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International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963

A STOCHASTIC STUDY OF A FOUR SPECIES SYNECOSYSTEM WITH BIO-ECONOMIC YIELDING OF BOTH
VICTIM & KILLER
Mellacheruvu Naga Srinivasu 1 , Konda Shiva Reddy 2 , Ayyavu Sabarmathi 3
1, 3

2

School of Advanced Sciences, VIT University, Vellore, T.N., India
Deptt. of Mathematics, Anurag Group of Institutions, Hyderabad, A.P., India

ABSTRACT
This paper investigates the stochastic stability of a four species syn-ecosystem with bionomic harvesting of both
victim (prey) and killer (predator) species. The necessary conditions for the existence of positive solutions are
obtained at different steady states. Moreover we analyzed the local and global stability of deterministic model.
The likelihood of survival of bio-economic balance is being conversed. We investigated the inhabitant intensities
of fluctuations around the interior equilibrium due to noise. We also discussed the physical significance of the
population variances on stability. Finally, we carried out the given numerical simulations to visualize the
analytical results using Matlab.

KEYWORDS:

Commensal, steady state, Routh-Hurwitz criteria, global stability, bionomic harvesting,
stochastic perturbation, Fourier transforms.

I.

INTRODUCTION

There has been a rising curiosity in the study of harvesting and randomly fluctuating driving forces in
a prey-predator-host-commensal system. It is observed in nature that species do not exist in solitary.
While species are in the presence of harvesting and randomly fluctuating driving forces, they fight for
food, space and are predated by other species. Consequently it is more of natural significance to
consider the effect of interaction between species when we study the dynamical behaviour of
conventional syn-ecosystem models. So a suitable mathematical model is required to study the effect
of harvesting and noise on the interacting species.
Moreover ecology is the study of the inter-relationship between creatures and their surroundings. As it
is usual that when there are two or more species live in a common territory, they interact with each
other in dissimilar ways. Mathematical modeling has been playing an important role for the last half a
century in explaining several phenomena that are concerned with individuals or groups in nature.
Lotka [1] and Volterra [2] established theoretical ecology momentously and opened new epochs in the
filed of life and biological sciences. The Ecological interactions can be broadly classified as
Ammensalism, Commensalism, Competition, Mutualism, Neutralism, Predation and Parasitism which
are based on studies carried out by researchers on the inter-relationship between creatures.
It is noteworthy to mention that the general concept of modeling has been presented in the treatises of
Meyer [3], Cushing [4], Kapur [5, 6], Srinivas [7] who studied competitive ecosystem of two species
and three species with limited and unlimited resources. On the other hand Laxminarayan and Pattabhi
Ramacharyulu [8] studied prey-predator ecological models with partial cover for the prey and
alternate food for the predator. At the same time, Archana Reddy [9] and Bhaskara Rama Sharma [10]
investigated diverse problems related to two species competitive systems with time delay by
employing analytical and numerical techniques, Phani Kumar [11] studied some mathematical models
of ecological commensalism and Ravindra Reddy [12] discussed on the stability of two mutually
interacting species with mortality rate for the second species. Further Srilatha [13, 14] and Shiva

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Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
Reddy [15] studied stability analysis of three and four species. Hari Prasad and Pattabhi
Ramacharyulu [16-21] discussed on the stability of a three and four species syn-ecosystems. The
present authors [22-25] investigated the stability of three species and four species with stage structure,
optimal harvesting policy and stochasticity. Hari prasad [26], Kar [27], Carletti [28] and Nisbet [29]
inspired us to do the present investigation on the analytical and numerical approach of a emblematical
four species syn-ecosystem.
The paper is organized as follows: Section 1 is a major part that deals with the introduction we have
just seen. It records our research and previous research that has been carried out on the synecosystems. Section 2 describes the mathematical model (2.1)-(2.4). Further we analyze the stability
of deterministic model (3.1)-(3.4) in section 3 that consists of four sub-sections on steady states, local
stability and global stability. In section 4, we compute analytical estimates of the population variances
of the model (2.1)-(2.4). Physical significances of population variances are given in section 5. The
given computer simulations in section 6 helps us to validate the theoretical results. Section 7 is based
on the conclusions and section 8 highlights the future scope of the present work.

II.

