristianstad 55,95 14,1 Supplementary Table 3: Factor loadings, eigenvalues and explained variance of the PCs used to compute 4-dimensional hypervolumes of native and invasive populations of the Wasp spider. Variable BIO1 BIO2 BIO3 BIO4 BIO5 BIO6 BIO7 BIO8 BIO9 BIO10 BIO11 BIO12 BIO13 BIO14 BIO15 BIO16 BIO17 BIO18 BIO19 PET_HE_bio1 PET_HE_bio4 PET_HE_bio5 PET_HE_bio6 PET_HE_bio7 PET_HE_bio10 PET_HE_bio11 PC1 -0.91 -0.92 -0.75 0.14 -0.97 -0.72 -0.29 -0.16 -0.85 -0.95 -0.81 0.76 0.69 0.78 -0.74 0.70 0.78 0.89 0.41 -0.98 -0.81 -0.96 -0.94 -0.85 -0.97 -0.95 Eigenvalues 16.4 Explained Variance [%] 62.9 PC2 -0.34 0.06 -0.56 0.90 0.00 -0.64 0.85 0.50 -0.45 -0.05 -0.55 -0.53 -0.51 -0.37 -0.04 -0.53 -0.41 -0.01 -0.76 -0.14 0.26 0.01 -0.23 0.18 -0.01 -0.24 PC3 -0.07 -0.12 0.15 -0.34 -0.22 0.04 -0.33 -0.18 -0.01 -0.19 0.04 -0.35 -0.37 -0.25 0.23 -0.36 -0.27 -0.23 -0.31 -0.11 -0.48 -0.24 0.09 -0.45 -0.23 0.06 PC4 0.16 -0.13 0.04 -0.16 0.03 0.22 -0.25 0.66 -0.10 0.10 0.17 -0.05 -0.23 0.27 -0.37 -0.21 0.26 0.24 -0.30 0.02 0.03 -0.02 -0.03 0.00 0.00 -0.02 5.1 19.5 1.7 6.5 1.2 4.6