View of Influence of Noise Effect on a Peculiar Ecostystem

Influence of Noise Effect on a Peculiar Ecostystem

*

K.V.L.N.Acharyulu

¹

, G.Basava Kumar

² ,

K.Nagamani

³

& I.Rajeswari

⁴

1,2,3,4

Faculty, Department of Maths, Bapatla Engineering College (Autonomous)

Bapatla-522102, Andhra Pradesh, India.

(

^*

Corresponding Author:

[email protected] or [email protected])

ABSTRACT

The paper intends to investigate a peculiar ecosystem. It contains Ammensal and Enemy species with limited resources. Mortality and immigration are both applied on Ammesnal Species. Global stability is identified by choosing suitable Liapunov’s function. Stochastic Analysis has been employed. Series solutions are provided with the help of Homotopy perturbation method.

Key words:

Ammensal, Enemy Species, Stability, Local Stability, HPM, global stability, Gaussian white noise.

1. Introduction

Mathematical Modelling may be viewed as a interdisciplinary concept that deals with the relationship of mathematics and other disciplines. An analytical practice deals with many facets of the everyday environment. The phases are described as i).Defining the issue in the real world particularly in the biological or medical or social sense.(ii) prediction model formulation. (iii) Solving math problems that can emerge when evaluating the model.(iv) Development of analysis techniques and similar computer programmes for the computations involved.(v) To clarify and to see the outcomes in the context of the original issue and to convey this knowledge to all needy people. With an effective procedure of knowledge processing, mathematical models have been valuable methods in biological investigations. If these models are built and used appropriately, they can give insight into the interactions between the physical factors and the mechanism that influences the structure being examined. A major component in the design of experiments and in the analysis of results may be the subsequent interplay between experimental inquiry and the theoretical model.

There are two kinds of mathematical structures in general: Deterministic and Stochastic. The formulation of the processes in deterministic models relies on various axioms / hypotheses to be considered due to the relevant system biology and these may be provided in the form of autonomous or non- autonomous, ordinary / partial (linear or nonlinear) differential or integro-differential equations. In general, stochastic models are probabilistic models. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu[5-18]

examined local and global ecosystem equilibrium with multiple dimensions. In the earlier work, local stability was conducted for various Ammensal-Enemy eco-systems with diverse tools. The present investigation focuses primarily on establishing the global stability, Local stability and series solution and the authors investigated various ecological models for their stability. Many Research scholars [1-4] and Mathematicians [19-31] extended their significant contributions to this modelling field.

1.1 Notations:

This is an evolutionary environment where Ammensal and Enemy species live together. It is believed that all interacting ecological species are continuously harvested (migrated or immigrated) by depending upon available natural resources. Here the Ammensal species is effected by mortality and strengthened by harvesting

(i).X represents the density of Ammensal species at natural growth rate a

1.

(ii).Y stands for the density of the Enemy species, (iii).h

1

(=a

11

H

1

) is the harvesting of Ammensal species, (iv).

_i

(

ⁱ

)

ii

K a

= a be the carrying capacity of Ammensal Species .

(v).

¹²

11

( a )



= a

be the Ammensalism's coefficient.

Assume that the parameters described above are positive.

2. Constriction of Mathematical Model

The rate of growth equation for the Ammensal species with constant rates of mortality and harvesting:

2

11

(

1



1

)

= − − − +

dX a K X X XY H

dt (2.1)

The rate of growth equation for the Enemy species :

22

(

2

)

dY a Y K Y

dt = − (2.2)

In this model , the interior point is obtained as

2

1 1 2 2 1

* * * *

4 2

4 ( ) ( )

( , ) where ,

2

H K K K K

E x y x + +  −  + y K

= =

3. Global Stability of The System by Lyapunov Property Liapunov has developed a valuable tool to efficiently assess global stability.

Theorem (4.1): The constituted special ecosystem (2.1)-(2.2) is globally asymptotically stable at the positive equilibrium(x

^*

,y

^*

).

