Reachability of a Soil Phosphorus Target That Satisfies Agricultural Production and Water Quality Goals

Phosphorus fertilization has supported remarkable improvements in agricultural productivity but also degraded water quality. Watershed simulation models have been broadly instrumental to crafting phosphorus management responses. However, simulation-based studies rely on predesigned watershed scenarios (e.g., initial conditions and management actions) and are blind to outcomes that might only emerge from unseen scenarios. Meanwhile, efforts to restore water quality have routinely failed. In contrast to simulation-based methods, here we implement optimal control and reachability methods that describe watershed phosphorus trajectories for any initial condition and fertilizer strategy. The trade-off is that these new methods require simplification of the system's dynamics. For a two-pool phosphorus model, we define a dual management target where (a) plant-available phosphorus satisfies crop demand but (b) total phosphorus losses meet water quality goals. From this target, we compute backwards-reachable sets that indicate the minimum time in which the target can be reached from all initial conditions. For a typical watershed in the U.S. corn belt, we find that it will take at least 42 years to reach the joint agricultural and water quality target. We show that the optimal (time-minimizing) fertilizer rate strategy drives a roundabout trajectory toward the target where soil phosphorus violates the crop demand threshold during the interim time. However, we find that even small, short-term agricultural sacrifices can profoundly hasten progress toward the long-term, joint target of agricultural productivity and water quality. These results and methods complement traditional simulation-based studies and provide watershed managers with a richer characterization of uncertainty and management options.