CHAPTER 6
MODEL FOR THE EFFECTS OF INDUSTRIALIZATION,
POPULATION, PRIMARY - SECONDARY TOXICANTS
ON DEPLETION OF FORESTRY RESOURCE
INTRODUCTION
The environmental problems in India are growing rapidly. Industrial pollution,
soil erosion, deforestation, rapid industrialization, urbanization and land degradation
are all worsening problems. Over- exploitation of the country resources be it land or
water, the industrial process has resulted in environmental degradation of resources.
Airborne pollutants can be classified broadly into two categories: primary and
secondary. Primary pollutants are those that are emitted into the atmosphere by a
source such as fossil fuels combustion from power plants, vehicle engines and
industrial production, by combustion of biomass from agriculture and land clearing
purposes, and by natural processes. Secondary pollutants are formed within the
atmosphere when primary pollutants react with sunlight, oxygen, water and other
chemicals present in the air.
Atmospheric processes, including oxidation procedures, particle formation and
equilibria, determine the fate of primary emission and, in most cases, the secondary
product of these processes are the more important ones concerning their effects on
human health and the quality of the environment,(Shukla and Dubey, [1996] ). So, the
pollutants in both of their forms are a serious threat for the survival of the resource
145
biomass and exposed population and in order to regulate these pollutant wisely, we
must assess the risk of the resource biomass and populations exposed to pollutants.
Therefore, it is important to study the effects of pollutants on a resource-dependent
biological population by making use of mathematical models.
In view of the above considerations, in this chapter, a nonlinear mathematical
model is proposed and analyzed for the survival of a resource-dependent biological
population in the presence of two toxicants (primary and secondary). It is assumed
that the density of the primary toxicant is increased by the population and
industrialization in the environment and the secondary toxicant formed from primary
toxicant, is more toxic. This situation is modeled by a system of five ordinary
differential equations. Stability theory of nonlinear differential equations and a fourth
order Runge-Kutta method are used to analyze and predict the behavior of the model.
6.1 THE MATHEMATICAL MODEL
We consider an ecosystem where the resource biomass is being depleted due
to the pressure of industrialization, population and primary-secondary toxicants in the
environment. The system is assumed to be governed by the following differential
equations:
rB 0 B 2
dB
 rB N B 
 IB,
dt
K B TP , TS 
r N2
dN
 rP B N  P 0
  1 IN ,
dt
M TP , TS 
dTP
 QI , N    0TP  1 BT P   2 NTP  gTP ,
dt
146
(6.1.1)
dTS
 gTP   1TS  1 BTS   2 NTS ,
dt
dI
I

 r1 I 1     I B   2 I N .
dt
 L
B0  0, N 0  0, TP 0  0, TS 0  0, I 0  0.
In model (6.1.1), B is the density of resource biomass, N is the density of the
population, TP and TS are the densities of primary and secondary toxicants into the
environment. I is the density of industrialization. It is assumed that the dynamics of
the resource biomass, population and industrialization are governed by logistic type
equations. It is also assumed that the growth rate of the resource biomass decreases
with an increase in density of the population while its carrying capacity decreases
with an increase in environmental concentration of primary-secondary toxicants. It is
further assumed that growth rate of the population and industrialization increases as
the density of resource biomass increases.  is the depletion rate coefficient of the
resource biomass due to the industrialization and  is the corresponding growth rate
coefficient of industrialization due to resource biomass. It is also considered that the
emission of primary toxicant into the environment is industrialization and population
dependent and the secondary toxicant which is transformed from the primary toxicant,
is more toxic. The positive constant g is the transformation rate coefficient of primary
toxicant into secondary toxicant in the environment.  1 and  2 are the growth rate
coefficients of industrialization and population respectively due to their interaction. r1
is the intrinsic growth rate coefficient of industrialization. 1 ,  2 and 1 ,  2 are the
depletion rate coefficients of primary and secondary toxicants due to resource
biomass and population, respectively.  0 and  1 are the natural washout rate
147
coefficients of the primary and secondary toxicants from the environment,
respectively. The constant, 0    1, is a fraction, which represent the magnitude of
transformation of primary toxicant into secondary toxicant.
Model (6.1.1) is derived from following assumptions:
(H1): The function rB N  denotes the specific growth rate of resource biomass which
decreases as N increases. Hence we take
rB 0  rB 0  0,

rB  N   0 for N  0.
(H2): The function K B TP , TS  represents the maximum density of resource biomass
which the environment can support in the presence of primary and secondary
toxicants, and it also decreases as TP and TS increases. Hence we take
K B 0,0  K B 0  0,
K B TP , TS 
 0,
TP
K B TP , TS 
0
TS
for TP  0, TS  0.
(H3): The function rP B  denotes the growth rate coefficient of the population and it
increases as the resource biomass density increases. Hence we take
rP 0  rP 0  0,

rP B   0
for B  0.
(H4):The function M TP , TS  represents the maximum density of population which
the environment can support in the presence of primary and secondary toxicants, and
it also decreases as TP and TS increases. Hence we take
M 0,0  M 0  0,
M TP , TS 
 0,
TP
M TP , TS 
 0 for TP  0, TS  0.
TS
(H5): The function QI , N  is the rate of introduction of toxicant into the environment
which increases as I and N increase. Hence we take
148
QI , N 
QI , N 
 0,
 0 for I  0, N  0.
I
N
Q0,0  Q0  0,
To analyze the model (6.1.1), we need the bounds of dependent vatiables involved.
For this we find the region of attraction in the following lemma.
6.2 BOUNDEDNESS OF SOLUTIONS
Lemma (6.2.1): Suppose that assumptions (H1) - (H5) hold. Then all solutions of
system (6.1.1) are bounded within the region ,
  B, N , TP , TS , I : 0  B  K B 0 ,0  N  N a  ,0  TP  TS  Qm , 0  I  I a  
where Qm 
QI a , N a 

,   min  0  g  g ,  1 .
Proof: Proof is analogous to the proof of lemma (3.1.1) of chapter 3.
6.3 EQUILIBRIUM ANALYSIS
The system (6.1.1) may have eight nonnegative equilibria in the B  N  TP  TS  I




Q0
Q0 g
Q0
 Q0 g
space, namely E1  0,0,
,
,0 , E2  0,0,
,
, L ,
 0  g  1 ( 0  g ) 
 0  g  1 ( 0  g ) 








~
~ ~ ~
~ ~
~ ~
~ ~
~
ˆ ˆ ˆ ˆ
E3 0, N , TP , TS ,0 , E4  0, N , TP , TS , I , E5 Bˆ ,0, TˆP , TˆS ,0 , E6  Bˆ , Nˆ , TˆP , TˆS ,0 ,






   
E 7 B,0, TP , TS , I and E * B * , N * , TP* , TS* , I * . The existence of E1 and E2 is obvious.
We prove the existence of the other equilibria.


