View of A Complete Study on a Substantial Ecological Model

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A Complete Study on a Substantial Ecological Model

*K.V.L.N.Acharyulu¹, G.Basava Kumar², M.N.Srinivas³, I.pothu Raju⁴

1,2,4Faculty, Department of Maths , Bapatla Engineering College (Autonomous),

Bapatla-522102, Andhra Pradesh, India.

3Faculty, Department of Maths, Vellore Institute of Technology (VIT), Vellore-632 014,Tamilnadu ,India.

(^*Corresponding Author Email: [email protected] or [email protected])

ABSTRACT

The paper is aimed to explore a substantial ecosystem by conceptualizing the model with a Prey-Ammensal, a Predator and an Enemy to the Prey-Ammensal.Limited resources are considered for all species. Perturbation analysis is carried out for identifying the existence of the ecological model. The studies of local stability and global stability established the behavior and nature of the ecosystem. Diffusion analysis and Stochastic Analysis are employed. The Homotopy perturbation approach is used to evaluate the series solutions. Whenever possible, the appropriate graphs are illustrated.

Key words:

Local Stability, Global Stability, HPM, Gaussian White Noise, Diffusion Analysis 1. Introduction

The analysis of the diversity and distribution of various species using the same resources in the same ecosystem may also be referred as ecology. The Local stability and Global stability of various ecological models were established by K.V.L.N.Acharyulu and N.ch. PattabhiRamacharyulu in multifarious aspects [5- 9]. Eminent Scholars, Mathematicians [1-4] and Lotka A.J [10] defined and executed useful principles to identify the nature of various ecological models.

1.1 Notations Adopted for the Significant Ecosystem:

In this model, N1(t) represents Prey-Ammesnal species population strength with the natural growth rate a1.N2(t) stands for the predator’s population strength striving of the Prey-Ammensal (N1) with the natural growth rate a2.N3(t) denotes the Population Strength of the enemy to the Prey-Ammesnal(N1) with the natural growth rate a3.aii refers the rate of decrease of Ni due to insufficient resources of Ni,i =1,2,3.a12 can be represented as the rate of decrease of the Prey-Ammesnal(N1) due to inhibition by the predator (N2).a13

refers the rate of decrease of the Ammensal (N1) due to competitive inhibition from enemy (N3).a21 stands for the rate of increase of the predator (N2) due to its successful attacks on the Prey-Ammesnal species (N1).The carrying capacities of Ni ,i = 1, 2, 3 are represented by Ki : ai/aii.Theco-efficient of Prey- Ammensalism is denoted by α= a13/a11.The co-efficient of Prey-Ammesnal suffering rate is represented by P=a12/a11.The co-efficient of predator consumption of the Prey-Ammesnal is labeled by Q=a21/a22

All these variables and considered parameters are assumed as non-negative.

2. Basic Equations The equations for the significant ecosystem are constituted as (i) The rate of growth equation for Prey-Ammensal species (N1):

dN1

dt = a11N1( K1– N1– PN2 –



N3 ) (2.1)

(ii) The rate of growth equation for predator (N2):

dN2

dt = a22 N2 ( K2 – N1 + Q N1 ) (2.2)

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(iii) The rate of growth equation for enemy (N3):

dN3

dt = a33N3( K3 – N3 ) (2.3)

Here The co-existent state ^E^*

(

^{N ,N ,N}¹ ² ³

)

^{exists at}

(viii) ₁ K¹ PK² αK³ ₂ QK +K¹ ² QαK³ ₃ ₃

N = ; N = ; N =K

1+PQ 1+PQ

− − −

(4) With the terms and conditions K₁PK₂+K and K₃ ₃K₁/Q

3.Analysis of Stability at Co-Existent State Lemma (3.1):

11 1 3 2 11 1 3 2 1 3 2

11

22 1 2 3 22 1 2 3

3 33

a (K α K PK ) a P(K α K PK ) α(K α K PK )

