Stability of a Prey-Predator Model with SIS Epidemic Disease in Predator Involving Holling Type II Functional Response

IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 2 Ver. III (Mar - Apr. 2015), PP 38-53 www.iosrjournals.org

DOI: 10.9790/5728-11233853 www.iosrjournals.org 38 | Page

Stability of a Prey-Predator Model with SIS Epidemic Disease in

Predator Involving Holling Type II Functional Response

Ahmed Ali Muhseen1, and Israa Amer Aaid

2,

1 Assistant Lecturer, Ministry of Education, Rusafa\1, Baghdad-Iraq, 2College of Business and Economics, Iraqi University, Baghdad-Iraq,

Abstract: In this paper, a prey-predator model with infectious disease in predator population involving

Holling type II functional response is proposed and studied. The existence, uniqueness and boundedness of the

solution of the system are studied. The existence of all possible equilibrium points is discussed. The local

stability analysis of each equilibrium point is investigated. Finally further investigations for the global

dynamics of the proposed system are carried out with the help of numerical simulations.

Keywords: eco-epidemiological model, SIS epidemic disease, prey-predator model, stability analysis, Holling

type II functional response.

I. Introduction Mathematical models are divided into two main sections called the first ecology models where

interested in studying the interactions between members of a particular community, for example, (humans or

animals, etc.) and known model (lotka - volterra) [1,2] the basis for many of the studies in this field. The second

type is called epidemiological models that care about the study and analysis of the spread of infectious diseases

between humans and animals and the SIR is the basis model for this type of models has been by Kermack and

McKendric in 1927 [3]. In more studied showed that combines both these types where named eco-

epidemiological models, for example, in 1986 Anderson and May [4] were the first who merged the above two

fields, ecological system and epidemiology system, they formulated a prey-predator model with infectious

disease spread among prey by contact between them. In the subsequent time many researchers proposed and

studied different prey-predator models with disease spread in prey population [5-8]. In addition to the above

there are many investigations about prey-predator model with disease in the predator population. Haque [9]

proposed a prey-predator model includes a Susceptible-Infected-Susceptible (SIS) parasitic infection in the predator population with linear functional response and nonlinear disease incidence rate. Haque and Venturino

[10] considered a prey-predator model with SI epidemic disease spread in predators involving linear functional

response. Das [11] studied a prey-predator model with SI epidemic disease in predators included Holling type-II

as a functional response. Venturino [12] proposed and analyzed prey-predator model with SIS disease in

predators included linear functional response and linear disease incidence. Haque and Venturino [13] considered

a prey-predator model with SI epidemic disease spread in predators included ratio-dependent functional

response and linear disease’s incidence rate. Dahlia [14] studied a prey-predator model with SIS epidemic

disease in prey. In this paper we proposed and analyzed a mathematical model describing prey-predator model

having SIS epidemic disease in the predator population involving Holling type-II functional response with the

disease transmitted between the predator species by contact.

II. The Mathematical Model Consider an eco-epidemiological model consisting of two preys and two predators is proposed for

study. Let )(TX denotes to the density of the first prey population at time T , and )(TY is the density of the

second prey population at timeT . However, the predator is divided in to two classes namely, infected )(TZ and

susceptible )(TW , here )(TZ and )(TW represent the population density at time T for the susceptible and

infected predator respectively. Now in order to formulate the above model mathematically the following

assumptions are considered:

1. The preys )(TX and )(TY grow logistically in the absence of predation with intrinsic growth rates

2,1;0 iri with carrying capacity 0K and 0L respectively.

2. The predators )(TZ and )(TW consume the prey )(TX according to Lotka-Vollterra functional responses

and Holling type II functional response with attack rates 0m , 0p and conversion rates 01 m ,

01 p respectively, While 0b is the half saturation constant. Further, it is assumed that the predator

Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…


)(TZ consume the prey )(TY according to Lotka-Vollterra functional responses with attack rate 0n and

conversion rate 01 n respectively.

3. The disease transmitted within the population of predator only by contact between the predator individuals,

according to simple mass action law with contact infected rate 0 .