~ ~ ~
Existence of E3 0, N , TP , TS ,0 :
~ ~
~
In this case, N , TP and TS are the positive solutions of the following equations.


~
~ ~
N  M TP , TS ,
(6.3.1)
149
~
TP 
 
~
Q 0, N
~
 f 1 ( N ), say,
~
0 2N  g
(6.3.2)
~
~  gf1 ( N )
~
TS 
~  f 2 ( N )., say,
1   2 N
(6.3.3)
~
~
~
It is noted that from equation (6.3.2) and (6.3.3) that TP and TS are functions of N
~
only. To show the existence of E 3 , we define a function F1 ( N ) as follows
~
~
~
~
F1 ( N )  N  M ( f1 ( N ), f 2 ( N )).
(6.3.4)
From equation (6.3.4), we note that
 Q0
 gQ0 
  0.
F1 (0)  M 
,
  0  g  1 ( 0  g ) 
 Q(0, N a )

 gQ(0, N a )
  0.
F1 N a   N a  M 
,
  0  g   2 N a ( 0  g   2 N a )( 1   2 N a ) 
 
~
~
~
Thus there exists a root N in the interval 0  N  N a given by F1 N  0.
 ~
Now, the sufficient condition for E 3 to be unique is F1 ( N )  0, where
 ~
 ~
 ~
F1 ( N )  1  M 1 f 1 ( N )  M 2 f 2 ( N )  0.
(  g )Q2  Q0 2
 ~
where f1 ( N )  0
,
~
( 1  1 N  g ) 2
(6.3.5)


~  ~
~
 ~  g ( 1   2 N ) f1 ( N )   2 f1 ( N )
f 2 (N )
.
~
( 1   2 N ) 2
~
~
~
With this value of N , the value of TP and TS can be found from equations (6.3.2) and
(6.3.3), respectively. This completes the existence of E3 .
~
~ ~
~ ~
~ ~
~
Existence of E 4  0, N , TP , TS , I  :


~
~
~ ~
~ ~
~
~
In this case N , TP , TS and I are the solutions of the following equations:
150
~
~
~
rP 0 N
~
rP 0 


I
 0,
~
1
~ ~
~
M (TP , TS )
(6.3.6)
~
~
~
~
~ ~
~
~
~~
~
~
Q( I , N )   0TP   2 NTP  gTP  0,
(6.3.7)
~
~
~
~
~
~~
~
 gTP  1TS   2 NTS  0,
(6.3.8)
~
~
~
L(r   2 N )
~
I
 g1 ( N ), say,
r
Using the value of
(6.3.9)
~
~
I from equation (6.3.9) in equations (6.3.7) and (6.3.8), we
obtain
~
~ ~
~
~
~
Q( g1 ( N ), N )
~
~
TP 

g
(
N
), say,
~
2
~
0  2N  g
(6.3.10)
~
~
~
~
~  gQ2 ( N )
~
TS 

g
(
N
), say,
~
3
~
1   2 N
(6.3.11)
~
~
~ ~
~
~
It is noted from equations (6.3.9), (6.3.10) and (6.3.11) that I , TP and TS are functions
~
~
~
~
of N only. To show the existence of E 4 , we define a function F2 ( N ) as follows
~
~
~
~
~
~
~
~
~
~
F2 ( N )  rP0 N  (rP 0   1 g1 ( N )) M ( g 2 ( N ), g 3 ( N )).
(6.3.12)
From equation (6.3.12), we note that
 QL,0 gQL,0 
  0.
F2 0  rP 0   1 L M 
,




g



g
1
0
 0

F2 N a   rP 0 N a  rP 0   1 g1 N a M g 2 N a , g 3 N a   0
where g1 ( N a ) 
L(r   2 N a )
Q( g1 ( N a ), N a )
 gg 2 ( N a )
, g2 (Na ) 
, g3 (N a ) 
.
r
0  2 Na  g
1   2 N a
~
~
~
~
~
~
Thus there exists a root N in the interval 0  N  N m given by F2  N   0.
 
151
~
 ~
Now, a sufficient condition for E4 to be unique is F2 ( N )  0, where
~
~
~

~
~
~
~
 ~
F2 ( N )  rP 0  L 1 2 M ( g 2 ( N ), g 3 ( N ))  (rP 0   1 g1 ( N )
r
~
~
 ~
 ~
(M 1 g 2 ( N )  M 2 g 3 ( N ))  0.
(6.3.13)
(  g )(Q1 L 2  Q2 r )  Q( L,0)r 2
~
 ~
where g 2 ( N )  0
,
~
~
( 0   2 N  g ) 2
~
 ~
g3 (N ) 
~
~
~
~
~
~
g ( 1   2 N ) g 2 ( N )   2 g 2 ( N )

~
~
( 1   2 N ) 2

.
~
~
~
~
~ ~
~
With this value of N , values of I , TP and TS can be found from equations (6.3.9),
(6.3.10) and (6.3.11), respectively. This completes the existence of E4 .


Existence of E5 Bˆ ,0, TˆP , TˆS ,0 :
In this case Bˆ , TˆP , TˆS are the solutions of the following equations


Bˆ  K B TˆP , TˆS ,
TˆP 
Q0
 0   1 Bˆ  g
(6.3.14)
 h1 ( Bˆ ), say,
(6.3.15)
 gh1 ( Bˆ )
TˆS 
 h2 ( Bˆ ), say,
ˆ
 1  1 B
(6.3.16)
It is noted from equations (6.3.15) and (6.3.16) that TˆP and TˆS , are functions of B̂
only. To show the existence of E 5 , we define a function F3 ( Bˆ ) as follows
F3 ( Bˆ )  Bˆ  K B .(h1 ( Bˆ ), h2 ( Bˆ )).
(6.3.17)
From equation (6.3.17), we note that
152
 Q0
gQ0 
  0.
F3 0   K B 
,
  0  g  1  0  g  
F3 K B 0   K B1
Q0
 gQ0
 K B2
 0,
 0  g  1 K B 0
( 1  1 K B 0 )( 0  g  1 K B 0 )
Thus there exists a root B̂ , in the interval 0  Bˆ  K B 0 , given by F3 ( Bˆ )  0.

