Chapter 5
MODELLING THE EFFECT OF INDUSTRIALIZATION AND
TOXICANT ON A RESOURCE- DEPENDENT
COMPETING SPECIES
INTRODUCTION
Environmental pollution by various industries and other human activities is
one of the most important socio-ecological problems. The presence of variety of
toxicants in the environment is a threat to the survival of exposed populations,
including mankind. Uncontrolled contribution of toxicant to the environment has led
many species to extinction and several others to be on the verge of extinction. It is
well known fact that resource plays a significant role in the development of any
country. But it is being depleted by increased industrialization, over growth of
population (particularly in the third world countries) and associated pollution ( Shukla
et al.[1989]; Dubey and Dass, [1999] ).
We know that species do not exist alone in nature. They interact with other
species in their surroundings for their survival. Competition between species may be
one kind of interaction (Albrecht et al.,[1974] ) and occurs naturally between living
organisms which co-exist in the same environment when limited amount of resources
are available, and several species depend on these resources. Thus, each of the species
competes with the others to gain access to the resources. As a result, species less
suited to compete for the resources must either adapt or die out. For example, animal
compete over water supplies, food, and mates, etc. Humans compete for water, food,
96
and mates as well, though when these needs are met deep rivalries often arise over the
pursuit of wealth, prestige, and fame.
In view of above considerations, in this chapter, a nonlinear mathematical
model is proposed and analyzed for the survival of resource dependent competing
species (such as human beings competing for fuel, fodder for cattle required for milk
production, medicine, food, etc.) where both the competing species and its resource
are affected by the constant emission of toxicant into the environment as well as by
population pressure augmented industrialization. This situation is modeled by the
system of five ordinary differential equations. Stability theory of nonlinear differential
equations and fourth order Runge-Kutta method are used to analyze and predict the
behavior of the model.
5.1 THE MATHEMATICAL MODEL
The following system of differential equations is considered to study the effect
of toxicants emitted into the environment from various external sources as well as due
to industrialization on resource-dependent competing species present in the ecological
system.
r B2
dB
 r I B  0
  1 BN 1   2 BN 2 ,
dt
K I , T 
2
r N
dN 1
 r1 B N1  10 1  N1 N 2   1 N1T ,
dt
K1
2
r N
dN 2
 r2 B N 2  20 2  N 1 N 2  1 N 2T ,
dt
K2
r I2
dI
 r3 N1 , N 2 I  30  1  b BI ,
dt
K3
dT
 QI    0T   0 BT   1 N 1T   2 N 2T .
dt
97
(5.1.1)
B0  0, N1 0  0, N 2 0  0, I 0  0, T 0  0.
In model (5.1.1), B (t ) is the density of resource biomass, N1 (t ) and N 2 (t ) are
the densities of first and second competing species, respectively, I (t ) is the density of
industrialization pressure, and T (t ) is the concentration of toxicants present in the
environment under consideration.
Model (5.1.1) is derived under following assumptions:
(H1): The function r (I ) denotes the specific growth rate of resource biomass which
decreases as I increases. Hence we take
r 0  r0  0,
r ( I )  0 for I  0.
(H2): Resource biomass grows logistically and its carrying capacity K I , T 
decreases with the increase in densities of industrialization and toxicants present in
the environment. Hence we take
K 0,0  K 0  0,
K I , T 
K I , T 
 0,
0
I
T
for I  0, T  0.
(H3):  1 ,  2  0 are the depletion rates coefficients of the resource biomass due to the
first and second competing species respectively.
(H4): The densities of both competing species are assumed to be governed by logistic
equations with carrying capacities
K1  0 and K 2  0, and their growth rates
coefficients r1 ( B) and r2 ( B) respectively, which increases as the resource biomass
density increases. Hence we take

r1 0  r10  0, r1 B   0

r2 0  r20  0, r2 B   0 for B  0.
(H5):  ,   0 are the competitive rate coefficients, 1 , 1  0 are the depletion rates
coefficients of first and second competing species due to uptake of toxicant from
environment.
98
(H6): The density of industrialization is also assumed to be governed by logistic
equations with carrying capacities K 3  0, and growth rate coefficient r3 N1 , N 2 
which increases as the density of competing species increases. Hence, we take
r3 0,0  r30  0,
The constant
r3 N1 , N 2 
r N , N 
 0, 3 1 2  0 for N1  0, N 2  0.
N 2
N1
1  b  0
is the part of resource biomass which is used for
industrialization.
(H7): The function QI  is the rate of introduction of toxicant into the environment
which increases as I increase. Hence we take
Q0  Q0  0,
QI   0
for I  0.
(H8):  0  0 is the natural depletion rate of toxicant from the environment and the
positive constants  0 ,  1 and  2 are the rates of depletion of toxicant in the
environment due to uptake of toxicant by resource biomass and competing species,
respectively.
It is noted that even when Q  0, model (5.1.1) remains meaningful as in this case,
the toxicant is emitted not only by external sources but also by population pressure
augmented industrialization.
To analyze the model (5.1.1), we need the bounds of dependent variable involved. For
this we find the region of attraction in the following lemma.
5.2 BOUNDEDNESS OF SOLUTIONS
Lemma (5.1.1): Suppose that assumptions (H1) - (H8) hold. Then all solutions of
system (5.1.1) are bounded within the region ,
  B, N1 , N 2 , I , T : 0  B  K 0 , 0  N1  N1m , 0  N 2  N 2m , 0  I  I m , 0  T  Tm 
99
where N 1m 
Tm 
Q I m 
0
K
K1
K
r1 K 0 , N 2 m  2 r2 K 0 , I m  3 r3  N 1m , N 2 m   1  b K 0 ,
r10
r20
r30
.
Proof: Proof is analogous to the proof of lemma (3.1.1) of chapter 3.
5.3 EQUILIBRIUM ANALYSIS
The system (5.1.1) may have sixteen nonnegative equilibria in the B, N1 , N 2 , I , T



Q  
Q( K 3 ) 
, E2 B ,0,0,0, T , E3 0, N1 ,0,0, T ,
space namely E0  0,0,0,0, 0 , E1  0,0,0, K 3 ,


0 
0





ˆ ˆ
ˆ
E5  Bˆ , Nˆ 1 ,0,0, Tˆ ,


E4 0,0, Nˆ 2 ,0, Tˆ ,


0, N ,0,I,T,


~ ~
~
E6 B,0, N 2 ,0, T ,
 
 
  

   
E8 B, N1 , N 2 ,0, T , E9 B, N1 ,0, I , T , E10 B,0, N 2 , I , T ,




 ,
E12 B,0,0, I, T
E13
1


~
~
~ ~
~
~
E7  0, N1 , N 2 ,0, T ,




   
E11 0, N1 , N 2 , I , T ,


  
E14  0,0, N 2 , I , T , E * B* , N1* , N 2* , I * , T * .


The existence of E0 and E1 is obvious. We prove the existence of other equilibrium
points.
Existence of E 2 B ,0,0,0, T  :
In this case, B and T are the positive solutions of the following equations.
B  K 0, T   0,
T 
(5.3.1)
Q0
.
0   0B
(5.3.2)
Equating the value of T , in equation (5.3.1) and then we define a function J (B ), as
follows
100


Q0
.
J ( B )  B  K  0,
 0   0B 
(5.3.3)
From (5.3.3), we note that
 Q
J (0)   K  0, 0
 0
J (K0 ) 

  0.

q2Q0
 0.
0   0 K0
Thus there exist a root B , in the interval 0  B  K 0 , such that J ( B )  0.
Now, the sufficient condition for E 2 , to be unique is J ( B )  0, where
J ( B ) 

  0 B   q2Q0 0
2
0

  0B 
2
0
 0.
Knowing the value of B , we obtained value of T  0 from equation (5.3.2). This
completes the existence of E2 .


Existence of E3 0, N1 ,0,0, T :
In this case N1 and T are the solutions of the following equations:
r10 
T 
r10 N1
 1T  0.
K1
Q0
 0   1 N1
(5.3.4)
(5.3.5)
.
Equating the value of T , in equation (5.3.4) and then we define a function F ( N1 ), as
follows
F ( N 1 )  r10 
r10 N 1
Q0
 1
.
K1
 0   1 N1
(5.3.6)
From (5.3.6), we note that
101
F (0)  r10 
 0 Q0
 0.
0


Q0 r10
  0.
F ( N1m )   r11 K 0  1
 0 r10   1 K1r1 ( K 0 ) 

Thus there exist a root N 1 , in the interval 0  N1  N1m , such that F ( N1 )  0.
Now, the sufficient condition for E 3 , to be unique is F ( N1 )  0. where
r
Q0 1
F ( N1 )   10   1
 K1
( 0   1 N1 ) 2


  0.


With this value of N 1 , value of
T can be found from equation (5.3.5). This
completes the existence of E3 .