1+PQ 1+PQ a 1+PQ

a Q(QK +K Qα K ) a (QK +K Qα K )

1+PQ 1+PQ 0

o 0 K a

If

− − − − − − − −

 − 

 

− − −

 

 =  

 

 − 

 

 

Then the system is stable only when (α +β) > 4α β(1+pq)₁ ² ₁ ,(α +β) =4α β(1+PQ)¹ ² ¹

2

1 1

( + ) 4 (1+ pq)

Proof: The characteristic roots are :

 =

−

k3 a33,

2

1 1 1

(α +β)± (α +β) 4α β(1+pq)

λ= 2

− −

where (5) Case (i): When(α +β) > 4α β(1+pq)₁ ² ₁

In this Case(i), real and negative roots are obtained. Hence, It is stable.

Case(ii): When(α +β) =4α β(1+PQ)₁ ² ₁

In this case(ii), real and negative roots are found. Therefore, It is stable.

Case (iii): When( ₁+ )²4 ₁ (1+ pq)

Here, the roots which are complex with negative real part occurred, thus the state is stable.

Lemma(3.2):

If be the jocobian matrix (mentioned above) then the solution curves are representing asymptotical stability.

Proof: The solution curves are obtained as

λ t1

1 10 1 3 33 1 30 1 20 1 3 33

1

1 2 1 3 33

(λ +β)[U (λ +K a ) αα U ] α PU (λ +K a )

U = e

(λ λ )(λ +K a )

− −

−

3 33

2 K a t

2 10 2 3 33 1 30 1 20 2 3 33 λ t 3 33 1 30

2 1 2 3 33 1 3 33 2 3 33

(λ +β)[U (λ +K a ) αα U ] α PU (λ +K a ) (β K a )αα U

+ e e

(λ λ )(λ +K a ) (λ +K a )(λ +K a )

− − − − −

−

U2 = (λ +α )(λ +β)(λ +K a )+(λ +K a )βQU¹ ¹ ¹ ¹ ³ ³³ ¹ ³ ³³ ¹⁰ βQ αα U¹ ³⁰ ^{λ t}¹ (λ λ )(λ +K a ) e

−

11 1 3 2 22 1 2 3

1

a (K αK PK ) a (QK +K QAK )

α = ,β=

1+PQ 1+PQ

− − −

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3 33

2 K a t

2 1 2 2 3 33 2 3 33 10 1 30 λ t 1 30

2 1 2 3 33 1 3 33 2 3 33

(λ +α )(λ +β)(λ +K a ) + (λ +K a ) β Qu α α βQ u αα βQ u

+ e e

(λ λ )(λ +K a ) (λ +K a )(λ +K a )

− − −

− U3 = U30 e–K3a33t

(2.4) Geometrical interpretation:

The system is asymptotically stable in the three cases (i). U 10 > U20 >U30, (ii).U 10 >U30 >U20 & (iii).U 20 >

U30 > U10.The solution curves are illustrated in Fig.1 to Fig.3

4. Analysis of Stability in Different Aspects

Stability Analysis in terms of Local and Global Stabilities of considered significant ecosystem by R-H criteria, Lyapunov Theorem, Diffusive analysis, Stochastic Analysis is discussed with the following theorems.

Theorem (4.1):

11 1 12 1 13 1

21 2 22 2

33 3

is Jocbian matrix with the valid condition 0

0 0

s

a N a N a N

If a N a N

a N a

− − − 

 

 = − 

 − 

 

 

Fig.2: Case (ii): U 10 >U30 > U20

Fig.1: Case (i): U 10 > U20 >U30

Fig.3: Case (iii): U 20 > U30 > U10

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2 2

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

K −N −PN −N = K −N +QN = K −N = then the corresponding system is Locally stable at

( )

*

1 2

Coexistence Equilibrium tateS E N ,N Proof:

