CHAPTER 3
MODEL FOR THE EFFECT OF ENVIRONMENTAL
POLLUTION ON THE SURVIVAL OF RESOURCE DEPENDENT
POPULATION WITH TIME DELAY
INTRODUCTION
Due to the advancement in technology and industrialization at rapid pace,
large amount of toxicant (pollutant) enter into both aquatic and terrestrial environment
and effect the biological population as well as their resource seriously. When a
resource-biomass is exposed to toxicant, the toxicant interferes with resource
metabolism. Due to this, the intrinsic growth of the biomass may be adversely
affected. The effect of toxicant on the resource-biomass is sometime greater than the
consumer species. It may lead to depletion of resources such as forestry, fishery,
fertile topsoil, crude oil, mineral, etc.. In extreme cases the extinction of species may
also occur affecting the bio-diversity of the ecosystem (Freedman and Shukla, [1991],
Shukla and Dubey, [1997], Shukla et al. [2009]). Therefore, it is important to study
the effect of pollutants on biological population as well as on resource-biomass on
which it depend. The uptake of environmental toxicant by the resource-biomass may
not be instantaneous. Therefore, the consideration of delay effect in such situation is
very logical and close to real system. It is common observation that introduction of
delay in the system lead to unstable behavior of the system.
Keeping these in view, a delay differential model is proposed to study the
effect of toxicant on a biological population as well as on resource-biomass. The
50
model assumes that toxicant affect the carrying capacity of both the biological
population and their resource biomass. The time delay in the uptake of toxicant by the
resource biomass reduces its growth rate which implicitly affects the dynamics of
biological population. The model has been mathematically analyzed and numerical
simulations have been carried out to study their local and global stability.
3.1 MATHEMATICAL MODEL
We consider an ecosystem, where the survival of a biological population
dependent on a resource is affected by a pollutant present in the environment. It is
assumed that the dynamics of population and resource-biomass is governed by the
logistic equation. The following system of differential equation can be written as:
r B2
dB
 r ( N ) B- 0
,
dt
K (T )
rN 0 N 2
dN
 rN ( B)N ,
dt
M (T )
(3.1.1)
dT
 Q(t )   0T   1 BT   BU  1 1U ,
dt
dU
  1U   1 BT  BU   0 0T .
dt
B(0)  0, N (0)  0, T (0)  0, U (0)  0.
In model (3.1.1), B (t ) is the density of resource-biomass, N (t ) is the density of
biological population, T (t ) is the concentration of pollutant (toxicant) present in the
environment and U (t ) is the uptake concentration of the pollutant taken up by the
resource-biomass from the environment. Q(t ) is the rate of introduction of the
pollutant into the environment, which may be constant, zero or periodic.  0 is the
depletion rate coefficient of pollutant from the environment, perhaps from biological
51
transformation,
chemical
hydrolysis,
volatization,
microbal
degradation
or
photosynthetic degradation, and a fraction  0 of it may re-enter into the resource
biomass with the uptake of the pollutant. 1 is the natural depletion rate coefficient of
U (t ) due to ingestion and depuration of the pollutant, and a fraction 1 of it may re-
enter into the environment due to recycling. Also, the uptake concentration of the
pollutant may decrease with the rate coefficient  , due to resource biomass and a
fraction  of which may re-enter into the environment.  1 is the depletion rate
coefficient of environmental pollutant due to its uptake by the resource-biomass.
Now, we assume that environmental pollutant does not affect the growth of
the resource-biomass, but the pollutant after entering into the biomass gets converted
to a substance which is toxic to the resource-biomass and consequently the growth
rate of the resource biomass decreases. This conversion causes a delay in the
depletion of the resource-biomass. Hence, we have
t
W (t )   f (t  p)U ( s )ds
(3.1.2)

where,   0 is a constant, f (t ) is a non-negative continuous function called a weight
function or Kernel (Rao, 1992) and it satisfies,
t
 f (t  p)dp  g
0
.
(3.1.3)

Where g 0 is a positive constant.
In equation (3.1.2), W (t ) may be thought as the concentration of toxic substance
which has been formed by the conversion of U (t ) due to some metabolic changes. If
we consider the weight function f (t ) in the form (Dubey and Hussain, [2004]):
52
f (t  p)  e
α0 (t  p )
,
 0  0.
Then equation (3.1.2) reduces to
t
W (t ) 

e
α0 (t  p )
U ( s)ds.

