Growth rates on conditioned soil (r_i,j) were derived from plant biomass on PSF plots (input data) in 2017 and/or 2018 (Kulmatiski et al. 2011). ‘Neutral’ growth rates (ν_i) were set to be growth rates on ‘home’, ‘away’, or across all PSF plots. Growth rates were calculated from final biomass on different soil types. For example for P_i,j, r_i,t = (P_i,j/P₀)^1/T-1, where T=52 time steps, P₀ = g m^-2.

Two models (equation 1 or 2), three sources of input data (2017 only, 2018 only, or 2017 then 2018), and three ‘neutral’ growth rates (home, away, all plots) produced 18 Null and 18 PSF model simulations. Carrying capacities were defined as the mean ± two standard deviations of total plant biomass (Κ) in diversity-productivity plots or plant species biomass (κ) across all PSF plots. Models were run for three 52-time step iterations (t). To simulate harvest, each 53rd time step, plant biomass was set to 1 % of P_i,t. Simulations were performed in R (R Core Development Team, 2015).

We estimated the net biodiversity effect based on calculations proposed by Loreau & Hector (2001), which estimate the yield increase ΔY of a plant community compared to the combined performance of plant species in monocultures. We further used equations to partition the net biodiversity effect (ΔY) into selection and complementarity effects (Loreau & Hector, 2001).

To describe species richness effects, we fit random intercept (linear mixed) and linear mixed models in R (R Core Development Team, 2015) using lme4 (Schielzeth & Forstmeier, 2009; Bates et al, 2015; Schmid et al., 2017). Due to low establishment in 2015, we analyzed species biomass data by plot and year (i.e., the sum of spring and fall harvests as g dry mass m^-2) for 2016 and 2017. Analyses followed those of Roscher et al. (2008) on the pre-existing diversity productivity experiment. The maximum likelihood method was used to find the ‘best’ random model among random intercepts such as species composition (com), year and their interaction (Roscher, Schumacher, Weisser et al., 2007; Roscher et al., 2008). The effects of block (soil gradient; Huston & McBride, 2002; Weisser et al., 2017) and the interaction of block and Year (block:Year) were included as random intercepts for the pre-existing experiment, but not for the current experiment which was located in one area. From these random intercept models with no fixed effects, we extracted fixed effects of interest (Schmid et al., 2017). The contrast of monoculture and polyculture (MP) and the linear contrast of species richness (SR) were fixed effects (MP+SR). When analyzing mechanisms of the diversity-productivity relationship (selection and complementarity), the model was fit without the contrast of mono- and polycultures. Mixed models were fit with the maximum likelihood method to derive statistical significance of fixed effects from likelihood ratio tests (Χ ²; Roscher et al., 2016). To avoid pseudoreplication, replicate plots were averaged prior to analyses so that the diversity productivity dataset had 300 samples (100 species compositions x three years). To compare species richness effects between experiments and simulations, a second random intercept model was used with the fixed effects model: data+SR+data:SR and random intercept model: com+com:Year where data is the data source (i.e., observed, PSF, or Null model predictions).