The Dynamics of Four Species Food Web Model With Stage Structure

8/16/2019 The Dynamics of Four Species Food Web Model With Stage Structure

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INTERNATIONAL JOURNAL OF TECHNOLOGY ENHANCEMENTS AND EMERGING ENGINEERING RESEARCH, VOL 4, ISSUE 3 14ISSN 2347-4289

Copyright © 2016 IJTEEE.

3. When the top predator exists in the third level, it is

assumed that the top predator consumes both thepreys in the first level according to Lotka-Volterra typeof the functional response with maximum attack rates

01 c and 02 c for )(1 t N and )(2 t N respectively,

while it attacks the immature predator at the secondlevel with maximum attack rate 0 . Further, it is

assumed that there is enter-specific competitionbetween the mature predator and top predator with

intensity of competition rates 01 and 02

respectively. Finally both the predators (maturepredator and top predator) are decay exponentially with

natural death rates 01 d and 02 d respectively in

the absence of their food.

According to these assumptions the dynamics of the abovedescribed food web system can be formulatedmathematically with the following set of differentialequations:

52

542535522451135

41541

4222411134

533422241113

5224222122

22

5114112111

11

)1()1(

1

1

N d

N N N N e N N ce N N cedT

dN

N d N N

N N bem N N ben N dT

dN

N N N N N bme N N bnedT

dN

N N c N N b N N a L

N sN

dT

dN

N N c N N b N N a K

N

rN dT

dN

….…..(1)

Here 0)0(1 N , 0)0(2 N , 0)0(3 N , 0)0(4 N and

0)0(5 N . Note that the above model contains 23 positive

parameters in all, which makes the analysis of the systemvery difficult. So, in order to reduce the number ofparameters and determine which parameters represent thecontrol parameters, the following dimensionless variablesare used.

r

d u

bu

a

ecu

r

Kecu

r

d u

cu

a

ebu

r

e Kbu

r

bu

cu

r u

ba

ebu

r

K eu

c

cu

b

bu

r

Kau

La

r u

r

su N

r

c x N

r

b x

N r

x N r

a x

K

N xrT t

218

1

217

1

4216

3115

114

1

113

1

2212

1111

110

198

11

227

16

1

25

1

24

23

1215

154

14

3321

21

1

,,,

,,,,

,,,,

,,,,

,,,,

,,,,

Accordingly, the dimensionless of system (1) becomes

)(

)(

)1()1(

)(

)(

)1(

)()1(

5

5185417535521651155

44145413

421241113104

3539384274163

2

52542421322212

1514121111

f

xu x xu x xe x xu x xudt

dx

f xu x xu

x xum x xun xudt

dx

f x xu xu x xmu x xnudt

dx

f

x xu x xu x xu xu xudt

dx

f x x x x x x x xdt

dx

……(2)

Here ,),,,,( 54321T x x x x x 0)0(1 x , 0)0(2 x

0)0(3 x , 0)0(4 x and 0)0(5 x . Clearly, the interaction

functions 4321 ,,, f f f f and 5 f of system (2) are continuous

and have continuous partial derivatives on the state space

}.0)0(,0)0(,0)0(

,0)0(,0)0(:{

543

2155

x x x

x x R R

Hence these functions are Lipschizian on 5 R and then the

solution of the system (2) with nonnegative initial conditionexists and is a unique. Further, all the solutions of system

(2) which initiate in 5 R are uniformly bounded as shown in

the following theorem.

Theorem (1): All the solutions of system (2), which initiate

in 5 R , are uniformly bounded.

Proof: From the first equation of system (2) we get:

)1( 111 x x

dt

dx

Then according to the comparison theorem [18], the abovedifferential inequality gives that

1)(suplim 1

t xt

, hence 1)(1 t x ; 0t

Similarly, from the second equation of system (2) we obtain

that

22

1)(suplim

ut x

t

,hence2

2

1)(

ut x ; 0t

Now define the function

)()()()()()( 54321 t xt xt xt xt xt M and then take the

time derivative of )(t M along the solution of system (2

gives:

M H dt

dM H M

dt

dM


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Where },,,1min{ 181411 uuu , 211 22 xu x H with

108 uu . Now, it is easy to verify that the solution ofthe above linear differential inequalities can be written

t e

H M

H t M

0)(

Where ))0(),0(),0(),0(),0(( 543210 x x x x x M ,So that

H t M

t

)(suplim 0;)( t H t M

.

Thus all solutions are uniformly bounded and the proof iscomplete.

■

3. Existence of equilibrium pointsIt is observed that, system (2) has at most elevenbiologically feasible equilibrium points, namely

10,...,2,1,0; i E i . The existence conditions for each of these

equilibrium points are derived in the following. Thevanishing equilibrium point )0,0,0,0,0(0 E and the axial

equilibrium points )0,0,0,0,1(1 E and )0,0,0,,0(2

12 u

E

always exist. The first two species equilibrium point

)0,0,0,,( 213 x x E , where

213

211

)1(

uuu

uu x

, 12 1 x x (3a)

exists under one set of the following sets of conditions

13 uu & 1

2 u (3b)

Or

13 uu & 12 u (3c)

The second two species equilibrium point

)ˆ,0,0,0,ˆ( 514 x x E , with

,ˆ15

181

u

u x and 15 ˆ1ˆ x x (4a)

exists under the condition

1518 uu (4b)

The third two species equilibrium point ),0,0,,0( 525 x x E

,

where

)1( 225

15 xu

u

u x

and16

182

u

u x

(5a)


16182 uuu (5b)

Moreover, the first three species equilibrium poin

)0,,,0,( 4316 x x x E where

14

118

63

118106

1481

1

)1(,)1(

x x

x xu

nu x

uununu

uu x

(6a)

exists if the following condition holds

118106148 )1( uununuuu (6b)

