National Conference on the Caspian forests of Iran
“Past, Current, Future”
University of Guilan, Rasht, Iran, April 26-27, 2017
A general dynamic function for the basal area of individual trees derived
from a production theoretically motivated autonomous differential equation
P. Lohmander1*
1- Professor, Optimal Solutions, cooperating with Linnaeus University, Sweden
*Corresponding author’s E-mail: [email protected]
Abstract
A general dynamic function for the basal area of individual trees has been derived from
a production theoretically motivated autonomous differential equation. The differential
equation is
dx
 a x 1  cx  , a  0, c  0, 0  x  c 1 and the general dynamic
dt
function is: x(t ) 
  x0 c  1  a

e
  x0 c  1 


 x
c 0
  x0

2

 1


2
c  1  a ct 
 e  1

c  1 

ct
.
Keywords: Dynamic function, differential equation, basal area, forest growth
Introduction
The mathematical methods utilized in this paper are well presented by Braun (1983)
and Simmons (1972). Now, fundamental biological production theory will suggest a
relevant differential equation. We start with a small tree. Consider a stem segment, of
height H , of the tree. The stem segment is cylindrical with diameter D1 . The leaves
cover a cylinder with diameter D2 . D2   D1 ,   1. The sun light reaches the tree from
2
the side. V is the volume of the stem segment. x is the basal area. x  ( / 4) D1 .
V  Hx . Volume increment is proportional to the photo synthesis level, P , which in
turn is proportional to the sun light projection area on the leaves, A . A  HD2 . We
may conclude that:
National Conference on the Caspian forests of Iran
“Past, Current, Future”
University of Guilan, Rasht, Iran, April 26-27, 2017
dV
dx
dx
H
 P  A  D2  D1  x . Hence,
 x
or
dt
dt
dt
dx
 a x , a  0 . As the size of the tree increases, the production efficiency declines.
dt
Furthermore, the value of  is often lower for large trees than for small trees. A relevant
dx
 a x 1  cx  a  0, c  0, 0  x  c 1 .
function considering this is:
dt
Mathematical model development and analysis
dx
 ax 0.5  bx1.5
dt
a  0, b  0, 0  x 
a
b
dx
b
 a x 1  cx  , c   0, 0  x  c 1
dt
a
1
dx  a dt . Integration gives
x 1  cx 
ln

 
  at  k . Let us investigate the left hand side, called Z .
c x  1  ln
c x 1
0
c
dZ

dx
c

dZ

dx 2 x
dZ

dx
 c 


c x  1  2 x 
c
1

1
1
dx   a dt  k0 .
x 1  cx 



c x 1

2 x

1

 c 


c x  1  2 x 
1


c x 1
. Then,


1
. This confirmes that the integration was correct.
x 1  cx 
 c x 1 
ln 

 c x  1   at  k which leads to ln  c x  1   cat  k


1
0
c
 c x 1 
Let y 
 


c x 1  c x  1
dZ

dx 2 x c x  1 c x  1
 gy  1 
x , g  c , h  ga . Then, ln 
  ht  k1 .
 gy  1 
k1  ck0 .

National Conference on the Caspian forests of Iran
“Past, Current, Future”
University of Guilan, Rasht, Iran, April 26-27, 2017
k
Let K  e 1 . We get the simplified expression:
gy  1
 Ke ht which can
gy  1


ht
ht
be transformed to: gy  1  Keht  gy  1 or g 1  Ke y  1  Ke and
y
1  Keht
Keht  1
.
which gives the desired equation
y

x

g 1  Keht 
g  Keht  1
 Ke
x(t ) 
c  Ke
ht
ht
 1
2
 1
2
.
 Ke  1
c  Ke  1
0
Let us determine K . We utilize the initial condition: x0  x(0) .
which leads to
K
x0 c  1
x0 c  1
x0 c  K  1  K  1 . Hence,

x0 
0

x0 c  1 K  x0 c  1 and finally
. Now, we know how to determine K . Later, the sign and magnitude of K
will be needed in the analysis. Do we know the sign of K ?