4. The infected predator )(TW is recovered and they become susceptible )(TZ again with recover rate

constant 0 .

5. The predator populations (susceptible and infected) decrease due to the natural death rates 2,1;0 ii .

6. The disease in infected predator )(TW may causes mortality with a constant mortality rate represented

by 03 .

Moreover the dynamics of the above model can be represented by the following set of nonlinear first order

differential equations:

WWWZdT

dW

ZWWZZYnXmdT

dZ

nYZYrdT

dX

mXZXrdT

dX

Xb

XWp

LY

Xb

pXW

KX

)(

)1(

)1(

32

111

2

1

1

(1)

Note that the above model contains (16) positive parameters in all, which makes mathematical analysis of the

system very difficult. So in order to reduce the number of parameters and determined which parameter

represents the control parameter, the following dimensionless variable are used:

Trt 1 , KXx ,

LYy ,

1rmZz ,

Kr

pWw

1

Accordingly, system (1) can be rewritten in the following non dimensional form:

),,,()(

),,,(

),,,()1(

),,,()1(

4131211101

9

387654

232

11

wzyxwfwwwwwzwxw

wxw

dt

dw

wzyxzfzwwwwzwyzwxzwdt

dz

wzyxyfyzwyywdt

dy

wzyxxfxw

xwxzxx

dt

dx

(2)

Here 0)0( x , 0)0( y , 0)0( z and 0)0( w with the following constants represent the non dimensional

parameters

Kbw 1 ,

1

2

2 r

rw ,

mnw 3 ,

1

1

4 r

Kmw ,

1

1

5 r

Lnw ,

p

kw

6 ,

prmKw1

7 ,

1

1

8 rw

,

1

1

9 r

pw ,

mw

10 ,

111 r

w , 1

2

12 rw

,

1

3

13 rw

.

It has been observed that the non dimensional system (2) contains (13) parameters only, while the original

system (1) contains (16) parameters. Obviously the interaction functions 321 ,, fff and 4f of the system (2) are

continuous and have continuous partial derivatives on the state space 4R , therefore these functions are

Lipschizian on its domain 4R and then the solution of system (2) with non negative initial condition exists and

is unique. In addition, all the solutions of system (2) which initiate in 4R are uniformly bounded as shown in

the following theorem.

Theorem 1: All the solutions of the system (2), which initiate in R4 are uniformly bounded provided that

4

79

w

wwB (3a)



Where

131211 wwwB (3b)

Proof: Let ))(),(),(),(( twtztytx be any solution of the system (2). Since

),1( xxdt

dx )1(2 yyw

dt

dy

Thus by solving these differential inequalities:

0,1)(1)(lim ttxtxSupt

0,1)(1)(lim ttytySupt

Now, consider the function:

wzyxwwzyxUw

w

w

w

9

4

3

5

4),,,(

Then the time derivative of )(U along the solution of the system (2) is:

Usdt

dU

Where

3

52

42w

wwws

and

4

79,,,1.min 82 w

wwBww

Clearly, is positive constant under the sufficient condition (3a). By comparing the above differential

inequality with the associated linear differential equation, we obtain:

0)()(.lim ttUtUSup sst

hence, all the solutions of system (2) that initiate in R4 are confined in the region

}:),,,{(9

4

3

5

44

s

w

w

w

wwzyxwURwzyx under the given condition, thus these solutions are

uniformly bounded, and then the proof is complete. ■

III. The Existence Of Equilibrium Points In this section, the existence of all possible equilibrium points of system (2) is discussed. It is observed that,

system (2) has at most ten equilibrium points, namely )0,0,0,0(0 E , )0,0,0,1(xE , )0,0,1,0(yE ,

)0,0,1,1(xyE always exist. While the existence of other equilibrium points are shown in the following:

The equilibrium point )0,,0,( zxExz , where

4

8

w

wx ,

4

81w

wz (4)

exists uniquely in the 2. RInt of xz plane provided that:

14

8 w

w (5)

The equilibrium point )0,~,~,0( zyEyz , where

5

8~w

wy ,

53

852 )(~ww

wwwz

(6)

exists uniquely in the 2. RInt of yz plane provided that:

