Objective (ii) for the problem is
"(ii) 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?"
In my code, I have just one top up application, at day x.
My guess is that one is to maximise;
profit= yield*saleprice - nitrogenTotal*Nprice
There are studies on N-effects in dry land wheat farming. Because data is mostly related to final yield, and most commonly with a single application of fertilizer, the better known results are for this, specifically with other variables (ET evapotranspiration, etc.) fixed, the yield Y is approximately quadratic in N, Y=a0 + a1*N +a2*N^2, a0 and a1 positive, a2 negative. Might one be able to do something similar with biomass B (roots? stem and leaves) and more than one application of N, say just one more? Assuming yes, and quadratic responses at both day0 and dayx one is led to optimising biquadratics as opposed to the single application which had the farmers optimising using a single quadratic. bquadratic
Some background info,esp. on soil chemistry, etc. is given at soilchemn
GK wonders if, depending on weather/rainfall, it might be reasonable to apply urea (and?) fertilizer in 2 applications.
Suppose the amount M for a single application is calculated, for example, from the quadratic in
A simple N calculator for achieving water-limited yield of wheat crops
Zvi Hochman, Francois WaldnerProceedings of the 20th Agronomy Australia Conference, 2022 Toowoomba Qld
www.agronomyaustraliaproceedings.org
The suggestion is to apply say 4M/5 just before seeding and M/5 within 3 weeks after seeding, the exact day x depending on rain.
For the 2nd application, a dry day for the application with the right amount of rain the next day would be good.
(and?) gemini said: "While granular urea is dominant, the use of liquid Nitrogen (UAN, often branded as Flexi-N or similar) is also common, particularly as a flexible in-season or in-furrow option."
Urea takes a bit of time to be converted to nitrate and ammonia/ammonium for uptake by plants.
Nitrates leach out in heavy rainfall events.
Nitrogen fertilizer is most effective for stem growth.
Nitrogen helps with protein in grain but premiums for high protein are not huge.
GK has read:
1. If repeat fertilizer is done, better no later than 6 weeks after germination.
2. Unless very heavy early rain has leached N below root zone, not much, if any, gain in split applications over same amount all at once.
3. It has been reported that the profit as a function of N-inputs has a quite flat maximum.
ASIDE. Other applications of biquadratics, but relatively few, if any focussing on their critical points. biquadraticsgen