Existence of E 4 0,0, Nˆ 2 ,0, Tˆ :

Analogous to the existence of E 3 , we can show the existence of E 4 0,0, Nˆ 2 ,0, Tˆ

ˆ ˆ
ˆ
Existence of E5  Bˆ , Nˆ 1 ,0,0, Tˆ  :


ˆˆ ˆˆ ˆˆ
In this case B, N1 , T are the solutions of the following equations
ˆ
ˆ

r0 K 1  r10  r11 K  0, Tˆ    1Tˆ 


ˆ

  f  Tˆˆ , say,
Nˆ 1 
1 
ˆˆ 
 

r0 r10   1 r11 K 1 K  0, T 


ˆ
r0   1 f1  Tˆ 
ˆ
  K  0, Tˆˆ   f  Tˆˆ , say,
Bˆ 


2 


 
r0
(5.2.7)
(5.2.8)
ˆ
ˆˆ
ˆ ˆ
Q0   0Tˆ   0 Bˆ Tˆ   1 Nˆ 1Tˆ  0.
(5.2.9)
ˆ
ˆ
ˆ
It is noted from equations (5.2.7) and (5.2.8) that Nˆ 1 and Bˆ are functions of Tˆ only.
ˆ
To show the existence of E 5 , we define a function F1 (Tˆ ) as follows
102
ˆ
ˆ
ˆ ˆ
ˆ ˆ
F1 (Tˆ )  Q0   0Tˆ   0Tˆf 2 (Tˆ )   1Tˆf1 (Tˆ ).
(5.2.10)
From equation (5.2.10), we note that F1 0  Q0  0.
Q
F1  0
 0

Q
   0
0


Q
  1 f1  0


 0


Q
   0 f 2  0

 0

   0,



 Q 
Q
r0 K 1  r10  r11 K  0, 0    1 0
0
Q 
 0 

where f1  0  
 Q 
 0 
r0 r10   1 r11 K 1 K  0, 0 
 0 




and
Q 
r0   1 f1  0 
Q 
  0  K  0, Q0 .
f 2  0  
  
r0
0 
 0 

ˆ
ˆ
ˆ
Thus there exists a root Tˆ in the interval 0  Tˆ  Tm , given by F1 (Tˆ )  0,
ˆ
Now, the sufficient condition for E 5 to be unique is F1(Tˆ )  0, where
ˆ
ˆ
ˆ
ˆ  ˆ
ˆ  ˆ
F1(Tˆ )    0   0 f 2 (Tˆ )   1 f1 (Tˆ )   0Tˆf 2 (Tˆ )   1Tˆf1 (Tˆ )   0.


(5.2.11)
r r K (r q ( K  r )  r01 )  r0 r11 K 0 K12 11
ˆ
.
where f1(Tˆ )  0 10 1 11 2 1 1 0
2
ˆ




 r0 r10   1r11 K1 K  0, Tˆ  



ˆ
ˆ
ˆ
ˆ
 q K (r  r K (0, Tˆ )   1Tˆ )  1 K (0, Tˆ ) f1(Tˆ )
ˆ
f 2(Tˆ )  q 2  1 2 1 10 11

.
ˆˆ
r
0
r0 r10   1 r11 K 1 K (0, T )
ˆ
ˆ
ˆ
With this value of Tˆ , value of Nˆ 1 and Bˆ can be found from equations (5.2.7) and
(5.2.8). This completes the existence of E5 .


~ ~
~
Existence of E6 B,0, N 2 ,0, T :


~ ~
~
Analogous to the existence of E 5 , we can show the existence of E6 B,0, N 2 ,0, T .
103
~
~
~ ~
~
~
Existence of E7 (0, N1 , N 2 ,0, T );
~
~ ~
~ ~
~
In this case, N1 , N 2 , T are the solutions of the following equations
~
~
~
 r r  r   T~
 K 2   1K 2T 

10 20
20  1
~
~
~
~


  f 3 (T ), say,
N1  K1 
r10 r20  k1 k 2




(5.2.12)
~
~
 r  f (T~ )   T~ 
~
~
~
20
3
1

 K  f (T~
N2 
), say,
2
4


r20


(5.2.13)
~
~
~
~
~ ~
~
~ ~
~
Q0   0T   1 N1T   2 N 2T  0.
(5.2.14)
~
~
~
~
It is noted from equation (5.2.12) and (5.2.13) that N 1 and N 2 are functions of
~
~
~
~
T only. To show the existence of E 7 , we define a function F2 (T ) as follows
~
~
~
~
~
~
~ ~
~
~ ~
~
F2 (T )  Q0   0T   1T f 3 (T )   2T f 4 (T ).
(5.2.15)
From equation (5.2.15), we note that F2 0  Q0  0.
Q
F2  0
 0

Q
   0
0


Q
 1 f3  0


 0


Q
   2 f 4  0

 0

   0,


 Q  K  r r   r  Q  K 2 0   1K 2 Q0 
.
where f 3  0   1  10 20 0 20 1 0
r10 r20  k1k 2
 0  0 

 Q   r   f 3Q0  1Q0 
 K 2 .
f 4  0    20 0
r20 0
 0  

~
~
~
~
~
~
Thus, there exists a root T in the interval 0  T  Tm given by F2 (T )  0.
~
~
Now, the sufficient condition for E7 to be unique is F2 ( T )  0, where
~
~
~
~
~
~
~
~
~  ~
~
~  ~
~
F2(T )    0   1 f 3 (T )   2 f 4 (T )   1T f 3 (T )   2T f 4 (T )   0.


104
(5.2.16)
~
K (1 K 2  r20 1 )
~
,
where f 3(T )  1
r10 r20  K1 K 2
~
~
~
K 2 ( 1  f 3(T )
~
f 4(T )  
.
r20
~
~
~
~
~
~
With this value of T , value of N 1 and N 2 can be found from equations (5.2.12) and
(5.2.13). This completes the existence of E7 .


  

Existence of E8 B, N1 , N 2 ,0, T :
   
In this case, B, N1 , N 2 , T are the solutions of the following equations



r0 B
r0 
   1 N1   2 N 2  0,
K 0, T
(5.2.17)

 r10 N1


r1 ( B) 
 N 2   1T  0,
K1
(5.2.18)

 r20 N 2


r2 ( B) 
 N1  1T  0,
K2
(5.2.19)


 
 
Q0   0T   0 BT   1 N1T   2 N 2T  0.
(5.2.20)
 
From the equation (5.2.18) and (5.2.19), we have





 
r20 r1 ( B)   1 r20T  K 2 r2 ( B)  1 K 2T
N1  K1
 g1 ( B, T ),
r10 r20  K1 K 2
say,
(5.2.21)





 
r10 r2 ( B)  K1r1 ( B)  1 r10T  1 K1T
N2  K2
 g 2 ( B, T ),
r10 r20  K1 K 2
say,
(5.2.22)


Using values of N1 and N 2 from (5.2.21) and (5.2.22) in equations (5.2.17) and
(5.2.20) respectively, we get
 



 

 
G1 ( B, T )  r0 K (0, T )  r0 B   1 K (0, T ) g1 ( B, T )   2 K (0, T ) g 2 ( B, T )  0,
(5.2.23)
 


  

 
G2 ( B, T )  Q0   0T   0 BT   1Tg1 ( B, T )   2Tg 2 ( B, T )  0.
(5.2.24)
From (5.2.23), we note the following



when T  0, then B  Ba , where
105

S11 Ba  S12  0,
where
S11  r0 (r10 r20  K1 K 2 )   0 K 0 K1 (r20 r11  K 2 r21 )   2 K 0 K 2 (r10 r21  K1r11 ),
S12  r0 K 0 (r10 r20  K1 K 2 )  1 K 0 K1r20 (r10  K 2 )   2 K 0 K 2 r10 (r20  K1 ).


Let S ( Ba )  S11 Ba  S12 .
S (0)  S12  0,
S ( K 0 )  S11 K 0  S12  0.



Thus there exists a root Ba in the interval 0  Ba  K 0 given by S ( Ba )  0.

Now, the sufficient condition for E8 to be unique is S ( Ba )  0, where

S ( Ba )  S11  0.


G1
 1 K (0, T ) K1
 2 K (0, T ) K 2
(r11r20  r21 K 2 ) 
  r0 
r10 r20  K1 K 2
r10 r20  K1 K 2
B
(r10 r21  K1r11 ),
(5.2.25)

 
G1
 1 K (0, T ) K1
(1 K 2   1 r20 ),
  q 2 r0   1 q 2 g1 ( B, T ) 
r10 r20  K1 K 2
T
(5.2.26)
Now, from (5.2.25) and (5.2.26) we get
G1


B

T
 
G1
T
B

B
It is clear that   0, if either
T
(i)
G1
G1
  0 and
  0, or
B
T
(ii)
G1
G1
  0 and
  0,
B
T
(5.2.27)
106
From (5.2.27), we note the following



when B  0, then T  Tb , where
2

H11Tb  H12Tb  H13  0,
where
H11   1 K1 (1r20  1 K 2 )   2 K 2 (r10 1  1 K1 ),
H12  [ 0 (r10 r20  K1 K 2 )   1 K1r20 (r10  K 2 )   2 K 2 r10 (r20  K1 )].
H 13  Q0 (r10 r20  K1 K 2 ).



