Thanks for the highlight!!

Our reachability methodology describes outcomes from any initial condition for the optimal fertilizer strategy - complementing traditional scenario-based simulations that require predefined IC's and actions but can handle complex models. (7/7)

Adding a second objective to minimize temporary crop nutrient deficit, we show that small short-term deficits can significantly hasten progress toward the joint, long-term target. (6/7)

The time-minimizing ("optimal") path to the target takes 42 years, but includes a temporary, intermediate phase with crop nutrient deficit. Without allowing any temporary crop deficit, it will take 66 years. At status quo fertilizer rates, it would take 423 years. (5/7)

Through the computation of "backwards-reachable sets", we identify the minimum time needed to reach the target from any possible initial condition. This approach complements scenario-based simulations where predefined initial conditions and management actions are required. (4/7)

We then derive the fertilizer rate strategy that minimizes the time needed to redirect the soil phosphorus system to achieve this joint management target. The time-minimizing strategy is “bang-bang”, meaning it switches between the minimum and maximum allowable rates. (3/7)

As a preliminary, we build a simple, two-pool phosphorus model and define a soil phosphorus target that simultaneously supports both agricultural productivity and water quality goals. (2/7)