Differentiate the above equation w.r.t ‘t’, we get
dW
 U   0W .
dt
Thus, the inclusion of delay in the uptake of environment pollution by the resource
leads to the system (3.1.1) as:
r B2
dB
 r ( N ,W ) B  0
,
dt
K (T )
rN 0 N 2
dN
 rN ( B) N ,
dt
M (T )
dT
 Q(t )   0T   1 BT   BU  1 1U ,
dt
(3.1.4)
dU
  1U   1 BT  BU   0 0T ,
dt
dW
 U   0W .
dt
B0  0, N 0  0, T 0  0, U 0  0, and W 0  0.
In system (3.1.4), W (t ) may be thought of as the concentration of the toxic substance
which has been formed by the conversion of U (t ), due to some metabolic change. We
assume
that
the
growth
rate
of
resource-biomass
is
affected
by W and
population N , where as carrying capacity is affected by environmental pollutant T .
 is the growth rate coefficient of W (t ) which is assumed to be proportional to the
concentration of U (t ) and  0 is the natural depletion rate coefficient of W (t ) .
53
Model (3.1.4), is derived under following assumptions.
(H1): The function r ( N ,W )  0, is the specific growth rate of the resource-biomass
which decreases as the density of population and delay in the uptake of pollutant
increases, i.e.,
r 0,0  r0  0,
r ( N,W )
r ( N,W )
 0,
 0 for N  0, W  0.
N
W
(H2): The function K (T ) is the carrying capacity of the resource-biomass which the
environment can support in the presence of environmental pollutant, and it decreases
as the environmental concentration of the pollutant increases, i.e.,
K (0)  K 0  0, K (T )  0 for T  0.
(H3): The function rN ( B)  0 is the specific growth rate of biological population
which increases as the density of resource-biomass increases, i.e.,
rN (0)  rN 0  0, rN ( B)  0 for B  0.
(H4): The function M (T ) is the carrying capacity of the population which the
environment can support in the presence of environmental pollutant, and it decreases
as the environmental concentration of the pollutant increases, i.e.,
M 0  M 0  0,
dM
 0, for T  0.
dT
We analyze model (3.1.4) for three values of Q (t ), namely,
(i) Q(t )  Q0  0, (ii) Q(t )  0, and (iii) Q(t ) periodic.
To analyze the model (3.1.4), we need the bounds of dependent variable involved. For
this we find the region of attraction in the following lemma.
54
3.2 BOUNDEDNESS OF SOLUTION
Lemma (3.1.1): Suppose that assumptions (H1) - (H4) hold. Then all solutions of
system (3.1.4) are bounded within the region 
Q 

  B, N , T ,U ,W : 0  B  K 0 , 0  N  M 0 , 0  T  U  W  0 , where  0  1,
 

1  1, 1 (1  1 )   and   min  0 1   0 ,  1 1  1    ,  0 .
Proof: It is assumed here that all the initial values of variables considered in model
(3.1.4) belongs to the region  and are positive.
From the first equation of system (3.1.4), we get

dB
B 
.
 r0 B1 
dt
 K0 
This implies that, 0  B  K 0 .
From the second equation of model (3.1.4), we get

dN
N 
.
 rN 0 N 1 
dt
 M0 
This implies that, 0  N  M 0 .
From the third, fourth and fifth equation of model (3.1.4), we get
dT dU dW


 Q0   0T (1   0 )  U  1 1  1       0W
dt
dt
dt
 Q0   T  U  W ,
Where   min  0 1   0 ,  1 1  1    ,  0  .
This implies that, 0  T  U  W 
Q0

.
Hence, the lemma follows.
55
3.3 EQUILIBRIUM ANALYSIS
Case I: Constant introduction of pollutant, i.e., Q(t )  Q0  0.
The model (3.1.4) has four non-negative equilibrium points in the B  N  T  U  W


Q0
Q0 0
Q0 0
, E1 0, N , T ,U ,W ,
space namely, E0  0,0,
,
,
 0 1   01   1 1   01   0 1 1   01  



 


E2 B,0, T ,U ,W , E * B* , N * , T * ,U * ,W * . The existence of E0 is obvious. We will
prove the existence of E1 , E2 and E * as follows.