Let H (Tb )  H 11Tb2  H 12Tb  H 13 ,
H (0)  H 13  0,
Q
H  0
 0

Q
  H 11  0

 0
2

Q
  H 12 0  H 13  0.
0

 Q


Thus there exists a root Tb in the interval 0  Tb  0 given by H (Tb )  0.
0

Now, the sufficient condition for E8 to be unique is H (Tb )  0, where


H (Tb )  2 H 11Tb  H 12  0.



 0T K 2
G2
 1T K1
(r11r20  r21 K 2 ) 
   0T 
r10 r20  K1 K 2
r10 r20  K1 K 2
B
(r10 r21  K1r11 ),
(5.2.28)


 
 
G2
 1T K1
(1 K 2   1 r20 )   2 g 2 ( B, T )
   0   0 B   1 g1 ( B, T ) 
r10 r20  K1 K 2
T

 2T K 2

( 1 K1  1 r10 ),
r10 r20  K1 K 2
Now, from (5.2.28) and (5.2.29) we get
107
(5.2.29)
G2


B

T
 
G2
T

B

G
G2
B
It is clear that   0, since 2  0 and
  0.
T
B
T
(5.2.30)
under the conditions r10  K 2 , r20  K1 , 1 K1  1r10 , and 1 K 2  1r20 .
 
Thus the two isoclines (5.2.23) and (5.2.24) intersects at a unique ( B, T ) if in addition


to conditions (5.2.27) and (5.2.30), the inequality Ba  Tb holds. Knowing the value




of B and T , we get N1 and N 2 can be calculated from equations (5.2.21) and (5.2.22).
This completes the existence of E8 .


 
 
Existence of E9 B9 , N 1 ,0, I , T :
   
In this case, B, N1 , I , T are the solutions of the following equations

r(I ) 


r0 B
    1 N1  0,
K (I ,T )
(5.2.31)

 r N

r1 ( B)  10 1   1T  0,
K1
(5.2.32)



r I
r3 ( N1 ,0)  3 0  1  b B  0,
K3
(5.2.33)


 
 
Q( I )   0T   0 BT   1 N 1T  0.
(5.2.34)
From equations (5.2.33) and (5.2.34) respectively, we get
 K


I  3 (r30  r31 N1  (1  b) B), say
r30
(5.2.35)
 

 
Q( f 5 ( B, N1 ))
T

  f 6 ( B, N ), say.
 0   0 B   1 N1
(5.2.36)
108


Equating the values of I and T , from equations (5.2.35) and (5.2.36) in equation
(5.2.31) and (5.2.32) respectively, we get
 
 
 
 


V1 ( B, N )  K ( f 5 ( B, N 1 ), f 6 ( B, N 1 ))( r ( f 5 ( B, N 1 )   1 N 1 )  r0 B  0,
(5.2.37)

 
 r N
 
V2 ( B, N1 )  r1 ( B)  10 1   1 f 6 ( B, N1 )  0,
K1
(5.2.38)
From (5.2.37), we note the following



when N  0, then B  Bb , where


2
v11 Bb  v12 Bb  v13  0,
where
v11 
a1 K 32 (1  b) 2 (q1  q 2 r30Q1 )
,
r302
v12 
K 3 (1  b)(q1  q2 Q1 )(a1 K 3 (1  r30 )  r0 )  a1 (q2 Q0  K 0 )
,
r30
v13  (r0  a1 K 3 )( K 0  (q1  q2 Q1 ) K1  q2 Q0 ).



Let V ( Bb )  v11 Bb2  v12 Bb  v13 ,
V (0)  v13  0,
V ( K 0 )  v11 K 02  v12 K 0  v13  0.



Thus there exists a root Bb in the interval 0  Bb  K 0 given by V ( Bb )  0.

Now, the sufficient condition for E 9 to be unique is V ( Bb )  0, where


V ( Bb )  2v11 Bb  v12  0.
 
 
 
 

V1
   K ( f 5 ( B, N1 ), f 6 ( B, N1 ))( a1 f 5( B, N1 ))  (r ( f 5 ( B, N1 )   1 N1 )
B
 
 
(q1 f 5( B, N1 )  q 2 f 6( B, N1 ))  r0 ,
109
(5.2.39)
 
 
 
 

V1

   K ( f 5 ( B, N1 ), f 6 ( B, N1 ))( a1 f 5 ( B, N1 )   1 )  (r ( f 5 ( B, N1 )   1 N1 )
N1
 
 
(q1 f 5( B, N 1 )  q 2 f 6( B, N1 )).
(5.2.40)
Now, from (5.2.39) and (5.2.40) we get
V1


N 1
B
  
V1
N 1

B

B
It is clear that   0, if either
N1
(i)
V1
V1
  0 and
  0, or
B
N1
(ii)
V1
V1
  0 and
  0,
B
N1
(5.2.41)
From (5.2.38), we note the following



when B  0, then N 1  N 1a , where


w11 N a2  w12 N 1a  w13  0,
where
w11  r10 1 , w12  r10 1 K1  r10 0 
1Q1 K 3 r31
r30
, w13  r10 0 K1  1Q0 
1Q1 K 3



Let W ( N1a )  w11 N a2  w12 N 1a  w13 ,
W (0)  w13  0,
2
K

K

K
W  1 r1 ( K 0 )   w11  1 r1 ( K 0 )   w12 1 r1 ( K 0 )  w13  0.
r10
 r10

 r10



K
Thus there exists a root N 1a in the interval 0  N1a  1 r1 ( K 0 ), given by
r10
110
r30
,



W
(
N
W ( N 1a )  0. Now, the sufficient condition for E 9 to be unique is
1a )  0, where


W ( N1a )  2w11 N 1a  w12  0.


 


V2
 ( 0   0 B   1 N1 )Q1 K 3 (1  b)  Q( f 5 ( B, N1 )) 0 r30 

  r1( B)  1 



2


B
r
(



B


N
)
30
0
0
1 1


(5.2.42)


 
r10 
(



B


N
)
K
r

Q
(
f
(
B
, N1 )) 1r30 
V2
 0

0
1 1
3 31
5
  



.

 2
K


1
N1
r30 ( 0   0 B   1 N1 )


(5.2.43)
Now, from (5.2.42) and (5.2.43) we get
V2


N 1
B
.
  
V2
N 1

B

B
It is clear that   0, if either
N1
(i)
V2
V2
  0 and
  0, or
B
N1
(ii)
V2
V2
  0 and
  0.
B
N1
(5.2.44)
 
Thus the two isoclines (5.2.37) and (5.2.38) intersects at a unique ( B, N1 ) if in


addition to conditions (5.2.41) and (5.2.44), the inequality Bb  N1a
holds. Knowing




the value of B and N1 , we get I and T can be calculated from equations (5.2.35) and
(5.2.36). This completes the existence of E9 .
   
Existence of E10 B,0, N 2 , I , T  :
   
Analogous to the existence of E 9 , we can show the existence of E10 B,0, N 2 , I , T .
111


   
Existence of E11 0, N1 , N 2 , I , T :
   
In this case, N1 , N 2 , I , T are the positive solutions of the following equations



r10 N1
r10 
 N 2   1T  0,
K1
(5.2.45)

  



K 2 K1 r10 1  r20 1    1 K1  1 r1 0 N1
N2 
 s1 N1 , say
K1 (1 K 2   1 r20 )


 K



3
I 
r3 N1 , s1 ( N1 )  s 2 ( N1 ), say
r3 0

T
(5.2.46)
(5.2.47)


Q( s 2 ( N1 ))

  s3 ( N1 ), say.
 0   1 N1   2 s1 ( N1 )
(5.2.48)
 

It is noted from equations (5.2.46), (5.2.47) and (5.2.48) that N 2 , I and T are the

functions of N1 only. In order to show the existence of E11 , we define a function

F3 ( N 1 ) as follows




r10 N1
F3 ( N1 )  r10 
 s1 ( N1 )   1 s3 ( N1 ).
K1
From equation (5.2.49), we note that
F3 (0)  r10  s1 (0)  1 s3 (0)  0.
where
s1 (0) 
K 2 (r10 1  r201 )
,
1 K 2  1r20
s3 (0) 
Q0 r30  Q1 K 3 r3 (0, s1 (0))
.
r30 ( 0   2 s1 (0))

K

K

K

F3  1 r1 ( K 0 )   r11 K 0  s1  1 r1 ( K 0 )    1 s 2  1 r1 ( K 0 )   0,
 r10

 r10

 r10


K
 K  K r ( K )  r201  r10 r11 K 0  
where s1  1 r1 ( K 0 )   2 1 1 1 0
,
r10 (1 K 2  1r20 )
 r10

112
(5.2.49)
K

s3  1 r1 ( K 0 )  
 r10

K

K 3  K1
r3 
r1 ( K 0 ), s 2 
r1 ( K 0 )  

r30  r10
 r10

.
 K1

K1
0  1
r1 ( K 0 )   2 s1 
r1 ( K 0 ) 
r10
 r10

Q0  Q1



K
Thus there exists a root N1 in the interval 0  N1  1 r1 ( K 0 ), given by F3 ( N 1 )  0.
r10

Now, the sufficient condition for E11 to be unique is F3( N 1 )  0, where



r
 
  
F3( N1 )   10  s1 ( N1 )  1 s3 ( N1 )  0.
 K1

(5.2.50)

K (r    1 K1
 
where s1 ( N1 )   2 10 1
,
K1 (1 K 2  1r20 )




 
 

X (Q2 K 3 (r31  r32 s1 ( N1 ))  Q( K 3 r3 ( N 1 , s1 ( N1 )))( 1   2 s1 ( N1 ))
 
s3 ( N1 ) 
,
r30 X 2


X   0   1 N1   2 s1 ( N 1 ).