85 ww (7)

The equilibrium point )0,ˆ,ˆ,ˆ( zyxExyz exists in RInt 3. of -xyz plane, where

5342

85253 )(ˆ

wwww

wwwwwx

(8a)

5342

84342 ][ˆ

wwww

wwwwwy

(8b)



5342

8542 ][ˆ

wwww

wwwwz

(8c)

exists uniquely in the 3. RInt of xyz plane provided that:

},.{max548

www (9a)

854

www (9b)

The equilibrium point ),,0,( wzxExzw

exists in RInt 3. of -xzw plane, where

101

91

)(

)(

wxw

xwBBwz

and

xwBwwwBwwww

xwBBwxwww

)]([][

])([

9610761071

9148

(10)

While, x

represents the positive root of each of the following equation.

0432

23

1 qxqxqxq (11)

where:

107961 )( wwwBwq

][][

1)]([

9461071

196107210

9

wBwBwwww

wwBwwwqw

wB

][]

1][

1])[[(

984

16107

9610713

10

9

10

wBwBw

wBwww

wBwwwwq

w

wB

wB

]1)([ 8610711410

BwBwwwwwqwB

So by using Descartes rule of signs, Eq. (11) has a unique positive root say x

provided that one set of the

following sets of conditions hold:

0q and 0q 0,q 421 (12a)

0q and 0q 0,q 431 (12b)

0q and 0q 0,q 421 (12c)

0q and 0q 0,q 431 (12d)

Therefore, by substituting x

in Equation (10), system (2) has a unique equilibrium point in the 3. RInt of

-xzw plane given by ),,0,( wzxExzw

, provided that

},.{max, 9610748 wBBwwwxww

(13a)

Or,

},.{min, 9610748 wBBwwwxww

(13b)

Finally the coexistence equilibrium point ),,,( *wzyxExyzw exists in 4. RInt , where

1021

9310231021

)(

)]([][

wwxw

xwBwwwBwwwwy

(14a)

101

91

)(

)(

wxw

xwBBwz

(14b)

31102

912

)(

)])([

Bxwww

xwBBwBw

(14c)

Where

2

9153

54811022

)][(

))((

xwBBwww

wxwwxwwwB



xwBwwwBwwwwB )]([][

96107610713

While, x represents the positive root of each of the following equation:

0542

33

24

1 QxQxQxQxQ (15)

where:

)]([ 961071021 wBwwwwwQ

])(

)([

)()21(

)]([

49

61071102

9101

9610722

wwB

Bwwwwww

wBww

wBwwwwQ

])()([

)(

)]()1([

][

)2)1(2(

])[[(

5398521

9

9110

6107

9110

96107213

wwwBwwww

wB

wBww

Bwww

wBww

wBwwwwwQ

)(2

)]2)2([

][

]])[[[(

9531

9110

6107221

10961072214

wBBwww

wBww

Bwwwww

BwwBwwwwwQ

])([

]][[

535810221

1061072315

BwwwwwwBw

BwBwwwwwQ

So by using Descartes rule of signs, Equation (15) has a unique positive root say x provided that one set of the

following sets of conditions hold:

0Q and 0Q0,Q 0,Q 5321 (16a)

0Q and 0Q0,Q 0,Q 5421 (16b)

0Q and 0Q0,Q 0,Q 5431 (16c)

0Q and 0Q0,Q 0,Q 5431 (16d)

0Q and 0Q0,Q 0,Q 5421 (16e)

0Q and 0Q0,Q 0,Q 5321 (16f)

Therefore, by substituting x in Equations (14a), (14b) and (14c), system (2) has a unique equilibrium point in

the 4. RInt by ),,,( *wzyxExyzw , provided that

},.{max93102

wBBwww (17a)

00 32 BandB (17b)

Or,

00 32 BandB (17c)

IV. The Stability Analysis In this section the stability locally analysis of the above mentioned equilibrium points of system (2) are

investigated analytically. The Jacobian matrix of system (2) at each of these points is determined and then the

eigenvalues for the resulting matrix are computed, finally the obtained results are summarized in the following:

The Jacobian matrix of system (2) at the equilibrium point )0,0,0,0(0 E can be written

as 4,3,2,1,;][)( 4400 jicEJJ ij , where 111 c , 222 wc , 833 wc , 734 wc , Bc 44 and zero

otherwise. Then the eigenvalues of 0J are:

0101



0202 w (18)

0803 w 004 B

Therefore, the equilibrium point 0E is a saddle point.

