Mechanistic modelling of plant growth is a relatively recent research field, and has gained an increasing interest with the sophistication of the description of plant-environment interactions in crop models. Contemporary agricultural engineering searches for safe methods of raising crop yields, using a combination of knowledge from a number of sciences. Mathematical modelling of agribiological processes is performed as a part of agricultural engineering. In this paper, mathematical modelling was applied to plant growth and observed that it was theoretically possible to derive an optimal control method by the proposed models.

1. Mathematical Modelling to Predict Plant Growth in Terms Light Exposure and Leaf Area
Mathematical Modelling to Predict Plant Growth in Terms
Light Exposure and Leaf Area
Yalda Elbegi
Lecturer, Department of Mathematics Education Faculty, Jawzjan University, Sheberghan, Afghanistan
Email: yaldaelbegi01@gmail.com
Mechanistic modelling of plant growth is a relatively recent research field, and has gained an
increasing interest with the sophistication of the description of plant-environment interactions in
crop models. Contemporary agricultural engineering searches for safe methods of raising crop
yields, using a combination of knowledge from a number of sciences. Mathematical modelling of
agribiological processes is performed as a part of agricultural engineering. In this paper,
mathematical modelling was applied to plant growth and observed that it was theoretically
possible to derive an optimal control method by the proposed models.
Keywords: mathematical model, plant growth, leaf area, light exposure
INTRODUCTION
Plants are highly plastic in their potential to adapt to
changing environmental conditions. For example, they can
selectively promote the relative growth of the root and the
shoot in response to limiting supply of mineral nutrients
and light, respectively. This phenomenon is referred to as
balanced growth or functional equilibrium. Mathematical
modelling of agribiological processes are performed as a
part of agricultural engineering and its current activity
(Michalek, 2008). Since, application of modern prediction
methods in agriculture may bring notable financial
advantages, mathematical modelling answers the
challenges related thereto.
Despite numerous publications which describe the plant
growth and development processes, there is still no
agreement as to the value and usefulness of calculation
models in the developmental biology or agriculture. The
discussion concerns mainly the possibilities of application
of research methods, including mathematical deduction, to
the data with biological nature (Keller, 2002). However,
despite a low number of negative opinions, it is estimated
that modelling and simulation constitute and will continue
to constitute one of the most important elements in the
description and prediction of plant growth and
development (Room et al. 1996; Minorsky, 2003; Niklas,
2003).
Classical growth models, inter alia, logisitc, Gompertz and
Richards equations are widely and often used for
description of biological processes (Werker and Jaggard,
1997). Biologists prefer the most often used models, such
as logistic functions, Gompertz or monomolecular, which
have only three parameters (if it is assumed that the
minimum value is 0) (Zeide, 1993; Birch, 1999).
METHODOLOGY
Model 1
The light falling on a sunlit crop at t minutes after sunrise
has intensity I given by 𝐼 = 𝐼 𝑚 𝑠𝑖𝑛 (
𝜋𝑡
ℎ
) cal
12
min −−
m
where mI is the maximum daily intensity and h is the day
length (min).
So, the total daily incoming light energy (S) can be
determined as follows:
𝑆 = ∫ 𝐼
ℎ
0
𝑑𝑡 = [𝐼 𝑚 𝑐𝑜𝑠 (
𝜋𝑡
ℎ
)]
0
ℎ
= 𝐼 𝑚
ℎ
𝜋
[1 − 𝑐𝑜𝑠 𝜋]
𝑆 = 2𝐼 𝑚
ℎ
𝜋
If L is the leaf weight per square metre of ground and A is
the leaf area per square metre, ( )LA= is termed the
Research Article
Vol. 7(1), pp. 143-145, March, 2020. © www.premierpublishers.org. ISSN: 2375-0499
International Journal of Statistics and Mathematics

2. Mathematical Modelling to Predict Plant Growth in Terms Light Exposure and Leaf Area
Elbegi Y. 144
specific leaf area. A is called the leaf area index and is
important because if 1A , all leaves receive direct
sunlight while if A>1, only the leaves in the top layer
receive direct sunlight. Hence,
Sunlit leaf area index





=
1,1
1,
A
AA
AS
Conversion of sunlight into growth materials is governed
by a law, rkIP −=
Where P is the rate of growth per square metre
( )12
min −−
gm , r the leaf respiration rate per square metre,
I the light intensity and k is a constant conversion factor.