The characteristic equation of A is  −I =0 i.e

11 1 12 1 13 1

21 2 22 2

33 3

0 0

a N a N a N

a N a N

a N



− − − −

− − =

− −

(

^a³³^N³

) (

^{a N}¹¹ ¹

)(

^a²²^N²

)

^{a a}¹² ²¹^{N N}¹ ² ⁰

 = − − ^ + + + ^ =

( ) ⁽ ⁾ ( )

3 2

11 1 22 2 33 3 1 2 11 22 12 21 33 3 11 1 22 2

.

i e = + a N +a N +a N +^N N a a +a a +a N a N +a N

( )

33 1 2 3 11 22 12 21

a N N N a a a a

+ + 

By arranging Routh Array, all the elements in the first column are positive.

1 2 3

1 3

1

1 0 , 0,X X 1.X 0 & 0

Those are X X

X

  −   whereX₁ =a N₁₁ ₁+a N₂₂ ₂ +a N₃₃ ₃

( ) ( )

2 1 2 11 22 12 21 33 3 11 1 22 2

X =N N a a +a a +a N a N +a N

,X2 =a N N N33 1 2 3

(

a a11 22+a a12 21

)



² ²

1 2 3

11 22 1 11 12 21 1 11 22 2 12 21 2 1 2

1 11 1 22 2 33 3

1. 1

X X X

a a N a a a N a a N a a N N N

X a N a N a N

− = + +  + + + 



2 2 2 2

11 33 1 11 33 3 1 3 22 33 2 22 33 3 2 3 2 11 22 33 1 2 3 0

a a N a a N N N a a N a a N N N a a a N N N

   

+ +  + +  + 

The system is therefore locally stable under the R-H criterion at E^*( N 1, N2)

Theorem(4.2): The positive equilibrium^E^*

(

^{N N N}¹^, ²^, ³

)

of system (2.1)-(2.3) is globally asymptotically stable

only when

( )

1 1 1 ¹ 1 2 2 2 ² 2 2 2 2 ²

1 2 2

ln N ln N ln N (4.0)

V t N N N l N N N l N N N

N N N

        

= − −  +  − −  +  − −   ,

1, 2 0

where l l  .

Proof: The suitable Lyapunov function ^{V t}( )is defined as (4.0) for verifying the global stability at the interior equilibrium point ^{E N ,N ,N}^*

(

¹ ² ³

)

^.

3 3 3

1 1 1 2 2 2

1 2

1 2 3

N N dN

N N dN N N dN

dV l l

dt N dt N dt N dt

 

 −   −  −

=  +   +  

     

1 1 1 11 1 12 2 13 1 3 1 2 2 2 22 2 21 1

( ) ( ) ( ) ( )

= − − − − + − − −

dV N N a a N a N a N N l N N a a N a N dt

2( 3 3) ( 3 33 3)

+l N −N a −a N

2 2 2

13 13

12 21 12 21

11 1 1 22 2 2 33 3 3

22 22

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

2 2 2 2 2 2

a a

a a a a

dV a N N a N N a N N

dt a a

 

 − + + + − + + + − + + − 

  ⁰

dV

 dt 

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Theorem (4.3):

The diffusive equation system is constituted as

2

1 2 1

1 1 11 1 12 1 2 13 1 3 1 2

N N

a N a N a N N a N N D

t u

 

= − − − +

  (4.1)

2

2 2 2

2 2 22 2 21 2 1 2 2

N N

a N a N a N N D

t u

 = − − + 

  (4.2)

2

3 2 3

3 3 33 3 3 2

N N

a N a N D

t u

 = − + 

  (4.3)

where D₁,D₂,D₃represents the constant diffusion coefficients of N1,N2,N3 respectively

Then all the Eigen values of the system (4.1)-(4.3) are having negative parts if and only if the conditions 0

A , C0, AB C− 0 hold

where A=a N₁₁ ₁+a N₂₂ ₂+a N₃₃ ₃+K²(D₁+D₂+D₃);