The Jacobian matrix of system (2) at the equilibrium point )0,0,0,1(xE can be written

as 4,3,2,1,;][)( 44 jidEJJ ijxx , where 11311 dd , 1

114

1

wd 222 wd , 8433 wwd ,

734 wd , Bdw

w

1441

9 and zero otherwise. Hence, the eigenvalues of xJ are:

011

x , 022 wx , 843

wwx and Bw

wx

11

9

4 (19)

Clearly, xE is a saddle point.

Now, the Jacobian matrix of system (2) at the equilibrium point 0,0,1,0yE can be written

as 4,3,2,1,;][)( 44 jieEJJ ijyy , where 111 e , 222 we , 323 we , 8533 wwe , 734 we ,

Be 44 and zero otherwise. The eigenvalues of yJ are:

11y , 022

wy , 853wwy ,

By 4

(20)

Hence, yE is a saddle point.

Now, the Jacobian matrix of system (2) at the equilibrium point 0,0,1,1xyE can be written

as 4,3,2,1,;][)( 44 jifEJJ ijxyxy , where

11311 ff , 1

114

1

wf , 222 wf , 323 wf , 85433 wwwf , 734 wf and Bf

w

w

1441

9 and

zero otherwise. Therefore, the eigenvalues of xyJ are given by:

111

yx , 0222 wyx , 85433

wwwyx , Bw

wyx

11

9

44 (21)

Consequently, xyE is locally asymptotically stable in the 4R if the following condition is satisfied.

854 www (22a)

Bw

w

11

9 (22b)

However, xyE will be saddle point in the 4R if we reversed of the above condition.

Theorem 3: Assume that the equilibrium point )0,,0,( zxExz of system (2) exists. Then it is locally

asymptotically stable in 4R if one of the following sets of conditions hold

Bzwxw

xw

101

9 (23a)

zww 32 (23b)

Proof: Note that, it is easy to check that the Jacobian matrix of system (2) at the equilibrium point xzE can be

written 4,3,2,1,;][)( 44 jigEJJ ijxzxz

xgg 1311 , xw

xg1

14 , zwwg 3222 , zwg 431 , zwg 532 , zwwg 6734 ,

Bzwgxw

xw

10441

9 and zero otherwise. Therefore the characteristic equation can be written in the form:

0])[( 322

13

44 AAAg

Consequently, either

Bzwxw

xwzx

101

9

11



Or 0322

13 AAA (24a)

Here

3122133

311322112

22111 )(

gggA

ggagA

ggA

(24b)

and

31131122112211

321

)( ggggggg

AAA

(25)

So, by substituting the values of ,ijg and then simplifying the resulting terms we obtain:

)( 321 zwwxA , )( 3243 zwwzxwA and ]))([( 43232 zxwxzwwzwwx

According to the Routh-Hurwitz criterion all the roots (eigenvalues) of third order equations have real parts

provided that 3,1;0 iAi and 0 . Note that, 3,1;0 iAi and 0 provided that condition (23b) holds.

Further 011zx provided that condition (23a) holds. Therefore all the eigenvalues of xzJ have negative real

parts under the give conditions and hence the equilibrium point xzE is locally asymptotically stable, which

completes the proof. ■

Theorem 4: Assume that the equilibrium point )0,~,~,0( zyEyz of system (2) exists. Then it is locally


Bzw ~10 (26a)

1~ z (26b)

Proof: Note that, it is easy to check that the Jacobian matrix of system (2) at the equilibrium point yzE can be

written:

4,3,2,1,;][)( 44 jihEJJ ijyzyz

zh ~111 , ywh ~222 , ywh ~

323 , zwh ~431 , zwh ~

532 , zwwh ~6734 , Bzwh ~

1044 and zero

otherwise. Therefore the characteristic equation can be written in the form:

0]~~~

)[~

( 322

13

44 CCCh


Bzwzy ~~1011

Or 0~~~

322

13 CCC (27a)

here

3223113

322322112

22111 )(

hhhC

hhhhC

hhC

(27b)

and

32232222112211

321

)( hhhhhhh

CCC

(28)

So, by substituting the values of ,ijh and then simplifying the resulting terms we obtain:

)~1(~21 zywC , )~1(~~

533 zzywwC and ]~~)~~1)(~1[(~5322 zywwywzzyw

Again due the Routh-Hurwitz criterion all the roots (eigenvalues) of third order equations have real parts

provided that 3,1;0 iCi and 0 . Note that, 3,1;0 iCi and 0 provided that condition (26b) holds.

Further 0~

11zy provided that condition (26a) holds. Therefore all the eigenvalues of yzJ have negative real

parts under the give conditions and hence the equilibrium point yzE is locally asymptotically stable, which

completes the proof. ■



Theorem 5: Assume that the equilibrium point )0,ˆ,ˆ,ˆ( zyxExyz of system (2) exists. Then it is locally

asymptotically stable in 4R if the following condition hold:

Bzwxw

xw

ˆ10ˆ

ˆ

1

9 (29)

Proof: Note that, it is easy to check that the Jacobian matrix of system (2) at the equilibrium point xyzE can be

written:

4,3,2,1,;][)( 44 jikEJJ ijxyzxyz

xkk ˆ1311 , xw

xkˆ

ˆ14

1 , ywk ˆ222 , ywk ˆ323 , zwh ˆ431 , zwk ˆ532 , zwwk ˆ6734 ,

Bzwkxw

xw

ˆ10ˆ

ˆ44

1

9 and zero otherwise. Therefore the characteristic equation can be written in the form:

0]ˆˆˆ)[ˆ( 322

13

44 KKKk


Bzwxw

xwzyx

ˆˆ

10ˆ

ˆ

1

9

111

Or 0ˆˆˆ32

21

3 KKK (30a)

here

3122133223113

3113322322112

22111 )(

kkkkkkK

kkkkkkK

kkK

(30b)

and

322322

31131122112211

321

)(

kkk

kkkkkkk

KKK

(31)

So, by substituting the values of ,ijh and then simplifying the resulting terms we obtain:

ywxK ˆˆ 21 , )(ˆˆˆ 42533 wwwwzyxK and ]ˆˆ[ˆ]ˆˆ[ˆ 5322

2422 zwwxwywzwywx


provided that 3,1;0 iKi and 0 . Note that, 3,1;0 iKi and 0 always as shown above. Further

0ˆ111zyx provided that condition (29) holds. Therefore all the eigenvalues of xyzJ have negative real parts

under the give condition and hence the equilibrium point xyzE is locally asymptotically stable, which completes

the proof. ■

Theorem 6: Assume that the equilibrium point ),,0,( wzxExzw

of system (2) exists. Then it is locally


zww

32 (32a)

10

1091

767

1

2

4 21)(

)()(,.min

w

wwww

wwzww

xwzxwxww

(32b)

zww

67 (32c)

Proof: Note that, it is easy to check that the Jacobian matrix of system (2) at the equilibrium point xzwE can be

written:

4,3,2,1,;][)( 44 jilEJJ ijxzwxzw

2

1 )(11 1

xw

wxl

, xl

13 , xw

xl

1

14 , zwwl

3222 , zwl

431 , zwl

532 , z

wwl

7

33

,

zwwl

6734 , 2

1

91

)(41

xw

wwwl

, wwl

1043 and zero otherwise. Therefore the characteristic equation can be

written in the form:

0])[( 322

13

22 LLLl




zwwwzx

32111

Or 0322

13 LLL

(33a)

here

)()(

)(

413343311441134311343

43344114311333112

33111

llllllllllL

llllllllL

llL

(33b)

and

)(

)())((

3114343343

1334141141331131132211

321

lllll

lllllllllll

LLL

(34)