For a crop canopy of leaf area index, A, area SA receives
direct sunlight and has the growth given above while the
remainder may be considered to have I = 0.
Assuming no growth or loss during the night, one can
obtain an expression for the daily increase W in weight
per square metre of crop.
𝛥𝑊 = 𝛥𝑊(𝑆𝑢𝑛𝑙𝑖𝑡) + 𝛥𝑊(𝑈𝑛𝑙𝑖𝑡)
𝛥𝑊 = 𝐴 𝑆 ∫ (𝑘𝐼 𝑚 𝑠𝑖𝑛
𝜋𝑡
ℎ
− 𝑟) 𝑑𝑡 +
ℎ
0
(𝐴 − 𝐴 𝑆) ∫ −𝑟
ℎ
0
𝑑𝑡
𝛥𝑊 = 2𝐴 𝑆
ℎ𝑘
𝜋
𝐼 𝑚 − 𝐴𝑟ℎ
Using the fact that 𝑆 = 2𝐼 𝑚
ℎ
𝜋
𝛥𝑊 = 𝐴 𝑆 𝑘𝑆 − 𝐴𝑟ℎ
The leaf weight L is a function of total weight only. The
allometric ratio  is defined thus:
𝛼 =
Proportional increase in leaf weight
Proportional increase in total weight
so 𝛼 =
𝛥𝐿 𝐿⁄
𝛥𝑊 𝑊⁄
hence,  is also a function of W only.
One can obtain an expression for the increase jL in day
j in terms of 𝐿𝑗−1, 𝑊𝑗−1, 𝛼𝑗−1, and parameters k, S, r, A and
SA as follows:
𝐿𝑗 = 𝐿𝑗−1 + 𝛥𝐿𝑗−1
𝛼𝑗−1 =
𝛥𝑊𝑗−1
𝐿𝑗−1
×
𝑊𝑗−1
𝛥𝑊𝑗−1
=
𝛥𝐿𝑗−1 𝑊𝑗−1
𝐿𝑗−1(𝑆𝐴 𝑆 𝑘 − 𝐴𝑟ℎ)
𝛥𝐿𝑗−1 =
𝛼𝑗−1 𝐿𝑗−1(𝑆𝐴 𝑆 𝑘 − 𝐴𝑟ℎ)
𝑊𝑗−1
Now 𝐿𝑗 = 𝐿𝑗−1 +
𝛼𝑗−1 𝐿𝑗−1(𝑆𝐴 𝑆 𝑘 − 𝐴𝑟ℎ)
𝑊𝑗−1
so Δ𝐿𝑗 = 𝐿𝑗 − 𝐿𝑗−1
=
𝛼𝑗−1 𝐿𝑗−1(𝑆𝐴 𝑆 𝑘 − 𝐴𝑟ℎ)
𝑊𝑗−1
Model 2
In the initial stages of growth, the allometric ratio may be
assumed constant and the root weight is assumed
negligible in comparison to the leaf weight. One can show
that the initial leaf growth is exponential.