2

11 1 22 2 33 3 22 2 11 1 33 3 ( 22 2( 1 3) 11 1( 2 3) 33 3 1 2))

B=a N a N +a N a N +a N a N +K a N D +D +a N D +D +a N D +D

4

1 3 2 3 1 2 12 1 21 2

( ) ;

K D D D D D D a N a N

+ + + +

2 2 2 2

33 3 3 12 1 21 2 11 1 1 22 2 2 33 3 3

( )( ) ( )( )( )

C= a N +K D a N a N + a N +K D a N +K D a N +K D

The set of equations (4.1)-(4.3) is a diffusion system with the conditions on N₁(u, ),t N₂(u, )t andN₃(u, )t in 0 u L L, 0 as below

1(0, ) 1( , ) 2(0, ) 2( , ) 3(0, ) 3( , ) N t N L t N t N L t N t N L t 0

t t t t t t

 = = = = = =

     

To discuss the system's steady state, the system(4.1)-(4.3) can be linearized by puttingN₁=N₁+v₁,

2 2 2

N =N +v ,N₃ =N₃+v₃ and we obtain

2

1 1

11 1 1 12 1 2 13 1 3 1 2

v N

a N N a N N a N N D

t u

 = − − − + 

 

(4.4)

2

2 2

22 2 2 21 2 1 2 2

v N

a N N a N N D

t u

 = − + + 

  (4.5)

2

3 3

33 3 3 3 2

v N

a N N D

t u

 = − + 

 

(4.6) Let the solutions of the system (4.4)- (4.6) be in the form of

1( , ) 1 ^tcos

v u t =e^ ku;v u t₂( , )=₂e^^tcosku v u t₃( , )=₃e^^tcosku. Then the model becomes

2 1( ) 11 1 1 12 1 2 13 1 3 1(K N )1

v t = −a N N −a N N −a N N +D

(4.7)

2 2 ( ) 22 2 2 21 2 1 2(K N )2

v t = −a N N +a N N +D (4.8)

2 3( ) 33 3 3 3(K N )3

v t = −a N N +D (4.9)

The characteristic equation of the variational matrix of the system (4.7)-(4.9) is in the form of

3 2

0

A B C

 +  + + = (4.10)

where A=a N₁₁ ₁+a N₂₂ ₂+a N₃₃ ₃+K²(D₁+D₂+D₃);

2

11 1 22 2 33 3 22 2 11 1 33 3 ( 22 2( 1 3) 11 1( 2 3) 33 3 1 2))

B=a N a N +a N a N +a N a N +K a N D +D +a N D +D +a N D +D

4( )

K D D D D D D a N a N

+ + + +

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2 2 2 2

33 3 3 12 1 21 2 11 1 1 22 2 2 33 3 3

( )( ) ( )( )( )

C= a N +K D a N a N + a N +K D a N +K D a N +K D

By the Routh-Hurwitz criterion, all the Eigen values of (4.10) have negative parts if and only if A0, 0

C , AB C− 0.

Theorem(4.4): If the interior equilibrium point

(

^{N N N}¹^, ²^, ³

)

of the system without diffusion is globally stable, then the corresponding uniform steady state of the diffusive model (4.1)-(4.3) under zero flux boundary conditions is also globally asymptotically stable.

Proof:- Let us define the function ₁ ₁ ₂ ₃

0

( ) ( , , )

R

V t =



V N N N du

1 2 3

1 2 3 1 1 1 1 2 2 2 2 3 3 3

1 2 3

( , , ) ( ) ln(N ) ( ) ln(N ) ( ) ln(N )

V N N N N N N l N N N l N N N

N N N

 

   

= − − +  − − +  − − 

     Now we

differentiateV₁w.r.to t along withN1,N2,N3 of the diffusive model (4.1)-(4.3) we get

1 1 2 3

1 1 2

1 2 3

0

( ) . . .

