The equation of production at crop level.
We have seen that crop based models may express the biomass growth par unit m² as follow:

dW = LUE . PAR . (1 - exp ( -k LAI ) )
where:

LUE is the Light Use Efficiency,
LAI the Leaf Area Index
PAR the Photosynthetic Active Radiation incident on the crop (per m²)

Expression of the crop model at the single plant level.

Let d define the plant density.
Then Sd = 1 / d represents the potential available surface at the single plant level.
Let also S_l define the total surface of functionnal leaves. We can then write:

LAI = S_l / S_d

The production equation can thus be expressed as:

dW = LUE . PAR . S_d(1 - exp ( -k S_l / S_d ) )

Now, at low density, we introduce S_p, a virtual surface area such as

S_p = F(S_d)
with F(S_d) -> S_d when the crop cover gets closed (at significant density value).

Moreover, we can integrate those equations on the duration of a growth cycle.
Assuming that the leaf area remins constant during a growth cycle, we obtain the first GreenLab equation, defining the fresh biomass production as follow:

Q(n) =	E(n) . Sp	(	1 - exp ( -k	Sl(n)	)	)	(equation 1)
	r			Sp






where:

Q(n) is the fresh biomasss produced at cyle n
E(n) is an agregate variable standing for the Environmental ressource conditions, typically such as LUE and PAR
r represents the water resource, proportional to the Water Use Efficiency inverse (WUE)
k stands for the BeerLaw extinction coefficient
S_p stands for the projection surface,as defined higher
S_l(n) stands for the total functional leaf surface