So, by substituting the values of ,ijl and then simplifying the resulting terms we obtain:

z

ww

xw

wxL

7

2

1 )(1 1

zxw

wwww

xwx

xw

ww

xw

w

wzww

wzwwwxL

2

1

2

971

1

2

1

91

2

1

)(104

)(10

)(673 1)(

and

xw

zxw

z

wwzww

xw

wxw

x

xw

wxww

xw

wz

wxw

z

ww

xw

w

ww

zww

zxw

x

1

4767

2

112

1

91

2

1

7

7

2

1

)(10

67)()(

)(4

)(

)(1

1

1


provided that 3,1;0 iLi and 0 . Note that, 01 L under the condition (32b), while conditions (32a)-

(32c) ensure the positivity of 3L (i.e. 03 L ) and 0 . Further 0111wzx

provided that condition (32a)

holds. Therefore all the eigenvalues of xzwJ have negative real parts under the give condition and hence the

equilibrium point xzwE is locally asymptotically stable, which completes the proof.

Theorem 7: Assume that the positive equilibrium point ),,,( *wzyxExyzw of system (2) exists and let the

following inequalities hold:

21 xww (35a)

zwwwzwxwwww691101971

(35b)

2

1

121 72

4xw

wwww (35c)

wwwww 722

35 2 (35d)

Then it is locally asymptotically stable in the 4. RInt .

Proof. It is easy to verify that, the linearized system of system (2) can be written as

UEJ xyzwdtdU

dtdX )(

here twzyxX ),,,( and tuuuuU ),,,( 4321 where xxu1 , yyu2 , zzu3 and wwu4 .

Moreover,



2

1

111xw

wxa , xa13 ,

xw

xa1

14 , ywa 222 , ywa 323 , zwa 431 , zwa 532 ,

z

wwa 7

33 , zwwa 6734 , 2

1

91

41

xw

wwwa , wwa 1043 and zero otherwise. Now consider the following

positive definite function

www

xwu

z

u

y

u

x

uV

91

1

2

4

2

3

2

2

2

1

2

)(

222

It is clearly that 44: RRV and is a continuously differentiable function so that 0),,,( wzyxV and

0),,,( wzyxV otherwise. So by differentiating V with respect to

time t , gives

dt

du

www

xwu

dt

du

z

u

dt

du

y

u

dt

du

x

u

dtdV 4

91

14332211 )(

Substituting the values of 4,3,2,1; idt

dui in the above equation, and after doing some algebraic manipulation;

we get that:

43

3235

222314

21

91

10167

2

2

37

2

1

)(

)1(1

uu

uuww

uwuuwu

ww

wxw

z

zww

z

uww

xw

wdtdV

Obviously, due to conditions, we get that

43

)(

2

222

2

21

91

1019167

37

37

2

1

1

uu

uw

u

zww

zwxwwwzww

z

uww

z

uww

xw

wdtdV

Clearly 0dt

dV, therefore the origin and then xyzwE is locally asymptotically stable point in the 4. RInt and

hence the proof is complete. ■

V. Numerical Simulation In this section the global dynamics of system (2) is investigated numerically. The objectives are

confirming our analytical results and discuss the role of the existence of disease in the intermediate predator

population on the dynamical behavior of the system. For the following set of hypothetical, biologically feasible,

set of parameters, definitely different set of hypothetical parameters can be chosen also, system (2) is solved

numerically starting at different initial points as illustrated in Figure (2).

0001.0;1.0;1.0;3.0;1.0

1.0;1.0;3.0;3.0;2.0;6.0;6.0;5.0

131211109

87654321

wwwww

wwwwwwww (36)

Note that from now onward we will use solid line to describes the trajectory of the prey x ; the dashed

line to describes the trajectory of prey y ; the doted line to describes the trajectory of infected predator z ; the

dash dot to describes the trajectory of susceptible top predator w .

The effect of varying the half saturation constant rate of the susceptible first prey 1w on the dynamics

of system (2) is studied and then the trajectories of the system (2) are drawn in figures (3a)-(3d), for the values

of 9.0,7.0,5.01 w with the other parameter fixed as given in equation (36).