R v N v N v N

V t du I I

N t N t N t

       

=



  +  +   = + where

2

2 2

3

1 2

1 2 1 2 2 2 3 2

1 2 3

0 0

and

R R

N

N N

dv v v v

I du I D D D du

dt N u N u N u

       

=



=



   +   +   

Using the established result of B.Dubey&J.Hussain [1],

2

2 2

2 2 2

3

1 2

2 1 2 2 2 3 2

1 2 3

0 0 0

R R R

N

N N

v v v

I D du D du D du

N u N u N u



 

        

= −



    −



    −



   

2

2 2

3 3

1 1 2 2

2 1 2 2 2 3 2

1 2 3

0 0 0

R R R

N N

N N N N

I D du D du D du

N u N u N u



   

   

= −



   −



   −



   It is observed that ifI₁0 then dV¹ 0

dt 

Hence globally asymptotically stable of the system is obtained.

5. Stochastic Analysis

The following system of non-linear ordinarily differential equations establishes the model equations for the constructed significant ecosystem as

(i) The rate of growth equation for Prey-Ammensal species (N1):

1 2

1 1 11 1 12 1 2 13 1 3 1 1

dN ( )

dt =a N −a N −a N N −a N N +   t (5.1) (ii) The rate of growth equation for predator species (N2):

2 2

2 2 22 2 12 1 2 2 2

dN ( )

dt =a N −a N +a N N +   t (5.2)

(iii) The rate of growth equation for enemy species (N3):

3 2

3 3 33 3 3 3

dN ( )

dt =a N −a N +   t (5.3)

Let  ₁, ₂&₃are real constants,  = 

( )

t  1

( ) ( ) ( )

t ,2 t ,3 t is a 3D Gaussian white noise process satisfying Ei

( )

t =⁰ i=^{1, 2,3}

Where _{i j}_, is the Kroneckersymbol ; is the  −dirac function.

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3 3

1 1 2 2 dN du

dN du dN du

, ,

dt = dt dt = dt dt = dt From equation (5.1)

* * * * *

1

13 1 3 13 1 13 3 13 1 1

du ( )

dt = −a u u −a u q −a u s −a s q +   t (5.4) From equation (5.2)

* 2 * *2 * * * *

2

2 2 2 22 2 22 2 22 21 1 2 21 1 21 2 21 2 2

du 2 ( ) (5.5)

dt ==a u +a p −a u − a u p −a p +a u u +a u p +a u s +a s p +   t From equation (5.3)

* 2 * *2

3

3 3 3 33 3 33 3 33 3 3

du 2 ( )

dt ==a u +a q −a u − a u q −a q +   t (5.6) The like part of (5.4),(5.5)&(5.6) is given by

* * *

1

11 1 12 2 13 3 1 1

du ( )

dt = −a u s −a u s −a u s +  t

(5.7)

* * 3 *

2

22 2 21 1 2 2 33 3 2 2

du

du ( ) & ( )

dt = −a u p +a u p +  t dt = −a u q +  t (5.8)& (5.9) Taking Fourier technique of (5.7),(5.8)&(5.9),we get

Eq (5.7) ^^{ }¹^{( )}^{ =}¹

(

ⁱ^⁺^{a s u}¹¹ ^*

)

¹^{( )}^ ⁺â¹²^s*û²^{( )}^ ⁺â¹³^s*û³^{( )}^ ^(5.10)

Eq (5.8)   2 2( ) =

(

i+a p u22 ^*

)

2( ) −a21p* ( )u1

(5.11)

Eq(5.9) ^{  }³ ³^{( )}^ ⁼

(

ⁱ^⁺^{a q u}³³ ^*

)

³^{( )}^ ^(5.12)

The matrix form of (5.10),(5.11)&(5.12) isM( )u( )  = ( ) (5.13) Here

11 12 13

21 22 23

31 32 33

M M M

M( ) M M M ;

M M M



 

 

=  

 

 

1 2 3

( )

( ) ( ) ;

( ) u

u u

u



 



 

 

=  

 

 

1 1 2 2 3 3

( )

( ) ( )

