**Impact of Diffusion on a Special Ecosystem **

*******K.V.L.N.Acharyulu**^{1}**,****G.Basava Kumar**^{2}**, I.Pothuraju**^{3 }**& S.V.Vasavi**^{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 **

In this paper, the authors aim to discuss a special ecosystem with constant rates of mortality and harvesting for Ammensal Species. Local stability is established at interior equilibrium point. Spatiotemporal Analysis has been carried out. Global stability is identified at the interior point. The impact of diffusion on this model is traced.

**Key words: **

Ammensal , Enemy Species, Stability, Ruth-Hurwitz(R-H) stability criterion, Diffusion, HPM, Spatiotemporal Analysis

**1. Introduction **

Biomathematics, also called as mathematical biosciences, is a multidisciplinary field with a wide and exponentially increasing literature distributed across the last century through numerous disciplines.

Mathematicians, physicists, computational scientists, ecologists, medical scientists, demographers, several researchers have made contributions to it. In mathematical biosciences, the main aim is to gain an overview and better understanding about the challenges of real life. For the last few decades, Mathematicians research the merits and demerits of mathematical simulation, mathematical techniques. It has been the foundation of the modern growth of scientific research.

Mathematical modelling in the modern sciences of real life situations is an effort in the language of mathematics to define and explain certain instances of everyday life. Although the reach of mathematical modelling is in many sectors expanding and deepening, it is not only confined to the usage of already proven mathematical techniques. It is very important to notice that one of the main tasks of mathematicians employed in fields such as life, medical and social sciences is to create new mathematical strategies that deal with complicated problems that exist in nature and in our routine as well. Situations are also very difficult in life sciences. As such, before attempting to devise a new mathematical model, one must have some experience regarding the scenario. If a model is formulated, by using an appropriate mathematical technique, the effects may be noticed and the findings are contrasted. Further modifications to the model are indicated by the inconsistencies between theoretical assumptions centered on the model and real life observations.

K.V.L.N.Acharyulu and N.Ch.Pattabhi Ramacharyulu examined different cases of ecological Ammensal models [5-18] and also investigated various ecological models for their stability in manifold dimensions. Many Research scholars [1-4] and Mathematicians[19-31] extended their significant contributions in this modelling field.

A real life problem may seldom be converted into a mathematical problem with all its generality.

Even though it can be converted in such a way, it might not be feasible to satisfactorily solve the resulting mathematical problem. Therefore, it will be appropriate to 'simplify' or 'idealize' or 'approximate' the problem with another problem, taken as a relevant model of the initial problem and mathematically interpreted and solved at the same time. All the basic characteristics of the issue will be kept in this

idealisation phase, giving up those characteristics that are not inevitable or so relevant to the condition under investigation.

**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 with constant mortality rate. Here

(i).X is the density of Ammensal species with natural growth rate a1, (ii).Y is the density of the enemy species with natural growth rate a2, (iii).h1=a11H1 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 Ammensalism's coefficient.

(vi).m= Decrease of Ammensal species due to harvesting.

Assume that the parameters described above are positive.

** 2. Constriction of Mathematical Model **
The equations for the special ecosystem are presented as below:

The rate of the growth for Ammensal Species with constant rates of mortality and harvesting

2

11( 1 1)

*dX* *a* *K X* *X* *XY* *H*

*dt* = − − − + (2.1)

Equation for the growth rate of the Enemy species

22 ( 2 )

*dY* *a Y K* *Y*

*dt* = − (2.2)

**2.1. Normal State: **

θ φ * *

1 2 3 4

E (0,0),E (x ,0), E (0,y ) ,E (x ,y ) are eqilibrium po

Here ints

The stability of the system around the internal equilibrium *E x*_{4}( ^{*},*y*^{*})point is being studied

### ( )

2

1 2 1 1 2

* *

2

( ) 4 ( )

where ' Co-existe

2 nce state

+ + − +

= *K* *K* *H* *K* *K* =

*only* *x* *y* *K*

**3. Local Stability **^{E x y}^{4}

### (

^{}

^{,}

^{}

### )

**by Routh-Hurwitz Stability Criterion**

Local stability at equilibrium point implies that if you bring the system next to the point anywhere, it can shift itself in any period of time to the equilibrium point. Global equilibrium implies that from every possible starting stage, the system would arrive at the equilibrium point. In general, Routh-Hurwitz stability criterion is the best tool to discuss the Local stability and Lyapunov Theorem is an effective method to establish Global Stability.

