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After reading this article you will learn about the growth expression and equation of plant growth.
Growth curve are helpful to an understanding of the general pattern of plant development. The plant is a product of both its genetic constitution (potential for maximum growth) and its environment (variable).
Therefore, growth may be expressed as follows:
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G = f(x1, x2, x3, x4, …,xn)
where, G = Some measure of plant growth and x1, x2, x3, x4, …, xn the various growth factors.
Again, if all but one of the growth factors is present in adequate amounts, an increase in the quantity of this limiting factor will generally result in increasing plant growth as follows:
G = f(x1) x2, x3, x4, …, xn
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However, this is not a simple linear relationship.
Liebig’s law of minimum:
In 1862 Justus von Liebig, a German chemist states that “Every field contains a maximum of one or more and a minimum of one or more nutrients. With this minimum, be it lime, potash, nitrogen, phosphoric acid, magnesia or any other nutrient, the yields stand in direct relation. It is the factor that governs and controls … yields. Should this minimum be lime … yield … will remain the same and be no longer even though the amount of potash, silica, phosphoric acid etc. ….be increased a hundred fold.”
This law can be simply stated as follows – “Even if all but one of the essential elements be present, the absence of that one constituent will render the crop barren.”
Mitscherlich’s law:
E.A. Mitscherlich (1909) was among the first to quantify the relationship between plant growth response and the addition of a growth factor. He developed two laws from his research works which are as follows:
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(i) Mitscherlich’s Law of Physiological Relationships:
“Yield can be increased by each single factor even when it is not present in the minimum as long as it is not present in the optimum.”
(ii) Mitscherlich’s Growth Law and Equation:
Growth Law: “Increase in yield of a crop as a result of increasing a single growth factor is proportional to the decrement from the maximum yield obtainable by increasing the particular growth factor.”
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Equation:
Mitscherlich developed an equation that related growth to the supply of plant nutrients. He observed that when plants were supplied with adequate amounts of all but one nutrient their growth was proportional to the amount of this one limiting element that was supplied to the soil.
Plant growth increases with the application of more amounts of nutrient element, but not in direct proportion to the amount of that nutrient element applied. Mathematically it can be expressed as follows:
dy/dx=(A-y)C
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where, dy is the increase in yield resulting from an increment, dx of the growth factor x, A is the maximum possible yield resulted from the optimum supply of all growth factors.
y is the yield obtained after any given quantity of the factor x has been applied and C is the proportionality constant (considered as efficiency coefficient or factor).
Spillman’s equation:
W.J. Spillman expressed the relation as follows:
y = M (1 – Rx)
where y is the amount of growth produced by a given quantity of the growth factor x, x is the quantity of the growth factor, M is the maximum possible yield and R is a constant.
By combining it with Mitscherlich equation, Spillman developed a reduced form of equation as follows:
y = A(1 – 10-cx)
y = A-A . 10-cx
or A-y = A. 10-cx
or log (A – y) = log A + (-cx) log 10 [taking log in both the sides]
or log (A – y) = log A – cx . 1
or log (A – y) = log A – c(x)
where, y is the yield produced by a given quantity of the growth factor x, A is the maximum possible yield and c is a constant which depends on the nature of the growth factor.
The percentage of maximum yield and increase in the unit of growth factor shows the following relationship:
Baule Unit:
Units of a growth factor are designated as a “baule” according to the German mathematician, Baule. Baule suggested that the unit of fertilizer, or any other growth factor, be taken as that amount necessary to produce a yield that is 50 per cent of the difference between the maximum possible yield and the yield before that unit was applied.
It is evident that plants require different absolute amounts of nitrogen, phosphorus and potassium, but the amount in pounds of each (i.e. nitrogen or phosphorus or potassium) required to produce a yield that is 50 per cent of the maximum possible is called 1 baule unit.
According to this concept, 1 baule of a growth factor is equivalent to 1 baule of any other growth factor in terms of growth-promoting ability. The values of the baule unit in pounds per acre of N, p2O5 and K2O are 223, 45 and 76 respectively.