( )

 

  

 

  

 

 =  

  

 

 

* * *

11 12 13

* *

21 22

* 33

where M( ) 0

0 0

i a s a s a s

a p i a p

i a p



 



− + 

 

= − + 

 + 

 

2 2

detM( ) R( ) I( )

 = +

2 2 * 2 * 2 * * * * * * * 2

22 11 33 11 22 33 12 21 33

detM( ) (  a p  a s  a q a a a s p q a a a s p q )

 = − − − + +

3 * * * * * * * * 2

11 22 12 21 22 33 11 33

(  a a s p a a s p a a p q a a s q )

+ − + + +

( )  

¹

5.13 ( ) ( ) ( )

From u  = M  ⁻ +  

 

(1,1) (1,2) (1,3)

11 21 31

1 (2,1) (2,2) (2,3)

12 22 32

(3,1) (3,2) (3,3)

13 23 33

M ( ) M ( ) M ( )

where M( ) ( ) 1 M ( ) M ( ) M ( )

M( )

M ( ) M ( ) M ( )

T T T

CF CF CF

k

  

    

   

−

 

 

= =

 

 

1 2

2 2

(2, ) (2, )

3 3

2 1 2 2

1 1

, ,

2 ( ) 2 ( )

CF i CF i

i i

u i u i

i i

M M

Now d d

M M

   

   

 

 

= − = −

=

 

 =

 

 ³

(3, ) 2 3

2 3

1

2 ( )

CF i i

u i

i

M d

 M 

 

 

= −

=

 



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(1,1) 2 * 2 2 2 * * 2

11^CF ( ) (a a22 33 ) (a22 a33 ) ;

M ^  p q   p q

 = − + +

2 2

(2,1) * 2 2 * 2 (3,1)

12^CF ( ) (a a21 33 ) (a21 ) , 13^CF ( ) 0;

M ^  = q + p M ^  =

(1,2) 2 * * 2 2 * 2

21^CF ( ) (a a12 33 ) (a12 ) ;

M ^  = s q + s

2 2

(2,2) * * 2 2 2 * * 2 (3,2)

22^CF ( ) (a a11 33 ) (a11 a33 ) ; 23^CF ( ) 0;

M ^  = s q − + s + q M ^  =

2 2

(1,3) * 2 * * 2 (2,3) * * 2

31^CF ( ) (a13 ) (a a13 22 p ) ; 32^CF ( ) (a a21 13 ) ;

M ^  = s + s M ^  = s p

(3,3) 2 * * * * 2 2 2 * * 2

33^CF ( ) (a a11 12 a a12 21 ) (a11 a22 )

M ^  = s p + s p − + s + p

1 2

(1) : 0 & 0 Case If  =  =

1 2

* 2 * * 2 * * 2

2 3 13 13 22 2 3 21 13

2 2

( ) ( p ) ( p )

1 1

2 ( ) , 2 ( )

u u

a s a a s a a s

then d d

M M

    

   

 

− −

 + 

=



=



3

* * * * 2 2 2 * * 2

3 11 12 12 21 11 22

2

( p p ) ( p )

1

2 ( )

u

a a s a a s a s a

d M

 

 

 



−

 

  + − + + 

=



1 3

(2) : 0 & 0 Case If  =  =

1

* * 2 * 2

2 2 12 33 12

2

( q ) ( )

1

2 ( )

u

a a s a s

then d

M

  

 



−

 +

=



2 3

* * 2 2 2 * * 2

2 2 11 33 11 33 2

2

( q ) ( a q )

1 , 0

2 ( )

u u

a a s a s

d M

 

  

 



−

 − + +

=



=

1

* * 2 2 2 * * 2

2 1 22 33 22 33

2 3 2

( q ) ( a q )

(3) : 0 & 0 1

2 ( )

u

a a p a p

Case If then d

M

 

 

 



−

 − + +

 =  = =



2 3

* 2 * 2

2 1 21 33 21 2

2

( q ) ( )

1 ; 0

2 ( )

u u

a a a p

d M

   

 



−

 +

=



=

Clearly the steadiness of populations for smaller estimations of mean square vacillations is pointed out by the population variances.

