Statistical analyses
We used mean values from June 10 to August 27 for all variables in the following statistical analyses (Table S1). Relationships among phytoplankton and zooplankton biomasses, specific production rate and fish abundance were examined by correlation analysis. To test differences in phytoplankton and zooplankton community composition among the treatments and between the two ponds, permutational multivariate analysis of variance (PERMANOVA) was performed by the adonis() function in R package “vegan” (Oksanen et al. 2018). In this test, we used 999 permutations and the Euclidean distance both for phytoplankton and zooplankton communities as an index of dissimilarity in the community.
We applied mean phytoplankton carbon biomass, zooplankton carbon biomass, fish abundance, specific production rate and fraction of edible phytoplankton for P , H , θ , μ , andαedi in eq. (9), respectively. Forαnut , we focused on phosphorus since freshwater limnetic ecosystems are primarily phosphorus limited (Schindler 1974; Smith a& Schindler 2009) and since growth of zooplankton is affected by relative phosphorus in algae (Frost et al. 2006, Urabe et al. 2018). Specifically, we used the carbon to phosphorus ratio of seston as a surrogate for αnut because this ratio has been generally used in theories of ecological stoichiometry (Sterner & Elser 2002). Thus, we expected lower H/P ratios at larger values of seston carbon to phosphorus ratio. To examine effects of these explanatory variables on the H/P ratio, a simple regression analyses was performed. Then, after checking multicollinearity among the explanatory variables by variance inflation factors (Kennedy 2008), we fitted these data to eq. (9) using a lm function of R 3.2.1 (R core team, 2018) with the examination of Akaike’s information criterion. In this analysis, 95% confidence intervals of the regression coefficients were estimated using bootstrapping with a residual resampling procedure (Moulton & Zeger 1991) and 1999 replicates. Since eq. (9) indicates ana priori effect direction of a given variable, we estimated upper or lower one-tailed 95% confidence intervals (100 and 5 percentiles) for the explanatory variables according to negative or positive effects predicted by eq. (9). Effect sizes of these explanatory variables were assessed using standardized regression coefficients of the multiple regression. Finally, to examine whether effects of explanatory variables on the H/P mass ratio were independent of each other and significant, we performed partial regression analysis with residual leverage plot according to Sall (1990) using leveragePlots() in R package “car” (Fox &bWeisberg 2011).