MATHEMATICAL MODEL

In this present paper, we assume the presence of randomly fluctuating driving forces on the growth of
the species Si , i  1, 2,3, 4 at time ‘ t ’ of a conventional syn-eco system. The table (2.1) exhibits some
of the real examples of the present syn-eco system. The figure (2.1) represents the system where four
species are living together with the following suppositions: (i) The system comprises of a prey ( S1 ),
predator ( S 2 ) two hosts S3 and S 4 (ii) S1 is prey of S 2

(ii) S1 is commensal of S3 (iii) S 2 is

predator of S1 (iv) S 2 is commensal of S 4 (v) S3 is host of S1 and (vi) S 4 is host of S 2 which
results the following stochastic system with ‘additive noise’. Let x(t ) , y (t ) , z (t ) and w(t ) be the
population densities of species S1 , S 2 , S3 and S 4 respectively at time instant ‘ t ’. Let a1 , a2 , a3 and

a4 be the natural growth rates of species S1 , S 2 , S3 and S 4 respectively. Keeping these in view and
following [26-29], the dynamics of the stochastic system may be governed by the following nonlinear
differential equations:

dx
(2.1)
 a1 x  a11 x 2  a12 xy  a13 xz  q1E1 x  1 1 (t )
dt
dy
(2.2)
 a2 y  a22 y 2  a21 xy  a24 yw  q2 E2 y  2 2 (t )
dt
dz
(2.3)
 a3 z  a33 z 2  3 3 (t )
dt
dw
(2.4)
 a4 w  a44 w2   4 4 (t )
dt
In the above model a11 , a22 , a33 and a44 are self inhibition coefficients of species S1 , S 2 , S3
and S 4 respectively. a12 is the interaction coefficient of S1 due to S 2 , a21 is the interaction
coefficient of S 2 due to S1 , a13 is coefficient of commensal for S1 due to the host S3 , a24 is the
coefficient of commensal for S 2 due to the host S 4 , K1 , K 2 , K 3 and K 4 are the carrying capacities
of species S1 , S 2 , S3 and S 4 respectively, where K1  a1 / a11 ; K2  a2 / a22 ; K3  a3 / a33 ,

K4  a4 / a44 . q1 , q2 are the catchability coefficients of species S1 , S 2 respectively. E1 , E2 are the
efforts applied to harvest the species S1 , S 2 respectively. 1 ,  2 ,  3 and  4 are real constants and

  t    1 (t ), 2 (t ), 3 (t ), 4 (t ) is a four dimensional Gaussian white noise process agreeable

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International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
E  i  t   0; i  1, 2,3, 4

(2.5)

E  i  t  j  t    ij  t  t   ; i  j  1, 2,3, 4

(2.6)

where  ij and  are Kronecker and Dirac delta functions respectively. In addition to the variables

x , y , z , w the model parameters a1 , a2 , a3 , a4 , a11 , a22 , a33 , a44 , a12 , a21 , a13 , a24 are alleged
to be non negative constants.

Figure 2.1: represents schematic idea of the four species living together
Table 2.1 represents some real examples of the system (2.1)-(2.4)

III.

Example

S1

S2

S3

S4

1

Beetles

Small fish

phytoplankton

Zooplankton

2

Small bird

Man

Cow

Dog

3

Bugs

Insects

Rabin bird

Squirrel

4

Rabbit

Wolf

Bushes

Shelter-tree

5

Goat

Tiger

E-coli

Soil bacteria

STABILITY ANALYSIS OF DETERMINISTIC MODEL

In the absence of randomly fluctuating driving forces on the growth of the species, the model system
(2.1)-(2.4) reduces to

dx
 x  a1  q1E1    a11 x  a12 y  a13 z  
dt
dy
 y  a2  q2 E2    a22 y  a21 x  a24 w  
dt
dz
 z  a3  a33 z 
dt
dw
 w  a4  a44 w
dt
Throughout our study let us suppose that
a1  q1E1  0 and a2  q2 E2  0
3.1 Steady States

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(3.1)
(3.2)
(3.3)
(3.4)
(3.5)

Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
In this section, we present the basic outcomes on the nonnegative equilibriums of the model (3.1)(3.4) namely L0 (0, 0, 0) , L1 ( x , y ,0,0) , L2 ( x , y , z ,0) and L3 ( x* , y* , z* , w* ) which are attained
by solving x  y  z  w  0
Case (i): L0 (0, 0, 0) : The population is extinct and this always exists.
Case (ii): L1 ( x , y ,0,0) : Here x and y are positive solutions of x  0 and y  0 . We get

y

1
a  a  q E   a  a  q E 
 a12 a21  a22 a11   21 1 1 1 11 2 2 2 

(3.1.1)

