Moderator: Alaric Korb (Royal Agricultural Society of Western Australia)

1. The Context

Western Australian agriculture operates on ancient, highly weathered soils (often sandy and distinctively heterogeneous) under a “dryland” system where water is the primary limiting factor. Farmers face a critical dilemma:

• Too little fertilizer: Yield gaps and economic loss.
• Too much (or wrong timing): Nitrogen leaches rapidly through sandy profiles beyond the root zone, becoming an economic loss and an environmental hazard (acidification and water table contamination).
• The Twist: Unlike European or North American systems, WA rainfall is highly variable. A strategy that works in a “decile 5” (average) year might be disastrous in a “decile 2” (dry) year.

2. The Mathematical Challenge 

We are asking the study group to move beyond simple “yield curve” static calculations. We need a dynamic model that couples fluid dynamics in porous media with stochastic decision theory.

The problem can be split into two layers for the mathematicians:

1. The Forward Problem (Physics): Modelling the vertical and horizontal transport of solutes (Nitrate/Ammonium) through a heterogeneous soil profile using coupled Partial Differential Equations (PDEs).
2. The Inverse/Control Problem (Optimization): Determining the optimal function  (fertilizer application over time and space) that maximizes profit while satisfying strict environmental constraints, under stochastic rainfall forcing.

3. Objectives for the Study Group

• Develop a reduced-order dynamical model that approximates the transport of Nitrogen through WA specific soil types (e.g., duplex soils: sand over clay) without the heavy computational cost of full hydrological simulations (like HYDRUS).
• Solve a Stochastic Optimal Control problem: Determine the optimal “stopping time” or “trigger points” for top-up fertilization. Question: Given a probabilistic forecast of rain in the next 14 days, should the farmer apply N now, wait, or abandon application?
• Quantify “Leaching Risk”: creating a probability density function for nutrient loss below the root zone  based on soil hydraulic conductivity and rainfall variance.