We work out the variational matrix about

* * *

11 1 11

* 22 2

( 2 )

0 ( 2 )

*a* *k* *x* *y* *a x*

*J* *a* *k* *y*

− + + −

= −

* 1 *

11 * 11

* 22

( )

0 ( )

*a* *x* *H* *a x*

*J* *x*

*a* *y*

− + −

=

−

Characteristic equation of J at *E*_{4} is

* 1 *

11 * 11

* 22

( )

0

0 ( )

*a* *x* *H* *a x*

*x*

*a* *y*

− + − −

=

− −

*2

2 * 1 1

22 11 *H* * *x* 11 22 *H* **x* 0

*a y* *a* *a a*

*x* *x*

^{} ^{} ^{+} ^{}^{} ^{} ^{+} ^{}

+ + + =

Let _{22} ^{*} _{11} *H*^{1}_{*} *x*
*A* *a y* *a*

*x*

+

= +

*2 1

11 22 *

*H* *x*
*B* *a a*

*x*

+

=

By using R-H stability criteria,
Clearly _{22} ^{*} _{11} *H*^{1} _{*} *x* 0

*A* *a y* *a*
*x*

+

= +

*2 1

11 22 *

and *H* *x* 0

*B* *a a*

*x*

+

=

All the elements (coefficients) in the first column of Routh array are positive.

Hence by R-H stability Criteria, it can be stated that the model is locally stable at interior equilibrium point

2

1 2 1 1 2 * *

2 4

( ) 4 ( )

, . ( , )

2

*K* *K* *H* *K* *K*

*K* *i e E x y*

+ + − +

**4. Spatiotemporal Analysis **

Due to the development and implementation of innovative mathematical tools enabling the analysis of broad spatiotemporal databases, spatiotemporal data analysis is emerging the research fields.

Spatiotemporal models emerge as data is gathered over time and space, with at least one spatial and one temporal property. An occurrence in a spatiotemporal dataset represents a spatial and temporal process that occurs at a time t and x. Other uses for spatiotemporal research cover genetics, geography, meteorology, medicine, and including transport

Spatiotemporal data processing involves all temporal and spatial similarities are taken into considera tion.Evaluation of the time and space measurements of data bringstremendous uncertainty to the data proces sing method .In such complicated computational situations. Spatiotemporal Analysis can help us to analize the models.

Here we considered an ecological system where Ammensal and enemy are living together. It is assumed that Ammensal species is harvested with constant mortality rate at a constant rate.

Let us consider the diffusive equation system as

2 2

11( 1 1) 1 2

*X* *X*

*a* *K X* *X* *XY* *H* *D*

*t* *s*

= − − − + +

(4.1)

2

22 ( 2 ) 2 2

*Y* *Y*

*a Y K* *Y* *D*

*t* *s*

= − +

(4.2)
In this*D*1,*D*2 represent the constant diffusion coefficients of the Ammensal & Enemy.

The set of equations (4.1)-(4.2) is a diffusion system with the conditions on *X s t*( , ),*Y s t*( , )in0 *u* *L L*, 0
and

(0, ) ( , ) (0, ) ( , )

*X* *t* *X L t* *Y* *t* *Y L t* 0

*t* *t* *t* *t*

= = = =

** **

(4.3)

The system (4.1)-(4.3) can be linearized by positioning the system to address its steady condition. In terms of the inner steady state, the linear component of the system (4.1) - (4.2) is accomplished as

2

* *

11 11 1 2

*X* *X*

*a Xx* *a* *Yx* *D*

*t* *s*

= − − +

^{ } ^{(4.4) }

2

*Y* * *Y*

*a y Y* *D*

= − + _{` } _{(4.5)}

by assuming *x*=*x*^{*}+*X* and *y*= +*y*^{*} *Y*. Let the solution of the system (4.4)-(4.5) be the form
( , t) 1 * ^{t}*cos

*X s* =*e*^{} *ku*, *Y s t*( , )=_{1}*e*^{}* ^{t}*cos

*ku*; where

_{1},

_{2}are amplitudes and k is the wave number of the solution.

*X*,

*Y*are propagations of populations. Corresponding to the diffusive system (4.4)-(4.5), the characteristic equation is

2 *A* *B* 0

+ + = (4.6)

where*A a x*= _{11} ^{*}+*a y*_{22} ^{*}+*k D*^{2}( _{1}+*D*_{2});*B a a x y*= _{11 22} ^{* *}+*k a D x*^{2}( _{11} _{1} ^{*}+*a D y*_{22} _{2} ^{*})+*D D k*_{1} _{2} ^{4}.