 
x0  0  c  0 

x c  1  0 . Let us investigate the sign of
x0 c  1 . We
assume that the value of x0 makes sure that the increment is strictly positive.
dx
 a x 1  cx  . Then, we know that:
dt
1  cx0  0  cx0  1 
c x0  1 
x0 c  1  0
As a result, we know that K  0 . Do we know the something about K ?
K
 1
 1
 lim K (   )  1 .
0    x0 c  1 . K (   ) 
 0
 1
 1
dK   1    1


2
d
  1
dK
2

 0.
d   12
With this information, we know that K  1 . Now, we may determine x(t ) as an explicit
function of x0 and the parameters.
National Conference on the Caspian forests of Iran
“Past, Current, Future”
University of Guilan, Rasht, Iran, April 26-27, 2017
 Ke
x(t ) 
c  Ke
ht
x(t ) 
ht
 1
2
 1
2
 K
  x0 c  1  a

e
  x0 c  1 


 x
c 0
  x0

x0 c  1
 ha c 
x0 c  1
2

ct
 1


2
c  1  a ct 
 e  1

c  1 

 Ke  1
Now, the dynamic properties of x(t ) 
c  Ke  1
2
ht
ht
2
will be determined.


dx 
2 Khc
  Keht  1 Keht  12   Keht  12  Keht  1

4
dt  c 2  Keht  1 



ht
ht


dx  2 Khc  Ke  1 Ke  1 

 Keht 1   Keht  1
4
2
ht

dt 
c  Ke  1






 
 4 Khc Ke ht 2  1
 
dx 

4
dt 
c 2  Ke ht  1




We already know that K  1 . Hence,
 Ke

ht 2

 1  0 . As a result, we find that
 d  Keht  12 


1 

ht


dt
1  ht  1
2
Ke

1
hK


Ke 
 
lim x(t )  
 

2
ht
t 
ht
1
c




2
c
Ke

1
hK


dc Ke  1
h 0
c 1  ht 
 

K 1
 Ke 


dt


1
Hence, we know that, as t   , x(t ) monotonically converges to c .
dx
 0.
dt
National Conference on the Caspian forests of Iran
“Past, Current, Future”
University of Guilan, Rasht, Iran, April 26-27, 2017
Results
A general dynamic function for the basal area of individual trees has been derived from
a production theoretically motivated autonomous differential equation. The dynamic
properties have been determined and monotone convergence has been proved.
Discussion
The production theoretically motivated autonomous differential equation has recently
been tested with regression analysis, using two sets of forest data from the Caspian
forests. Detailed reports will soon be available. The parameters obtained the expected
signs and the t-values indicated very high precision. The adjusted R2-values, the very
high F-values and the lack of unexplained trends in the residual graphs, indicated a
relevant model. The functional form has been found to work very well with empirical
data from Iran and hopefully it will be possible to test it also on forest data from other
regions of the world. It is possible to use the model also under the influence of some
forms of size dependent competition, via adjustments of the parameters (a, b) . Further
developements in this direction should be expected. Presently, different forest research
projects with a strong focus on empirical investigations are active, in Sweden and in
Iran. Hopefully, the derived analytical model will be useful in these projects.
Conclusions
Differential equations can be very useful tools in forest management optimization. It is
however important that they have theoretical and empirical support. Future efforts
should be directed towards applications of the suggested and derived function within
forest management problems in several regions of the world.
Acknowledgments
The author is grateful to Associate Professor Soleiman Mohammadi Limaei who
arranged several forest excursions and meetings in Iran. Furthermore, he arranged
meetings and cooperation with seriously interested PhD students with detailed forest
data from the Caspian forests. Professor Johan Bergh, at Linnaeus University in
Sweden, coordinates a very interesting project on optimal continuous cover forestry and
creates a very positive research environment. Thank you!
References
Braun, M., 1983, Differential equations and their applications, Applied mathematical
sciences, 3 edition, Springer Verlag, New York, 546p.
Simmons, G.F., 1972, Differential equations with applications and historical notes,
McGraw-Hill, New York, 465p.