The second three species equilibrium point

)0,~,~,~,0( 4327 x x x E where

)~1(~

),~1(~~

,)1(

~

222

14

22228

173

128107

1482

xuu

u x

xu xuu

umu x

uumumu

uu x

(7a)


1281071482 )1( uumumuuuu (7b)

The third three species equilibrium point

)~~,0,0,

~~,~~( 5218 x x x E , where

215

152151653

151518532

15

216181

~~~~1~~

,)()(

)()(~~

,

~~~~

x x x

uuuuuuu

uuuuuu x

u xuu x

(8a)

exists if the following condition holds

1615

1815

152151653

15151853

)()(

)()(0

uu

uu

uuuuuuu

uuuuuu

(8b)

The top predator free equilibrium point

)0,,,,( 43219 x x x x E

,which is given by


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5/20


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Straightforward computation shows that all the eigenvalues

of )( 2 E J have negative real parts if the following conditions

hold:

1482

107128

18216

2

)1(

1

uuu

umuuum

uuu

u

(14e)

Hence 2 E is locally asymptotically stable. However, it is a

saddle point otherwise. The Jacobian matrix of system (2)

at 3 E can be written as

)(

0000

000

000

0

0

)(

55

4410

27168

254422123

1111

3

ijd

d

d u

xmu xnuu

xu xu xuu xu

x x x x

E J

…….(15a)

Here

1821611555

1421211144 ,)1()1(

u xu xud

u xum xund

Then the characteristic equation of )( 3 E J is given by

0)( 18216115212212 u xu xu B B A A …..(15b)

where

22111 xuu x A and 0)( 213212 x xuuu A under the

second condition of the existence of 3 E . While

])1()1[( 8142121111 uu xum xun B ,

22111482 x xuu B

with 1061181 )1( unuuun , 1071282 )1( umuuum .

Therefore the eigenvalues can be written as:

18216115

221

1

2

2

1

1

5

43

21

42

1

2,

42

1

2,

u xu xu

B B B

A A

A

x

x x

x x

(15c)

Accordingly, it is easy to verify that all these eigenvalueshave negative real parts if the following conditions aresatisfied

148212111

1482211

18216115

)1()1( uu xum xun

uu x x

u xu xu

(15d)

Hence, 3 E is locally asymptotically stable. However, it is a

saddle point otherwise.

The Jacobian matrix of system (2) at 4 E can be written as

)ˆ(

0ˆˆˆ

0ˆ00

0ˆ00

000ˆ0

ˆˆ0ˆˆ

)(

517555161815

4410

168

22

1111

4

ijd

xu xe xuuu

d u

xunu

d

x x x x

E J

….(16a)

Here 5513122 ˆˆˆ xu xuud 1451311144 ˆˆ)1(

ˆ u xu xund

The characteristic equation of )( 4 E J is given by

0ˆˆˆˆˆˆˆˆ)ˆˆ( 21221222 B B A Ad (16b)

Where 11 ˆˆ x A and 118152 ˆ)(

ˆ xuu A , while

1115131481 ˆ)1(ˆˆ xun xuuu B and

110611851381482 ˆ])1[(ˆˆ xunuuun xuuuu B . Therefore

the eigenvalues are:

221

1

2211

55131

ˆ4ˆ2

1

2

ˆˆ,ˆ

ˆ4

ˆ

2

1

2

ˆˆ

,ˆ

ˆˆˆ

43

51

2

B B B

A A A

xu xuu

x x

x x

x

(16c)

Hence, all these eigenvalues have negative real parts if thefollowing conditions are satisfied

51381481106118

55131

ˆˆ])1[(

ˆˆ

xuuuu xunuuun

xu xuu (16d)

Thus, 4 E is locally asymptotically sable in the5 R

however, it is a saddle point otherwise.

The Jacobian matrix of system (2) at 5 E can be written as

)(

0

000

000

0

00001

)(

51755516515

4410

278

252422123

52

5

ijd

xu xe xu xu

d u

xmuu

xu xu xuu xu

x x

E J

…..(17a)


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Here 1451321244 )1( u xu xumd

. So the characteristic

equation of )( 5 E J is given by:

021221211 B B A Ad

(17b)

Where 2211 xuu A

and 521652 x xuu A

, while

2125131481 )1(ˆ

xum xuuu B

,

210712851381482 ])1[(ˆ xumuuum xuuuu B

. Thus the

eigenvalues of )( 5 E J can be written as:

221

1

221

1

52

42

1

2,

42

1

2,

1

43

52

1

B B B

A A A

x x

x x

x x

x

(17c)

Now straightforward computation shows that all theeigenvalues of )( 5 E J have negative real parts provided

that the following conditions are satisfied

51381482107128

52

)1(

1

xuuuu xumuuum

x x

(17d)

Hence 5 E is locally asymptotically stable in the 5 R ,

however it is a saddle point otherwise. The Jacobian matrix

of system (2) at 6 E can be written as

)(

0000

)1()1(

0000

0

)(

55

4134410412411

391684746

22

1111

6

ijd

d

xud u xum xun

xu xnuu xmu xnu

d

x x x x

E J

....(18a)

Here

14111444413122 )1(, u xund xu xuud

184173511555 u xu xe xud .