Existence of E1 0, N , T ,U ,W :
Here N , T ,U and W are obtained by solving the following equations:
N  M (T ),
(3.3.1)
T
Q0
,
 0 1   01 
(3.3.2)
U 
Q0 0
,
 1 (1   01 )
(3.3.3)
W
Q0 0
.
 0 1 1   01 
(3.3.4)
From equations (3.3.1), (3.3.2), (3.3.3) and (3.3.4) it is clear that
N  0, T  0, U  0, and W  0 .
Here we shall show the existence of E * , and the existence of E2 can be concluded
from the existence of E * .


Existence of E * B* , N * , T * ,U * ,W * :
In this case B* , N * , T * ,U * and W * are the solutions of the following equations:
56
r ( N *,W *) 
rN ( B*) 
r0 B *
 0,
K (T *)
(3.3.5)
rN 0 N *
 0,
M (T *)
(3.3.6)
Q0   0T * 1 B * T *  B *U * 1 1U *  0,
(3.3.7)
  1U * 1 B * T * B *U *  0 0T *  0,
(3.3.8)
U *  0W *  0.
(3.3.9)
From equations (3.3.6), (3.3.7), (3.3.8) and (3.3.9) we get
T* 
Q0 ( 1  B*)
 h1 ( B*),
f ( B*)
(3.3.10)
N* 
rN ( B*)M (h1 ( B*))
 h2 ( B*),
rN 0
(3.3.11)
U* 
Q0 ( 1 B *  0 0 )
 h3 ( B*),
f ( B*)
(3.3.12)
W* 
 h3 ( B*)
 h4 ( B*).
0
(3.3.13)
Where f ( B*)   0 1 1   01    1 1 B * 1  1   B *  0 1  0    1 1   B *2 .
It is noted from equations (3.3.10), (3.3.11), (3.3.12), and (3.3.13) that T *, P*,U *
and W * are the functions of B * only. In order to show the existence of E * , we
define a function F (B*) as follows:
F B *  r0 B * r h2 B *, h4 B *K h1 B *.
(3.3.14)
From (3.3.14), we note that
F 0  r h2 0, h4 0K h1 0  0.
where h1 (0) 
Q0
 0,
 0 (1   01 )
h2 (0)  M 0 
57
M 1Q0
 0, and
 0 (1   01 )
h3 (0) 
Q0 0
 0.
 0 1 (1   01 )
Also from (3.3.14), we have
F K 0   r0 K 0  r h2 K 0 , h4 K 0 K h1 K 0   0,
where h1 ( K 0 ) 
h4 ( K 0 ) 
Q0 ( 1  K 0 )
 0,
f (K 0 )
h2 ( K 0 ) 
(3.3.15)
rN ( K 0 ) M (h1 ( K 0 ))
 0, and
rN 0
Q0 (1 K 0   0 0 )
 0.
 0 f (K 0 )
Thus there exist a root B * in the interval 0  B*  K 0 , such that F ( B* )  0.
Now, the sufficient condition for E * to be unique is F ( B*)  0 where



F ( B*)  r0  (r1h2 ( B*)  r2 h4 ( B*)) K h1 B *  r (h2 ( B*), h4 ( B*)) K1h1 ( B*)  0.
Q  f ( B*)  ( 1  B*) f ( B*)

where h1 ( B*)  0
,
f 2 ( B*)

r M (h1 ( B*))  rN ( B*)M 1 h1 ( B*)

h2 ( B*)  N 1
,
rN 0
Q0  f ( B*)1  (1 B * Q0 0 ) f ( B*)

h4 ( B*) 
, and
 0 f 2 ( B*)
f ( B*)  1 1 1  1   0 1  0   2 1 1   B * .
With this value of B* , values of T * , N * ,U * and W * can be found from (3.3.10),
(3.3.11), (3.3.12) and (3.3.13), respectively. This completes the existence of E * .
3.4 STABILITY ANALYSIS
3.4.1 Local Stability
To discuss the local stability of system (3.1.4), we compute the variational
matrix of system (3.1.4). The entries of general variational matrix are given by
58
differentiating
the
right
hand
side
of
system
(3.1.4)
with
respect
to
B, N , T , U and W , i.e.
2r0 B

r N ,W   K (T )


rN 1 N
M E   
   T  U
1



1T  U

0

0
 K1
r0 B 2
2
K (T )
 M1
rN 0 N 2
2
M (T )
  0   1 B 
0
 1 B   0 0
0
0
 r1 B
rN ( B) 
2rN 0 N
M (T )
0
0
 B  1 1
  1  B 


 r2 B 

0 
.
0 

0 
  0 
The variational matrix M ( E0 ) at the equilibrium point E0 is given by
 Q0 0

