2.4. Global predictions and future projections of SHB pupal performance

We used performance curves of pupal survival rates (S(TM)) and development time (D(T)) to predict both processes separately based on current soil temperature and moisture conditions. To account for the strong geographic differences in diurnal variation of soil temperature and the corresponding consequences for pupal performance we predicted performance for each of the 3‐hr intervals per month and integrated them in a second step by averaging. We then used Equation (3.1) to calculate the composite index of pupal performance per month, whereas we used the maximum of S(TM)/D(TM) across all grid cells and months (global maps for monthly survival rate, development time and pupal performance are provided in Figures [Link], [Link], [Link]). Monthly pupal performance was further condensed in two ways: (a) it was averaged across the months per grid cell assuming that pupal performance accumulates across the varying conditions within a year; and (b) by extracting the highest level of performance across the months per grid cell assuming that invasiveness depends on maximum performance during shorter periods.

To assess the predictive ability of the model in general and to discriminate the relevance of mean annual climatic conditions from short‐term optimal conditions for pupal performance, we used actual reported georeferenced occurrences in the native and invaded range (Table S3). We included only established populations by focusing on observations of 3 years or longer. Since these data represent presence‐only data, we used the continuous Boyce index (Boyce, Vernier, Nielsen, & Schmiegelow, 2002; Hirzel, Lay, Helfer, Randin, & Guisan, 2006) to assess the quality of our predictions. This index varies from −1 (worse than expected by chance) to 0 (not better than expected by chance) to 1 (perfect predictions). The Boyce index compares the predicted frequency distribution of evaluation points with their expected frequency based on the distribution within a selected area and is thus sensitive to the spatial extent of the selected area. To overcome a potential bias by selecting a too large area, we analysed occurrences in different regions separately by calculating convex hulls for three areas in Africa, three in North America and one in Australia with sufficient data points. Further, we considered some uncertainty in the georeferences of the observations and used the maximum of pupal performance across all grid cells within a 25 km buffer. For evaluation of the Boyce index, we used an analogy to the categorisation recommended by Landis and Koch (1977) for the true skill statistic, which also ranges between −1 and 1, the following: excellent, Boyce index > 0.75; good, 0.40 < Boyce index < 0.75; and poor, Boyce index < 0.40.

Future projections of pupal survival rates, development time and pupal performance were calculated analogous to current conditions but for means of comparison, we used the same value of max[S(TM)/D(TM)] for Equation (3.1) as for current conditions.

All analyses were performed in the statistical environment R (R Core Team, 2016) using the packages colorRamps (Keitt, 2012), ecospat (Broennimann, Di Cola, & Guisan, 2016), gdalUtils (Greenberg & Mattiuzzi, 2015), gtools (Warnes, Bolker, & Lumley, 2015), maptools (Bivand & Lewin‐Koh, 2016), minpack.lm (Elzhov, Mullen, Spiess, & Bolker, 2016), ncdf4 (Pierce, 2015), raster (Hijmans, 2017), RColorBrewer (Neuwirth, 2014), rgdal (Bivand, Keitt, & Rowlingson, 2016) and sp (Pebesma & Bivand, 2005).