Now, the sufficient condition for E 5 to be unique is F3 ( Bˆ )  0, where

F3 ( Bˆ )  1  K B1
Q0 1
gQ0 ( g1  1 0   1 1 )
 K B2
 0.
2
ˆ
( 0   1 B  g )
( 0   1 Bˆ  g ) 4
(6.3.18)
With this value of B̂ , value of TˆP and TˆS , can be found from equations (6.3.15) and
(6.3.16), respectively.
ˆ ˆ ˆ ˆ
Existence of E6 ( Bˆ , Nˆ , TˆP , TˆS ) :
ˆ ˆ ˆ ˆ
In this case, Bˆ , Nˆ , TˆP , TˆS are the solutions of the following equations
ˆ
rB ( Nˆ ) 
ˆ
rB 0 Bˆ
 0,
ˆ ˆ
K B (TˆP , TˆS )
(6.3.19)
ˆ
rP ( Bˆ ) 
ˆ
rP 0 Nˆ
 0,
ˆˆ ˆˆ
M (TP , TS )
(6.3.20)
ˆ
ˆ
ˆˆ
ˆˆ
ˆ
Q(0, Nˆ )   0TˆP  1 Bˆ TˆP   2 Nˆ TˆP  gTˆP  0,
ˆ
ˆ
ˆˆ
ˆˆ
 gTˆP   1TˆS  1 Bˆ TˆS   2 Nˆ TˆS  0.
(6.3.21)
(6.3.22)
From the equation (6.3.21) and (6.3.22), we have
ˆ
TˆP 
ˆ
Q(0, Nˆ )
ˆ ˆ
 d1 ( Bˆ , Nˆ ), say,
ˆ
ˆ
 0   1 Bˆ   2 Nˆ  g
153
(6.3.23)
ˆ ˆ
 gd1 ( Bˆ , Nˆ )
ˆˆ
ˆ ˆ
TS 
 d 2 ( Bˆ , Nˆ ), say.
ˆ
ˆ
 1  1 Bˆ   2 Nˆ
(6.3.24)
Using values of TP and TS from (6.3.23) and (6.3.24) in equations (6.3.19) and
(6.3.20) respectively, we get
ˆ ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ
F4 ( Bˆ , Nˆ )  (rB 0  rB1 Nˆ ) ( K B 0  K B1d1 ( Bˆ , Nˆ )  K B 2 d 2 ( Bˆ , Nˆ ))  rB 0 Bˆ  0,
(6.3.25)
ˆ ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ
F5 ( Bˆ , Nˆ )  (rP 0  rP1 Bˆ ) ( M 0  M 1d1 ( Bˆ , Nˆ )  M 2 d 2 ( Bˆ , Nˆ ))  rP 0 Nˆ  0,
(6.3.26)
From (6.2.25), we note the following
ˆ
ˆ
ˆ
when Nˆ  0, then Bˆ  Bˆ a , where
ˆ
d11 Bˆ a2  d12 Ba  d13  0,
where
d11  1 1 K B 0 , d12  K B 0 (1 0  1 1  g1  Q0 1 K B1 )  1.
d13  K B 0 ( 1 0  g 1 )  K B1Q0 1  Q0 gK B 2 .
ˆ
ˆ
Let D1 ( Bˆ a )  d11 Bˆ a2  d12 Ba  d13 ,
D1 (0)  d13  0,
D1 ( K B 0 )  d11 K B 0  d12 K B 0  d13  0.
ˆ
ˆ
 ˆ
Thus there exists a root Bˆ a in the interval 0  Bˆ a  K 0 given by D1 ( Bˆ a )  0.
ˆ
 ˆ
Now, the sufficient condition for Bˆ a to be unique is D1 ( Bˆ a )  0. where
ˆ
 ˆ
D1 ( Bˆ a )  2d11 Bˆ a  d12  0.
F4
ˆ
 ˆ ˆ
 ˆ ˆ
 [( rB 0  rB1 Nˆ ) ( K B1 d1 ( Bˆ , Nˆ )  K B 2 d 2 ( Bˆ , Nˆ ))  rB 0 ],
ˆˆ
B
154
(6.3.27)
F4
ˆ ˆ
ˆ ˆ
ˆ
 ˆ ˆ
 ˆ ˆ
 rB1 K (d1 ( Bˆ , Nˆ ), d 2 ( Bˆ , Nˆ ))  rB ( Nˆ )( K B1 d1 ( Bˆ , Nˆ )  K B 2 d 2 ( Bˆ , Nˆ )) 
ˆˆ


N
(6.3.28)
Now, from (6.3.27) and (6.3.28) we get
F4
ˆˆ
ˆˆ
B
  N .
ˆ
F4
Nˆ
ˆ
Bˆ
It is clear that
ˆ
Bˆ
 0, if either
ˆˆ
N
(i)
F4
F4
 0 and
 0, or
ˆ
ˆ
Bˆ
Nˆ
(ii)
F4
F4
 0 and
 0,
ˆˆ
ˆ
B
Nˆ
(6.3.29)
From (6.3.26), we note the following
ˆ
ˆ
ˆ
when Bˆ  0, then Nˆ  Nˆ b , where
ˆ
ˆ
v11 Nˆ b2  v12 Nˆ b  v13  0,
where
v11  M 0 2  2  M 1Q2  2 ,
v12  M 0 (  2 0   2 1  g 2 )  M 1 (Q0  2  Q2 1 )  M 2 gQ2  1.
v13  M 0 ( 0 1  g 1 )  M 1Q0 1  M 2 gQ0 .
ˆ
ˆ
ˆ
Let D2 ( Nˆ b )  v11 Nˆ b2  v12 Nˆ b  v13 ,
D2 (0)  v13  0,
D2 ( N a )  v11 N a2  v12 N a  v13  0,
ˆ
ˆ
ˆ
Thus there exists a root Nˆ b in the interval 0  Nˆ b  N a given by D2 ( Nˆ b )  0.
155
ˆ
 ˆ
Now, the sufficient condition for Nˆ b , to be unique is D2 ( Nˆ b )  0. where
ˆ
 ˆ
D2 ( Nˆ b )  2v11 Nˆ b  v12  0.
F5
ˆ ˆ
ˆ ˆ
ˆ
 ˆ ˆ
 ˆ ˆ
 rP1 M (d1 ( Bˆ , Nˆ ), d 2 ( Bˆ , Nˆ ))  rP ( Bˆ )( M 1 d1 ( Bˆ , Nˆ )  M 2 d 2 ( Bˆ , Nˆ )),
ˆˆ
B
(6.3.30)
F5
ˆ
ˆ ˆ
ˆ ˆ
 [( rP 0  rP1 Bˆ ) ( M 1 d1 ( Bˆ , Nˆ )  M 2 d 2 ( Bˆ , Nˆ ))  rP 0 ]  0.
ˆ
Nˆ
(6.3.31)
Now, from (6.3.30) and (6.3.31) we get
ˆ
Bˆ
ˆ
Nˆ
F5
ˆˆ
  N .
F5
ˆ
Bˆ
ˆ
Bˆ
It is clear that
 0, if either
ˆˆ
N
(i)
F5
F5
 0 and
 0, or
ˆ
ˆ
Bˆ
Nˆ
(ii)
F5
F54
 0 and
 0,
ˆˆ
ˆˆ
B
N
(6.3.32)
ˆ ˆ
Thus the two isoclines (6.3.25) and (6.3.26) intersects at a unique ( Bˆ , Nˆ ) if in
ˆ
ˆ
addition to conditions (6.3.29) and (6.3.32), the inequality Bˆ b  Nˆ b
holds. Knowing
ˆ
ˆ
ˆ
ˆ
the value of Bˆ and Nˆ , we get TˆP and TˆS , can be calculated from equations (6.3.23)
and (6.3.24). This completes the existence of E6 .