Hence the characteristic equation of )( 6 E J is given by

0])[)(( 322

13

5522 A A Ad d ……(18b)

Where

][

;);(

3142113

21331124433111

Rd Rd A

R Rd d Ad d d A

With

413343313

433444332411444111 ;;

d d d d R

d d d d Rd d d d R

While

31424433133111321 )()( Rd Rd d Rd d A A A A

So the eigenvalues in the 2 x and 5 x -directions are given

by

1841735115

44131

5

2 ;

u xu xe xu

xu xuu

x

x

(18c)

However the other three eigenvalues represent the roots othe third order polynomial in Eq. (18b), which have negative

real parts if and only if 01 A , 03 A and 0 . So

straightforward computation shows that all the eigenvalues

of )( 6 E J have negative real parts if the following conditionsare satisfied:

31424433133111

106118

148

11

141

1841735115

44131

)()(

)1(,

)1(min

Rd Rd d Rd d A

unuuun

uu

un

u x

u xu xe xu

xu xuu

…..(18d)

So, 6 E is locally asymptotically stable, however, it is saddle

point otherwise. The Jacobian matrix of system (2) at 7 E

can be written as

)~

(

~0000

~~)1(~)1(

~~~~

~~0~~0000~~1

)(

55

41310412411

392784746

252422123

42

7

ijd

d

xuu xum xun

xu xmuu xmu xnu

xu xu xuu xu

x x

E J

…..(19a)

Here

1421244 ˆ)1(~

u xumd 184173521655~

ˆˆ~

u xu xe xud .

The characteristic equation of )( 7 E J is written as:

0]~~~~~~

)[~~

)(~~

( 322

13

5511 A A Ad d ……(19b)

Where

]~~~~

[~

;~~~~~

);~~~

(~

3242223

21332224433221

Rd Rd A

R Rd d Ad d d A

423343323

433444332422444221

~~~~~;

~~~~~;

~~~~~

d d d d R

d d d d Rd d d d R


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While

32424433133221321~~~

)~~

(]~~~

[~~~~~

Rd Rd d Rd d A A A A

Therefore the eigenvalues in the 1 x and 5 x -directions are

given by

1841735216

42

~~~~

,~~1~

5

1

u xu xe xu

x x

x

x

(19c)

However the other three eigenvalues represent the roots ofthe third order polynomial in Eq. (19b), which have negative

real parts if and only if 0~1 A , 0

~3 A and 0

~ . So

straightforward computation shows that all the eigenvalues

of )( 7 E J have negative real parts if the following conditions

are satisfied:

24433133221324

107128

148

12

142

1841735216

42

~)

~~(]

~~~[

~~~

)1(,

)1(min~

~~~

~~1

Rd d Rd d A Rd

umuuum

uu

um

u x

u xu xe xu

x x

……(19d)

So, 7 E is locally asymptotically stable, however, it is saddle

point otherwise. Now, since the stability analysis of theremaining equilibrium points of system (2), usinglinearization method, became more complicated, thereforewe will study them with the help of Lyapunov method. In thefollowing we will start first to specify the region of global

stability of the equilibrium points 7,,2,1; i E i .

Theorem (2): Assume that 1 E is locally asymptotically

stable in 5 R and the following conditions hold

119164

5615912511755

)1(

)()1()1(

uuuun

enuuuuumuuuenm

…..(20a)

1065118515109 )1( uuneuuenuuu (20b)

15149151191465 )1( uuuuuunuune (20c)

21

2

1 )1(4

x

(20d)

Where155

1611551

uu

uuuu and161155

16212

uuuu

uuu

. Then the

equilibrium point 1 E is globally asymptotically stable.

Proof: Consider the following function

5544

33221115211 )ln1()...,,(

xr xr

xr xr x xr x x x L

Here 5,,2,1; ir i are positive constants to be determined.

It is easy to see that

),,()...,,( 515211 R RC x x x L in addition ,0)0,0,0,0,1(1 L while

0)...,,( 5211 x x x L ,5

51 ),...,( R x x and

)0,0,0,0,1(),...,( 51 x x . Further more by taking the derivative

with respect to the time and simplifying the resulting termswe get that

518551155154175134

41144535593

3104835216552

421247342

4111463122212

2121213212111

)()(

)()(

)()(

])1([

])1([

)()()1(

xur x xur r x xur ur

xr ur x xer ur

xur ur x xur ur

x xumr mur ur

x xunr nur r xuur

xur r x xur r xr dt dL

Now by choosing the positive constants 5,,2,1; ir i as

follows

155

15119

561594

159

53

155

1621

1,

)1(

,,,1

ur

uuun

enuuur

uu

e

r uu

u

r r

and then substituting them in the above equation , we get

415119

6515914

315119

5615910

159

85

42)1(

)()1()()1(

51855415

17

15119

6515913

222121155

16321

1

1)1(

)(

)1(

)(

)1(

)(

11)1(

151195

55615912755169411

xuuun

uneuuu

xuuun

unuuuu

uu

ue

x x

xuu x xu

u

uuun

uneuuu

x x x xuu

uu x

dt

dL

uuuun

uenuuuumuumeuuuun

Now, due to the boundedness of the logistic term

]1[ 2221 x x by the 21 4 , then its easy to verify tha

dt

dL1 is negative definite under the sufficient conditions

(20a)-(20d). Hence the solution of system (2) will approach

asymptotically to 1 E from any initial point satisfies the

above condition and then the proof is complete.■

Theorem(3):Assume that )0,0,0,,0( 22 x E

;

22

1

u x

islocally asymptotically stable in 5 R then it is a

globally asymptotically stable provided that the following

119164

5615912511755

1

11

uunuu

enuuuuumuuuenm

……..(21a)

1065118515109 1 uuneuuenuuu (21b)


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151495216119414655 1 uuuu xuuuunuuune

(21c)

18216 u xu

(21d)

155

1621

2

1

4 uu

uuu

(21e)

Here

155

21631551

uu

xuuuu

and

2163155

1552

xuuuu

uu

.

Proof: Consider the following functions

554433

2

22222115212 ln,.......,,

xc xc xc

x

x x x xc xc x x x L

Where 5,....,1, ici are positive constants to be determined.