6. Series Solutions by Homotopy Perturbation Method The series solutions are obtained by HPM as

1( ) 1 ( 1 11 1 12 2 13 3) 1

N t = c + a −a c −a c −a c c t

(

a1 2a c11 1 a c12 2 a c13 3

)

(a1 a c11 1 a c12 2 a c13 3)

+ − − − − − − 12( 2 22 2 21 1) 2 13

(

3 33 3

)

3 1 ²

2 a a a c a c c a a a c c c t

− − + − −  +

( )



a1−2a c11 1−a c12 2−a c c13 3 1

⁽

a1−2a c11 1−a c12 2−a c13 3

⁾

( )

1 11 1 12 2 13 3 12 2 22 2 21 1 2 13 3 33 3 3

(a −a c −a c −a c )−a a( −a c +a c c) −a a −a c c 

 ( )

1 11 1 11 1 12 2 13 3 1 12 2 22 2 21 1 2 13 3 33 3 3

2c a (a a c a c a c c) a a( a c a c c) a a a c c 

− − − − + − + + − 



1 11 1 12 2 13 3 12 1 2 2 22 2 21 1 2 22 2 21 1

(a −a c −a c −a c )−a c c (a −2a c +a c) (a −a c +a c )

 ( )( ) 

³

21( 1 11 1 12 2 13 3) 1 13 1 3 2 33 3 3 33 3 3 ...

6 a a a c a c a c c a c a a c a a c c t

+ − − − − − − +

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

( 2 2 22 2 21 1)( 2 22 2 21 1) 21( 1 11 1 12 2 13 3) 1 2 ²

2 a a c a c a a c a c a a a c a c a c c c t

+ − + − + + − − − 

 

21( 1 11 1 12 2 13 3) 1 2 2( 2 22 2 21 1) 21 1( 1 11 1 12 2 13 3)

a a a c a c a c c c a a c a c a c a a c a c a c

+ − − − + − + − − −

 

22( 2 22 2 21 1) 2 21 1 2 ( 1 2 11 1 12 2 13 3) ( 1 11 1 12 2 13 3)

a a a c a c c a c c a a c a c a c a a c a c a c

− − + + − − − − − −

( ) 

³

12( 2 22 2 21 1) 2 13 3 3 33 3 ...

6 a a a c a c c a c a a c  t

− − + − −  +

( ) ( )( )

²

3( ) 3 3 33 3 3 3 2 33 3 3 33 3 3

2 N t =c + a −a c c t+ a − a c a −a c c t

(

3 33 3

) (

3 3 2 33 3

)

² 2 33

(

3 33 3

)

3



³ ...

6 a a c c  a a c a a a c c t

+ −  − − − +

7.Conclusions

On the basis of this study, we draw the following conclusions on an important three species significant ecosystem:

(i).The asymptotic stability of the system in various cases is asserted by using geometrical interpretation.

(ii).Local stability of the system is observed at interior equilibrium state by Routh-Hurwitz criterion (iii).Global Stability is also established by constructing suitable Lyapunov function.

(iv).Diffusion and Stochastic Analysis addressed effectively the stability of the system.

(v).Homotopy perturbation method efficiently extracted the series solutions of the three species ecological model.

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, N.Ch. Pattabhi Ramacharyulu (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.

[6]. K.V.L.N.Acharyulu and N.Ch. Pattabhi Ramacharyulu (2010), On The Stability of an Ammensal- Enemy Harvested Species Pair with Limited Resources, Int. J. Open Problems Compt. Math (IJOPCM), 3(2),241-266.

[7]. 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(6),49-58.

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

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

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

[10]. Lotka A.J.(1925), Elements of Mathematical Biology, Williams and Wikkins, Baltimore,USA