   
Existence of E 7 B,0, TP , TS , I :
   
In this case B, TP , TS , I are the solutions of the following equations.
156


rB 0 B
rB 0    I  0,
K B TP , TS
(6.3.33)




Q( I ,0)   0TP   1 BTP  gTP  0,
(6.3.34)





gTP   1TS  1 BTS  0,
(6.3.35)


 I
r1 1 -   B  0.
 L
(6.3.36)
From equation (6.3.34), (6.3.35) and (6.3.36) we get,



 B 
  e1 ( B), say.
I  L1 
r1 




Q e1 ( B),0
TP 
 e2 ( B), say,

 0  1 B  g

(6.3.37)

(6.3.38)



ge2 ( B)
TS 
  e3 ( B), say.
 1  1 B
(6.3.39)
 

From equations (6.3.37), (6.3.38) and (6.3.39) we note that TP , TS and I are functions


of B only. To show the existence of E 7 , we define a function F6 ( B ) as follows





F6 ( B)  rB 0 B  (rB 0  e1 ( B)) K B (e2 ( B), e3 ( B)).
(6.3.40)
From equation (6.3.40), we note that
 QL,0 gQL,0 
  0.
F6 (0)  rB 0  L K B 
,
  0  g  1  0  g  
F6 K B 0   rB 0 ( K B1e2 ( K B 0 )  K B 2 e3 ( K B 0 ))  e1 ( K B 0 ) K B e2 K B 0 , e3 K B 0   0.
where
 K B 0
e1 ( K B 0 )  L1 
r1


,

e2 ( K B 0 ) 
ge2 ( K B 0 )
Q0  Q1e1 ( K B 0 )
.
, e3 ( K B 0 ) 
 1  1 K B 0
 0  1 K B 0  g



Thus there exists a root B, in the interval 0  B  K B 0 , given by F6 ( B)  0.
157

Now, the sufficient condition for E7 to be unique is F6 ( B)  0, where




L
 
 
K B (e2 ( B), e3 ( B))  (rB 0  e1 ( B)) ( K B1e2 ( B)  K B 2 e3 ( B)).
where F6 ( B)  rB 0  
r
 Q L ( 0  g )  1r1Q( L,0)
where e2 ( B)  1
,

r1 ( 0  1 B  g ) 2
 
e3 ( B ) 


g
 
 2 (e2 ( B)( 1  B)  e2 ( B) 1 ).
( 1  1 B)

 

With this value of B, value of I , TP , and TS can be found from equations (6.3.37),
(6.3.38) and (6.3.39) respectively. This completes the existence of E7 .


Existence of E * B * , N * , TP* , TS* , I * .
In this case, B* , N * , TP* , TS* , I * are the solutions of following equations.
rB ( N * ) 
rB 0 B*
 I *  0,
*
*
K B (TP , TS )
(6.3.41)
rP ( B* ) 
rP 0 N *
  1 I *  0,
*
*
M (T , T )
(6.3.42)
P
S
Q( I * , N * )   0TP*   1 B*TP*   2 N *TP*  gTP*  0,
(6.3.43)
gTP*   1TS*  1 B*TS*   2 N *TS*  0,
(6.3.44)
 I* 
  B*   N *  0.
r1 1 
2

L


(6.3.45)
From the equation (6.3.45), we have
I* 


L
r1  B*   2 N *  s1 ( B* , N * ), say.
r1
With this value of I *, and from the equation (6.3.43) and (6.3.44), we have
158
(6.3.46)
TP* 
TS* 
Q( s1 ( B* , N * ), N * )
 s 2 ( B* , N * ),
*
*
( 0  g   1 B   2 N )
 gs 2 ( B* , N * )
( 1  1 B*   2 N * )
 s3 ( B* , N * ),
say,
say.
(6.3.47)
(6.3.48)
Using values of I * , TP* and TS* in equations (6.3.41) and (6.3.42) respectively, we get
F7 ( B* , N * )  (rB ( N * )  s1 ( B* , N * )) K ( s 2 ( B* , N * ), s3 ( B* , N * ))  rB 0 B* , (6.3.49)
F8 ( B* , N * )  (rP ( B* )   1 s1 ( B* , N * )) M ( s 2 ( B* , N * ), s3 ( B* , N * ))  rP 0 N *. (6.3.50)
From (6.3.49), we note the following
when N *  0, then B *  Be* , where
3
2
m11 Be*  m12 Be*  m13 Be*  m14  0,
where

Q L

m12  rB 0  L  K B 0 1 1 rB 0   1  ( K B1 1  K B 2g 1 )  rB 0 ( 0 1   1 1  g1 )
 r1




QL
L 
m11    1 1 rB 0 
  K B 0 1 1  1  ( K B1 1  K B 2g 1 ),
r1 
r1


L



QL
   K B 0 ( 0 1  1 1  g1 )  K B1 Q0 1  Q1 L1  1 1 
r1
 r1






QL
 K B 2g  1Q0  Q1 L 1  1  ( 0  g ),
r1



159



Q1 L
1 
 K B 0 ( 0 1   1 1  g1 )  K B1  Q( L,0) 1 
r1


m13  rB 0  L 
  rB 0 1 ( 0  g )


Q
L
1
 K B 2g   1 (Q( L,0) 

 0 

r


1


L

   K B 0 1 ( 0  g )  K B1 1Q( L)  K B 2g (Q( L)( 0  g ),
 r1

m14  rB 0  L K B 0 1 ( 0  g )  K B1 Q( L) 1   K B 2g Q( L)( 0  g .
3
2
Let M 1 ( Be* )  m11 Be*  m12 Be*  m13 Be*  m14 ,
M1 (0)  m14  0,
M 1 ( K B 0 )  m11 K B3 0  m12 K B2 0  m13 K B 0  m14  0.
Thus there exists a root B *E in the interval 0  Be*  K B 0 given by M 1 ( Be* )  0.