Proof: Now construct suitable Liapunov function to address the global stability at interior equilibrium

* *

4( , )

E x y

( ) ( ⁽

^*

⁾

^*

^{(ln ln}

^*

⁾ ) (

¹

⁽

^*

⁾

^*

^{(ln ln}

^*

⁾ )

V t = x − x − x x − x + l y − y − y y − y , l

₁

 0 (3.1)

* *

1

dV x x dx y y dy

dt x dt l y dt

 −   − 

=  +  

   

 

* *

1

11 ( 1 H ) 1 22 ( 2 )

x x y y

a x k x y l a y k y

x



x y

 −    − 

=  − − − + +   −

* 2

* 2 * 2 * *

11 1 * 11

( )

( ) ( ) ( x x ) ( )( )

a x x y y H a y y x x

xx 

= − − − − − − − − −

( )

* 2

* 2 * 2 11 * 2 * 2

11 1 *

( )

( ) ( ) ( ) ( ) ( )

2

dV x x a

a x x y y H y y x x

dt xx



 − − − − − − − − + −

* 2 1 11 * 2 11

11 *

( ) ( ) 1

2 2

H a a

x x a y y

xx

 

   

 − −  + + − −  + 

* 2 1 11 * 2 11

11 *

( ) ( ) 1 0

2 2

dV H a a

x x a y y

dt xx

 

     

 −   −   + +   + −   +     

Provided

₁₁ ¹_* ¹¹ 0 1 ¹¹ 0

2 2

H a a

a and

xx

 

 + +   + 

   

   

The condition V (t) < 0 hol ds.hence the non- diffusive system asymptotis ically stable.

4. Stochastic Analysis of The Model

Stochastic analysis is based on the statistical calculus. This calculation was developed in order to resolve issues resulting from the theory of probability in which systems are driven along paths that cannot usually be separated. Stochastic analysis is a fundamental method of many modern probability theories with statistical inferences and is used in many fields of application from biology to physics.

The equations of special ecosystem with noise effect on (2.1)-(2.2) are

1 2

11

(

1 1

)

1(t)

dX a K X X XY H

dt = − − −  + +   (4.1)

22

(

2

)

2 2(t)

dY a Y K Y

dt = − +   (4.2)

here  

₁, ₂

stands for real constants , the Gaussian white noise effect : 

_i( )t ⁼[ ( ),



₁ t



₂( )]t

is in a two dimensional system with the conditions ( ( )) E 

_i

t = 0; i = 1, 2;

( ); 1, 2

v =  

ij

t t −  i = = j where 

_ij

is the Kronecker delta function;  is the Dirac –delta function.

By the concept of

Nisbet and Gurney [21],

Gaussian white noise effect at the interior equilibrium point

* *

4( , )

E x y is discussed by taking perturbations

* *

1 2

( ) ( ) and ( ) ( )

X t = u t + S Y t = u t + P

Hence, the model (4.1)-(4.2) reduces to the following linear system and The linear part of the system (4.1)-(4.2) is

1 1 *

11

(

1 2

)

1(t)

du a S u u

dt = − + +   (4.3)

2 2 *

22 2 2(t)

du a P u

dt = +   (4.4)

Taking the Fourier transform of (4.3) and (4.4) we get,

(

^*

)

^*

1 1

( ) i a S u

11 1

( ) a S u

22 2

( )

  

%

=  +

%

 +

%

 (4.5)

(

^*

)

2 2

( ) i a P u

22 2

( )

  

%

=  −

%

 (4.6)

Now, represent (4.5) and (4.6) in a standard matrix form as

M

( ) ( )

 u% = %

( ) (4.7)

where ( ) ^{( ) ( )}

( ) ( )

A B

M C D

 

  

 

=  

  ; ( )

¹

2

( ) ( ) u u

u

 



 

=  

 

% %

%

; ( ) ( ) ( )

1 1 2 2

  

    

 

=  

 

 

% %

%

;

* * *

11 22 22

( ) ; ( ) ; ( ) 0; ( )

A



=i



+a S B



=a S C



= D



=i



−a P

(4.8)