**Theorem (4.1): The interior equilibrium is locally asymptotically stable in the presence of diffusion if and **
only if *A*0and*B*0.

This theorem follows immediately by Routh-Hurwitz criteria.

* * 2

11 22 ( 1 2) 0

*A a x*= +*a y* +*k D* +*D* *B*=(*a a x y*_{11 22} ^{* *}+*k a D x*^{2}( _{11} _{1} ^{*}+*a D y*_{22} _{2} ^{*})+*D D k*_{1} _{2} ^{4}) 0
**Theorem (4.2): **

If the system at the interior equilibrium point is globally stable without diffusion.Also under zero flux boundary conditions, the resulting diffusive model (4.1)-(4.3)

is asymptotically stable globally by significantly enhancing the coefficients of diffusion.

**Proof: Let us define the function** 1

### ( )

0

( ) ,

*R*

*V t* =

###

*V X Y ds*

^{, }Now we differentiate V1 w.r.to t along with X and Y

1

1 2

0

*dV* *R* *V X* *V Y*

*ds* *I* *I*

*dt* *X* *t* *Y* *t*

=

###

+ = + (4.7)2 2

1 0 a 2 0 1 2 2 2

where ^{R}*dV* nd ^{R}*V* *X* *V* *X*

*I* *dx* *I* *D* *D* *ds*

*dt* *X* *s* *X* *s*

=

###

=###

+ ^{(4.8) }

Using the established result of B.Dubey & J.Hussain [2], we get

2 2

2 2

2 1 0 2 2 0 2

*R* *V* *X* *R* *V* *Y*

*I* *D* *ds* *D* *ds*

*X* *s* *Y* *s*

= −

###

−###

2 2

* *

1 0 2 2 0 2

*R**X* *X* *R**Y* *Y*

*D* *ds* *D* *ds*

*X* *s* *Y* *s*

= −

###

−###

(4.9) From (4.7), (4.8) and (4.9),

Clearly *I*_{1}0then *V t*_{1}( ) is negative, it could easily be observed.If*I*_{1}0, then it can be observed that by
enhancing the sufficiently large diffusion coefficients D1 and D2, *V t*_{1}( ) can be made negative.

**5. Global Stability of the Special Ecosystem **

Now we discuss the global stability of the considered Special Ecosystem in which the Ammensal Species is influenced by constant rate of mortality and harvesting at variable rate.

The equations are

(i).The rate of the growth for Ammensal Species

2

11( (1 ) 1 1)

*dX* *a* *m K X* *X* *X Y* *H*

*dt* = − − − − + (5.1)

(ii).Equation for the growth rate of the Enemy species

22 ( 2 )

*dY* *a Y K* *Y*

*dt* = − (5.2)

By taking perturbations(P1,P2) and after linearization, we get

1 1 2

11 1 11 2

(1 ) 2 2

*dP* *m K* *K*

*X* *a P* *a P X*

*dt*

− +

= − + −

2

22 2

*dP* *a P Y*
*dt* = −

The corresponding characteristic equation is

2 1 2 1 2

11 22 11 22

(1 ) (1 )

2 2 0

2 2

*m K* *K* *m K* *K*

*X* *a* *a Y* *X* *a a Y*

+^{}^{} +^{} ^{−} ^{+} ^{}^{} + ^{}+^{} +^{} ^{−} ^{+} ^{}^{} =

### The above equation is in the form

2### + *A* + = *B* 0

1 2

11 22

wher (1 )

2 2

e *m K* *K* 0

*A*=*X* + − +

###

*a*+

*a Y*

^{ }

^{(5.3) }

1 2

11 22

(1 )

2 0

2

*m K* *K*

*B*= *X* + − +

###

*Y a a*

^{ }

^{(5.4) }The criteria for the presence of the function of Liapunov are fulfilled.

### (

^{2}

^{2}

### )

1, 2 1 1 2 2

(P P ) 1

Now, we def ne 2

i *E* =2 *R P* + *S P P* +*TP* (5.5)

## (

^{22}

## )

^{2}

^{(1}

^{)}

_{2}

^{1}

^{2}

^{2}

^{11 22}

where

*m K* *K*

*a X* *X* *a a Y*

*R* *D*

− +

+ +

= (5.6)

11 22

*a a* *X Y*

*S* *D*

= −

(5.7)

2

2 2 2 2

1 2 1 2

11 11 11 22

(1 ) (1 )

4 2

2 2

*m K* *K* *m K* *K*

*X* *a* *a* *X* *X* *a a Y*

*T* *D*

+ − + + + + − +

= (5.8)

and*D*=*AB*0

It is obvious from the equations (5.3) and (5.4) that D>0 and R>0.