Now, the sufficient condition for B e* to be unique is M 1 ( Be* )  0. where
2

M 1 ( Be* )  3m11 Be*  2m12 Be*  m13  0,
s ( B* , N * )) K ( s ( B* , N * ), s ( B* , N * ))  r

1
2
3
B0


F7


 

*
B

 * *
 * * 
*
* *

 (rB ( N )  s1 ( B , N ))( K B1 s 2 ( B , N )  K B 2 s3 ( B , N )

(6.3.51)
(r  s  ( B* , N * )) K ( s ( B* , N * ), s ( B* , N * ))

B1
1
2
3


F7


 

*
N

 * *
 * * 
*
* *

 (rB ( N )  s1 ( B , N ))( K B1 s 2 ( B , N )  K B 2 s3 ( B , N )

(6.3.52)
Now, from (6.3.51) and (6.3.52) we get
F7
*
*
B
  N .
F7
N *
B *
160
It is clear that
B*
 0, if either
*
N
(i)
F7
F7
 0 and
 0, or
*
B
N *
(ii)
F7
F7
 0 and
 0.
B*
N *
(6.3.53)
From (6.3.50), we note the following
when B*  0, then N *  N e* , where
3
2
n11 N e*  n12 N e*  n13 N e*  n14  0,
where
n11  (rP 0 2  2  M 2gQ2 2 ),

Q L

Q L

n12  rP 0   1 L  M 0 2  2  M 1  1  2  2  Q2  2 )  M 2g  1  2 2  Q2 2 
 r1

 r1






 rP 0 ( 0  21   2 1  g 21 )  M 0 ( 0  2   2  2  g 2 )  M 1  2  Q0  Q1 L 2  1  Q2 
 r1






 M 2g 2  Q1 L 2  1 ,
 r1


n13  rP 0

 Q1 L

 2  1  Q ( L,  1 )  2 ) 
 M 0 (( 0  g )  2   2 1 )  M 1 
 r1


  1 L 



 M 2g ( 0  g ) Q1 L 2  Q2   Q ( L,0) 2 

 r



1







2

 1  Q2 
M 0 1 ( 0   2  g )  M 1 1  Q0  Q1 L
  L
 r1


  r  (  g ),
 1 2 
 P0 1 0
r1 

Q1 L 

 M 2g ( 0  g ) Q0  Q1  r 

1 



n14  rP 0   1 L M 0 ( 0  g ) 1  M 1 1Q( L,0)  M 2g ( 0  g )Q( L,0).
161
3
2
Let M 2 ( N e* )  n11 N e*  n12 N e*  n13 N e*  n14 ,
M 2 (0)  n14  0,
M 2 ( N a )  n11 N a3  n12 N a2  n13 N a  n14  0.
Thus there exists a root N e* in the interval 0  N e*  N a given by M 2 ( N e* )  0.

Now, the sufficient condition for N e* , to be unique is M 2 ( N e* )  0, where
2

M 2 ( N e* )  3n11 N e*  2n12 N e*  n13  0.
F8

 (rP1 ( B* )   1 s1 ( B* , N * )) M ( s 2 ( B* , N * ), s3 ( B* , N * ))
*
B


 (rP ( B* )   1 s1 ( B* , N * ))( M 1 s2 ( B* , N * )  M 2 s3 ( B* , N * )),
(6.3.54)
F8

  1 s1 ( B* , N * )) M (s 2 ( B* , N * ), s3 ( B* , N * ))  rP 0
N *


 (rP ( B* )   1 s1 ( B* , N * ))( M 1 s2 ( B* , N * )  M 2 s3 ( B* , N * )).
(6.3.55)
Now, from (6.3.54) and (6.3.55) we get
F8
N *
B *

.
F8
N *
B *
It is clear that
(i)
B*
 0, if either
N *
F8
F8
 0 and
 0, or
B*
N *
(6.3.56)
162
(ii)
F8
F8
 0 and
 0.
*
*
B
N
Thus the two isoclines (6.3.49) and (6.3.50) intersects at a unique ( B* , N * ) if in
addition to conditions (6.3.53) and (6.3.56), the inequality Be*  N e* , holds. Knowing
the value of B* and N * , we get I *, TP* and TS* , can be calculated from equations
(6.3.46), (6.3.47) and (6.3.48). This completes the existence of E * .
6.4. STABILITY ANALYSIS
6.4.1 Local Stability
To discuss the local stability of system (6.1.1), we compute the variational matrix of
system (6.1.1).The entries of general variational matrix are given by differentiating
the right hand side of system (6.1.1) with respect to B, N , TP , TS , and I i.e.

 v11


r N
M ( E )   P1

  1TP
  T
 1 S
 I
Q 2   2 TP
 rB 0 B 2 K B1
K B2 TP , TS 
 rP 0 N 2 M 1
M 2 TP , TS 
  0   1 B   2 N  g 
  2TS
g
 2I
0
 rB1 B
v 22
where, v11  rB N  

 B 


 1N 
,

Q1 
 ( 1  1 B   2 N )
0 

0
v33 
 rB 0 B 2 K B 2
K B2 TP , TS 
 rP 0 N 2 M 2
M 2 TP , TS 
0
2rB 0 B
2rP 0 N
 I , v22  rP B  
  1I ,
K B TP , TS 
M TP , TS 
 2I 
v33  r1 1    B   2 N .
L

The variational matrix M ( E1 ) at equilibrium point E1 is given by
163
rB 0
0
0
0


0
rP 0
0
0

 Q
   1Q0
Q2  2 0   0  g  0

M ( E1 ) 
 g
0  g
  0Q g
 Q g
 1 0
 2 0
g
 1
 1  0  g 
  1  0  g 

0
0
0
0
0
0 
Q1 
.

0

r1 
From M ( E1 ) , we note that characteristic roots namely, rB 0 , rP 0 and r1 are positive,
giving a saddle point which is stable in the TP  TS plane and unstable in the
B  N  I , space. Therefore, equilibrium point E1 is unstable.
The variational matrix M ( E2 ) at equilibrium point E2 is given by
0
0
0
 rB 0  L

rP1 N
rP 0   1 L
0
0

 Q
   1Q0
Q2  2 0   0  g  0
M E 2   
 g
0  g
  0Q g
 Q g
 1 0
 2 0
g
 1
 1  0  g 
  1  0  g 

L
 2L
0
0
0 
0 
Q1 
,

0 

 r1 
From M ( E1 ) , we note that characteristic roots namely, rB 0  L, and rP 0   1 L are
positive, giving a saddle point which is stable in the TP  TS  I space and unstable
in the B  N , plane. Therefore, equilibrium point E 2 , is unstable.
The variational matrix M ( E3 ) at equilibrium point E 3 is given by
 
~
 rB N

~
 rP1 N

M ( E3 )  
~


T
1
P

~
  1TS

 0
0
~
rP 0 N

~ ~
M TP , TS
~
Q 2   2 TP
~
  2TS

0

0
~
 rP 0 N 2 M 1
~ ~
M 2 TP , TS
~
 0  2N  g

g
0
~
 rP 0 N 2 M 2
~ ~
M 2 TP , TS
0
~
 ( 1   2 N )
0
0


164



0 

~
 1N 

,
Q1 
0 
~
r1   2 N 
 
~
~
From M ( E3 ) , we note that characteristic roots namely, rB N and r1   2 N are
positive, giving a saddle point which is stable in the N  TP  TS space and unstable
in the B  I , plane. Therefore, equilibrium point E 3 is unstable.
The variational matrix M ( E4 ) at equilibrium point E4 is given by
~
~
~
~
r ( N
)  I
0
B