Hence the solution of (4.7) is given by

^u%

( )

^{ =K( )  %}

( ) ^,

where

^{K( )}

^

_{= }^M

( ) ^

^_⁻¹

^(4.9)

The solutions of (4.9) are given by

( )

²

( ) ( )

1

; 1, 2

i ij j

j

u



K

  

i

=

=



^% =

%

(4.10)

The spectrum of u i

_i

, = 1, 2 are given by ( )

²

( )

²

1

; 1, 2

ui j ij

j

S

 

K



i

=

=



=

Intensities of fluctuations of the component u

_i

, i = 1, 2 are provided by

2 2

2

1

1 ( ) ; 1, 2

2

ui j ij

j

K d i

   





= −

=   =

From (4.10), we obtain

1

2 2

2

1 2

1 ( ) ( )

2 ( ) ( )

u

D B

d d

M M

 

    

  

 

− −

 

 

=  + 

 

   

2

2 2

2

1 2

1 ( ) ( )

2 ( ) ( )

u

A C

d d

M M

 

    

  

 

− −

 

 

=  + 

 

   

where

M( )



=R( )



+iI( )

 ;

R

( ) 

= −(



²+a a S P11 22 ^{* *})

;

I

( ) 

=



(a S11 ^*−a P22 ^*)

If we take into consideration the noise effect on one of the species 

₁

= 0 or 

₂

= 0 then we have

1 2

* 2

2 2 22 2

1 2 2

( ) 1

0 and & 0

2 ( ) ( )

u u

If a S d

R I

    

  



−

= = =

 +

1 2

2 1 2 * 2 2

2 2 2 22

0 & 1 ( ) & 0

2 ( ) ( )

u u

If a P d

R I

     

  



−

 

= =  +  +  =

Population variances suggest population stability with smaller mean square fluctuations, whereas greater population variance values indicate population instability.

5. Series Solutions by Homotopy Perturbation Method (HPM)

HPM is a effective and valuable technique for discovering series solutions of non linear equations without a linearization procedure. He first implemented the process efficiently. HPM incorporates perturbation and homotopy processes. This approach will take advantage of traditional perturbation method thus avoiding constraints. In general, several mathematicians used this approach successfully to solve all kinds of linear and nonlinear equations in Science, Engineering and Technology.

By the definition of homotopy, the following structure can be designed as

1 1 1 2

1

X

0

( X

0

a

1 1

a

11 1

a

12 1 2

a H

11 1

) 0

 − +  +  +  +   − =

1 1 1 2

2

Y

0

( Y

0

a

2 2

a

22 2

) 0

 − +  −  +  =

( ) ( )

1 1,0 t and 2 2,0

Assume



=

 

=



t

(5.1)

The first approximations are taken into account as

1,0

( ) t

1

(0) X t

0

( )

1

and

2,0

( ) t

2

(0) Y t

0

( )

2

 =  = =   =  = =  (5.2)

2 3 4 5

1( )t 1,0( )t 1,1( )t 1,2( )t 1,3( )t 1,4( )t 1,5( ) ...t



=



+

 

+

 

+

 

+

 

+

 

+

(5.3)

2 3 4 5

2( )t 2,0( )t 2,1( )t 2,2( )t 2,3( )t 2,4( )t 2,5( ) ...t



=



+

 

+

 

+

 

+

 

+

 

+

(5.4)

Here 

_{i j},

(

i=1, 2 ; j=1, 2,3...

) ,to be decided by the substitution of(5.1),(5.2),(5.3)&(5.4) Now comparing the coefficient of various powers of  in the above approximations After simplification, various coefficients are obtained as below

1

: The coefficient of 

1 2

1,1 ( )t a1 1,0( )t a11 1,0( )t a12 1,0( )t 2,0( )t a H11 1 0



+



+



+

 

− =

1 2

2,1( )t a2 2,0( )t a22 2,0( )t 0



−



+



=

2: The coefficient of



1

1,2 ( )t a1 1,1( )t 2a11 1,0( )t 1,1( )t a12 1,0( )t 2,1( )t a12 1,1( )t 2,0( )t 0and