 

With this value of N 1 , value of N 2 , I and T can be found from equations (5.2.46),
(5.2.47) and (5.2.48), respectively. This completes the existence of E11.


 :
Existence of E12 B,0,0, I, T
 are the positive solutions of the following equations:
In this case, B, I, T
r (I) 
r0 B
 0,
)
K (I, T
(5.2.51)

I  (r30  (1  b) B ) K 3  u ( B), say,
1
r30
(5.2.52)

  Q0 r3 0  Q1 (r30  (1  b) B ) K 3  u ( B), say,
T
2
r3 0 ( 0   0 B)
(5.2.53)
It may be noted from equations (5.2.52) and (5.2.53) that I and T are the functions of
B only. To show the existence of E , we define a function F ( B) as follows
4
12
113
F4 ( B)  r (u1 ( B)) K (u1 ( B), u 2 ( B))  r0 B.
(5.2.54)
From equation (5.2.54), we note that

Q( K 3 ) 
  0.
F4 0  r ( K 3 ) K  K 3 ,
 0 

F4 K 0   r0 K 0  r u1 K 0 K u1 K 0 , u 2 K 0   0,
where u1 ( K 0 ) 
(r30  (1  b) K 0 ) K 3
Q  Q1u1 ( K 0 )
, u2 (K 0 )  0
.
r30
 0   0 K0
Thus there exists a root B in the interval 0  B  K 0 given by F4 B  0.
Now, the sufficient condition for E12 to be unique is F4 ( B)  0, where
K
q  Q (1  b) K 3  r30 0 Q( K 3 )
F4( B)  r0  r (u1 ( B))  3 q1 (1  b)  2 0 1

r30 ( 0   0 B) 2
 r30


K3
a1 (1  b) K (u1 ( B), u 2 ( B))  0.
r30
(5.2.55)
With this value of B, value of I and T can be found from equations (5.2.52) and
(5.2.53) respectively. This completes the existence of E12 .


 ,0, I, T
 :
Existence of E13 0, N
1
 , I, T
 are the solutions of the following equations
In this case, N
1
r10 
r10 
  0,
N1  1T
K1

(5.2.56)

I  K 3 r  r N


30
31 1  m1 ( N 1 ), say,
r3 0
(5.2.57)

  Q(m1 ( N1 ))  m ( N
 ), say.
T
2
1






 0   1 N1
(5.2.58)
114
It may be noted from equations (5.2.56) and (5.2.58) that I and T are the function of
 ) as
 only. In order to show the existence of E , we define a function F ( N
N
5
1
13
1
follows
 )  r  r10 N
   m ( N
 ).
F5 ( N
1
10
1
1 2
1
K1
(5.2.59)
From (5.2.59), we note that
F5 0  r10  1
Q( K 3 )
0
 0.

K

K

F5  1 r1 ( K 0 )    r11 K 0   1 m2  1 r1 ( K 0 )    0,
 r10

 r10


K
 r r Q( K 3 )  Q1 K 32 r31r1 ( K 0 )
where m2  1 r1 ( K 0 )   10 30
.
r30 ( 0   1 K1r1 ( K 0 ))
 r10


 in the interval
  K1 r ( K ) given by F T  0.
Thus there exists a root N
0 N
5
1
1
1
0
r10
 )  0,
Now, the sufficient condition for to be unique is F5( N
where
1
 r10 1 ( 0 Q1 K 3 r31  r30 1Q( K 3 )) 

 )  
F5( N
 
  0.
1



 ) 2
K


(



N
1
0
1 1


(5.2.60)
 , value of I and T can be found from equations (5.2.56) and
With this value of N
1
(5.2.57), respectively.


  
Existence of E14 0,0, N 2 , I , T :
  
Analogous to existence of E14 , we can show the existence of E14 0,0, N 2 , I , T .


Existence of E * B* , N1* , N 2* , I * , T * :
In this case, B* , N1* , N 2* , I * , T * are the solutions of following equations
115
r ( I *) 
r0 B *
  1 N1*   2 N 2*  0,
K I *, T *
(5.2.61)
r1 0 N1*
r1 B * 
 N 2*   1T *  0,
K1
(5.2.62)
r2 0 N 2*
 N1*  1T *  0,
K2
(5.2.63)
r I*
r3 ( N1* , N 2* )  3 0  1  bB*  0,
K3
(5.2.64)
QI *   0T *  0 B * T *  1 N1*T *  2 N 2*T *  0.
(5.2.65)
r2 B * 
From equations (5.2.62), (5.2.63) and (5.2.64) respectively, we get
N1* 
K1
r2 r1 B *  K 2 r2 B *  1 K 2  1r2 0 T *
r10 r2 0  K1 K 2 0
= f 5 ( B*, T *),
N 2* 
(5.2.66)
K2
r1 r2 B *  K1r1 B *  1 K1  1r10 T *,
r10 r2 0  K1 K 2 0
= f 6 ( B*, T *),
I* 

(5.2.67)
K3
r31 K1
(r30 
(r20 r1 ( B*)  K 2 r2 ( B*)  (1 K 2  1r20 )T *)
r30
r10 r20  K1 K 2
r32 K 2
(r10 r2 ( B*)  K1r1 ( B*)  (1 K1  1r10 )T *)  (1  b) B*).
r10 r20  K1 K 2
= f 7 . ( B*, T *),
(5.2.68)
putting the value of N1* , N 2* and I * in equations (5.2.61) and (5.2.65), we get
F6 ( B*, T *)  Q( f 7 ( B*, T *))   0T *  0 B * T * ( 1 f 5 ( B*, T *)
  2 f 6 ( B*, T *))T *
(5.2.69)
116
F7 ( B*, T *)  (r ( f 7 ( B*, T *))   1 f 5 ( B*, T *)   2 f 6 ( B*, T *)K ( f 7 ( B*, T *), T *)
 r0 B * .
(5.2.70)
From (5.2.69), we note the following
when T *  0, then B*  Be* , where
x11 Be*  x12  0,
where
x11 
Q1 K 3 Be*
r31K1 (r20r11  K 2 r21 )  r32 K 2 (r10r21  K1r11 ),
r30 (r10r20  K1 K 2 )
x12  Q( K 3 ) 
Q1 K 3
r31K1 (r20r10  K 2 r20 )  r32 K 2 (r10r20  K1r10 ).
r30 (r10r20  K1 K 2 )
Let X 1 ( Be* )  x11 Be*  x12 ,
X 1 (0)  x12  0,
X 1 ( K 0 )  x11 K 0  x12  0.
Thus there exists a root B e* in the interval 0  Be*  K 0 given by X 1 ( Be* )  0.
Now, the sufficient condition for E * to be unique is X 1 ( Be* )  0, where
X 1 ( Be* )  x11  0,
F6
 Q( f 7( B*, T *))   0T *  1T * f 5( B*, T *)   2T * f 6( B*, T *).
B *
(5.2.71)
F6
 Q( f 7( B*, T *))   0   0 B *  1 f 5 ( B*, T *)   1T * f 5( B*, T *)
T *
  2 f 6 ( B*, T *)   2T * f 6( B*, T *).
Now, from (5.2.71) and (5.2.72) we get
117
(5.2.72)
F6
B *
  T *
F6
T *
B *
It is clear that
B *
 0, if either
T *
(i)
F6
F6
 0 and
 0, or
B *
T *
(ii)
F6
F6
 0 and
 0,
B *
T *
(5.2.73)
From (5.2.70), we note the following
when B*  0, then T *  Te* , where
(r ( f 7 (0, T *))   1 f 5 (0, T *)   2 f 6 (0, T *)K ( f 7 (0, T *), T *)  0
Let X 2 (Te* )  (r ( f 7 (0, Te* ))   1 f 5 (0, Te* )   2 f 6 (0, Te* ) K ( f 7 (0, Te* ), Te* ),