~
~
~

rP 0 N 2
~
r
N


P1
~
~ ~
~
M  TP , TS 




~
~
~
~
M (E4 )  
  1TP
Q 2   2 TP

~
~
~
~



T


T
1 S
2 S


~
~
~
~
I
 2I


0
0
~
~
~2
~
 rP 0 N M 1
 rP 0 N 2 M 2
~
~
~ ~
~
~ ~
~
M 2  TP , TS 
M 2  TP , TS 




~
~
   0   2 N  g 
0


~
~
g
 ( 1   2 N )
0
0






,
Q1 


0 
~
~
r1 I 


L 
0
~
~
 1N
~
~
~
~
From M ( E4 ) , we note that characteristic root namely, rB ( N )  I is positive, giving a
saddle point which is stable in the N  TP  TS  I space and unstable in the B ,
direction. Therefore, equilibrium point E4 is unstable.
The variational matrix M ( E5 ) at equilibrium point E 5 is given by

rB 0 Bˆ 2

ˆ ˆ
 K B T P , TS

0
M ( E5 )  
ˆ
   1TP
   Tˆ
1 S

0



 rB1 Bˆ

rP Bˆ
 rB 0 Bˆ 2 K B1
K 2 Tˆ , Tˆ
B


P
S

0
Q2   2TˆP
  Tˆ
  0   1 Bˆ  g
0
0
g
2 S
 rB 0 Bˆ 2 K B 2
K 2 Tˆ , Tˆ

 Bˆ 

B
P
S
0
0 
,
0
Q1 
 ( 1  1 Bˆ )
0 

0
r1  Bˆ 




From M ( E5 ) , we note that characteristic roots namely, rP Bˆ
and r1   Bˆ are
positive, giving a saddle point which is stable in the B  TP  TS space and unstable in
the N  I , plane. Therefore, equilibrium point E 5 is unstable.
165
The variational matrix M ( E6 ) at equilibrium point E6 is given by
ˆ

rB 0 Bˆ

 K  Tˆˆ , Tˆˆ 
B P
S 




ˆ

rP1 Nˆ

M ( E6 ) 


   1TˆˆP

ˆ
  1TˆS

0

ˆ
 rB1 Bˆ
ˆ
rP 0 Nˆ

ˆ ˆ
M  TˆP , TˆS 


ˆˆ
Q 2   2 TP
ˆ
  2TˆS
ˆ
 rB 0 Bˆ 2 K B1
ˆ ˆ
K B2  TˆP , TˆS 


ˆˆ 2
r N M
ˆ
 rB 0 Bˆ 2 K B 2
ˆ ˆ
K B2  TˆP , TˆS 


ˆˆ 2
r N M
ˆ ˆ
M 2  TˆP , TˆS 


ˆ ˆ
M 2  TˆP , TˆS 


 d11
0
g
 d 22
0
0
P0
0
ˆ
ˆ
where d11    0  1 Bˆ   2 Nˆ  g ,


1
P0
2





ˆˆ

 1N
,



Q1

0

ˆ
ˆ
r1  Bˆ   2 Nˆ 
ˆ
 Bˆ
ˆ
ˆ
d 22    1  1 Bˆ   2 Nˆ .


From M ( E6 ) , we note that characteristic root namely,
ˆ
ˆ
r1  Bˆ   2 Nˆ is positive,
giving a saddle point which is stable in the B  N  TP  TS space and unstable in the
I direction. Therefore, E6 is unstable.
The variational matrix M ( E7 ) at equilibrium point E7 is given by


rB 0 B
 

K
T
B
P , TS

0



M ( E7 ) 


T
1

P
  1TS



I





 rB 0 B 2 K B1
 rB1 B
 
K B2 TP , TS


rP B   1 I
0


Q 2   2 TP   0   1 B  g

  2TS
g

 2I
0





 rB 0 B 2 K B 2
 
K B2 TP , TS
0

 B 

0 

0
Q1 ,

 ( 1  1 B)
0 

r1 I 
0

L 






From M ( E7 ) , we note that characteristic root namely, rP B   1 I is positive, giving
a saddle point which is stable in the B  TP  TS  I space and unstable in the N
direction. Therefore, E7 is unstable.
166
The variational matrix M (E*) at equilibrium point E * is given by








*
M (E )  








rB 0 B*
K B TP* , TS*

2

 rB 0 B * K B1
K 2 T *,T *
 rB1 B*
B
rP 0 N *
M T *,T *

P
S

2
  1TP*
Q2   2TP*
 1TS*
  2TS*
g
I *
 2I*
0


P
S

 rP 0 N * M 1
M 2 T *,T *
 
Q I * , N * 

rP1 N *
P
S
2
 rB 0 B* K B 2
K 2 T *,T *
B

P
S

2
 rP 0 N * M 2
M 2 T *,T *

P
0
TP*

gTP*
TS*
0
S


*
 B 



1N * 


Q1 ,



0 

*
rI 
 1 
L 
In the following theorem we show that E * is locally asymptotically stable.
Theorem 6.4.1: In addition to assumptions (H1) – (H5), let the following inequalities
hold
rP1 N *   1TP*  1TS*  I * 
rB 0 B*
,
K B (TP* , TS* )
rB1 B*  Q2 -  2TP*   2TS*   2 I * 
rP 0 N *
,
M (T * , T * )
P
(6.4.1)
(6.4.2)
S
2
2
K B1
M1
Q( I * , N * )
rB 0 B* 
rP 0 N *  g 
,
K B2 (TP* , TS* )
M 2 (TP* , TS* )
TP*
(6.4.3)
2
2
K B2
M2
gTP*
*
*
rB 0 B 
rP 0 N 
,
K B2 (TP* , TS* )
M 2 (TP* , TS* )
TS*
(6.4.4)
r1 I *
*
*
B   1 N  Q1 
.
L
(6.4.5)
then E * is locally asymptotically stable.
167
Proof: If inequalities (6.4.1) – (6.4.5) hold, then by Gerschgorin’s theorem
(Lancaster and Tismenetsky, 1985), all eigenvalues of M (E * ) have negative real
parts and interior equilibrium E * is locally asymptotically stable.
6.4.2. Global Stability
Theorem (6.4.2): In addition to the assumption (H1) – (H5), let rB ( N ), rP ( B),
K B TP , TS  , M TP , TS  and Q ( I , N ) satisfy the conditions 0  rB N   1 ,
0  -rP B    2 , M n  M TP , TS   M 0 , K m  K B (TP , TS )  K B 0 ,0  
0
M
 m2 ,
TS
K
K
Q
Q
M
  3 ,0 
  4 , 0   B  k1 , 0   B  k 2 , 0  
 m1, , in  for
I
N
TP
TS
TP
some positive constants 1 ,  2 ,  3 ,  4 , k1 , k 2 , K 0 , K m , M 0 , M n , m1 , m2 .
(6.4.6)
Then if the following inequalities hold
rB 0
rP 0
,
4 K (T * , T * ) M (T * , T * )
B
P
S
P
S
1   2 2  1
(6.4.7)