+



+

 

+

 

+

 

=

1

2,2 ( )t a2 2,1( )t 2a22 2,0( )t 2,1( )t 0



−



+

 

=

3: The coefficient of



1 2

( )t a ( )t 2a ( )t ( )t a ( )t



+



+

 

+



12 1,0

( )

2,2

( )

12 1,1

( )

2,1

( )

12 1,2

( )

2,0

( ) 0 a  t  t a  t  t a  t  t and

+ + + =

1 2

2,3( )t a2 2,2( )t 2a22 2,0( )t v2,2( )t a22 2,1 ( )t 0



−



+



+



=

4: The coefficient of



1

1,4 ( )t a1 1,3( )t 2a11 1,0( )t 1,3( )t 2a11 1,1( )t 1,2( )t



+



+

 

+

 

12 1,0

( )

2,3

( )

12 1,1

( )

2,2

( )

12 1,2

( )

2,1

( )

12 1,3

( )

2,0

( ) 0 a v t v t a v t v t a v t v t a v t v t and

+ + + + + =

1

2,4 ( )t a2 2,3( )t 2a22 2,0( )t 2,3( )t 2a22 2,1( )t 2,2( )t 0



−



+

 

+

 

=

Now

_{1,1 1 1,0 11} ²_{1,0 12 1,0 2,0 11 1}

0 0 0 0

( ) ( ) ( ) ( ) ( )

t t t t

t a t dt a t dt a t t dt a H dt



= −

 

−

 

−

  

+



( )

²

1,1 t ( a1 1 a11 1 a12 1 2 a H t11 1)

    

 = − − − +

2

2,1 2 2,0 22 2,0

0 0

( ) ( ) ( )

t t

t a t dt a t dt



=

 

−

 

2 2,1( )t (a2 2 a22 2 )t

  

 = −

( )

1,2 1 11 1 12 2 1,1 12 1 2,1

0 0

( ) 2 ( ) ( )

t t

t a a a t dt a t dt

 = − −  −    −   

( )

^{2 2 2}

1,2( ) 1 2 11 1 12 2 ( 1 1 11 1 12 1 2 11 1) 12 1( 2 2 22 2)

2

t a a a a a a a H a a a t

           

 = − − − − − − + − − 

( )

2,2 2 22 2 2,1

0

( ) 2 ( )

t

t a a t dt

 = −   

( ) ( )

^{2 2}

2,2

[

2

2

22 2

(

2 2 22 2

)]

2

t a a a a t

   

 = − −

1,3 1 1,2 11 1,0 1,2 11 ²1,1

0 0 0

( ) ( ) 2 ( ) ( ) ( )

t t t

t a t dt a t t dt a t dt



= −

 

−

  

−

 

12 1,0 2,2 12 1,1 2,1 12 1,2 2,0

0 0 0

( ) ( ) ( ) ( ) ( ) ( )

t t t

a  t  t dt a  t  t dt a  t  t dt

−  −  − 

( )

1,3 1 11 1 12 2 1,2

0

( ) 2 ( )

t

t a a a t dt

   

 = − − −



11 1,1 12 2,1 1,1 12 1 2,2

0 0 0 0

( ) ( ) ( ) ( )

t t t t

a



t dt a



t dt



t dt

  

t dt

 

− +  −



 



 

( ) ( )



²

1,3( )t a1 2a11 1 a12 2 a1 2a11 1 a12 2 ( a1 1 a11 1 a12 1 2 a H11 1)



^

 



     

 =  − − −  − − − − − − +

2 2 2

12 1

(

2 2 22 2

)

11

(

1 1 11 1 12 1 2 11 1

)

12

(

2 2 22 2

)

a  a  a    a a  a  a   a H a a  a  

− −   − − − − + + − 

( ) 

³

2 2

1 1 11 1 12 1 2 11 1 12 1 2 22 2 2 2 22 2

( ) 2 ( )

6 a  a  a   a H a  a a  a v a  t

− − − + − − −

( )