0
0
0
 r0  r2   1    
0 1
0 1

0
rN 0
0
0

Q

0
0   0 1 1
M ( E 0 )    1  0 
 0 1   01 

Q0
    
0  0 0   1
1
0

 0 1   01 

0
0
0


From
M ( E0 ), we
note
that
two
characteristic

0 

0 

0 .

0 

  0 
roots
namely,
rN 0
and


 Q0 0
 r0  r2
, are positive giving a saddle point which is stable in the




1



0
1
0
1


T  U  W space and unstable in the B  N plane. Therefore, E0 is unstable.
The variational matrix M ( E1 ) at equilibrium point E1 is given by
 r ( N ,W )


rN 1 N

M ( E1 )  
  T  U
 1
  1 T  U

0

0
 rN 0
0
0
2
M
N
 2 1 rN 0 N
0
M T 
M (T )
0
 0
1 1
0
 0 0
 1
0
0

59
0 

0 

.
0 

0 
  0 
From M ( E1 ), we note that one characteristic root, r ( N ,W ) is positive giving a saddle
point which is stable in the N  T  U  W , space and unstable in the B , direction.
Therefore, E1 is unstable.
The Variational matrix M E2  at equilibrium point E2 is given by

r B
  0

K (T )

0
M E 2   
  1 T  U

  1 T  U

0

 r1 B

rN ( B )
K1
2
K 2 (T )
0
r0 B
0
 ( 0   1 B )
0
 1 B   0 0
0
0

 r2 B 


0
0 
.
 B  1 1
0 

 ( 1  B )
0 

  0 
0
From M ( E2 ), we note that one characteristic root, rN (B ) is positive giving a stable in
the B  T  U  W , space and unstable in the N , direction. Therefore E 2 , is unstable.
The Variational matrix M (E * ) at equilibrium point E * is given by

r B*
  0

K (T * )


rN 1 N *
*

M (E ) 

  1T *  U *

*
*
  1T  U

0

 r1 B*
rN 0 N *

M (T * )
K1


K 2 (T * )
M1
M 2 (T * )
r0 B*
rN 0 N *
0
 ( 0   1 B*)
 B*   
0
0
0
1
2
0
0

 r2 B* 


0
0 
.

B*  1 1
0 

 ( 1  B*)
0 

  0 
0
2
The conditions for E * to be locally asymptotically stable is given in the following
theorem.
Theorem (3.4.1): In addition to the assumptions (H1) - (H4). If the following
inequalities hold
60
rN 1 N *  2 1T *    1U * 
r1 B* 
rN 0 N *
,
*
M (T )
K1
r0 B* 
K (T * )
2
r0 B*
,
*
K (T )
(3.4.1)
(3.4.2)
M1
2
2
M (T * )
2
rN 0 N *   0 0   0 ,
(3.4.3)
B*  1 1     1  B* ,
(3.4.4)
r2 B*   0 .
(3.4.5)
then E * is locally asymptotically stable.
Proof: If inequalities (3.4.1) – (3.4.5) 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.
3.4.2 GLOBAL STABILITY
Theorem (3.4.2): In addition to the assumption (H1) – (H4), let r ( N ,W ), rN ( B),
K (T ) and M (T ) satisfy the conditions 0  
0  rN ( B)   3 ,
K m  K (T )  K 0 ,
r N ,W 
r N ,W 
 1 , 0  
 2 ,
N
W
0   K (T )  k ,
0  -M T   m.
M m  M T   M 0 ,
(3.4.6)
In  for some positive constants 1 ,  2 ,  3 , k , K 0 , m, M 0 . Then if the following
inequalities hold
1   3 2  1
r0
rN 0
2 K (T * ) M (T * )
,
(3.4.7)
61
2

m  2 rN 0
 rN 0 M 0 2  
( 0  1 B*),

M m  3 M (T *)