It is easy to see that ,,,....., 51512 R RC x x L and 00,0,0,,0 22 x L

while 551512 ,....,;0,...., R x x x x L

and .0,0,0,,0,..., 251 x x x

. Further more by taking the

derivative with respect to the time and simplifying theresulting terms, we get that

518554175134

525242425216552

421247342

2132310483

535593511551

4111463121321

4144

2

222121112

1

1

1

xuc x xucuc

x xuc x xuc x xucuc x xumcmucuc

x xuc xucuc

x xecuc x xucc

x xuncnucc x xucc

xuc x xuuc x xcdt

dL

So by choosing the constants 5,.....,2,1, ici as follow

155

15119

651594

159

53

155

1621

1,

1

,,,1

uc

uuun

uneuuc

uu

ec

uu

ucc

Thus by substituting these constants in the above equation,we get that

515

2161821

155

163

5415

17

15119

6515913

315119

65159101185

421

11

4151195

216119465159145

2

22155

16211211

2

1

1

11

1

1

1

151195

56515912755169411

xu

xuu x x

uu

uu

x xu

u

uuun

uneuuu

xuuun

uneuuuuuen

x x

xuuuun

xnuuuuuneuuuu

x xuu

uuu x x

dt

dL

uuuun

uuneuuumuumeuuuun

Now, due to the boundedness of the logistic term

]1[ 1211 x x by the 21 4 , then its easy to verify that

dt

dL2 is negative definite under the sufficient conditions

(21a)-(21e). Hence the solution of system (2) will approachasymptotically to 2 E from any initial point satisfies the

above condition and then the proof is complete.■


stable in 5 R . Then, it is a globally asymptotically stable

provided that the following conditions hold.

12655161194

15129511755

11

11

uuuemnuuuun

uuuumuuuenm

……(22a)

10

118565159

14

21196411511965

1)1(1

u

uuen

uneuu

u

xuuuun xuuun

unu (22b)

18216115 u xu xu (22c)

5

1615212

5

16315 4

u

uuuu

u

uuu

(22d)

Proof: Consider the following functions

5544332

12222

1

11111513

ln

ln,....,

xc xc xc x

x x x xc

x

x x x xc x x L

Where 5,....,1, ici are positive constants to be determined

It is easy to verify that ,,,....., 51513 R RC x x L and 00,0,0,, 213 x x L while 0,...., 513 x x L for al

551,...., R x x and 0,0,0,,,..., 2151 x x x x . Moreover bytaking the derivative with respect to the time and simplifyingthe resulting terms, we get that


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424211144

525211185

51155154175134

535593310483

421247342

41114631

52165522

22212

22113212

1113

)1(

)1(

x xuc xcuc

x xuc xcuc x xucc x xucuc

x xecuc xucuc

x xumcmucuc

x xuncnucc

x xucuc x xuuc

x x x xucc x xcdt

dL

So by choosing the positive constants as below

1,

1

,,,

5119

651594

9

53

5

162151

cuun

uneuuc

u

ec

u

ucuc

and then substituting these constants in the above

equation, we get that

521611518

5417119

6515913

41

1

3119

65159101185

421

11

22115

16315

222

5

162121115

3

1

1

1

1195

2164115511965159145

1195

65159125755169411

x xu xuu

x xuuun

uneuuu

x

xuun

uneuuuuuen

x x

x x x xu

uuu

x xu

uuu x xu

dt

dL

uuun

xuu xuuuununeuuuu

uuun

uneuuuumuumeuuuun

So, by using condition (22d) we obtain that

521611518

5417119

6515913

3119

65159101185

41

1

421

11

2

225

16211115

3

1

1

1

1195

2164115511965159145

1195

65159125755169411

x xu xuu

x xuuun

uneuuu

xuun

uneuuuuuen

x

x x

x xu

uuu x xu

dt

dL

uuun

xuu xuuuununeuuuu

uuun

uneuuuumuumeuuuun

Now its easy to verify thatdt

dL3 is negative definite under

the sufficient conditions (22a)-(22c). Hence the solution of

system (2) will approach asymptotically to 3 E from any

initial point satisfies the above condition and then the proois complete. ■


stable in 5 R , then it is globally asymptotically stable

provided that the following conditions hold:

12655161194

15129511755

1111

uuuemnuuuunuuuumuuuenm

(23a)

52161611155 ˆˆ xuuu xuu (23b)

6510

859115159

15914

1711511965

)ˆ(1

)ˆ(1

uneu

u xuuenuu

uuu

u xuuunune

(23c)

211515

18ˆˆ x x x

u

u (23d)


5

555554433

221

11111514

ˆlnˆˆˆˆˆ

ˆˆ

lnˆˆˆ,....,

x

x x x xc xc xc

xc x

x x x xc x x L

Where 5,....,1,ˆ ici are positive constants to be determined

It is easy to see that ,,,....., 51514 R RC x x L and 0

ˆ,0,0,0,ˆ 514 x x L while 0,...., 514 x x L for al 551,...., R x x and .ˆ,0,0,0,ˆ,..., 5151 x x x x Further moreby taking the derivative with respect to the time andsimplifying the resulting terms, we get that

51854517511144518541114631

421247342

5216552212115162

535593310483555

2113255111551

5417513422212

2111

4

ˆˆˆˆˆˆˆ

ˆ1ˆˆˆ

1ˆˆˆ

ˆˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆ

xuc x xuc xcuc

xuc x xuncnucc

x xumcmucuc

x xucuc xuc xc xuc

x xecuc xucuc xec

x xcuc x x x xucc

x xucuc xuuc x xcdt

dL

Therefore by choosing the positive constants as below

155

15119

651594

159

53

155

1621

1ˆ,

1ˆ

,ˆ,ˆ,1ˆ

uc

uuun

uneuuc

uu

ec

uu

ucc


11/20


12/20



39

539 xu

e xu

(25c)

417

4115 xu

x xu (25d)

417125431375416134

417125341375

1

1

xuuum x xuumu xuuu

xuuum x xuumu

..(25e)

1582

46 uu xnu (25f)

413

117111714152

13

171115

11

xu

xuunuuu

u

uunu

(25g)

413

11711171482

413

171016

1

xu

xuunuuu

xu

uu xnu

...(25h)


552

4442

333

221

11111516

22

,....,

xc x xc

x xc

xc x

x L x x xc x x L n

Where 5,....,1, ici are positive constants to be determined.