1
a  a  q E   a  a  q E 
 a12 a21  a22 a11   22 1 1 1 12 2 2 2 
a
a  q2 E2
For x to be positive, we must have 22  2
a12 a1  q1 E1
x

(3.1.2)
(3.1.3)

Case (iii): L2 ( x , y , z ,0) : Here x , y and z  are positive solutions of x  0 y  0 and z  0 . We
get x 

1

 a22 a3a13  a33a22  a1  q1E1   a33a12  a2  q2 E2  
a33  a12 a21  a22 a11  
1

 a21a3a13  a33a21  a1  q1E1   a33a11  a2  q2 E2 
a33  a12 a21  a22 a11  
a
z  3
a33
a
a  q2 E2
For x to be positive, we must have 22  2
a12 a1  q1 E1
y 

(3.1.4)
(3.1.5)
(3.1.6)
(3.1.7)

Case (iv): L3 ( x* , y* , z* , w* ) (The interior equilibrium): Here x , y  , z  and w are positive solutions
of x  0 , y  0 , z  0 and w  0 .

a3
a33
a
w  4
a44

We get z  

(3.1.8)
(3.1.9)

x* 

 a44 a13a3a22  a33a44 a22  a1  q1E1   a12 a33a24a4 
1


a33a44  a12 a21  a22 a11  
 a12 a33a44  a2  q2 E2  

(3.1.10)

y* 

 a44 a13a3a21  a11a33a24 a4  a33a44 a21  a1  q1E1  
1


a33a44  a12 a21  a22 a11  
 a33a44 a11  a2  q2 E2  

(3.1.11)

For x* to be positive we must have the following:

a3 a12 a33a24

a4 a44 a13a22
a22 a2  q2 E2

a12 a1  q1 E1

(3.1.12)
(3.1.13)

The similar work has been carried out by ChaoLiu [30].
3.2. Local Stability
We now analyze the local stability of the interior steady state [31]. The Variational matrix of the
system (3.1)-(3.4) at L3 ( x* , y* , z* , w* ) is

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Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963

 a11 x*

a y*
J   21
 0

 0

a12 x*

a13 x*

a22 y
0
0

0
a3
0

*

0 

a24 y* 
0 

a4 

(3.2.1)

The characteristic equation of (3.2.1) is in the form of  4  A 3  B 2  C  D  0 (3.2.2)
where A  a11 x  a3  a4  0 , B  a11a22 x y  a11a3 x* y   a11a4 x y   a3a4  0 ,

C  a11a22 a4 x y  a11a22a3 x* y  a21a3 x y   a21a4 x y   0 ,
D  a11a22 a3a4 x y  a21a3a4 x* y*  0 .
The system is locally asymptotically stable if all the eigen values of the above characteristic equation
have negative real parts. By Routh-Hurwitz criteria, it follows that all eigen values
of (3.2.2) have negative real parts if and only if A  0, C  0, D  0, C ( AB  C )  A2 D ,

D( ABC  A2 D  C 2 )  0 . Hence L3 ( x* , y* , z* , w* ) is locally asymptotically stable.
3.3. Global Stability
Now we discuss the global stability [32] of the equilibrium points

L1 ( x , y ,0,0) and

L3 ( x* , y* , z* , w* ) of the system (3.1)-(3.4).
Theorem (3.3.1): The Equilibrium point L1 ( x , y ,0,0) is globally asymptotically stable.
Proof: let us consider the following Lyapunov function


 y 

 x 
V ( x, y )   x  x  x ln     l1  y  y  y ln   
 x 

 y 

Differentiating V w.r.to ‘ t ’ we get
 y  y  dy
dV  x  x  dx

 ;
  l1 
dt  x  dt
 y  dt
dV
2
2
  a11  x  x   l1a22  y  y    l1a21  a12  x  x  y  y 
dt



By choosing l1  a12 / a21 , dV / dt    a11  x  x  
2



a22 a12
2
y  y < 0
a21


Hence the equilibrium point L1 ( x , y ,0,0) is globally asymptotically stable.
Theorem (3.3.2): The interior equilibrium point L3 ( x* , y* , z* , w* ) is globally asymptotically stable
2
2
if 4a21a22  a12a24
and 4a11  a13
.

