be regarded as a random variable Xi,j distributed as a Bernoulli trial with probability pi,j. Second, the predicted butterfly species richness Yi in square i was ...
SUPPORTING INFORMATION The need for large-scale distribution data to estimate regional changes in species richness under future climate change APPENDIX S3: Projections of future changes in butterfly species richness with local and continental models. We used Generalized Additive Models (GAM) (Hastie & Tibshirani, 1987) with 4 splines (k = 5) to build the species distribution models for each butterfly species in the biomod2 R-package (Thuiller et al., 2016). From these models, we derived the probabilities of occurrence (i.e. climatic suitability) for each species in each 10-km resolution square of the three countries (SW Finland, Belgium/Netherlands and NE Spain) under present-day climate (1971-2000) and under future climate change scenarios (2021-2050 and 2051-2080). In each country, butterfly species richness was calculated for each combination of time period and climate change scenario using both the local and continental models. First, each individual model produced a probability pi,j of species j to be present in square i. The presence of species j in square i can be regarded as a random variable Xi,j distributed as a Bernoulli trial with probability pi,j. Second, the predicted butterfly species richness Yi in square i was calculated as the sum of the probabilities of occurrence of all species in that square, instead of converting the probabilities of occurrence to binary presence-absence predictions and subsequently adding up the predicted presences for each species (Hill et al., 2009). This approach does not have the tendency to overpredict species richness (Dubuis et al., 2011), it does not require an arbitrary cut-off value and it prevents biasing the richness to the extremes when a large number of species have a probability close to the cut-off value. We only used the species that are currently present in the different countries to estimate species richness for the future time periods, thereby excluding new native species that can potentially colonize the countries. The log transformed butterfly species richness in square i was then modelled with a linear model for each country separately with the following parameterization:
log 𝑌! = 𝛽𝐿 + 𝛶𝐿𝑇 + 𝜀! L is a binary variable indicating the modelling approach (‘local’ or ‘continental’). T is a factor variable with the different combinations of time period and climate change scenario (‘1971-2000’, ‘2021-2050Bambu’, ‘2021-2050-Gras’, ‘2021-2050-Sedg’, ‘2051-2080-Bambu’, ‘2051-2080-Gras’, ‘2051-2080-Sedg’). The formula has no intercept. 𝛽 is the coefficient of the modelling approach variable (L) and 𝛾 is the coefficient of the interactions between L and T. Two examples are provided below: Continental models and in 1971-2000: log 𝑌! = 𝛽!"#$%#&#$'( + 𝜀! Continental models under BAMBU scenario in 2021-2050: log 𝑌! = 𝛽!"#$%#&#$'( + 𝛾!"!#!!"#"!!"#$% + 𝜀! We chose this parameterization, as it is a convenient way to obtain the ratios of butterfly species richness between the present-day situation and the future climate change scenarios. As butterfly species richness (Yi) was modelled using the log scale, the relative change in butterfly species richness between the present-day and the future situations was simply the exponentiation of the interaction coefficients (𝛾). The confidence interval of this ratio was the exponentiation of the confidence intervals of the interaction coefficient.
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The hypothesis we tested here was that there is no difference in butterfly species richness changes between the local and the continental models, i.e. the 𝛽 coefficients for the local and the continental models (averaged over the three climate change scenarios) are the same. We tested for differences between local and continental models in the projected butterfly richness changes from 1971-2000 to 2021-2050, from 1971-2000 to 2051-2080 and from 1971-2000 to 2021-2080. Since the residuals of the linear models (εi) showed a strong spatial autocorrelation in all countries (up to 150-300 km), we applied generalized least squares with a spatial correlation structure using the nlme Rpackage (Piñheiro et al., 2009). References Dubuis, A., Pottier, J., Rion, V., Pellissier, L., Theurillat, J.P. & Guisan, A. (2011) Predicting spatial patterns of plant species richness: a comparison of direct macroecological and species stacking modelling approaches. Diversity and Distributions, 17, 1122-1131. Hastie, T. & Tibshirani, R. (1987) Generalized additive models: some applications. Journal of the American Statistical Association, 82, 371-386. Piñheiro, J., Bates, D., DebRoy, S., Sarkar, D. & R Core team (2009) nlme: Linear and Nonlinear Mixed Effects Models. Thuiller, W., Georges, D., Engler, R. & Breiner, F. (2016) biomod2: Ensemble Platform for Species Distribution Modeling.
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