According to the above figures, it is clear that as the half saturation constant rate of susceptible of first

prey 1w increase the solution of system (2) approaches asymptotically to the coexistence equilibrium point.

However decreasing .1w

Now the effect of varying the growth rate of the susceptible second prey 2w on the dynamics of

system (2) is studied and then the trajectories of the system (2) are drawn in figures (4a)-(4d), for the values of

2.0,1,8.0,6.02 w with the other parameter fixed as given in equation (36).

According to the above figures, it is clear that as the growth rate of susceptible of second prey 2w

increase the solution of system (2) approaches asymptotically to the coexistence equilibrium point. However

decreasing 2w , say 2.00 2 w , the solution of system (2) approaches asymptotically to xzwE equilibrium

point.

The effect of varying the attack of second prey by the susceptible of predator 3w on the dynamical

behavior of system (2) is studied. The system (2) is solved numerically different values of the attack rate

9.0,7.0,6.03 w keeping other parameters fixed as given in equation (36), and then the solutions of system (2)

are drawn in figures (5a)-(5c).

Obviously, as the attack rate of second prey by the susceptible of predator 3w increase the value the

first prey species started increase while the value of other species are decrease and the system (2) still has

asymptotically stable coexistence equilibrium point.

The effect of varying the conversion rate from first prey by susceptible predator 4w on the dynamical

behavior of system (2) is investigated by choosing 8.0,5.0,3.04 w keeping other parameter fixed as given in

equation (36) and then the solutions of the system (2) are drawn in figures (6a)-(6b).

From the above figures, it is observed that, as the conversion rate between first prey and susceptible

predator 4w increases the all prey species starting decreases and all predator species increases but the system

still has an asymptotically stable coexistence point.

Now, the dynamical behavior of system (2) under the effect of varying the contact infection rate

106, ww is investigated. The system (2) is solved numerically for the set of parameters values given by equation

(36) with 7.0,4.0,3.0106 ww and then the trajectories of the system (2) are drawn in figures (7a)-(7c).

Again, the system (2) has an asymptotically stable coexistence equilibrium point. In addition it is

observed that, as the contact infection rate increases the second prey species and infected predator species

started increase while the value of the first prey species and susceptible predator species decrease.

Now, the effect of varying recovery rate 117 ww on the dynamical behavior of parameters values

given in equation (36) with 5.0,3.0,1.0117 ww and then the results are shown in figures (8a)-(8c).

Clearly, from the above figures, it is observed that increasing the value of recovery rate causes decreasing in the value of the second prey species and infected predator species while the values of the first prey

species and susceptible predator species increasing. And then the system (2) has approaches asymptotically to

the xyzE equilibrium point.

VI. Conclusions And Discussion In this paper, an eco-epidemiological model has been proposed and analyzed to study the dynamical

behavior of a Holling-type II prey–predator model with the disease in predator species. The model consists of

fore non-linear autonomous differential equations that describe the dynamics of fore different population’s namely first prey (X), second prey (Y), susceptible predator (Z), infected predator (W). In order to confirm our

analytical results and understand the effect of varying the infection rate 10,6, iwi and recovery

rate 11,7, iwi , on the dynamical behavior of the system (2), system (2) has been solved numerically for

different sets of initial points and different sets of parameters and the following observations are made:

1. For the set of hypothetical parameters values given by Equation (36), the system (2) approaches

asymptotically to globally stable point )17.0,62.0,37.0,06.0(xyzwE .

2. For the value of the half saturation constant rate of susceptible of first prey 1w increase the solution of

system (2) approaches asymptotically to the coexistence equilibrium point and increase in values of zx, and

w while decrease value of y .

3. For the values of the growth rate 2w increase then the system (2) still approaches to coexistence



equilibrium point and the values of zy, and w increase while the value of x decrease.

4. The value of attack rate parameter 3w increase then the system (2) still approaches to xyzwE equilibrium

point. And the value of x increase but the values of zy, and w decrease.

5. For increasing the value of conversion rate 4w leads to increase in the values of wz, while decreasing in

the values of yx, species.