It is easy to see that ,,,....., 51516 R RC x x L and 00,,,0, 4316 x x x L while 0,...., 516 x x L for all

551,...., R x x and 0,,,0,,..., 43151 x x x x x . Further more

by taking the derivative with respect to the time andsimplifying the resulting terms, we get that

544134175242124432735355393

5241343311463

5115515216552

42412437342

2

33832132211

441141141

52393

2

441114144

5114433104163

22212211211

2

1116

1

1

1

1

x x xucuc x xumc

x x xmuc x xec xuc

x xuc x x x x xnuc

x xucc x xucuc

x x xumc xmucuc

x xuc x xuc xuc

x x x x xuncc

x xuc x x xuncuc

x xc x x x xuc xnuc

xuuc x xc x xc x xcdt

dL

Now by choosing the positive constants as below

1,,, 53413

174

5

162151 cc

xu

uc

u

ucuc

and then substituting these constants in the above equationand using the conditions (25a),(25f)-(25h), we get that

42

4121413

1737

12113

17374

5

16

5353939511524

4

17

21151152215

16213

5

1615

2

)44(4132

111114173328

2

)44(4132

1111141711

2

15

2

)33(2

1811

2

156

x x

xum xu

u xmu

umu

u xmuu

u

u

x xe xu xu x xu x x

u

xuu xu xuuu

u x xu

u

uu

x x xu

xunuu x xu

x x xu

xunuu x x

u

x xu

x xu

dt

dL


dL6 is negative definite unde

the sufficient conditions (25b)-(25e). Hence the solution o


initial point satisfies the above condition and then the proois complete. ■ Note that the stated sub region in the abovetheorem represents the basin of attraction of the equilibrium

point 6 E .


stable in 5 R , then it is globally asymptotically stable in the

sub region of 5 R that satisfies the following conditions:

1155

2163155~

xuu

xuuuu

(26a)

12

142

1 um

u x

(26b)

39

539~

xu

e xu

(26c)

417

4216~~

xu

x xu (26d)

417114313641513

4171134136

~1~~~1~

xuun x xunu xuu

xuun x xunu (26e)

5

16821247

~

u

uuuu xmu (26f)

4135

2171217141621

2

13

1712

5

164

~1

1

xuu

xuumuuuuu

mu

uu

u

uu

(26g)


13/20



413

2171217148

2

413

171027

~1

~

xu

xuumuuu

xu

uu xmu

(26h)


552

4442

333

2

2222211517

~~

2

~~

2

~

~~~~~,...,

xc x xc

x xc

x x L x x xc xc x x L n

where 5,....,1,~ ici are positive constants to be determined.

It is easy to see that ,,,....., 51517 R RC x x L and 00,~,~,~,0 4327 x x x L while 0,...., 517 x x L for all

551,...., R x x and 0,~,~,~,0,..., 43251 x x x x x . Further more


44224124

5355393393

413634114

41143631

2441445216552

233834433104

4422422

22212

5115513322473

24421245252

24134

5441341754433273

213211232111

7

~~~1~

~~~~

~1~

~1~~~~

~~~~

~~~~~

~~~~~

~~~~~~

~1~~~~

~~~~~~

~~~~~~

x x x x xumc

x xec xuc xuc

x x xnuc xunc

xunc xnucc

x xuc x xucuc

x xuc x x x xuc

x x x xuc x xuuc

x xucc x x x x xmuc

x x xumc x xuc xuc

x x xucuc x x x x xmuc

x xucc x xucc xcdt

dL


1~~,~~,~,~ 53

413

174

5

162151 cc

xu

uc

u

ucuc

and then substituting these constants in the above equationand using the conditions (26b),(26f)-(26h), we get that

413641114

~13

17

113

17113

~615

5353~

93952~

1624

4~17

215

1631512

~

5

16315115

2

4~

44

~132

212114173

~3

2

8

2

4

~

44~132

21211417

2~

252

1621

2

3~

32

82

~2

52

16217

x x xnu xun xu

u

nu

uu xnuu

x xe xu xu x xu x x

u

x xu

uuu x x

u

uuu xu

x x xu

xumuu x x

u

x x xu

xumuu

x xu

uuu

x xu

x xu

uuu

dt

dL



the sufficient conditions (26a), (26c)-(25e). Hence the

solution of system (2) will approach asymptotically to 7 E

from any initial point satisfies the above condition and thenthe proof is complete. ■ Theorem (9): Assume that the third three species

equilibrium point 8 E exists, then it is a globallyasymptotically stable in 5 R , if the following conditions hold

muuuneuuuun

uuuumuuuenm

11

11

12655161194

15129511755 (27a)

10

598115

145

517524119

14

1151191465

~~165159

~~~~1~~1

u

xuuuen

uu

xuu xuuun

u

xuuunuune

uneuu

(27b)

5

1615212

5

163155 4u

uuuu

u

uuuu

(27c)


5

55555

2

22222

4433

1

11111518

~~ln

~~~~~~~~

ln~~~~~~

~~~~~~

ln~~~~~~,...,

x

x x x xc

x

x x x xc

xc xc x

x x x xc x x L

where 5,....,1,~~ ici are positive constants to be determined

It is easy to see that ,,,....., 51518 R RC x x L and 0~~,0,0,~~,~~ 5218 x x x L while 0,...., 518 x x L for al

551,...., R x x and 52151~~,0,0,

~~,

~~,..., x x x x x . Further more


355510483

41114631

552216552

421247342

4517524211144

54175134535593

2

2221255111551

2211321

2

1118

~~~~~~~~

1~~~~~~

~~~~~~~~

1~~~~~~

~~~~~~~~~~~~~~

~~~~~~~~

~~~~~~~~~~~~

~~~~~~~~~~~~

x xecucuc

x xuncnucc

x x x xucuc

x xumcmucuc

x xuc xuc xcuc

x xucuc x xecuc

x xuuc x x x xucc

x x x xucc x xcdt

dL


1

~~,1

~~

,~~,

~~,~~

5119

561594

9

53

5

162151

cuun

enuuuc

u

ec

u

ucuc

Then substituting these constants in the above equationand using the condition (27c), we get that


