Fig. 1: Conceptual diagram of the data analyses performed and their hypothesised results.

We used growing season averages of shallow lake time series to calculate 5-year simple moving averages (SMA), which act as low-pass filter to extract long-term variation. We used the residuals of the 5-year SMA, calculated as the growing season observation minus the SMA for the same year, as high-pass filter to reveal short-term variation (A)¹². We evaluated different SMA lengths and found 5 years to be ideal (see methods and SI for details). We categorized the data into overlapping TN:TP “windows” to assess the link between TN, TP, and Chla within different ranges of TN:TP (B). Here, we used the natural logarithm of TN:TP ratios because ratios follow a log-normal distribution⁵⁹. For each TN:TP window, the data were randomly sampled 300 times using a hierarchical bootstrap procedure to preclude temporal dependence of the sampled data⁶². For each random sample, we calculated generalized linear models with gamma distribution (5-year SMA) or linear models (SMA residuals) for the relationship between Chla and TP and/or TN, and extracted R² and model coefficients (C). Model slopes and R² are shown in the main text, see Methods and SI for details on the bootstrap and other model coefficients. We hypothesize high TP model slopes, high R² and low concentrations of reactive pelagic P in the form of soluble reactive phosphorus (SRP) at higher TN:TP, and high TN model slopes, high R² and low concentrations of reactive pelagic N as nitrate-N at lower TN:TP (D).