2

k
  1Qm  rB 0 K B 0 1
2

Km


rB 0
 1
 0  g   1 B*   2 N * ,

*
*
4
K B (TP , TS )


k
 1Qm  rB 0 K B 0 2
2

Km

 1
rB 0
 
 1  1 B*   2 N * ,

*
*
3
K B (TP , TS )


2

rB 0
r1
,
3 K (T * , T * ) L
B
P
S
   2  1
(6.4.8)
(6.4.9)
(6.4.10)
2

rP 0
m 
1
  4   2 Qm  rP 0 N m 12  
( 0  g   1 B*   2 N * ),


*
*
4 M (T , T )
Mn 

P
S
168
(6.4.11)
2

rP 0
m  1
  2 Qm  rP 0 N m 22  
( 1  1 B*   2 N * ),


M n  3 M (T * , T * )

P
S
(6.4.12)
rP 0
r1
,
*
*
3 M (T , T ) L
 1   2 2  1
P
(6.4.13)
S
 g 2  1 ( 1  1 B*   2 N * )( 0  g   1 B*   2 N * ),
(6.4.14)
3
 32 
1 r1
( 0  g  1 B*   2 N * ),
3L
(6.4.15)
E * is globally asymptotically stable with respect to all solutions initiating in the
positive orthant .
Proof: Consider the following positive definite function about E *


B  
N  1
  TP  TP*
   N  N *  N * ln
V B, N , TP , TS , I    B  B*  B* ln


B*  
N*  2


2

1
I 
.
 (TS  TS* ) 2   I  I *  I * ln
*
2
I 

Differentiating V with respect to time t, we get




dTS  I  I *  dI
dTP
dV  B  B*  dB  N  N *  dN


 TP  TP*
 TS  TS*

.
dt  B  dt  N  dt
dt
dt  I  dt






Substituting values of
dI
dB dN dP1 dP2
and
from the system of equation (6.1.1)
,
,
,
dt
dt dt dt dt
in the above equation and after doing some algebraic manipulations and considering
functions,
 
 r N   r N *
B
B

,

 B N    N  N *

*

rB N ,
 
N  N *,
N  N*
169
(6.4.16)
 
 r B   r B *
P
 P
,
 P B    B  B *
  *
rP B ,
B  B*,
 
(6.4.17)
B  B*
 
 Q I , N   Q I * , N
, I  I *,

*

I I
 Q1 I , N   
 Q I * , N
,
I  I *,

I



(6.4.18)
1
1

 K T , T  
K B TP* , TS
 B P S
,

*
T

T
 B1 TP , TS   
P
P

K B TP* , TS
1
,
 K 2 T *,T
TP
 B P S


(6.4.19)




TP  TP* ,
TP  TP* ,
1
1



*
* *
 K B (TP , TS ) K B (TP , TS ) ,

TS  TS*
 B 2 (TP* , TS )  

K B TP* , TS*
1
,
 K 2 T *,T *
TS
 B P S



(6.4.20)


1
1

 M (T , T ) 
P
S
M (TP* , TS )

,

*
T

T
 P1 (TP , TS )  
P
P

M TP* , TS*
1
,
 M 2 T *,T
TP
P
S


TS  TS* ,


TS  TS*
TP  TP*,
(6.4.21)
TP  TP*
1
1



*
* *
 M (TP , TS ) M (TP , TS ) ,

TS  TS*
 P 2 (TP* , TS )  

M TP* , TS*
1
,
 M 2 T *,T *
TS
P
S





170
TS  TS* ,
(6.4.22)
TS  TS*
Q 2
  

 Q I *, N  Q I *, N *
,

*

*
N

N
I ,N  
 Q I *, N *
,

N

 