 K r (r  K 2 )   2 K 2 r10 (r20  K1 ) 
X 2 (0)  r0  a1 Z1  1 1 20 10
K 0  q1 Z1   0,
r10 r20  K1 K 2


where Z1  K 3 
K 3 (r31 K1r20 (r10  K 2 )  r32 K 2 r10 (r20  K1 )
.
r30 (r10 r20  K1 K 2
X 2 (Tm )  (r ( f 7 (0, Tm ))   1 f 5 (0, Tm )   2 f 6 (0, Tm ) K ( f 7 (0, Tm ), Tm )  0,
Thus there exists a root Te* in the interval 0  Te*  Tm , given by X 2 (Te* )  0.
Now, the sufficient condition for Te* to be unique is X 2 (Te* )  0. where
X 2 (Te* )  (r ( f 7(0, Te* ))   1 f 5(0, Te* )   2 f 6(0, Te* ) K ( f 7 (0, Te* ), Te* )
 (r ( f 7 (0, Te* ))   1 f 5 (0, Te* )   2 f 6 (0, Te* )( q1 f 7(0, Te* )  q 2 )  0,
F7
 (a1 f 7( B*, T *)   1 f 5( B*, T *)   2 f 6( B*, T *)) K ( f 7 ( B*, T *), T *)
B *
 (r ( f 7 ( B*, T *))   1 f 5 ( B*, T *)   2 f 6 ( B*, T *))q1 f 7( B*, T *)  r0
118
(5.2.74)
F7
 (a1 f 7( B*, T *)   1 f 5( B*, T *)   2 f 6( B*, T *)) K ( f 7 ( B*, T *), T *)
T *
 (r ( f 7 ( B*, T *))   1 f 5 ( B*, T *)   2 f 6 ( B*, T *))(q1 f 7( B*, T *)  q2 ).
(5.2.75)
Now, from (5.2.74) and (5.2.75) we get
F7
B *
  T *
F7
T *
B *
It is clear that
B *
 0, if either
T *
(i)
F7
F7
 0 and
 0, or
B *
T *
(ii)
F7
F7
 0 and
 0,
B *
T *
(5.2.76)
Thus the two isoclines (5.2.66) and (5.2.67) intersects at a unique ( B*, T *) if in
addition to conditions (5.2.73) and (5.2.76), the inequality Be*  Te* holds. Knowing
the value of B * and T *, we get N1* , N 2* and I * can be calculated from equations
(5.2.66), (5.2.67) and (5.2.68). This completes the existence of E * .
5.4 STABILITY ANALYSIS
5.4.1 Local Stability
To discuss the local stability of system (5.1.1), we compute the variational matrix of
the system (5.1.1). The entries of general variational matrix are given by
differentiating the right hand of system (5.1.1) with respect to B, N1 , N 2 , I , and T i.e.
119

 A1

 r11 N 1
M (E)  
 r21 N 2
(1  b) I

   0T
 1 B
A2

r q B2
  2 B   a1 B  0 1
( K ( I , T )) 2

 N 1
0
 N 2
A3
0
r31 I
r32 I
A4
  1T
  2T
Q1

r q B2 
  0 2


( K ( I , T )) 2 

  1 N1 
.
 1 N 2 

0

A5

Where
A1  r ( I ) 
2r0 B
2r N
  1 N1   2 N 2 , A2  r1 ( B)  10 1  N 2  1T ,
K (I , T )
K1
A3  r2 ( B) 
2r20 N 2
2r I
 N1  1T , A4  r3 ( N1 , N 2 )  30  (1  b) B,
K2
K3
A5   0   0 B   1 N1   2 N 2 .
The variational matrix M ( E 0 ) at equilibrium point E0 is given by
 r0

 0


M ( E0 )   0

 0
  0 Q0
 
0

0
Q
r10  1 0
0
0
0
Q
 1 0
0
r20 
0
0
0
0
 1 Q0
0
0
0
r30
 Q
 2 0
0
Q1
0 

0 


0 .

0 

0 

From M ( E 0 ) , we note that characteristic roots namely, r0 , r10 
1Q0
Q
, r20  1 0
0
0
and r30 are positive, giving a saddle point which is stable in the T , direction and
unstable in the B  N1  N 2  I space. Therefore, the equilibrium point E 0 , is
unstable.
The variational matrix M ( E1 ) at equilibrium point E1 is given by
120
 r(K 3 )

0



0
M ( E1 )  

 (1  b) K 3
  0 Q( K 3 )

0

0
 Q( K 3 )
r10  1
0
0
r31 K 3
 Q( K 3 )
 1
0
r20 
0
0
0
0
 1Q ( K 3 )
0
r32 K 3
 Q( K 3 )
 2
0
0
 r30
Q1
0 

0 


0 .

0 

0 

From M ( E1 ) , we note that characteristic roots namely, r ( K 3 ), r10 
and r20 
1Q( K 3 )
,
0
1Q( K 3 )
are positive, giving a saddle point which is stable in the I  T ,
0
plane and unstable in the B  N1  N 2 space. Therefore, the equilibrium point E1 , is
unstable.
The variational matrix M ( E2 ) at equilibrium point E2 is given by

r0 B

 K (0, T )
0

M ( E2 )  
0


0

   0T
 1B
2B
 S 
r1 ( B )   1T
0
0
0
r2 ( B )  1T
0
0
0
r30  (1  b) B
  1T
  2T
Q1
r0 q 2 B 2 


( K (0, T )) 2 
0

,
0


0

 ( 0   0 B ) 
where

r q B2 
.
S    a1 B  0 1
2 
(
K
(
0
,
T
))


From M ( E2 ) , we note that characteristic roots namely, r1 ( B )  1T , r2 ( B )  1T ,
and r30  (1  b) B
are positive, giving a saddle point which is stable in the B  T ,
plane and unstable in the N1  N 2  I space. Therefore, the equilibrium point E 2 , is
unstable.
The variational matrix M ( E3 ) at equilibrium point E 3 is given by
121
r0   1 N 1

 r N
 11 1
M ( E3 )  
0


0

   0T
0
0
0
 N 1
0
r20   N 1  1T
0
0
0
r30  r31 N 1
  1T
  2T
Q1

r10 N 1
K1
0


  1 N1 

.
0


0

 ( 0   1 N 1 )
0
From M ( E3 ) , we note that characteristic roots namely, r0  1 N1 , r20  N1  1T ,
and r30  r31 N1 are positive, giving a saddle point which is stable in the N1  T , plane
and unstable in the B  N 2  I space. Therefore, the equilibrium point E 3 , is
unstable.
The variational matrix M ( E4 ) at equilibrium point E4 is given by
r0   2 Nˆ 2

0


M ( E 4 )   r21 Nˆ 2


0

ˆ
   0T
0
ˆ
r10  N 2   1Tˆ
 Nˆ 2
0
  1Tˆ
0
0
0
0
r Nˆ
 20 2
K2
0
  Tˆ
0
r30  r32 Nˆ 2
Q1
2


0


 1 Nˆ 2 .


0

 ( 0   2 Nˆ 2 )
0
From M ( E4 ) , we note that characteristic roots namely, r0   2 Nˆ 2 , r10  Nˆ 2  1Tˆ ,
and r30  r32 N̂ 2 are positive, giving a saddle point which is stable in the Nˆ 2  T , plane
and unstable in the B  N1  I space. Therefore, the equilibrium point E 4 , is unstable.
The variational matrix M ( E5 ) at equilibrium point E 5 is given by
ˆˆ

 r0 B
 K (0, Tˆˆ )


ˆ
M ( E5 )   r11 Nˆ 1


0

0

   Tˆˆ
0

ˆ
  1 Bˆ
ˆ
r10 Nˆ 1

K1
0
ˆ

r0 q1 Bˆ 2
ˆˆ
ˆˆ

  2 B  a1 B 

ˆ
( K (0, Tˆ )) 2

ˆ
 Nˆ 1
0
H1
0
0
0
ˆ
  1Tˆ
ˆ
  2Tˆ
H2
Q1
122
ˆ2 

  r0 q 2 Bˆ 

ˆ
( K (0, Tˆ )) 2 



ˆ
  1 Nˆ 1 .


