Moderator: Alaric Korb (Royal Agricultural Society of Western Australia) and Melanie Roberts (Griffith University)

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.