(3.4.8)
2


Q
k
1 r0
 r0 K 0 2   1 0  K 0  
( 0   1 B*),



3 K (T * )
Km


(3.4.9)
2
Q  1 r0

( 1  B*),
 1    0  
  3 K (T * )

(3.4.10)
4
((   1 ) B *  0 0  1 1 ) 2  ( 0   1 B*) ( 1  B*),
9
(3.4.11)
2
3
 2   0 ( 1  B*),
1
2
 22   0
r0
K (T * )
(3.4.12)
(3.4.13)
,
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
  (T  T *) 2
   N  N *  N * ln
V B, N , T ,U ,W    B  B*  B* ln
 

*
*
B  
N  2

1
1
 (U  U *) 2  (W  W *) 2 .
2
2
Differentiating V with respect to time t, we get
dV  B  B*  dB  N  N *  dN
dT
dU
dW


 (T  T *)
 (U  U *)
 (W  W *)
.




dt
B
dt
N
dt
dt
dt
dt




Substituting values of
dB dN dT dU
dW
,
,
,
and
dt dt dt dt
dt
from the system of equation
(3.1.4) in the above equation and after doing some algebraic manipulations and
considering functions,
62
 r  N , W   r ( N *,W )
,

*
NN
1  N , W   

r
(
N
*,
W)

,

N
N  N *,
(3.4.14)
N  N *,
 r ( N *,W )  r ( N *,W *)
, W  W *,

*
W W
 2 ( N *,W )  
*
 r ( N *,W *) ,
W W

W
(3.4.15)
1
 1
 K (T ) 
K (T * )

,

 3 T   
T T*

*
- K (T ) ,
 K 2 (T * )

(3.4.16)
 r ( B)  r ( B* )
N
 N
,
*
 1 B   
BB

*
r ( B ),
1
 1
 M (T ) 
M (T * )

,

 2 (T )  
T T*

*
- M (T ) ,
 M 2 (T * )

T  T *,
T  T *,
B  B*,
(3.4.17)
B  B*,
T  T *,
(3.4.18)
T  T *,
we get
dV
1
1
  a11 ( B  B*) 2  a12 ( B  B*) ( N  N *)  a 22 ( N  N *) 2 ,
dt
4
2
1
1
  a 22 ( N  N *) 2  a 23 ( N - N *) (T  T *)  a33 (T  T *) 2 ,
2
3
1
1
  a33 (T  T *) 2  a31 (T  T *) ( B  B*)  a11 ( B  B*) 2 ,
3
4
1
1
  a11 (T  T *) 2  a14 ( B  B*) (U  U *)  a 44 (U  U *) 2 ,
4
3
63
1
1
  a 44 (U  U *) 2  a 43 (U  U *) (T  T *)  a33 (T  T *) 2 ,
3
3
1
1
  a 44 (U  U *) 2  a 45 (U  U *) (W  W *)  a55 (W  W *) 2 ,
3
2
1
1
  a55 (W  W *) 2  a51 (W  W *) ( B  B*)  a11 ( B  B*) 2 .
2
4
where
a11 
r0
K (T * )
,
a12  1 ( N ,W )  1 ( B),
a22 
rN 0
M (T * )
,
a23  rN 0 N 2 (T ),
a13  (r0 B 3 (T )   1T   B), a14   1T  U , a15   2 ( N * ,W ), a 44   1  B* ,
a34   B*  1 1   1 B*   0 0 ,
The sufficient condition for
a 45   ,
a33   0   1 B*,
a55   0 .
dV
to be negative definite are that the following
dt
inequalities hold
a122 
1
2
2
a11a 22 , a 23
 a 22 a33 ,
2
3
2
a 45