0

0


H3

where
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
H 1  r2 ( Bˆ )  Nˆ 1  1Tˆ , H 2  r30  r31 Nˆ 1  (1  b) Bˆ , H 3  ( 0   0 Bˆ   1 Nˆ 1 ).
ˆ
ˆ
ˆ
From M ( E5 ) , we note that characteristic roots namely, H1  r2 ( Bˆ )  Nˆ 1  1Tˆ
and
ˆ
ˆ
H 2  r30  r31 Nˆ 1  (1  b) Bˆ1 are positive, giving a saddle point which is stable in
the B  N1  T , space and unstable in the N 2  I plane. Therefore, the equilibrium
point E 5 , is unstable.
The variational matrix M ( E6 ) at equilibrium point E6 is given by
~

r0 B

~
 K (0, T )

0

M ( E6 ) 
~
 r21 N
2


0

~
   0T
~
 1 B
~
2B
Y1
~
 N 2
0
0
~
  1T
~
r20 N 2

K2
0
~
  2T
~
 ~
r0 q1 B 2

  a1 B 
~
( K (0, T )) 2

0
0
Y2
Q1
~

r q B2 
  0 2~ 

( K (0, T )) 2 


0
.
~
 1 N 2 


0

Y3

where
~
~
~
~
~
~
~
Y1  r1 ( B)  N 2  1T , Y2  r30  r32 N 2  (1  b) B, Y3  ( 0   0 B   2 N 2 ).
~
~
~
From M ( E 6 ) , we note that characteristic roots namely, Y1  r1 ( B )  N 2   1T
and
~
~
Y2  r30  r32 N 2  (1  b) B are positive, giving a saddle point which is stable in
the B  N 2  T , space and unstable in the N1  I plane. Therefore, the equilibrium
point E 6 , is unstable.
The variational matrix M ( E7 ) at equilibrium point E7 is given by
123
~
~
~
~
r   N


N
0
1 1
2
2

~

~
r11 N 1


M ( E7 )  
~
~
r21 N 2



0

~
~

  0T
0
~
~
r10 N 1

K1
~
~
 N 2
0
~
~
 N 1
~
~
r20 N

K2
0
~
~
  1T
0
~
~
  2T


~

~
0
  1 N1


.
~
~
0
 1 N 2


~
~ ~
~

r3 ( N 1 , N 2 )
0
~
~
~
~ 
Q1
 ( 0   1 N 1   2 N 2 )
0
0
~
~
~
~
From M ( E7 ) , we note that characteristic roots namely, r0  1 N1   2 N 2 and
~
~ ~
~
r3 ( N1 , N 2 ) are positive, giving a saddle point which is stable in the N1  N 2  T ,
space and unstable in the B  I plane. Therefore, the equilibrium point E 7 , is
unstable.
The variational matrix M ( E8 ) at equilibrium point E8 is given by


r0 B


 K (0, T )


 r11 N 1
M ( E8 )  


 r21 N 2

0

   T
0


 1B

r10 N 1

K1

 N 2
0

  1T

2B

 N 1

r20 N 2

K2
0

  2T

 
r0 q1 B 2
  a1 B 

( K (0, T )) 2

0
0
 
r3 ( N 1 , N 2 )
Q1


r0 q 2 B 2 
 
 

( K (0, T )) 2 



  1 N1 
,


 1 N 2 

0


 Z 




where Z   ( 0   0 B   1 N1   2 N 2 ).
~
~ ~
~
From M ( E8 ) , we note that characteristic root namely r3 ( N1 , N 2 ) is positive, giving a
saddle point which is stable in the B  N1  N 2  T , space and unstable in the I
direction. Therefore, the equilibrium point E 8 , is unstable.
The variational matrix M ( E9 ) at equilibrium point E 9 is given by
124


r0 B

 
 K (I ,T )



r
N
 11 1
M ( E9 )  
0



(
1

b
)
I



   T
0


 1B

r10 N1

K1
0

r31 I

  1T

2

 
r
q
B
  2 B   a1 B  0 1 

( K ( I , T )) 2


 N 1
0
P1

r32 I

  2T
0

r I
 30
K3
Q1

2


r
q
B
  0 2  
 

( K ( I , T )) 2 




  1 N1 
.
0


0



 P2






P

r
(
B
)


N


T
,
P

(



B


N
where 1
2
1
1
2
0
0
1 1 ).



From M ( E9 ) , we note that characteristic root namely P1  r2 ( B)  N1  1T is
positive, giving a saddle point which is stable in the B  N1  I  T , space and
unstable in the N 2 direction. Therefore, the equilibrium point E 9 , is unstable.
The variational matrix M ( E10 ) at equilibrium point E10 is given by


r0 B
 

 K (I ,T )

0


M ( E10 )   r21 N 2



 (1  b) I


   0T

 1 B

2B
D1
0
0

r31 I

  1T

r20 N 2

K2

r32 I

  2T

 
r0 q1 B 2
  a1 B 
 
( K ( I , T )) 2

0
0

r30 I

K3
Q1


r0 q 2 B 2 
 
  

( K ( I , T )) 2 


0


 1 N 2 .


0



 D2





where D1  r1 ( B)  N 2   1T , D2  ( 0   0 B   2 N 2 ).



From M ( E10 ) , we note that characteristic root namely, D1  r1 ( B)  N 2  1T is
positive, giving a saddle point which is stable in the B  N 2  I  T , space and
unstable in the N1 direction. Therefore, the equilibrium point E10 , is unstable.
The variational matrix M ( E11 ) at equilibrium point E11 is given by
125



r ( I )   N   N
1 1
2
2




r11 N 1




M ( E11 )  
r21 N 2



(
1

b
)
I






T
0

0
0

r31 I

  1T

 N 1

r20 N 2

K2

r32 I

  2T

r10 N 1

K1

 N 2




  1 N1




 1 N 2
.


0


 
 ( 0   1 N 1   2 N 2 )
0
0
0
0

r30 I

K3
Q1



From M ( E11 ) , we note that characteristic root namely r ( I )   1 N1   2 N 2 is positive,
giving a saddle point which is stable in the N1  N 2  I  T , space and unstable in the
B direction. Therefore, the equilibrium point E11 , is unstable.
The variational matrix M ( E12 ) at equilibrium point E12 is given by

r0 B


 
 K (I ,T )
0


M ( E12 ) 
0

 (1  b)I



   0T
  1 B
  2 B
 G1
r1 ( B)   1T
0
0
0
r2 ( B)  1T
r31I
r32I
  1T
  2T
0
r30I

K3
Q1

r0 q1 B2




where G1  a1 B 

K I,T

 
r0 q 2 B2 

( K (I, T)) 2 
0

.
0


0


 ( 0   0 B) 



2 


 and r ( B)   T

From M ( E12 ) , we note that characteristic root namely, r1 ( B)  1T
2
1
are positive, giving a saddle point which is stable in the B  I  T , space and unstable
in the N1  N 2 plane. Therefore, the equilibrium point E12 , is unstable.
The variational matrix M ( E13 ) at equilibrium point E13 is given by
126

r (I)   N
1 1



 r11 N 1

M ( E13 )  
0

 (1  b)I



   0T
From M ( E13 ) ,
we
0


r10 N
1
K1
0
r31I
  1T
note
that











 N
0
 1 N
1
1


   T
r20  N
0
0
.
1
1




r30I







r32 I

0

K3






 )
  2T
Q1
 ( 0   1 N
1 
0
characteristic
0
roots
0
namely,
 and
r (I)   1 N
1
   T are positive, giving a saddle point which is stable in the N  I  T ,
r20  N
1
1
1
space and unstable in the B  N 2 plane. Therefore, the equilibrium point E13 , is
unstable.
The variational matrix M ( E14 ) at equilibrium point E14 is given by


r ( I )   N
2
2


0


 r N
21 2
M ( E14 )  



 (1  b) I


   0T
0
0

r10  N 2   1T
0


r20 N 2
 N 2

K2


r31 I
r32 I


  1T
  2T
0
0
0

r I
 30
K3
Q1



0


 1 N 2 
.


0

 
 ( 0   2 N 2 )
From M ( E14 ) , we note that characteristic roots namely,
0

r (I)   1 N
1
and


r10  N 2  1T are positive, giving a saddle point which is stable in the N 2  I  T ,
space and unstable in the B  N1 plane. Therefore, the equilibrium point E14 , is
unstable.
In the following theorem we show that E * is locally asymptotically stable:
Theorem (5.4.1):
In addition to assumptions (H1) – (H8), let the following
inequalities holds
127
(C1  C 2  ) 2  C1C 2
(C3 r31 ) 2  C1C3
r10 r20
,
K1 K 2
(5.4.1)
r10 r30
,
K1 K 3
(5.4.2)


1 r
(C11   1T *) 2  C1 10  0   0 B*   1 N1*   2 N 2* ,
2 K1
(5.4.3)


2
 r B *q

r0
*  1
2
 0


T
 0   0 B*   1 N1*   2 N 2* ,
0
2
 K ( I *, T *)

