1
Mathematical Modelling
Worksheet 11
Modelling Crop Growth
1.
The light falling on a sunlit crop t minutes after sunrise has intensity I given
 t 
by I  I m sin   cal m 2 min 1 where I m is the maximum daily intensity
h
and h is the day length in minutes.
From this information we can determine the total daily incoming light energy, S.
h
S
h
0

h
 t 
I dt   I m cos   I m 1  cos  

 h  0

S  2I m
h

If L is the leaf weight per square metre of ground and A is the leaf area per square
A
metre,  
is termed the specific leaf area. A is called the leaf area index and is
L
important because if A  1 all leaves receive direct sunlight while if A>1 only the
leaves in the top layer receive direct sunlight. Hence
 A, A  1
Sunlit leaf area index AS  
1, A  1
Conversion of sunlight into growth materials is governed by a law, P  kI  r
Where


P = rate of growth per square metre gm 2 min 1 .
r = leaf respiration rate per square metre.
I = light intensity.
and k is a constant conversion factor.
For a crop canopy of leaf area index A, area AS receives direct sunlight and has the
growth given above while the remainder may be considered to have I = 0.
2
Assuming no growth or loss during the night, we can obtain an expression for the
daily increase W in weight per square metre of crop.
W Sunlit   W Unlit 
h
h
t

AS   kI m sin  r dt  A  AS   rdt
0
0
h


hk
2 AS
I m  Arh

Using the fact that S  2 I m
h

 AS kS  Arh
The leaf weight L is a function of total weight only. The allometric ratio  is the
ratio;
Proportion al increase in leaf weight
Proportion al increase in total weight
so  
L L
W W
 is also a function of W only.
We can obtain an expression for the increase L j in day j in terms of L j 1 , W j 1 ,  j 1 ,
and parameters k, S, r, A and AS .
L j  L j 1  L j 1
 j 1 
W j 1
L j 1 
L j 1

W j 1
W j 1

L j 1W j 1
L j 1 SAS k  Arh 
 j 1 L j 1 SAS k  Arh 
W j 1
Now L j  L j 1 
 j 1 L j 1 SAS k  Arh 
W j 1
so L j  L j  L j 1

 j 1 L j 1 SAS k  Arh 
W j 1
3
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.
We can show that the initial leaf growth is exponential.
dL LSAS k  Arh 

dt
W
 SAS k  rh A

A
A  L
  Sk  rh L
  Sk  rh L
L j 

dL
  Sk  rh dt
L 
 ln L    Sk  rh t  ln 
 L  e  Sk  rh t
L0   L0
L  L0 e  Sk  rh t
A graph can now be drawn of L against time for the first 20 days, assuming initial leaf
weight L0  0.25 gm 2
The parameters are given by;
  0.02m 2 g 1 , S  320calm 2 day 1 , h  850 min, r  0.001gm 2 min 1 , k  0.005 gcal 1
and   1.
Leaf growth
A graph of L against Time
140000000
120000000
100000000
80000000
60000000
40000000
20000000
0
1
6
11
Time (days)
16
4
3.
The model of (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
B sin  of the light (assumed vertical) is intercepted by this layer and a
proportion 1  B sin  is transmitted to the second layer. If there are n layers,
we can determine the sunlit area AS per square metre in terms of A, n and  .
B sin 
1  B sin  
B sin  1  B sin  
.
.
.
n
1  B sin  n1 B sin 
n
AS   1  B sin  
n 1
th
layer

B sin 
j 1
But B 
A
n
n
A


1  1  sin  
A
n


 AS  sin 
A
n


1  1  sin  
n


A


 AS  1  1  sin  
n


4.
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 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).
5

dy
y 

 ry 1 
dt
y
m 

y m dwe rt
y 
y m  dw 1  e rt


This equation can be represented graphically.
y
d
In an experiment the following data were obtained:
Density d
4
8
16
32
64
128
Yield per plot
14.7
21.8
35.5
39.0
42.9
52.5
This data is in agreement with the above model
60
Yield per plot
50
40
30
20
10
0
0
50
100
Density d
150