2,3 , 1,4(

Similarly



t



t) are calculated

The approximations of 4-terms are adequate, hence, we have

4

2 3 4

1 1, 1,0 1,1 1,2 1,3 1,4

1 0

( ) lim ( )

x

( ) ( ) ( ) ( ) ( ) ( )

x

X t t t t t t t t



          

→ =

= =  = + + + +

4

2 3 4

2 2, 2,0 2,1 2,2 2,3 2,4

1 0

( ) lim ( )

x

( ) ( ) ( ) ( ) ( ) ( )

x

Y t t t t t t t t



          

→ =

= =  = + + + +

The series solutions are derived with the help of HPM as

2

1 1 1 11 1 12 1 2 11 1

( ) ( )

X t =  + − a  − a  − a   + a H t

(

1

2

11 1 12 2

) (

1 1 11 1² 12 1 2 11 1

)

12 1

(

2 2 22 2²

)

²

2 a a  a  a  a  a   a H a  a  a   t

+ − −   − − − − + − − 

( )

 a

1

2 a

11 1

 a

12



2

 ( a

1

2 a

11 1

 a

12



2

) ( a

1 1

 a

11 1



²

a

12 1 2

  a H

11 1

)

+ − − −  − − − − − − +

2 2

12 1

(

2 2 22 2

)

11

(

1 1 11 1 12 1 2 11 1

)

a  a  a   a a  a  a   a H

− −  − − − − +



³

2 2 2

12

(

2 2 22 2

) (

1 1 11 1 12 1 2 11 1

)

12 1

(

2

2

22 2

) (

2 2 22 2

)

6 a a  a   a  a  a   a H  a  a a  a  a  t

− −  − − − +  − − −

( )



a1 2a11 1



a12



2 ^

(

a1 2a11 1



a12



2

) (

 a1 2a11 1



a12



2

)

+ − − −  − − −  − − −

2 2

1 1 11 1 12 1 2 11 1 12 1 2 2 22 2

( − a  − a  − a   + a H ) − a  ( a  − a  )  

2 2

11( 1 1 11 1 12 1 2 11 1) 12( 2 2 22 2 )

a a



a



a

 

a H a a



a



− − − − + − −

2 2

1 1 11 1 12 1 2 11 1 12 1 2 22 2 2 2 22 2

( a  a  a   a H ) a  ( a 2 a  ) ( a  a  )

 − − − +  − − − 

  

2 2

11 1 1 11 1 12 1 2 11 1 12 2 2 22 2

6a ( a a  a   a H ) 3a (a  a  )^

− − − − + − −

 

( a

1

2 a

11



1

a

12



2

)  ( a

1



1

a

11



1²

a

12

 

1 2

a H

11 1

)  a

12 1

 ( a

2



2

a

22



2²

)

 − − − − − − + − −

  

( )

²

2 2

12 1

(

2

2

22 2

) (

2

2

22 2

) (

2 2 22 2

)

22 2 2 22 2

a  ^ a a   a a  a  a   a a  a  ^

−  −  − −  − − 



⁴

2 2

12 2 22 2 2 2 22 2 1 1 11 1 12 1 2 11 1

3 ( 2 )( )( ) ...

24 a a a  a  a  a  a  a   a H t

− − − − − − + +

2

2 2

2 2 2 22 2 2 22 2 2 2 22 2

( ) ( ) ( 2 ) ( )

2 Y t =  + a  − a  t + a − a  a  − a  t

( )

 ^a

²

^{2 a}

²²

^

²

^{ ( a}

²

^{2 a}

²²

^

²

^{) ( a}

²

^

²

^a

²²

^

²²

^{)  a}

²²

^{( a}

²

^

²

^a

²²

^

^{22 2}

⁾  ^t ₆

³

+ −  − −  − −

( ) ( )



^{a2 a22}

^

^{2  a2 2a22}

^

^{2 (a2 2a22}

^

^{2) (a2}

^

^{2 a22}

^

^{22) a22(a2}

^

^{2 a22}

^

^{22 2) }

+ −  −  − − − − 



⁴

2 2

22 2 2 22 2 2 22 2 2 2 22 2

3 ( ) ( 2 ) ( ) ...