2 K ( I *, T *)


(5.4.4)


r
1
(C2 1   2T *) 2  C2 20  0   0 B*   1 N1*   2 N 2* ,
2
K2
(5.4.5)
r20 r30
,
K 2 K3
(C3 r32 ) 2  C 2 C3
(5.4.6)

r
1
Q32  C3 30  0   0 B*   1 N1*   2 N 2*
2 K3

(5.4.7)
where
C1 
1

r1 ( B* )
,
C2 
2

r2 ( B* )
,
C3 
r B * K ( I *)  r ( I *)K
0
2

( I *, T *)
.
(1  b) K ( I *, T *)
2
(5.4.8)
Then E * is locally asymptotically stable.
Proof: We first linearize the system (5.1.1) by using the following transformations
B  B*  b,
N1  N1*  n1 ,
N 2  N 2*  n2 ,
I  I *  i,
T  T*  .
(5.4.9)
where b, n1 , n2 , i,  are small perturbation around the positive equilibrium.
Then using the following positive definite function in the linearized version of the
model
W t  
1
2 B*
b2 
C
C1
C
1
2
2
n1  2 n2  3 i 2   2 ,
*
*
*
2
2 N1
2N 2
2I
we obtain
128
(5.4.10)
dW
1
1
  a11b 2  a12bn1  a 22 n12 ,
dt
4
4
1
1
  a 22 n12  a 23 n1 n2  a33 n22 ,
4
4
1
1
  a33 n22  a31n2 b  a11nb 2 ,
4
4
1
1
  a11b 2  a14bi  a 44i 2 ,
4
4
1
1
  a 44i 2  a 42 n1i  a 22 n12 ,
4
4
1
1
  a 22 n12  a 25 n1  a55 2 ,
4
4
1
1
  a55 2  a51b  a11b 2 ,
4
4
1
1
  a 22 n12  a 25 n1  a55 2 ,
4
4
1
1
  a55 2  a53 n2  a33 n22 ,
4
4
1
1
  a33 n22  a34 n2 i  a 44i 2 ,
4
4
1
1
  a 44i 2  a 45i  a55 2 ,
4
4
where
a11 
2r0
,
K I *,T *


 2r 
a 22  C1  10 ,
 K1 


a55   0   0 B*   1 N1*   2 N 2* ,

a31   2  C 2 r2 ( B* ),
 2r 
a33  C 2  20 ,
 K2 
 2r 
a 44  C3  30 ,
 K3 

a12  1  C1r1 ( B* ),
a14  r ( I * ) 
a23  C1  C2  ,
r0 B* K ( I * )
 C3 1  b ,
2
*
*
K (I ,T )
129

a 42  C3 r3 ( N1* ),


a 25   1C1   1T * ,

a34  C3 r3 ( N 2* ),
 r B* K (T * )

a51   0
  0T * ,
 K 2 (I *,T * )



a53  1C 2   2T *,
a45  Q3 .
Sufficient conditions for
dW
dt
to be negative definite are that the following
inequalities hold:
a122 
1
a11a 22 ,
4
(5.4.11)
2
a 23

1
a 22 a33 ,
4
(5.4.12)
2
a 31

1
a 33 a11 ,
4
(5.4.13)
a142 
1
a11a 44 ,
4
(5.4.14)
2
a 42

1
a 22 a 44 ,
4
(5.4.15)
2
a 25

1
a 22 a55 ,
4
(5.4.16)
2
a 51

1
a11a 55 ,
4
(5.4.17)
2
a53

1
a33 a55 ,
4
(5.4.18)
2
a34

1
a33 a 44 ,
4
(5.4.19)
2
a 45

1
a 44 a55 .
4
(5.4.20)
By choosing C1 , C 2 , C 3 as given by equation (6.4.8), we note that ine5ualities
(5.4.11), (5.4.13) and (5.4.14) are automatically satisfied. We further note that (5.4.1)
 (5.4.12), (5.4.2)  (5.4.15), (5.4.3)  (5.4.16), (5.4.4)  (5.4.17),(5.4.5)  (5.4.18)
130
(5.4.6)  (5.4.19), (5.4.7)  (5.4.20). This shows that W is Lyapunov function with
respect to E * , proving the theorem.
5.4.2 Global Stability
Theorem (5.4.2): In addition to assumptions (H1) - (H8), let r ( I ), K I , T , r1 B,
r2 ( B), r3 N1 , N 2  and QI  satisfy the following conditions in  :
K
K

 m1 , 0  
 m2 , 0  r I   1 , 0  r1 B    2 ,
I
T

r

r

(5.4.21)
0  r2 B    3 , 0  3   4 , 0  3   5 , 0  QI    6 .
N1
N 2
K m  K I , T   K 0 , 0  
For some positive constants K m , m1 , m2 , 1 ,  2 ,  3 ,  4 ,  5 ,  6 .
Then if the following inequalities hold in 
r0
r10
,
4 K I m , Tm  K1
(5.4.22)
r0
r20
,
4 K I m , Tm  K 2
(5.4.23)
1   2 2  1
 2   3 2  1
2


r0
r30
m
1
 1  r0 B* 12  1  b  
,




4
K
I
,
T
K
K
m
m
3
m


(5.4.24)


2
 * m2

r0
1
 r0 B
  0Tm  
 0   0 B*   1 N1*   2 N 2* ,
2



4
K
I
,
T
K
m
m
m


r10 r20
,
4 K1 K 2
   2  1
 42 
(5.4.26)
1 r10 r30
,
4 K1 K 3
(5.4.27)


r10
 0   0 B*   1 N1*   2 N 2* ,
4 K1
1   1Tm 2  1
 52 
(5.4.25)
1 r20 r30
,
4 K 2 K3
(5.4.28)
(5.4.29)
131


r20
 0   0 B*   1 N1*   2 N 2* ,
4 K2
1   2Tm 2  1
 62 
(5.4.30)


1 r30
 0   0 B*   1 N1*   2 N 2* ,
4 K3
(5.4.31)
E * is globally asymptotically stable with respect to all solutions initiating in the
positive orthant  .
Proof: Consider the following positive definite function about E *

N 
B  
  N1  N1*  N1* ln 1  
V B, N1 , N 2 , I , T    B  B*  B* ln

B*  
N1* 



 N  N *  N * ln N 2    I  I *  I * ln I   1 (T  T *) 2 .
2
2
2



N 2*  
I*  2

Differentiating V with respect to time t, we get
dV  B  B *  dB  N 1  N 1*  dN 1  N 2  N 2*  dN 2  I  I *  dI




 dt
 I  dt
dt  B  dt  N 1  dt  N 2









 T T*
dTdt .
Substituting the values of
dB dN1 dN 2 dI
dT
,
,
,
and
dt dt
dt dt
dt
from the system equation
(5.1.1) in the above equation and after doing some algebraic manipulations and
considering functions,
 r I   r ( I * )
, I  I *,

*
1  I    I  I

r ( I * ),
I  I *.
(5.4.32)
 r B   r ( B * )
1
1

, B  B* ,

*
 1 B    B  B

*

B  B*.
r1( B ),
(5.4.33)
132
 r B   r ( B * )
2
 2
, B  B* ,

*
 2 B   
BB

r2 ( B * ),
B  B*.

(5.4.34)
 Q I   Q ( I * )
,
I  I *,


 I    I  I *

*

I  I *.
Q ( I ),
(5.4.35)
1
 1
 K I , T  
K I *,T

,

 2 I , T   
I  I*

1
K ( I * , T )

,
 K 2 (I *,T )
I

 
I  I *,
(5.4.36)
I  I *.
1
1



*
* *
 K (I ,T ) K (I ,T ) ,

3 (I *,T )  
T T*

1
K ( I * , T * )

,
 K 2 (I *,T * )

T


T  T *,
(5.4.37)
T  T *.

 r N , N   r N *, N
3
1
2
, N1  N1*,
3 1 2
*

N1  N1
 1 N1 , N 2   
 r3 N1*, N 2
,
N1  N1*.

N1




 

 r N *, N  r N *, N *
2
3
1
2
 3 1
, N 2  N 2* ,
*

N2  N2
 2 N1* , N 2  
 r3 N1* , N 2*
,
N 2  N 2*.


N
2




(5.4.38)

we get
dV
1
1
  c11 ( B  B* ) 2  c12 ( B  B* )( N1  N1* )  c 22 ( N1  N1* ) 2 ,
dt
4
4
133
(5.4.39)
1
1
  c11 ( B  B* ) 2  c13 ( B  B* )( N 2  N 2* )  c33 ( N 2  N 2* ) 2 ,
4
4
1
1
  c11 ( B  B* ) 2  c14 ( B  B* )( I  I * )  c 44 ( I  I * ) 2 ,
4
4
1
1
  c11 ( B  B* ) 2  c15 ( B  B* )(T  T * )  c55 (T  T * ) 2 ,
4
4
1
1
  c 22 ( N 1  N1* ) 2  c 23 ( N1  N1* )( N 2  N 2* )  c33 ( N 2  N 2* ) 2 ,
4
4
1
1
  c 22 ( N1  N1* ) 2  c 24 ( N1  N1* )( I  I * )  c 44 ( I  I * ) 2 ,
4
4
1
1
  c 22 ( N1  N1* ) 2  c 25 ( N1  N1* )(T  T * )  c55 (T  T * ) 2 ,
4
4
1
1
  c33 ( N 2  N 2* ) 2  c34 ( N 2  N 2* )( I  I * )  c 44 ( I  I * ) 2 ,
4
4
1
1
  c33 ( N 2  N 2* ) 2  c35 ( N 2  N 2* )(T  T * )  c55 (T  T * ) 2 ,
4
4
1
1
  c 44 ( I  I * ) 2  c 45 ( I  I * )(T  T * )  c55 (T  T * ) 2 ,
4
4
where
c11 
r0
r
r
r
, c22  10 , c33  20 , c44  30 , c55   0   0 B*   1 N1*   2 N 2*
K I , T 
K1
K2
K3
c12   1  1 B ,
 

c13   2   2 B ,
c14  1 I   r0 B* 2 I , T   1  b 

c15   r0 B* 3 I * , T   0T , c 23     , c 24   1 N1 , N 2 , c 25   1   1T ,


c34   2 N1* , N 2 , c35  1   2T , c 45   I .
Then sufficient conditions for
dV
to be negative definite are that the following
dt
inequalities hold
134
c122 
1
c11c 22 ,
4
2
c 24