14/20



411951

2~~

119415~~

171~~

1511951

11951

14565159145

54171191

6515913

31191

56159105~~981151

4211951

561591251

11951

7551694111

2

2~~

25

16211~~

1158

xuuun

xuuun xu xuuuun

uuun

uuuneuuuu

x xuuun

uneuuu

xuun

enuuuu xuuuen

x xuuun

enuuuuum

uuun

uumeuuuun

x xu

uuu x xu

dt

dL



the sufficient conditions (27a)-(27b). Hence the solution ofsystem (2) will approach asymptotically to 8 E from any

initial point satisfies the above condition and then the proofis complete. ■ Theorem (10): Assume that the top predator free

equilibrium point 9 E exists, then it is a globally

asymptotically stable in the sub region of 5 R that satisfies

the following conditions:

39

539 xu

e xu

(28a)

18216115 u xu xu

(28b)

14212111 11 u xum xun

(28c)

5

16521212

9

4

u

uuuud (28d)

44152

149

4d ud (28e)

445

1621224

9

4d

u

uuud (28f)

4482

349

4d ud (28g)

1582

469

4uu xnu

(28h)

168212

47 9

4

uuuu xmu

(28i)

where

5

16315512

u

uuuud

,

13

1711151324

1

u

uunuud

,

135

171251613424

1

uu

uuumuuud

,

413

1710241371413634

xu

uu x xumu x xunud

and

413

212111141744

11

xu

xum xunuud


552

444

2

22222

2

333

1

11111519

2ln

2ln,...,

xc x xc

x

x x x xc

x xc

x

x x x xc x x L

where 5,....,1, ici

are positive constants to be determined

It is easy to see that ,,,....., 51519 R RC x x L and 00,,,, 43219 x x x x L

while 0,...., 519 x x L for al

551,...., R x x and 0,,,,,..., 432151 x x x x x x

. Furthe

more by taking the derivative with respect to the time andsimplifying the resulting terms, we get that

524134544134175

441141141

4422412442

2

33833322473

2

222123311463

5216552525211185

4433104273163

5115515355393393

2

4421241114144

2211321

2

1119

1

1

11

x xuc x x xucuc

x x x x xuncc

x x x x xumcuc

x xuc x x x x xmuc

x xuuc x x x x xnuc

x xucuc x xuc xcuc

x x x xuc xmuc xnuc

x xucc x xec xuc xuc

x x xumc xuncuc

x x x xucc x xcdt

dL


1,,, 53413

174

5

162151 cc

xu

uc

u

ucuc

Then substituting these constants in the above equationand using the condition (28c)-(28i), we get that

5353939

524

4

17521611518

2

338

1115

2

4444

338

2

4444

1115

2

338

225

1621

2

4444

225

1621

2

225

162111

159

33

33

33

33

33

33

x xe xu xu

x x x

u x xu xuu

x xu

x xu

x xd

x xu

x xd

x xu

x xu

x xu

uuu

x xd

x xu

uuu

x xu

uuu x x

u

dt

dL


15/20





the sufficient conditions (28a)-(28b). Hence the solution of


initial point satisfies the above condition and then the proofis complete. ■

Theorem (11): Assume that the positive equilibrium 10 E

exists, then it is a globally asymptotically stable in the subregion of 5 R that satisfies the following conditions:

39

5 xu

e (29a)

*513212111 11 xu xum xun (29b)

*55

*44

*33

*55

*44

*33

with,

OR

with,

x x x x x x

x x x x x x

(29c)

5

161521212

9

4

u

uuuuq (29d)

44152

149

4 quq (29e)

445

1621224

9

4q

u

uuuq (29f)

44*5982349

4q xuuq (29g)

*598152*

469

4 xuuu xnu (29h)

5

*5981621

2*47

9

4

u

xuuuuu xmu

(29i)

where5

3161512

u

uuuq , *4111514 1 xunuq ,

*4125

16424 1 xum

u

uuq , 10271634 u xmu xnuq

and 212111*51344 11 xum xun xuq .


2*33*3

*5

5*5

*55

*5

2*44

*4

*2

2*2

*22

*2

*1

1*111

*15110

2

ln

2ln

ln,...,

x xc

x

x x x xc

x xc

x

x x x xc

x

x x x xc x x L

where 5,....,1,* ici are positive constants to be determined.

It is easy to see that

,,,....., 515110 R RC x x L and 0,,,, *5*4*3*2*110 x x x x x L while 0,...., 5110 x x L for all

551,...., R x x and

*5*4*3*2*151 ,,,,,..., x x x x x x x . Further more by taking thederivative with respect to the time and simplifying theresulting terms, we get that

))((

))((

))((

)()1(

)1(

1

1

)(

332247*3

331146*3

*55

*1115

*5

*1

*55

*335

*539

*3

*55

*2216

*55

*2

554417*5413

*4

244

212*4

111*4513

*4

*44

*11

*411

*4

*1

*44

*22

*412

*44

*2

2*2221

*2

23359

*38

*3

*44

*3310

*427

*316

*3

*22

*113

*2

*1

2*11

*1

10

x x x x xmuc

x x x x xnuc

x x x xucc

x x x xec xuc

x x x xucuc

x x x xuc xuc

x x xumc

xunc xuc

x x x x xuncc

x x x x xumcuc

x xuuc x x xucuc

x x x xuc xmuc xnuc

x x x xucc x xcdt

dL

By choosing the positive constants as below

1,, *5*4*35

16*215*1 cccu

ucuc

Then substituting these constants in the above equationand using the condition (29b) and (29d)-(29i), we get that