2
a 44 a55 ,
3
a152 
1
a132  a11a33 ,
3
1
4
2
a142  a11a 44 , a 34
 a33 a 44 ,
3
9
1
a11a55 .
2
(3.4.19)
Now, from equation (3.4.6) and mean value theorem, we note that
1 ( N ,W )  1 ,  2 ( N *,W )   2 ,  3 (T ) 
k
m
, 1 ( B)   3 ,  2 (T )  2 .
2
Km
Mm
(3.4.20)
Further, we note that the stability conditions (3.4.7) - (3.4.13) as stated in theorem
(3.4.2), can be obtained by maximizing the left-hand side of inequalities (3.4.19).
This completes the proof of theorem (3.4.2).
Case II: When Q(t )  0, i.e., in the case of instantaneous emission of the pollutant
into the environment, the corresponding results can be obtained from case I by
64
substituting Q0  0. In particular, it is noted that the resource-biomass and the
population settle down to their respective equilibrium levels under certain conditions,
and their magnitude are greater than their respective magnitude in case I.
Case III: Periodic introduction of the pollutant into the environment, i.e.,
Q(t )  Q0  (t ),  t      t .
In this case, model (3.1.4) can be written in the vector matrix form as
dx
 A( x)  C (t ), x(0)  x 0 ,
dt
(3.4.21)
Where
 x1   B 
 B ( 0) 
 0 
x   N 
 N (0) 
 0 
 2  




x   x3    T  , x 0   T (0)  , C (t)   (t)  ,
   




 x 4  U 
U (0) 
 0 
 x5  W 
W (0)
 0 


r0 x12


r
x
,
x
x



2
5
1
K ( x3 )


rN 0 x 22


rN ( x1 ) x 2 

.
A( x) 
M ( x3 )


Q0   0 x3   1 x1 x3  x1 x 4  1 1 x 4 
   1 x 4   1 x1 x3  x1 x 4   0 0 x3 


x 4   0 x 5


Let
M*
be

the
variational
matrix
corresponding
to
the
positive

equilibrium E * B* , N * , T * ,U * ,W * . Then under an analysis similar to Freedman and
Shukla [1991], we can state the following two theorems.
Theorem (3.4.3): If M * has no eigenvalues with zero real parts, then system (3.1.4)
with Q(t )  Q0  (t ),  t      t  has a periodic solution of period  , ( B(t ,  ),
N (t ,  ), T (t ,  ), U (t ,  ), W (t ,  )) such that
Bt,0, N t,0, T t,0, U t,0, W t,0 
65
( B* , N * , T * ,U * ,W * ).
Theorem (3.4.4): If M * has no eigenvalues with zero real parts, then for sufficiently
small  , the stability behavior of system (3.1.4) is same as that of E * .
Moreover,
a
periodic
solution
up
to
order

can
be
computed
as
1


*t  t M *s
 M *
 M * M *s
xt ,  ,    x  e
C ( s)ds   e
I e
e
C
(
s
)
ds
 e
  O .
0


