Fig. 2.

S-map estimates of first-order partial derivatives (Jacobian elements) quantify the state-dependent nature of interactions among phosphorus, CHL, and DO under reoligotrophication. (A) Effect of phosphorus on CHL. Positive values of $\partial \text{chl}/\partial \text{TP}$ indicate that an increase in TP_lake leads to an increase in CHL. At high levels of TP_lake (e.g., 1980s when TP_lake > 60 µg⋅L⁻¹), the mean annual effect of TP_lake on CHL was essentially zero. There is little evidence of phosphorus ($\partial \text{CHL}/\partial {\text{TP}}_{\text{lake}}$ ∼ 0) until the TP_lake concentration drops below 40 µg⋅L⁻¹. (B) S-map analysis finds the effect of CHL on DO changes with reoligotrophication (TP_lake concentration). Negative values of $\partial {\text{DO}}_{\text{B}}/\partial \text{CHL}$ show that, as expected, a decrease in CHL is usually associated with increased water quality (increased DO_B). However, this is not always true, and at higher TP_lake, the average effect of additional CHL is essentially zero.