*55*33539

*55

*4417413

2

*44

44*11

15

2

*44

44*22

5

1621

2

*44

44*33

*598

2

*33

*598*

1115

2

*33

*598*

225

1621

2

*22

5

1621*11

1510

33

33

33

33

33

33

x x x xe xu

x x x xu xu

x xq

x xu

x xq

x xu

uuu

x xq

x x xuu

x x xuu

x xu

x x xuu

x xu

uuu

x xu

uuu x x

u

dt

dL



the sufficient conditions (29a) and (29c). Hence the solution

of system (2) will approach asymptotically to 10 E from any

initial point satisfies the above condition and then the proois complete. ■

5. Numerical Simulation:In this section, the dynamics behavior of system (2) isstudied numerically. The objectives of this study are


16/20



confirming our obtained analytical results and understandthe effects of some parameters on the dynamics of system(2). Consequently, the system (2) is solved numerically fordifferent sets of initial conditions and for different sets ofparameters. Recall that system (2) contains two enter-specific competitions interactions, the first one between thetwo preys at the first level while the second one betweenthe mature predator in the second level and the top

predator at the third level. Although, the competitiveexclusion principle states that “two species that compete forthe exactly same resources cannot stably coexist”; theexistence of predator makes the coexistence of all speciespossible. Therefore we can’t find hypothetical set of datasatisfy the coexistence of all the species together, ratherthan that we found the set of data that satisfy thecoexistence for four populations of them as given below.Moreover since we presents the conditions that make thesystem has an asymptotically stable positive equilibriumpoint analytically, hence still there is possibility to have sucha data. It is observed that, for the following set ofhypothetical parameters values, system (2) has an

asymptotically stable top predator free equilibrium point 9 E

as shown in Fig. (1).

25.0,1.0,9.0,3.0,15.0

,05.0,1,3.0,3.0,1.0

,44.0,1.0,3.0,5.0,3.0

,15.1,15.1,19.1,5.1,2.1

518171615

1413121110

9876

54321

euuuu

uuuuu

uuunmu

uuuuu

… …(30)

0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

(a)

Time

P o p u l a t i o n s

x1

x2

x3

x4

x5

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

(b)

Time

F i r s t p e r y

( x 1

)

started at 0.4

started at 0.6

started at 0.9

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

(c)

Time

S e c o n d p r e y ( x 2 )

started at 0.2

started at 0.4

started at 0.9

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

(d)

Time

I m m a t u r e p r e d a t o r ( x 3 )

started at 0.1

started at 0.9

started at 0.6

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

(e)

Time

M a t u r e p r e d a t o r ( x 4 )

started at 0.7

started at 0.1

started at 0.5

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

Time

T o p p r e d a t o r ( x 5

)

(f)

started at 0.7

started at 0.1

started at 0.5

Fig. 1: Time series of the solution of system (2) for data given by (30).

(a) The trajectories of all species starting at )5.0,5.0,6.0,9.0,9.0( . (b)

The trajectories of 1 x - species starting from three different initial

points. (c) The trajectories of 2 x - species starting from three

different initial points. (d) The trajectories of 3 x - species starting from

three different initial points. (e) The trajectories of 4 x - species starting

from three different initial points. (f) The trajectories of 5 x - species

starting from three different initial points.


17/20



However, for the data given by Eq. (30) with initial point)6.0,3.0,2.0,5.0,3.0( that different from those used in Fig. (1),

the trajectory of system (2) approaches asymptotically to

third three species equilibrium point 8 E as drawn in figure

(2).

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

Time


x1

x2

x3

x4

x5

Fig. 2: Time series of the solution of system (2), for the data given by

(30) with initial point )6.0,3.0,2.0,5.0,3.0( ,that approaches

asymptotically to )37.0,0,0,04.0,58.0(8 E

Obviously Fig. (1) and Fig. (2), show clearly the existenceof sub region of global stability (basin of attraction) for eachequilibrium points of system (2). This confirms our obtainedanalytical results present in the previous section. Indeed theinitial points used in Fig. (1) satisfy the conditions given intheorem (10), while the initial point used in Fig. (2) satisfiesthe conditions in theorem (9). Note that in order to discussthe effect of the parameters values of system (2) on thedynamical behavior of system (2), the system is solvednumerically for the data given in Eq. (30) with varying oneparameter each time. It is observed that, for the abovehypothetical data, the parameters values

18,12,10,7,6,5, iui , m and n don’t have qualitative effect

on the dynamical behavior of system (2) and the system stillapproaches to a top predator free equilibrium point 9 E ,

rather than that they have quantitative effect on the position

of 9 E . Now by varying the parameter 1u keeping the rest of

parameters values as in Eq. (30), it observed that for

14.11u system (2) approaches asymptotically to

)0,,,0,( 4316 x x x E , while for 75.132.1 1u the solution of

system (2) approaches asymptotically to

)0,~,~,~,0( 4327 x x x E , Further for 75.11 u the solution

approaches asymptotically to ),0,0,,0( 525 x x E

as shown

in the typical figure given by Fig. (3).

0 5000 10000 150000

0.5

1

(a)

Time


x1

x2

x3

x4

x5

0 5000 10000 150000

0.5

1

1.5

(b)

Time


x1

x2

x3

x4

x5

0 5000 10000 150000

0.5

1

1.5

1.8

(c)

Time


x1

x2

x3

x4

x5

Fig. 3: Time series of the solution of system (2) for the data given by

Eq. (30) with different values of 1u . (a) System (2) approaches to

)0,8.0,2.0,0,1.0(6 E for 1.11u . (b) System (2) approaches to

)0,8.0,2.0,1.0,0(7 E for 5.11 u . (c) System (2) approaches to

)7.0,0,0,3.0,0(5 E for 8.11u .

