There are various models at varying levels of complexity. Asking google for simple ones suggests using variants of the logistic equation, for plant biomass B(t).
Perhaps more realistically for this problem, it should incorporate Br roots, Bs stem, Bl leaves, Bc crop (the saleable part)
What google said:
A common starting point for growth modeling, before integrating all specific factors, is the logistic equation, which can be modified to include environmental limits:
dB/dt=rB(1−B/K)×f(J)×g(W)×h(NPK)
In this conceptual equation:
𝐵 is the plant biomass.
𝑡 is time.
𝑟 is the intrinsic growth rate.
𝐾 is the carrying capacity (maximum size).
𝑓(𝐽) response function for sunlight
𝑔(𝑊) response function for water
ℎ(𝑁𝑃𝐾) response function for N,P,K nutrients
A value of 0 means the resource is completely limiting, stopping growth; a value of 1 means the resource is non-limiting.
Numerical solution is easy undergraduate, e.g. in Mathematica but with f, g, h all one, and the sunlight simplified to same length days and no intermittency from cloud cover.
Result is totally unsurprising logistic curve with wiggles from no growth at night.
plantGrowthDE[r_, K_] := D[B[t], t] == r*B[t]*(1 - B[t]/K)*Max[0, Sin[2*Pi*t]^2];
tMax = 75;
ans = NDSolve[{plantGrowthDE[0.3, 2], B[0] == 0.01}, B[t], {t, 0, tMax}]
Plot[B[t] /. ans[[1]], {t, 0, tMax}]
plot over a smaller number of days, say 20, to see the day/night wiggles.
Before I present a simple variant of this (with N and water, perhaps water as stochastic), it is, I think, appropriate to mention there are many more detailed models.
My favorite if I were to explore (but I'm not - over to others) would be the free open-source
Python Crop Simulation Environment (PCSE)
https://pcse.readthedocs.io/en/stable/user_guide.html#background-of-pcse
Another widely used one is aquacrop (available, free open-source, in R, matlab and python).
Also free for non-commercial use and up on github is APSIM (from CSIRO, etc.)
Perhaps APSIM is written in C++/CL1 and .Net, but I haven't checked. I have looked at a few user comments - one from California - reported APSIM predictions of fertiliser response was not good. Might have been some user error, if the software isn't easy to use. However the paper mentioned at bquadratic seems to say APSIM is OK.
There has been (at least one!) instance where aquacrop and APSIM have been run with same inputs.
As a paid service to farmers APSIM is under "Yield Prophet"
https://www.yieldprophet.com.au/yp/Home.aspx
There are lots of other simulators/farm advisers:
SYN is from WA Ag. Dept from several decades ago. It is now in an app for iPad
https://nbroadacre.weebly.com
(I haven't found an account of the modelling underpinning SYN ...)
Modifying the logistic equation is easy, but I haven't any idea what the coefficients should be.
In the code (just inventing numbers for the coefficients) it has nitrogen increasing the rate but no real effect on the carrying capacity. But in the wheat context, when it dries out before harvest, perhaps this is OK.
odeNeffectsmma