24 a a  a  a a  a  a  t

− − − − +

6. Conclusions

Based on the study of Migrated Ammensal Model, the following Conclusions have been observed:

(i).Global Stability is achieved by constructing proper Lyapunov function. The necessary theorems for global stability are established.

(ii).The stochastic Analysis is employed successfully for identifying the impact of smaller mean square fluctuations on the stability.

(iii).The series solutions with possible higher degrees are derived.

Acknowledgments

The authors are grateful to Dr.Komaravolu Chandrasekharan, A great and renounced mathematician from Bapatla, India for his unforgettable and most valuable contributions in the field of Mathematical Research.

The authors are also thankful to Dr.V.Damodara Naidu, Principal, Bapatla Engineering College, Bapatla

and the Management, Bapatla Education Society for their constant encouragement and valuable support.

References

[1]. B.Dubey and J.Hussian (2000), Modelling the interaction of two biological species in a polluted environment, J. Math. Anal. Appl., 246,58–79.

[2]. D.Del-Castillo-Negrete, B.A. Carreras, V. Lynch (2002),Front propagation and segregation in a reaction–

diffusion model with cross-diffusion, Phys. D 168, 45–60.

[3]. G.C. Cruywagen, J.D. Murray, P.K. Maini (1997), Biological pattern formation on two-dimensional spatial domains: a nonlinear bifurcation analysis, SIAM J. Appl. Math. 57 ,1485–1509.

[4]. J.D. Murray (1993), Mathematical Biology, Springer-Verlag, Berlin.

[5]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2010), AnAmmensal-Enemy Specie Pair With Limited And Unlimited Resources Respectively-A Numerical Approach, Int. J. Open Problems Compt.

Math (IJOPCM), 3(1),73-91.

[6]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2010), An Enemy- Ammensal Species Pair With Limited Resources –A Numerical Study, Int. J. Open Problems Compt. Math (IJOPCM), 3(3), 339-356.

[7]. K.V.L.N.Acharyulu and N.Ch. PattabhiRamacharyulu (2011), Mortal Ammensal and an Enemy Ecological Model with Immigration for Ammensal Species at a Constant Rate, International Journal of Bio-Science and Bio-Technology,3(1),39-48.

[8]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2011),An Immigrated Ecological Ammensalism with Limited Resources”- International Journal of Advanced Science and Technology,27(1), 87-92.

[9]. K.V.L.N.Acharyulu and N.Ch.Pattabhi Ramacharyulu (2011), A Numerical Study on an Ammensal - Enemy Species Pair with Unlimited Resources and Mortality Rate for Enemy Species”- International Journal of Advanced Science & Technology,30,13-24.

[10]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2011), An Ecological Ammensalism with Multifarious restraints- A Numerical Study” International Journal of Bio-Science and Bio-Technology, 3(2),1-12.

[11]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2011), Multiple Constraints in Ecological Ammensalism- A Numerical Approach, Int. J. Advance. Soft Comput. Appl., 3(2),1-15.

[12]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2012),On the Carrying capacity of Enemy Species, Inhibition coefficient of Ammensal Species and Dominance reversal time in An Ecological Ammensalism - A Special case study with Numerical approach, International Journal of Advanced Science and Technology, 43,49-58.

[13]. K.V.L.N.Acharyulu , N.Rama Gopal ,P.Prasanna Anjaneyulu and P. Rama Mohan (2012), Global Stability Analysis of A Three Level Ecological Ammensalism with Four Species, International Journal of Mathematical Sciences, Technology and Humanities,1(2),838-847.

[14]. K.V.L.N.Acharyulu, N.Phani Kumar, G.Bhargavi & K.Nagamani, Ecological Harvested Ammensal Model- A Homotopy Analysis, International Journal of Scientific and Innovative Mathematical Research,3(12),27-35.