Proof: To find the condition for global stability at L3 ( x* , y* , z* , w* ) , we construct the Lyapunov
function

V ( x, y, z, w)  ( x  x* )  x* ln( x / x* )   l1 ( y  y* )  y* ln( y / y* )   l2 ( z  z * )  z * ln( z / z * ) 

l3 (w  w* )  w* ln(w / w* ) 
where l1 , l2 and l3 are positive constants.

 dV / dt    x  x*  / x   dx / dt   l1  y  y*  / y   dy / dt   l2  z  z*  / z   dz / dt 

















l3  w  w* / w  dw / dt  ;





(dV / dt )  x  x* a11 x  x*  a12 y  y*  a13 z  z * 

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©IJAET
ISSN: 22311963






















 l1 y  y* a22 y  y*  a21 x  x*  a24 w  w* 
l2 z  z* a33 z  z*   l3 w  w* a44 w  w*  ;
a
1
1
By choosing l1  12 ; l2 
; l3 
a21
a33
a44



 a x  x*
11
(dV / dt )   



  X T AX
 a11

*
 xx 
 0

*

y y 
where X  
;
A

 a
 z  z* 
 213


*

 w  w 
 0




2










 z  z    a a / a   y  y 


  a a / a   y  y  w  w    z  z    w  w  

 a13 x  x*

*

*

12 22

*

12 24

0
a12 a22
a21


 a13
2

0

0

1

a12 a24
2a21

0

2

21

*

*

2

*

2

21



a12 a24 

2a21 

0 

1 

0

The system is globally stable if the derivative of Lyapunav’s function V is negative definite, that is if
the matrix A is positive definite, that is if the principal minors of A (say) M i , i  1, 2,3, 4 are
2
2
positive. The principle minors are positive if 4a21a22  a12 a24
and 4a11  a13
. Hence the system is
globally stable in the above parametric domain.

3.4. Bionomic Equilibrium
It is the combination of biological balance and economic balance. In section (3.1), we have conversed
about the biological balance which is given by x  y  z  w  0 . When the total profit obtained by
selling the yielded biomass equals the total cost utilized in yielding it, then we say that the bionomic
balance achieved. Let c1 be the constant harvesting cost of species S1 per unit effort and c2 be the
constant harvesting cost of species S 2 per unit effort. Let p1 be the constant price of species S1 per
unit biomass and p2 be the constant price of species S 2 per unit biomass. The revenue at any time is
given by
A  x, y, z, w, E1 , E2    p1q1 x  c1  E1   p2q2 z  c2  E2
(3.4.1)
Now if c1  p1q1 x and c2  p2 q2 y , then the economic rent obtained from the fishery becomes
negative and the fishery will be closed. Hence for the existence of bionomic equilibrium, it is
assumed that c1  p1q1 x and c2  p2 q2 y .
(3.4.2)
The bionomic equilibrium  ( x) ,( y) ,( z) ,( w) ,( E1 ) ,( E2 )  is the positive solution of

x  y  z  w  A  0.

(3.4.3)

By solving (3.4.3) we get,

( x)  c1 /( p1q1 );
( y)  c2 /( p2 q2 );
( z )  a3 / a33 ;
(w)  a4 / a44 ;

(3.4.4)
(3.4.5)
(3.4.6)
(3.4.7)

( E1 )  (1/ q1 )  a1  (a11c1 ) /( p1q1 )  (a12c2 ) /( p2 q2 )  (a13a3 ) / a33 

 E2   (1/ q2 ) a2  (a22c2 ) /( p2q2 )  (a21c1 ) /( p1q1 )  (a24a4 ) / a44 
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(3.4.8)
(3.4.9)

Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
( E1 )  0 when  a1  (a13a3 ) / a33 ]  [(a11c1 ) /( p1q1 )  (a12c2 ) /( p2q2 )  .

( E2 )  0 when  a2  (a21c1 ) /( p1q1 )  (a24 a4 ) / a44 ]  [(a22c2 ) /( p2 q2 )  .

(3.4.10)
(3.4.11)

If ( E1 )  ( E1 ) and ( E2 )  ( E2 ) , then the total cost utilized in harvesting the species population
would exceed the total revenues obtained from the ecological system. Hence some people would be in
loss and naturally they would withdraw their participation from the system.
Hence ( E1 )  ( E1 ) and ( E2 )  ( E2 ) cannot be maintained indefinitely.
If ( E1 )  ( E1 ) and ( E2 )  ( E2 ) , then the ecological system is more profitable, and hence in an
open access system, it would attract more and more people. This will have an increasing effect on the
yielding effort. Hence ( E1 )  ( E1 ) and ( E2 )  ( E2 ) cannot be continued indefinitely.