1
1
1
1
1
2
2
2
2
c 22 c 44 , c 25
 c 22 c55 , c34
 c33c 44 , c35
 c33c55 , c 45
 c 44 c55 .
4
4
4
4
4
c132 
1
c11c33 ,
4
c142 
1
c11c 44 ,
4
c152 
1
c11c55 ,
4
2
c 23

1
c 22 c33 ,
4
(5.4.40)
Now, from (5.4.21) and mean value theorem, we note that
1 I   1 ,  2 I , T  
m1
m
,  3 ( I *, T )  22 , 1 B    2 ,  2 B    3 ,
2
Km
Km
 1 N1 , N 2    4 ,  2 ( N1* , N 2 )   5 ,  I    6 .
(5.4.41)
Further, we note that the stability conditions (5.4.22) - (5.4.31) as stated in theorem
(5.4.2), can be obtained by maximizing the left- hand side of inequalities (5.4.40).
This completes the proof of theorem (5.4.2).
Remark. In model (5.1.1), the industrialization acts as a predator on the resource
biomass as well as on the competing species with an alternative resource, i.e., even in
the absence of resource biomass and the competing species the industrialization grows
logistically due to its dependency on some other alternative resource. If the
industrialization is assumed to act as a predator that completely depends upon the
resource biomass and the competing species, then the fourth equation of model (5.1.1)
may be replaced by
dI
 0 I  1 I 2  1  b BI  r30 N1 I  r31 N 2 I ,
dt
(5.4.42)
where 0  0 is the natural depletion rate coefficient of the industrialization and
1  0 is the intra-specific interference coefficient. In case of abundance of resource
and competing species, 1 may be taken to be zero. If we analyze model (5.1.1), when
the fourth equation of the model is replaced by (5.4.42), we note that the equilibrium
135

Q K 3  

corresponding to E1  0,0,0, K 3 ,
 0 

does not exist. Other fifteen equilibrium
points similar to that of the original model (5.1.1) exist under certain modified
conditions. However, the magnitudes of the equilibrium level get changed. In
particular, it is noted that the equilibrium level of industrialization is less and the
equilibrium level of resource biomass is more in comparison to the previous case. It
has also been found that if the concentration of pollutant increases unabatedly, then
the densities of resource biomass, competing species and industrialization decreases
faster as compared to the previous case. Stability condition for the positive
equilibrium to be locally and globally asymptotically stable can be obtained in a
certain manner as in theorem (5.4.1) and (5.4.2).
5.5 NUMERICAL SIMULATION AND DISCUSSION
To facilitate the interpretation of our mathematical findings by numerical simulation,
we integrated system (5.1.1) using fourth order Runge-Kutta method. We take the
following particular form of the functions involved in the model (5.1.1):
r I   r0  a1 I ,
r1 B   r10  r11 B, r2 B   r20  r21 B,
r3 N1 , N 2   r30  r31 N1  r32 N 2 ,
K I , T   K 0  q1 I  q2T ,
QI   Q0  Q1 I .
(5.5.1)
Now we choose the following set of values of parameters in model (5.1.1) and
equation (5.5.1).
r0  7, a1  0.006, K 0  15, q1  0.002, q 2  0.003,  1  0.03,  2  0.05, r10  4,
r11  0.02, K1  12,   0.006,  1  0.005, r20  6, r21  0.03, K 2  12,   0.006,
1  0.008, r30  7, r31  0.04, r32  0.03, K 3  20, b  .98,
Q0  10, Q1  .001,
 0  3,  0  0.3,  1  0.1,  2  0.08, 1  0.02,  2  0.05,  3  0.02,  4  0.03,
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5  0.02, 6  0.01, m1  0.001, m2  0.001, K m  15.
(5.5.2)
With the above values of parameters, we note that condition for the existence of E *
are satisfied and E * is stable.
B*  12.5073,
N1*  12.5127,
N 2*  12.5825,
I *  23.2226,
T *  1.1124.
It is further noted that all conditions of local stability (5.4.1) – (5.4.7) and global
stability (5.4.24) – (5.4.33) are satisfied for the set of values of parameters given in
(5.5.2).
Figures (1) – (3), shows the dynamics of resource biomass for different values
of a1 , q1 and q2 , with respect to time‘t’. It is analyzed from the figures that as the
density of industrialization and toxicant increase into the environment, the growth rate
as well as carrying capacity of resource biomass decreases. It is also noted that
initially resource biomass increases with time ‘t’, and after certain time it settle down
to its steady state, assuring the local stability of equilibrium point E * .
Figure 1, Variation of B , with time ‘t’for different values of a1 and
other values of parameters are same as in (5.5.2)
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Figure 2, Variation of B , with time for different values of q1 and
other values of parameters are same as in (5.5.2)
Figure 3, Variation of B , with time ‘t’ for different values of q 2 and
other values of parameters are same as in (5.5.2)
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Figures (4) and (5), shows the variation of N1 and N 2 , against time ‘t’ for different
values of Q1 . From these plot, we note that as Q1 , (the rate of formation of toxicant
due to industrialization) increases into the environment, the density of both competing
species decreases. Also, the effect of parameter Q1 , on the density of toxicant is
shown in figure (6). From this figure, we note that by a very small increment in the
value of Q1 , toxicant in the environment increases very rapidly as the time passes.
Figure 4, Variation of N1 , with time ‘t’ for different Q1 , and
other values of parameters are same as in (5.5.2)
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Figure 5, Variation of N 2 , with time ‘t’for different Q1 , and
other values of parameters are same as in (5.5.2)
Figure 6, Variation of T , Toxicant with time ‘t’ for different Q1 , and
other values of parameters are same as in (5.5.2)
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The variations of I , with time ‘t’ are plotted in figures (7) and (8) for different values
of r31 and r32 respectively. From these figures, it can be easily seen that by small
increment in values of these parameters density of industrialization grow rapidly.
From figure (9), we note that industrialization into the environment increases very
rapidly by small increment in growth rate of industrialization by resource biomass,
1  b.
This suggest that the resource biomass should be utilized appropriately by
industrialization otherwise it may driven to extinction. The variation of N1 and N 2 , in
the presence and absence of industrialization is plotted in figure (10). From this plot,
we can infer that presence of industrialization decreases the endemic level of both
competing species. It is due to the fact that due to presence of industrialization,
density of toxicant increases, which causes decrease in the density of competing
species.
Figure 7, Variation of I , with time ‘t’ for different values of r31 and
other values of parameters are same as in (5.5.2)
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Figure 8, Variation of I , with time ‘t’ for different values of r32 and
other values of parameters are same as in (5.5.2)
Figure 9, Variation of I , with time ‘t’ for different values of (1-b) and
other values of parameters are same as in (5.5.2)
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Figure 10, Variation of N1 with N 2 in the presence and absence of industrialization
for the set of parameters values given in (5.5.2)
5.6 Conclusion
In this chapter, a nonlinear mathematical model is proposed and analyzed to
study the survival of resource dependent competing species, where it is assumed that
competing species and its resource are affected by the toxicant emitted directly into
the environment from external sources as well as its concentration increases by
population pressure augmented industrialization. Equilibrium analysis has been found,
analytically as well as graphically, and it is observed that the nontrivial equilibrium is
locally as well as globally asymptotically stable under certain conditions. It is noted
that not only the concentration of toxicants emitted from external sources but also
formed by population pressure augmented industrialization plays an important role in
determining the stability of the system. It is obtained from the analysis that as the
cumulative rate of emission of the toxicant from external sources as well as its
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formation due to industrialization increases, the equilibrium density of both
competing species and its resource decreases. Also, it is found that as density of
industrialization increases into the environment, it reduces growth rate and carrying
capacity of resource biomass. Again, it is observed that initially with the enhancement
in density of industrialization, density of toxicant increases slowly, but after gradual
increment in density of industrialization, toxicant growth is very effective.
From the analysis, it is predicted that when both competing species are
affected by a toxicant in the absence of industrialization, the equilibrium densities are
greater than their values when toxicant is formed due to industrialization. Also, it is
obtained that the density of industrialization increases with increase in the density of
competing species and resource biomass.
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