 0

M
where I is the identity matrix.
The above results imply that a small periodic influx of pollutant causes a periodic
behavior in the system.
3.5 NUMERICAL SIMULATION AND DISCUSSION
To facilitate the interpretation of our mathematical findings by numerical
simulation, we integrated system (3.1.4) using fourth order Runge- Kutta method. We
consider the following particular form of the functions in the model (3.1.4).
r N , W   r0  r1 N  r2W ,
K (T )  K 0  K1 (T ),
rN B   rN 0  rN 1 B,
M (T )  M 0  M 1 (T ).
(3.5.1)
Now, we choose the following set of parameters in system model (3.1.4):
r0  9, r1  0.04, r2  0.01, K 0  20, K1  0.04, rN 0  2, rN1  0.01,
M 0  45, M 1  3, Q0  15,  0  8, 1  0.1,   0.1,   0.2,
1  0.2,  1  8,  0  0.08,   1,  0  0.2.
(3.5.2)
With the above values of parameters, it can be checked that conditions, (3.4.1) –
(3.4.5) for the existence of interior equilibrium E * are satisfied and is given by
B*  16.0636, N *  43.3951, T *  1.6104, U *  0.2447, W *  1.2234.
66
By choosing 1  .05,  2  .08,  3  .008, K m  12, M m  15, k  0.1, m  0.02
in theorem (3.4.2). It can also be verified that E * is globally asymptotically stable.
In the absence of toxic substance W , the following interior equilibrium point are
obtained corresponding to   8,  0  0, r2  0, and other parameters are same as
given in (3.5.2).
B*  16.0900, N *  43.4016, T *  1.6100, U *  0.244.
Figure (1), graphically illustrates the global stability of the interior equilibrium point
in B  W plane.
0.055
1
2
3
4
0.05
Toxic Substance ( W )
0.045
0.04
0.035
0.03
0.025
Equilibrium Point
0.02
0.015
0.01
0.005
4
6
8
10
12
Resource Biomass ( B )
14
16
18
Figure 1, Variation of the B with W for different initial starts
for the set of parameters values given in (3.5.2).
Figure (2), shows the variation of Resource-Biomass, B , with time ‘t’ for different
value of population growth rate rN 0 . From this figure, it can be easily seen that the
67
density of resource-biomass decreases with increase in population density. Figure (3),
shows the variation of Population, N , with time ‘t’ for different value of resource
biomass growth rate rN 1 . From this figure, it can be easily seen that the density of
population increases very rapidly with very slight increase in the resource biomass
density.
18
Resource Biomass ( B )
16
14
12
10
8
rN0=2
rN0=5
6
rN0=10
4
0
2
4
6
8
10
time ( t )
12
14
16
18
20
Figure 2, Variation of the B with time ‘t’ for different value of rN 0 ,
and other set of parameters values as given in (3.5.2).
68
50
45
Population ( N )
40
35
30
rN1=0.01
rN1=0.02
25
rN1=0.03
20
0
20
40
60
80
100
120
time ( t )
140
160
180
200
Figure 3, Variation of the N with time ‘t’ for different value of rN 1 ,
and other set of parameters are same as given in (3.5.2).
The variation of U with time ‘t’ for different emission rates of toxicant Q0 , is
displayed graphically in Figures (4). It is observed from the figure, that as density of
toxicant increases into the environment, it increases the uptake concentration of
toxicant U , by resource biomass. Figure (5), shows the variation of resource biomass
with time ’t’ in the presence and absence of toxic substance W . From this figure, it
has been noted that equilibrium level of resource biomass decreases in the presence
of toxic substance.
69
0.45
0.4
Uptake Concentration ( U )
0.35
0.3
0.25
0.2
0.15
Q0=15
Q0=20
0.1
0.05
Q0=25
0
20
40
60
80
100
120
time ( t )
140
160
180
200
Figure 4, Variation of the U with time ‘t’ for different value of Q0 ,
and other set of parameters values as given in (3.5.2).
18
17
Resource Biomass ( B )
16
15
14
13
12
11
10
Presence of toxic substance,( W )
Absence of toxic substance,( W )
9
8
0
2
4
6
8
10
time ( t )
12
14
16
18
20
Figure 5, Variation of the B with time ‘t’ for presence and absence of toxic
substance W , and other set of parameters values as given in (3.5.2).
70
3.6 CONCLUSION
In this chapter, a non-linear mathematical method is proposed and analyzed to
study the effect of environmental pollution dependent on resource-biomass with time
delay. It has been assumed that density of population depends partially on the
resource - biomass. The model has been completely analyzed using stability theory of
ordinary differential equations. An equilibrium analysis is presented and is found,
analytically as well as graphically, that the nontrivial equilibrium is locally as well as
globally asymptotically stable under certain conditions. It has been shown that in the
case of constant introduction of pollutant into the environment the magnitude of the
equilibrium level of the population increases with increase in resource-biomass
density and decrease with increase of toxicant concentration into the environment
(shown in figure (1), of chapter 2).
In our model, the concentration of environmental pollutant does not affect the
growth rates of resource-biomass and population directly. This pollutant when taken
up by the resource-biomass is being converted into some other chemical toxicant due
to some metabolic changes, which affect the growth rate of resource biomass. The
growth rate of resource biomass is also being reduced by increase in density of
population. From the analysis, it can be seen that the effect of time delay due to the
formation of the chemical toxicants decreasing the equilibrium level of resourcebiomass.
In the case of instantaneous introduction of pollutant into the environment, it
has been found that perhaps the concentration of pollutant was not enough to deplete
the resource-biomass and population. Hence, the pollutant will be washed out
completely and the resource-biomass as well as population would recover at its
71
original carrying capacity. The magnitude of densities of the population and resourcebiomass are larger than their respective densities in the case of constant introduction
of the pollutant. It has been also noted that a small periodic introduction of pollutant
into the environment induces a periodic behavior in the system.
72