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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…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 39 | Page
)(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)
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 40 | Page
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)
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 41 | Page
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
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 42 | Page
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
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 43 | Page
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
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 44 | Page
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
asymptotically stable in 4R if one of the following sets of conditions hold
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
Consequently, either
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. ■
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 45 | Page
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
Consequently, either
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
According to the Routh-Hurwitz criterion all the roots (eigenvalues) of third order equations have real parts
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
asymptotically stable in 4R if one of the following sets of conditions hold
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
Consequently, either
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 46 | Page
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
According to the Routh-Hurwitz criterion all the roots (eigenvalues) of third order equations have real parts
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,
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 47 | Page
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).
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 48 | Page
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
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 49 | Page
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 .
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DOI: 10.9790/5728-11233853 www.iosrjournals.org 50 | Page
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 .
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DOI: 10.9790/5728-11233853 www.iosrjournals.org 51 | Page
Figure (5): Time series of solutions of the system (2). (a) 6.03 wfor , (b) 7.03 wfor , (c) 9.03 wfor .
Figure (6): Time series of solutions of the system (2). (a) 3.04 wfor , (b) 5.04 wfor , (c) 8.04 wfor .
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
DOI: 10.9790/5728-11233853 www.iosrjournals.org 52 | Page
Figure (7): Time series of solutions of the system (2). (a) 3.0106 wwfor , (b) 4.0106 wwfor , (c)
7.0106 wwfor .
Staibilty Of A Prey-Predator Model With Sis Epidemic Disease In Predator Involving…
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Figure (8): Time series of solutions of the system (2). (a) 1.0117 wwfor , (b) 3.0117 wwfor , (c)
5.0117 wwfor .
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