On the other hand varying the parameter 2u keeping the

rest of parameters values as in Eq. (30), it observedthat for 04.179.0 2 u , the solution of system (2

approaches asymptotically to )0,~,~,~,0( 4327 x x x E , while fo

79.02 u the solution approaches asymptotically to

),0,0,,0( 525 x x E

. Moreover for the data given by Eq. (30)

with 51.13u , the solution of system (2) approaches

asymptotically to )0,,,0,( 4316 x x x E . In addition, varying

the parameter 4u in the range 23.14 u with other data as

in Eq.(30) the solution of system (2) approaches

asymptotically to )0,,,0,( 4316 x x x E too, however fo

05.14

u , it is observed that the solution of system (2

approaches asymptotically to the equilibrium poin

)0,~,~,~,0( 4327 x x x E . All these cases can be represented in

figures similar to those shown in Fig. (3), with slightlydifference in the position of equilibrium points. Similarly, fothe data given by Eq. (30) with one of the following ranges

at a time 12.18 u ; 54.09 u ; 11.011u ; 16.113 u

1.014 u ; 22.015 u ; 6.016 u ; 7.017 u or 7.05 e it is

observed that the trajectory of system (2) approachesasymptotically to the third three species equilibrium poin

)~~,0,0,

~~,~~( 5218 x x x E as explained in the typical figure

represented by Fig. (2) with slightly difference in the


18/20



position of point. Now, for the parameters values given inEq. (30) with varying the following three parameters

simultaneously 45.08 u , 2.014 u and 2.018 u , it is

observed that the solution of system (2) approachesasymptotically to the first two species equilibrium point

)0,0,0,,( 213 x x E as shown in the typical figure given by

Fig. (4).

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

Time


x1

x2

x3

x4

x5

Fig. 4: Time series of the solution of system (2), for the data given byEq. (30) with 5.08 u , 25.014 u and 3.018 u , that approaches

asymptotically to )0,0,0,01.0,9.0(3 E .

However, for the parameters values given in Equation (30)with varying the following two parameters simultaneously

3.15 u and 11.014 u , it is observed that the solution of

system (2) approaches asymptotically to the second two

species equilibrium point )ˆ,0,0,0,ˆ( 514 x x E as shown in

typical figure given by Fig. (5).

0 0.5 1 1.5 2

x 104

0

0.5

1

Time


x1

x2

x3

x4

x5


Eq. (30) with 5.15 u and 15.014 u , that approaches

asymptotically to )3.0,0,0,0,6.0(4 E .

Now, for the parameters values given in Eq. (30) withvarying the following three parameters simultaneously

9.02 u , 45.014 u and 35.018 u , it is observed that the

solution of system (2) approaches asymptotically to the

equilibrium point )0,0,0,,0(2

12 u

E as shown in the typical

figure given by Fig. (6).

0 0.5 1 1.5 2

x 104

0

0.6

1.2

Time


x1

x2

x3

x4

x5


Eq. (30) with 8.02 u , 5.014 u and 4.018 u , that approaches

asymptotically to )0,0,0,11.1,0(2 E .

Finally, for the parameters values given in Eq. (30) withvarying the following three parameters simultaneously

25.13 u , 4.014 u and 2.018 u , it is observed that the

solution of system (2) approaches asymptotically to theequilibrium point )0,0,0,0,1(1 E as shown in the typica

figure given by Fig. (7).

0 0.5 1 1.5 2

x 104

0

0.5

1

Time


x1

x2

x3

x4

x5


Eq. (30) with 3.13 u , 5.014 u and 4.018 u , that approaches

asymptotically to )0,0,0,0,1(1 E

6. Conclusions and discussionIn this paper, we proposed and analyzed an ecologicamodel that described the dynamical behavior of the foodweb real system. The model included five non-lineaautonomous differential equations that describe the

dynamics of five different populations, namely first prey),( 1 N second prey )( 2 N , immature predator )( 3 N , mature

predator )( 4 N and 5 N which is represent the top predator

The boundedness of system (2) has been discussed. Theexistence conditions of all possible equilibrium points areobtained. The local as well as global stability analyses ofthese points are carried out. Finally, numerical simulation isused to specific the control set of parameters that affect thedynamics of the system and confirm our obtained analyticaresults. Therefore system (2) has been solved numericallyfor different sets of initial points and different sets oparameters starting with the hypothetical set of data givenby Eq. (30), and the following observations are obtained.


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1) System (2) do not has periodic dynamic,instead of that the solution of system (2)approaches asymptotically to one of itsequilibrium point.

2) Decreasing the growth rate of the second prey,

1u , under a specific value leads to destabilized

9 E and the solution approaches to 6 E .

However increasing the value of 1u above aspecific value leads the system to approaches

to 7 E , Further increasing this parameter makes

the system approaches to 5 E .

3) Decreasing the value of intra specificcompetition between the individuals of second

prey, 2u , under a specific value leads the

system to approaches to 7 E , Further

decreasing this parameter makes the system

approaches to 5 E .

4) Increasing the parameter that describe theintensity of competition of the first prey to the

second prey, 3u , above a specific value leadsto destabilizing of 9 E and the solution

approaches to 6 E .

5) Decreasing the value of attack rate of mature

predator to the second prey species, 4u , under

a specific value leads the system to approaches

to 7 E , However increasing this parameter

above a specific value makes the system

approaches to 6 E .

6) Decreasing the value of growth rate of themature predator due to its feeding on the first

prey,11u , under a specific value makes the

solution of system (2) approaches

asymptotically to 8 E . The system has similar

behavior in case of decreasing 17u .

7) Increasing the value of gown up rate of the

immature predator, 8u , above a specific value

makes the solution of system (2) approaches

asymptotically to 8 E . The system has similar

behavior in case of increasing the value of 9u ,

13u , 14u , 15u , 16u or 5e .

8) Finally, varying the parameters values

18,12,10,7,6,5, iui , m and n don’t have

qualitative effect on the dynamical behavior ofsystem (2) and the system still approaches to a

top predator free equilibrium point 9 E ,

Keeping the above in view, all these outcomes depend onthe hypothetical set of parameters values given by Eq. (30),different results may be obtained for different sets of data.

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The Dynamics of Four Species Food Web Model With Stage Structure

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