IV.

STOCHASTIC ANALYSIS

In this section, we compute the population intensities of fluctuations (variances) of the system (2.1)(2.4) around the positive equilibrium L3 ( x* , y* , z* , w* ) due to noise, according to the method
introduced by Nisbet and Gurney [29] in 1982. The method was successfully applied by Prasenjit Das
[33] and M.N. Srinivas [34]. Now we assume the presence of randomly fluctuating driving forces on
the deterministic growth of the species Si , i  1, 2,3 of the system (3.1)-(3.4) at time ‘ t ’ which results
in the stochastic system (2.1)-(2.4) with ‘additive white noise’ process satisfying (2.5) and (2.6). Let
us consider the perturbation technique as follows:
Let x(t )  u1 (t )  S * ; y(t )  u2 (t )  P* ; z (t )  u3 (t )  T * ; w(t )  u4 (t )  U * ;
(4.1)

dx du1 (t ) dy du2 (t ) dz du3 (t ) dw du4 (t )

; 
; 
;

;
dt
dt
dt
dt
dt
dt
dt
dt

(4.2)

Using (4.1) and (4.2) in (2.1)-(2.4), we identify the respective linear system as

du1 (t )
 a11u1 (t )S *  a12u2 (t )S *  a13u3 (t )S *  1 1 (t )
dt
du2 (t )
  a22u2 (t ) P*  a21u1 (t ) P*  a24u4 (t ) P*   2 2 (t )
dt
du3 (t )
  a33u3 (t )T *  3 3 (t )
dt
du4 (t )
  a44u4 (t )U *   4 4 (t )
dt

(4.3)
(4.4)
(4.5)
(4.6)

Using Fourier transform methods on the linear system (4.3) - (4.6), we get
*
*
*

1 1 ( )  (i  a11S )u1 ( )  a12 S u2 ( )  a13 S u3 ( )
2 2 ( )  a21P*u1 ()  (i  a22 P* )u2 ()  a24 P*u4 ()
3 3 ( )  (i  a33T * )u3 ( )

4 4 ( )  (i  a44U )u4 ( )
*

(4.7)
(4.8)
(4.9)
(4.10)

The above system (4.7) - (4.10) can be represented in the matrix form as

A    u      

 A11 ( )

A21 ( )
where A    
 A31 ( )

 A41 ( )

1577

(4.11)

A12 ( )

A13 ( )

A22 ( )
A32 ( )
A42 ( )

A23 ( )
A33 ( )
A43 ( )

A14 ( ) 
 u1 ( ) 
 1 1 ( ) 



   ( ) 
A24 ( ) 
u ( ) 
;
; u     2
;      2 2
u3 ( ) 
 3 3 ( ) 
A34 ( ) 





A44 ( ) 
u4 ( ) 
  4 4 ( ) 

Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963

A11 ( )  (i  a11S * ); A12 ( )  a12 S * ; A13 ( )  a13 S * ; A14 ( )  0;
A21 ( )  a21 P* ; A22 ( )  (i  a22 P* ); A23 ( )  0; A24 ( )  a24 P* ;
A31 ( )  0; A32 ( )  0; A33 ( )  (i  a33T * ); A34 ( )  0;

(4.12)

A41 ( )  0; A42 ( )  0; A43 ( )  0; A44 ( )  (i  a44U * );
1

u     A     

Equation (4.11) can also be written as
Let  A  

1

 B( ) then B( ) 

Adj A  
A  

and u    B( )  

(4.13)

where A( )  R( )  iI ( )

R( )   4   2 a33a44T *U *   2 a22 a44 P*U *   2 a22 a33 P*T *   2 a11a44 S *U *
 2 a11a33 S *T *   2 a11a22 S * P*   2 a12 a21S * P*  a11a22 a33a44 S * P*T *U *
 a12 a21a33a44 S P T U
*

*

*

(4.14)

*

I ( )   3a11S *   3a22 P*   3a33T *   3a44U *   a22 a33a44 P*T *U *
 a11a33a44 S *T *U *   a11a22 a44 S * P*U *  a11a22 a33 S * P*T *
 a12 a21a33 S * P*T *   a12 a21a44 S * P*U *
We now depict some of the necessary preliminaries of the random population function. If the function
Y (t ) has a zero mean value, then the fluctuation intensity (variance) of its components in the
frequency interval ,   d  is SY ( )d , where SY ( ) is spectral density of Y and is defined as