[15]. K.V.L.N.Acharyulu & N.Phani Kumar (2015), Ecological Ammensalism-A Series Solution By Homotopy Perturbation Method, Acta Ciencia Indica, 41(4), 295-304.

[16]. K.V.L.N.Acharyulu, S.V.Vasavi & SK.Khamar Jahan (2016), A Homotopy Series Solution For A Competitive Ecological Model With Harvesting For Second Species, Acta Ciencia Indica, 42(3), 207-216.

[17]. K.V.L.N.Acharyulu (2017),Homotopy Analysis on Harvested Enemy Species with Variable Rate and Cover Protected Ammensal Species”, Acta Ciencia Indica, 43(3),205-213.

[18]. K.Kalyani, N. SeshagiriRao &K.V.L.N.Acharyulu (2014), A fixed point of self mapping in Hilbert space, Global Journal of Pure and Applied Mathematics , 10(6),783-786.

[19]. Lotka A.J.(1925), Elements of MathematicalBiology, Williams and Wikkins, Baltimore, USA.

[20]. M. Lakshmi Sailaja, K.V.L.N.Acharyulu, N.RamaGopal and P.Rama Mohan (2010), Ecological Ammensal Model With Reserve for One Species and Harvesting both The Species at Variable Rates, International Journal of Advances in Soft Computing and Its Applications, 3(3),264-281.

[21]. Nisbet RM, Gurney WSC.(1982), Modelling Fluctuating Populations. New York : John Wiley.

[22]. N.SeshagiriRao, K.Kalyani and K.V.L.N.Acharyulu (2014),Threshold results for host –Mortal Commensal ecosystem with limited resources, Global Journal of Pure and Applied Mathematics,10(6),787-791.

[23]. N.SeshagiriRao, K.Kalyani and K.V.L.N.Acharyulu (2014), Host-Monad Ecological Model With Phase Plane Analysis, International Journal of Applied Engineering Research, 9(23),21605-21610.

[24]. N.Seshagiri Rao, K.Kalyani and K.V.L.N.Acharyulu (2015), Threshold results of a Host -Mortal commensal ecosystem with a constant harvesting of the Commensal species, ARPN Journal of Engineering and Applied Sciences,10(2),802-805.

[25]. N.SeshagiriRao, K.V.L.N.Acharyulu and K.Kalyani (2015), Numerical study on ecological commensalism between two species with harvested Commensal, ARPN Journal of Engineering and Applied Sciences, 10(4),1548-1551.

[26]. N.Seshagiri Rao, K.V.L.N.Acharyulu & K.Kalyani (2015) , A Host-Mortal Commensal Species Pair With Limited Resources - A Numerical Study, International Journal of Bio -Science and Bio - Technology, 7(1),133-140.

[27]. N.Seshagiri Rao, K.V.L.N.Acharyulu and K.Kalyani(2015) , Phase Plane Analysis of A Host -Commensal Ecological Model, International Journal of Advanced Science and Technology,78, 59-66.

[28]. N.Seshagiri Rao, K.V.L.N.Acharyulu and K.Kalyani (2015), Chaotic behavior of host- Monad commensal species pair- A special numerical case study, ARPN Journal of Engineering and Applied Sciences, 10(14), 5976-5983.

[29]. N.Seshagiri Rao, K.V.L.N.Acharyulu & K.Kalyani (2015), A Host - Mortal Commensal Ecosystem With Harvesting Of Both The Species - Phase Plane Analysis, communicated to ARPN Journal of Engineering and Applied Sciences, 10(13),5490-5497.

[30]. N.Seshagiri Rao, K.V.L.N.Acharyulu and K.Kalyani (2015), Null Clines, Phase Planes Of Both Harvested Host-Mortal Commensal Eco-System, ARPN Journal of Engineering and Applied Sciences,10(14),5976- 5983.

[31]. N.Phani Kumar, K.V.L.N.Acharyulu, S.V.Vasavi & SK.Khamar Jahan (2015), A Series Solution of Ecological Harvested Commensal Model by Homotopy Perturbation Method, International Journal of Scientific and Innovative Mathematical Research,3(11),44-53.