N  N *,
(6.4.23)
N  N *,
we get
dV
1
1
  a11 ( B  B* ) 2  a12 ( B  B* )( N  N * )  a 22 ( N  N * ) 2
dt
4
4
1
1
  a11 ( B  B* ) 2  a13 ( B  B* )(TP  TP* )  a33 (TP  TP* ) 2 ,
4
4
1
1
  a11 ( B  B* ) 2  a14 ( B  B* )(TS  TS* )  a 44 (TS  TS* ) 2 ,
4
3
1
1
  a11 ( B  B* ) 2  a15 ( B  B* )( I  I * )  a55 ( I  I * ) 2 ,
4
3
1
1
  a 22 ( N  N * ) 2  a 23 ( N  N * )(TP  TP* )  a33 (TP  TP* ) 2 ,
4
4
1
1
  a 22 ( N  N * ) 2  a 24 ( N  N * )(TS  TS* )  a 44 (TS  TS* ) 2 ,
4
3
1
1
  a 22 ( N  N * ) 2  a 25 ( N  N * )( I  I * )  a55 ( I  I * ) 2 ,
4
3
1
1
  a33 (TP  TP* ) 2  a34 (TP  TP* )(TS  TS* )  a 44 (TS  TS* ) 2 ,
4
3
1
1
  a33 (TP  TP* ) 2  a35 (TP  TP* )( I  I * )  a55 ( I  I * ) 2 ,
4
3
where
a11 
rB 0
rP 0
, a12   B N    P ( B), a22 
, a23  - rP 0 N P1 (TP , TS ),
K B (TP* , TS* )
M (TP* , TS* )
a44   1  1 B*   2 N * , a55 
r1
, a15     , a24  rP 0 N P 2 (TP* , TS )   2TS ,
L
171
a35  Q1 I , N , a13  1TP  rB 0 B P1 (TP , TS ), a14  1TS  rB 0 B 2 (TP* , TS )
a 25   1   2 , a33   0  g   1 B*   2 N * , a34  g.
Then sufficient conditions for
dV
to be negative definite are that the following
dt
inequalities hold
a122 
1
1
1
1
1
2
2
a11a 22 , a132  a11a33 , a142  a11a 44 , a15
 a11a55 , a 23
 a 22 a33 ,
4
4
3
3
4
1
1
1
1
2
2
2
2
a 24
 a 22 a 44 , a 25
 a 22 a55 . a34
 a33 a 44 , a 35
 a 33 a 55 .
3
3
3
3
(6.4.24)
Now, from (6.4.6) and mean value theorem, we note that
 B N   1 ,  P B    2 ,  Q1 I , N    3 ,  Q 2 I *, N    4 ,  P1 TP , TS  
 P 2 (TP* , TS ) 
m2
Mn
2
,  B1 TP , TS  
k1
Km
2
,
 B 2 (TP* , TS ) 
k2
Km
2
.
m1
Mn
2
,
(6.4.25)
Further, we note that the stability conditions (6.4.7) - (6.4.15) as stated in theorem
(6.4.2), can be obtained by maximizing the left-hand side of inequalities (6.4.24). This
completes the proof of theorem (6.4.2).
6.5 NUMERICAL SIMULATIONS AND DISCUSSION
To facilitate the interpretation of our mathematical findings by numerical simulation,
we integrated system (6.1.1) using fourth order Runge - Kutta method. We take the
following particular form of the functions involved in the model (6.1.1):
rB N   rB 0  rB1 N , rP B   rP 0  rP1 B, K B TP , TS   K B 0  K B1TP  K B 2TS ,
M TP , TS   M 0  M 1TP  M 2TS , QI , N   Q0  Q1 I  Q2 N .
172
(6.5.1)
Now we choose the following set of values of parameters in model (6.1.1) and
equation (6.5.1).
rB 0  11, rB1  0.2, K B 0  12.2, K B1  0.1, K B 2  0.3,   0.01, rP 0  20, rP1  0.1,
M 0  10, M 1  0.1, M 2  0.2,  1  0.02, Q0  20, Q1  0.3, Q2  0.2,  0  14,  1  0.001,
 2  0.08, g  5,   0.5,  1  17, 1  0.6,  2  0.1, r1  9, l  5,   0.1,  2  0.2,
K m  0.001, k1  0.2, k 2  0.01, m1  0.02, m2  0.01, M n  1.3, 1  0.2,  2  0.1,
 3  1,  4  0.1.
(6.5.2)
With the above values of parameters, we note that condition for the existence of E *
are satisfied, and E * is given by
B*  9.6912, N *  10.3966, TP*  1.2140, TS*  0.1272, I *  6.6936.
(6.5.3)
It is further noted that all conditions of local stability (6.4.1) – (6.4.5), global stability
(6.4.7) – (6.4.15) are satisfied for the set of values of parameters given in (6.5.2).
In Figures (1) – (3), the primary and secondary toxicants for different rate of emission
of toxicants by natural sources or by industrialization and population Q0 , Q1 and Q2 ,
against time are plotted. From these plots, we can infer that as the rate of emission of
toxicants either directly or due to industrialization or by population increases,
equilibrium densities of both primary and secondary toxicants increases as expected.
These figures, show that the qualitative behaviors of both primary and secondary
toxicants are same whereas the quantitative behaviors of both toxicants are different
as the concentration of toxicants in the environment increases. The density of primary
toxicant is more in the environment than density of secondary toxicant by increase in
concentration of toxicants due to population and industrialization. Figure (4), shows
173
the dynamics of resource-biomass for different depletion rate coefficient of resource
biomass due to industrialization  , w.r.t time ‘t’. This shows that density of resourcebiomass decreases as  , increases. It is also noted that the resource-biomass density
initially increases w.r.t time t and after certain time it settle down to its steady state.
1.8
1.6
1.4
TP(Q0=30)
1
TP(Q0=20)
P
T ,T
S
1.2
0.8
0.6
TS(Q0=30)
TS(Q0=20)
0.4
0.2
0
0
10
20
30
40
50
time (t)
60
70
80
90
100
Figure 1, Graph of TP and TS versus t for different Q0 and other
values of parameters are same as in equation (6.5.2).
174
1.6
1.4
1.2
1
TP(Q1=0.8)
P
T ,T
S
TP(Q1=0.3)
0.8
0.6
0.4
TS(Q1=0.8)
TS(Q1=0.3)
0.2
0
0
20
40
60
80
100
time (t)
Figure 2, Graph of TP and TS versus t for different Q1 and other
values of parameters are same as in equation (6.5.2)
1.5
TP(Q2=0.6)
1
P
T ,T
S
TP(Q2=0.2)
0.5
TS(Q2=0.6)
TS(Q2=0.2)
0
0
10
20
30
40
50
time (t)
60
70
80
90
100
Figure 3, Graph of TP and TS versus t for different Q2 and other
values of parameters are same as in equation (6.5.2)
175
Figure 4, Graph of B versus t for different  and other values
of parameters are same as in equation (6.5.2).
10
9.8
Resource Biomass ( B )
9.6
9.4
9.2
Presence of Secondary toxicant, TS
Absence of Secondary toxicant, TS
9
8.8
8.6
8.4
8.2
8
0
1
2
3
4
5
time ( t )
6
7
8
9
10
Figure 5, Graph of B versus t for presence and absence of secondary toxicant and
other values of parameters are same as in equation (6.5.2).
176
Figure (5), shows the dynamics of resource biomass w.r.t time ‘t’ for presence and
absence of secondary toxicant. It is found that equilibrium level of resource biomass
is greater in the absence of secondary toxicant than in the presence of it. Figures (6)
and (7), shows the effects of K B1 and K B 2 , on the dynamics of resource-biomass w.r.t
time ‘t’. In both cases the carrying capacity of resource-biomass decreases with
increase in density of primary and secondary toxicant. These figures also show that
primary pollutant has more adverse effect on the resource-biomass carrying capacity
for a larger period than secondary toxicant. Similar behavior can be seen in Figures
(8) and (9), which is plotted between population and time ‘t’ for different values of
M 1 and M 2 , respectively.
Figure 6, Graph of B versus t for different k B1 and other values
of parameters are same as in equation (6.5.2).
177
Figure 7, Graph of B versus t for different k B 2 and other values
of parameters are same as in equation (6.5.2).
Figure 8, Graph of N versus t for different M1 and other values
of parameters are same as in equation (6.5.2).
178
Figure 9, Graph of N versus t for different M 2 and other values
of parameters are same as in equation (6.5.2).
6.6 CONCLUSION
In this chapter, a nonlinear mathematical model to study the effects of
industrialization, population, primary–secondary toxicants on depletion of forestry
resource is proposed and analyzed. It is assumed that primary toxicant is emitted into
the environment with a constant prescribed rate as well as its growth is increased by
increase in density of population and industrialization. Further, a part of primary
toxicant is transformed into secondary toxicant, which is more toxic, both affecting
the resource biomass and population simultaneously. Criteria for local stability,
instability and global stability are obtained by using stability theory of differential
equation. It is found that if the densities of industrialization and population increases,
then the density of primary toxicant become very large into the environment due to
179
which the densities of resource biomass and population decreases & it settle down at
its equilibrium level whose magnitude is lower than its original carrying capacity.
From the analysis it is also observed that in the absence of secondary toxicant, the
equilibrium level of resource biomass is more, than in the presence of it. It is also
found that due to high level of primary toxicant in the environment, it led in large
transformation of secondary toxicant, which decreases the densities of resource
biomass and population more than the case of single toxicant.
180