6. In addition it is observed that, the system (2) has an asymptotically stable coexistence equilibrium point, as

the contact infection rate increases the values of wy, species started increase while the value of zx,

species decrease.

7. It is observed that increasing the value of recovery rate causes decreasing in the value of wy, species while

the values of zx, species increasing. And then the system (2) has approaches asymptotically to the xyzE

equilibrium point.

Figure (1): Bloke diagram of our proposed model.

Figure (2): The solution of system (2) approaches asymptotically to the positive equilibrium point

)17.0,62.0,37.0,06.0(xyzwE for that data given by Eq. (36) starting from two different initial points (0.8, 0.7,

0.4, 0.2) and (0.1, 0.3, 0.8, 0.6) for sold line and dashed line respectively. (a) Trajectories of x . (b) Trajectories

of y . (c) Trajectories of z . (d) Trajectories of w .



Figure (3): Time series of solutions of the system (2). (a) 5.01 wfor , (b) 7.01 wfor , (c) 9.01 wfor .

Figure (4): Time series of solutions of the system (2). (a) 6.02 wfor , (b) 8.02 wfor , (c) 12 wfor , (d)

2.02 wfor .







Figure (7): Time series of solutions of the system (2). (a) 3.0106 wwfor , (b) 4.0106 wwfor , (c)

7.0106 wwfor .



Figure (8): Time series of solutions of the system (2). (a) 1.0117 wwfor , (b) 3.0117 wwfor , (c)

5.0117 wwfor .

References [1]. Murray, J.D.2002. Mathematical biology an introduction. Third edition. Springer-Verlag. Berlin Heidelberg.

[2]. Smith, J.M. 1974. Models in ecology. Cambridge university press. Great Britain.

[3]. May, R.M. 1974. Stability and complexity in model ecosystems.Princeton University Press. New Jersey.

[4]. Anderson, R.M. and May, R.M. 1986. The invasion and spread of infectious disease with in

[5]. Animal and plant communities,

[6]. Philos. Trans. R. Soc. Lond. Biol.

[7]. Sci. 314, pp. 533-570.

[8]. Haque, M., Zhen, J., and Venturino, E. 2009. Rich dynamics of Lotka-Volterra type predator-prey model system with viral disease

in prey species. Mathematical Methods in the Applied Sciences, 32, pp: 875-898.

[9]. Arino, O., Abdllaoui, A. El, Mikram, J., and Chattopadhyay, J. 2004. Infection in prey population may act as a biological control in

ratio-dependent predator-prey models. Nonlinearity, 17,pp: 1101-1116.

[10]. Chatterjee, S., Kundu, K., and Chattopadhyay, J. 2007. Role of horizontal incidence in the occurrence and control of chaos in an

eco-epidemiological system. Mathematical Medicine and Biology, 24,pp: 301-326.

[11]. Xiao, Y. and Chen, L. 2001. Modeling and analysis of a predator-prey model with disease in the prey. Mathematical Bioscences,

171, pp: 59-82.

[12]. Haque, M. 2010. A predator-prey model with disease in the predator species only. Nonlinear Analysis; RWA, 11(4),pp: 2224-2236.

[13]. Haque, M. and Venturino, E.2006. Increasing of prey species may extinct the predator population when transmissible disease in

predator species.HERMIS,7,pp: 38-59

[14]. Das, K.P.2011. A Mathematical study of a predator prey dynamics with disease in predator. ISRN Applied Mathematics,pp:1-16.

[15]. Venturino, E. 2002. Epidemics in predator-prey models: disease in the predators. IMA Journal of Mathematics applied in medicine

and biology, 19,pp: 185-205.

[16]. Haque,M. and Venturino,E.2007.An eco-epidemiological model with disease in predator, the ratio-dependent. Mathematical

Methods in the Applied Sciences, 30, pp: 1791-1809.

[17]. Dahlia, Kh., B., 2011. Stability of a prey-predator model with SIS epidemic model disease in prey. Iraq Journal of Science, Vol.52,

No. 4, pp: 484-493.

Documents

Stability of a Prey-Predator Model with SIS Epidemic Disease in Predator Involving Holling Type II Functional Response