Therefore, 1 baule of N = 1 baule of P2O5 = 1 baule of K2O
223 lbs. of N s 45 lbs. of P2O5 = 76 lbs. of K2O.
Determination of the value of the proportionality content c:
From the combined equation it is found that,
log A – log (A – y) = cx
or log A/A – y = cx
where, A = 100 (maximum yield)
y = 50 per cent of the maximum yield
x = baule unit.
When the nutrient supply is increased by 1 baule unit, putting the above value in the above formula we get,
A log A/A – y = cx
or log 100/100 – 50 = c (1) (...x = 1)
or log/50 =c
or log 2 = c
Therefore, c = 0.301
When conventional units of yield are used, the value of c varies with the particular growth factor i.e., the particular nutrient element. Mitscherlich found that the value of c was 0.122 for N, 0.60 for P2O5 and 0.40 for K2O.
The constant c gives an indication of whether the maximum yield level can be achieved by a relatively low or high quantity of the specific growth factor. When the value of c is small, a large amount of the nutrient element is required for plant growth and nutrition and vice versa.
Calculations of relative yields from an increment of a growth factor:
If A, the maximum yield, is considered to be 100 per cent, from the earlier combined equation developed by Spillman we get,
log (100 – y) = log 100 – 0.301 (x)
If none of the growth factor is available, that is, x = 0, then y = 0.
Again, if x = 1, then yield will be as follows:
log (100 – y) = log 100 – 0.301 (x)
or log (100 – y) = log 102– 0.301 (1)
or log (100 – y) =2 log 10 – 0.301
or log (100 – y) = 2 – 0.301
or log (100 – y)= 1.699
or log – y= Antilog of 1.699
or 100 – y= 50
or -y = -100 + 50
or -y = -50
... y = 50
So, with one unit of growth factor i.e. x = 1 results 50% of the maximum possible yield.
Further, when x = 2 i.e. two units of growth factors are present then yield will be as follows:
log (100 – y) = log 100-0.301 (2)
or log (100 – y) =2-0.602
or log (100 – y) = 1.398
or 100 – y= Antilog of 1.398
or 100 – y= 25
... y = 75
Therefore, with the successive increase in number of growth factors the per cent yield will increase @ 50 per cent from the addition of the preceding unit until a point will reach at which further increment of growth factor will have no consequence.
Increase in yield with more than one growth factor:
It has been demonstrated that when all growth factors but one, x, were present in optimum amounts, the addition of 1 unit of this factor x would produce a yield that was 50 per cent of the maximum possible yield. Suppose that all except two growth factors x1 and x2 were present in optimum amounts.
If 1 baule of each is added, the yield obtained will not be 50 per cent but 50 per cent times 50 per cent, or 25 per cent of the maximum. Again if all growth factors but three were present in optimum amounts, the simultaneous addition of 1 unit of each of the three would result in a yield that was 50 per cent x 50 per cent x 50 per cent or 12.5 per cent of the maximum.
The relationship is as follows:
where, x1, x2 and x3 are quantities of the growth factors to be added.
Bray’s nutrient mobility concept:
This concept has been proposed by R. Bray and his associates after modifying the concept developed by Mitscherlich-Baule-Spillman.
Bray’s concept states that:
“As the mobility of a nutrient in the soil decreases, the amount of that nutrient needed in the soil to produce a maximum yield (the soil nutrient requirement) increases from a variable net value, determined principally by the magnitude of the yield and the optimum percentage composition of the crop, to an amount whose value tends to be a constant.”
The magnitude of this constant is independent of the amount of crop yield, provided that the kind of plant, planting pattern and rate, and fertility pattern remain constant and that similar soil and seasonal conditions prevail.
Bray has modified the Mitscherlich equation to:
log (A – Y) = log A –C1 b – Cx
where, A, Y and x are maximum yield, yield obtained and amount of added fertilizer nutrient respectively.
b = amount of an immobile but available form of nutrients (like P and K)
C1 = constant or efficiency factor of b for yields
C = constant or efficiency factor of x
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