𝛥𝐿𝑗 =
𝑑𝐿
𝑑𝑡
=
𝛼𝐿(𝑆𝐴 𝑆 𝑘 − 𝐴𝑟ℎ)
𝑊
=
𝛼(𝑆𝐴 𝑆 𝑘 − 𝑟ℎ)𝐴
𝐴
𝐴 = 𝜌𝐿
= 𝛼(𝑆𝑘 − 𝑟ℎ)𝜌𝐿
= 𝛼(𝜌𝑆𝑘 − 𝜌𝑟ℎ)𝐿
∫
𝑑𝐿
𝐿
= ∫ 𝛼(𝜌𝑆𝑘 − 𝜌𝑟ℎ)𝑑𝑡
⇒ 𝑙𝑛 𝐿 = 𝛼(𝜌𝑆𝑘 − 𝜌𝑟ℎ)𝑡 + 𝑙𝑛 𝛽
⇒ 𝐿 = 𝛽𝑒 𝛼(𝜌𝑆𝑘−𝜌𝑟ℎ)𝑡
𝐿(0) = 𝐿0
𝐿 = 𝐿0 𝑒 𝛼(𝜌𝑆𝑘−𝜌𝑟ℎ)𝑡
Model 3
Model 2 can be improved by considering a crop of leaf area
index A where the leaves are at an angle  to the vertical.
The leaf canopy is assumed to consist of a number of
layers, each of leaf area index B. Hence, a proportion
sinB of the light (assumed vertical) is intercepted by
this layer and a proportion ( )sin1 B− is transmitted to
the second layer. If there were n layers, we can determine
the sunlit area SA per square metre in terms of A, n and
𝜃.
𝐵 𝑠𝑖𝑛 𝜃
(1 − 𝐵 𝑠𝑖𝑛 𝜃)
𝐵 𝑠𝑖𝑛 𝜃 (1 − 𝐵 𝑠𝑖𝑛 𝜃)
.
.
.
(1 − 𝐵 𝑠𝑖𝑛 𝜃) 𝑛−1
𝐵 𝑠𝑖𝑛 𝜃 (𝑛th
layer)
𝐴 𝑆 = ∑(1 − 𝐵 𝑠𝑖𝑛 𝜃) 𝑛−1
𝐵 𝑠𝑖𝑛 𝜃
𝑛
𝑗=1
But 𝐵 =
𝐴
𝑛
∴ 𝐴 𝑆 =
𝐴
𝑛
𝑠𝑖𝑛 𝜃
1 − (1 −
𝐴
𝑛
𝑠𝑖𝑛 𝜃)
𝑛
1 − (1 −
𝐴
𝑛
𝑠𝑖𝑛 𝜃)
∴ 𝐴 𝑆 = 1 − (1 −
𝐴
𝑛
𝑠𝑖𝑛 𝜃)
𝑛
In the production of any vegetable crop, there will be an
optimum density of planting, as below a certain separation
distance competition between plants for materials and light
will become excessive. In modelling the growth of plants in

3. Mathematical Modelling to Predict Plant Growth in Terms Light Exposure and Leaf Area
Int. J. Stat. Math. 145
such a situation, we can assume that the initial rate of
growth is exponential and that the rate of growth tends to
zero as the yield y (weight per unit area) approaches the
ultimate value y. We can use the assumptions to derive an
expression for the yield at time t given that the plants
initially have average weight w and are planted with
density d (number of plants per unit area).
𝑑𝑦
𝑑𝑡
= 𝑟𝑦 (1 −
𝑦
𝑦 𝑚
)
∴ 𝑦 =
𝑦 𝑚 𝑑𝑤𝑒 𝑟𝑡
𝑦 𝑚 − 𝑑𝑤(1 − 𝑒 𝑟𝑡)
Modern agricultural engineering searches for safe
methods of raising the quality of crop yields with the use of
an interdisciplinary combination of mathematics,
biophysics, agronomy, molecular biology and physics
(Ciesla et al., 2015). Since, application of modern
prediction methods in agriculture may bring notable
financial advantages, mathematical modelling answers the
challenges related thereto. In this paper, mathematical
modelling was applied to plant growth and observed that it
was theoretically possible to derive an optimal control
method by the proposed models.
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Accepted 18 February 2020
Citation: Elbegi Y. (2020). Mathematical Modelling to
Predict Plant Growth in Terms Light Exposure and Leaf
Area. International Journal of Statistics and Mathematics,
7(1): 143-145.
Copyright: © 2020 Elbegi Y. This is an open-access
article distributed under the terms of the Creative
Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium,
provided the original author and source are cited.