### ( )

2 2 1 2

22 11 22

2 2 2

(1 )

2 2

*A*

*m K* *K*

*a Y* *a a Y X*
*D* *RS T*

*so* *D* *D*

*l*

+ + − +

− =

### ( )

2 2 2

1 2 1 2

11 11 11 22

(1 ) (1 )

4 2

2 2

+ − + + + + − +

*m K* *K* *m K* *K*

*X* *a* *a X* *X* *a a Y*

*D*

### (

^{11 22}

### )

^{2}

2

^{}_{}

−

*a a X Y*
*D*

### ( )

2 2 2 2

0 0 . 0

*D* *RT* *S* *RT* *S* *i e S* *RT*

− − −

The function E(P ,P) at (5.5) is positive definite.

## (

^{2}

^{2}

## )

1 2

1 2

1 2

*dP* *dP* 0

*V* *V*

*P* *P*

*P dt* *P dt*

+ = − +

This is obviously a negative definite.

So, E(P1,P2) is a Liapunov’s function for the linear system.

Next we prove that E(P1,P2) is also a Liapunov’s function for the non-linear system.

Let γ1 and γ2 be two functions of X and Y defined by

### (

^{2}

### )

11 1 1

1( , )*X Y* *a* (1 *m K X*) *X* *X Y* *H*

= − − − − +

###

2( , )*X Y* *a Y K*22 2 *Y*

= −

1 2

1 2

We must now prove that *V* *V* is certainly negative.

*P* *P*

+

1 2

By taking Perturbations*X* = +*X* *P*and*Y* = +*Y* *P* in the above equations,weget

### ( )

1 1 2

1 2 11 1 11 2 1 1 2

1

(1 )

( , ) 2 ,

2

*dP* *m K* *K*

*P P* *a X* *P* *a* *X P* *P P*

*dt*

= = − + − + − +

### ( )

^{2}

1 1 2 11 1 11 1 2

where *P P*, = −*a P* −*a* *PP*
Similarly

### ( ) ( )

^{2}

2

1 2 22 2 2 1 2 2 1 2 22 2

2( , ) *dU* , where ,

*P P* *a Y P* *P P* *P P* *a P*

= *dt* = − + = −

From (5.5) _{1} _{2} _{1} _{2}

1 2

*V* and *V*

*RU* *SU* *S U* *T U*

*P* *P*

= + = +

### ( )

^{1}

^{2}

### ( )

1 2 1 2 11 1 11 2 1 1 2

1 2

(1 )

Now 2 ,

2
*m K* *K*

*V* *V*

*RP* *SP* *a X* *P* *a* *X P* *P P*

*P* *P*

^{} ^{−} ^{+} ^{}

+ = + − + − +

### (

^{S P}^{1}

^{T P}^{2}

### ) (

^{a Y P}^{22}

^{2}

^{}

^{2}

### (

^{P P}^{1}

^{,}

^{2}

### ) )

+ + − +

### ( ) ( )

2 2

1 2 1 2 1 2 1 1 2 1 2 2 1 2

1 2

Now *V* *V* ( , ) ( , )

*P* *P* *R P* *S P* *P P* *S P* *T P* *P P*

*P* *P*

+ = − + + + + +

1 2

Introducing polar coordinates *P* =cos & *P* =sin

### ( ) ( )

2

1 2 1 1 2 2 1 2

1 2

[( cos sin ) , ( cos sin ) , ]

*V* *V*

*R* *S* *P P* *S* *T* *P P*

*P* *P*

+ = − + + + +

Let us denote the largest of the numbers *R* ,*S T by* ,

### ( ) ( )

1 1, 2 and 2 1, 2

6 6

*P P* *P P*

2 2

2

1 2

1 2

for all sufficiently small

6 3

4 0

0 , *V* *V*

*P* *P*

^{} +^{} − + = −

E is thus a positive definite function according to the condition that

1 2

1 2

is negative definite

*V* *V*

*P* *P*

+

Thus the equilibrium state is “asymptotically stable” globally.

**6. Conclusions **

A research was extensively carried out on a special model of Mortal Ammensal and Enemy species. The following conclusions are put forth:

(i). Local stability is noticed by the Routh-Hurwitz criteria at the interior equilibrium point (v). Diffusion analysis addresses the system's stability.

(vi).Global stability is established at interior Point.

**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.

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