Data analysis
Linear mixed-effects models were used to investigate the effects of aridity, grazing, haying and their interaction on plant species richness and ecosystem functions. First, we tested the response of plant species richness to the fixed effects of aridity (as a continuous, linear predictor), grazing (as a binary predictor: 0 = no grazing or 1 = grazing), and haying (as a binary predictor: 0 = no haying or 1 = haying). These models included the two-way interactions between aridity and grazing or haying. None of the sites had both grazing and haying, and thus the models did not include the grazing × haying interactions. A random intercept was included for each site. The fixed effects model structure is shown in Table 1. Second, we tested the response of aboveground biomass and soil organic C to these same predictors, and also included plant species richness (continuous, linear predictor) and its interaction with aridity as fixed effects (see model structure in Table 1). The sites were classified into three land-use types (exclosure, grazing, or haying), and land-use was divided into the two factors grazing (yes or no) and haying (yes or no).
Piecewise structural equation modelling (SEM) was used to examine if aridity and land-use affect aboveground plant biomass and SOC through changes in plant species richness (Lefcheck 2016). SEM was performed with the specification of a conceptual model of hypothetical relationships, based on a priori and theoretical knowledge (Grace 2006). The initial model included the paths from aridity, grazing, and haying to plant species richness, aboveground biomass, and SOC, and from species richness to aboveground biomass and SOC. According to modification indices, the correlation between grazing and haying was added to the model. Results of linear mixed-effects models showed that there were significant interaction effects between aridity and grazing on species richness and between aridity and haying on aboveground biomass. The paths from aridity × grazing to species richness and from aridity × haying to aboveground biomass were added to the SEM model. The path from aridity × haying to aboveground biomass was not significant, and it was removed from the model. Grazing is binary (0 = no or 1 = yes), haying is binary (0 = no or 1 = yes), and the aridity x grazing interaction is the product of aridity and grazing. The Akaike information criterion (AIC) and Shipley’s test of d -separation were used to evaluate the fit of the models (Grace 2006; Lefcheck 2016), and the final test of d-separation (Fisher’s C = 4.51, P = 0.11) showed a good model fit. Response variables were log transformed to meet linear model assumptions where appropriate. Linear mixed-effects models and SEM were conducted using the “lme” and “piecewiseSEM” packages in R 4.0.2 (R Development Core Team 2013; Lefcheck 2016; Pinheiro & Sarkar 2016).