Y  

SY ( )  lim

T 

2

(4.15)

T

If Y has a zero mean value, the inverse transform of SY ( ) is the auto covariance function

1
CY ( ) 
2




 S  e d
i

(4.16)

Y



The corresponding variance of fluctuations in Y (t ) is given by

 Y 2  CY (0) 

1
2



S

Y

( )d

(4.17)



and the auto correlation function is the normalized auto covariance PY ( ) 

CY ( )
CY (0)

(4.18)

For a Gaussian white noise process, it is

E  i   j   
T 


S i j    ˆlim

1
T  Tˆ

 ˆlim


2


2

  E   t   t  e
i



2

1578

dt dt    ij

(4.19)



2

From (4.13), we have ui   
From (4.14) we have Sui   

 i ( t t  )

j

4


j 1

4

 B     , i  1, 2,3, 4
j 1

ij

j

Bij   , i  1, 2,3, 4
2

j

(4.20)
(4.21)

Vol. 6, Issue 4, pp. 1571-1584

International Journal of Advances in Engineering & Technology, Sept. 2013.
©IJAET
ISSN: 22311963
Hence by (4.17) and (4.21), the intensities of fluctuations of the variables ui , i  1, 2,3, 4 are given

1
2

by  ui 2 

4




j 1 

2

j

Bij ( ) d; i  1, 2,3, 4

(4.22)

That is, the variances of ui , i  1, 2,3, 4 are obtained as

u 2 
1

u

2
2

u

2

3

1

2







2
2
2
2



B
(

)
d



B
(

)
d



B
(

)
d



B
(

)
d

  1 11
;
2
12
3
13
4
14
















2
2
2
2


  1 B21 ( ) d    2 B22 ( ) d   3 B23 ( ) d    4 B24 ( ) d  ;













1 
2
2
2
2



  1 B31 ( ) d    2 B32 ( ) d   3 B33 ( ) d    4 B34 ( ) d  ;
2 







u 2 
4

1
2






1 
2
2
2
2



B
(

)
d



B
(

)
d



B
(

)
d


 4 B44 ( ) d 
  1 41
2
42
3
43



2 







(4.23)

X mn  iYmn
where Bmn ( ) 
; m, n  1, 2,3, 4
R( )  iI ( )
X11   2 a22 P*   2 a33T *   2 a44U *  a22a33a44 P*T *U * ;
Y11   3  a22 a33 P*T *  a33a44T *U *  a22 a44 P*U * ; X12   2 a12 S * ;
Y12  a12 a33 S *T *  a12 a44 S *U * ; X13   2 a13 S *  a13a22 a44 S * P*U * ;
Y13  a13a22 S * P*  a13a44 S *U * ; X14  a12 a24 a33 S * P*T * ; Y14  a12 a24 S * P* ;
X 21   2 a21P*  a21a33a44 P*T *U * ; Y21  a21a33 P*T *  a21a44 P*U * ;
X 22   2 a11S *   2 a33T *   2 a44U *  a11a33a44 S *T *U * ;
Y22   3  a33a44T *U *  a11a44 S *U *  a11a33S *T * ; X 23  a13a21a44 S * P*U * ;

Y23  a13a21S * P* ; Y24  a11a24 S * P*  a24a33 P*T * ; Y31  0; X 32  0; Y32  0;
X 33   2 a11S *   2 a22 P*   2 a44U *  a11a22a44 S * P*U *  a12a21a44 S *P*U * ;
Y33   3  a22 a44 P*U *  a11a44 S *U *  a11a22 S * P*  a12a21S * P* ;

X 34  0; Y34  0; X 41  0; Y41  0; X 42  0; Y42  0; X 43  0; Y43  0;
X 44   2 a11S *   2 a22 P*   2 a33T *  a11a22a33S * P*T *  a12a21a33S * P*T * ;
Y44   3  a22 a33 P*T *  a11a33S *T *  a11a22 S * P*  a12a21S *P* ;
Thus (4.23) becomes

u

1

1579

2












 1 X 112  Y112   2 X 12 2  Y12 2
1 
1


 2
2 
R ( )  I 2 ( ) 
 3 X 132  Y132   4 X 14 2  Y14 2







 
 d 
;
 
 

Vol. 6, Issue 4, pp. 1571-1584


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