Fig. 1 Schematic diagram of the grazing experiment treatments. Dark color treatments indicate sampled treatments. The grazing experiment treatments (each ca. 4.4 ha) were arranged in a randomized complete block design, which included four grazing intensity treatments with three repeats at each grazing intensity.
Vegetation sampling
Within each randomized block of different grazing intensities, a representative sample plot with the same terrain (5 m×5 m) was selected. The origin of the coordinate in each sample is defined as the upper left corner of the sample. Within this sample plot, a 1 m×1 m sample frame was placed 25 times in turn, and the precise spatial position ofS. breviflora and C. songorica in the sample is determined by the tape. In this fashion we measured the population density ofS. breviflora and C. songorica on 15 August 2019.
Data analysis
The population density of S. breviflora and C. songoricawere tested for homogeneity of variance (P < 0.05), and they all obeyed normal distribution. A generalized linear model (GLM) was used to test the effect of grazing intensity on the density ofS. breviflora and C. songorica . A Duncan test (Levene homogeneity test) was used to identify significant pairwise relationships among the different grazing treatments. By using several quadrats, we could also identify how sampling size may have influenced our results. To that end, we split the sample size of 5 m×5 m into 5 cm×5 cm, 10 cm×10 cm, 20 cm×20 cm, 25 cm×25 cm, 50 cm×50 cm and 100 cm×100 cm. Data was square-root transformed to meet a normal distribution, and variance analysis was used to compare the density changes of S. breviflora and C. songorica under different grazing intensities and across these increasing spatial scales.The bidirectional GLM method was used to test the effects of grazing intensity and spatial scale on the density of S. breviflora andC. songorica , and the generalized linear model was used to test the effect of spatial scale on the density of S. breviflora andC. songorica . In order to analyze the synergistic change of density with scale of S. breviflora and C. songorica in different grazing intensities, the scatter diagram of density ofS. breviflora and C. songorica indifferent grazing intensities was drawn, and the linear fitting (using MS data) was carried out (r 2 represents the determination coefficient). The analysis of variance was performed using SAS 9.4 (SAS Institute Inc.), and a threshold of P < 0.05 level was used to assay significant relationships. Assembly process and graphics rendering are completed in Sigmaplot 14.0 (Systat Software, 2011).
To study the inter-specific association of S. breviflora andC. songorica in desert steppe, the plot of 5 m×5 m was regarded as a plant community in this study. We used the Jaccard index to evaluate the strength and direction of inter-specific plant relationships (Guo et al., 2014). The Jaccard index is negatively correlated with competitiveness, and positively correlated with affinity. The calculation formula of Jaccard index is:
CJ =j /(a +b +j )
In the formula, CJ is the Jaccard index; j is the number of small quadrats with S. breviflora and C. songorica ;a is the number of small quadrats with only S. brevifloraand without C. songorica ; b is the number of small quadrats with only C. songorica and without S. breviflora . By repeating this analysis across the various spatial scales (ranging from 5 cm×5 cm to 100 cm×100 cm), we could evaluate if the strength and direction of these inter-specific relationship was impacted by sampling size.