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AdvancesinAppliedMathematics
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,2022,11(7),4323-4334
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117459
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DynamicBehaviorsofanDiffusion
IntraguildPredationModelwith
HuntingCooperation
YanFeng,XinyouMeng
∗
AppliedMathematicsDepartment,SchoolofScience,LanzhouUniversityofTechnology,Lanzhou
Gansu
Received:Jun.6
th
,2021;accepted:Jul.1
st
,2022;published:Jul.8
th
,2022
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,2022,11(7):4323-4334.
DOI:10.12677/aam.2022.117459
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Abstract
Therelationshipsamongspeciesincludemutualism,parasitism,competitionandpre-
dation,amongwhichpredationisindispensableinthebiologicalworld.Inthisway,
individualswillnotbefixedinacertainareaforalongtime,butmovetoaplace
conducive to their own growth.At this time, there is aphenomenon of diffusion in the
population.Therefore,basedontheinfluenceofmanyfactors,adiffusionintraguild
predationmodelwithhuntingcooperationisestablishedinthispaper.Firstly,the
existenceandstabilityofallequilibriumpointsarediscussedwithoutconsideringd-
iffusion.Secondly,wegetthetheorythatdiffusionwillleadtotheinstabilityofthe
model.Finally,theresultsareverifiedbynumericalsimulation.
Keywords
Diffusion,Intraguild,HuntingCooperation
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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Ú
ž
m
t
?
—
Ý
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⊂
R
N
(
N
≤
3)
´
3
˜
‡
ä
k
1
w
>
.
∂
Ω
k
.
˜
m
•
,
ν
´
>
.
∂
Ω
ü
{
•
þ
.
d
11
,d
22
Ú
d
33
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O
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n
«
+
g
*
Ñ
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ê
,
r
L
«
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]
«
+
Ñ
)
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.
d
0
i
s
´
n
«
+
g
,
k
Ç
;
α
1
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2
Ú
α
3
©
OL
«
ˆ
g
Ó
ö
ü
ž
m
SU
¯
K
•
Œ
ê
þ
;
β
0
i
s
´
ˆ
g
Ó
ö
Ó
¼
=
z
Ç
,
i
= 1
,
2
,
3.
ë
ê
c
L
«
+
S
Ó
ö
«
+
3
ö
L
§
¥
,
Ó
ö
ƒ
m
Ó
Ü
Š
r
Ý
.
X
Ú
¥
¤
k
ë
ê
þ
•
~
ê
.
½
Â
U
= (
X,Y,Z
)
T
,
D
= diag(
d
11
,d
22
,d
33
)
9
J
(
U
) =
J
1
(
U
)
J
2
(
U
)
J
3
(
U
)
=
rX
−
d
1
X
−
α
1
XY
−
α
2
XZ
β
1
XY
−
(
α
3
+
cZ
)
YZ
−
d
2
Y
β
2
XZ
+
β
3
(
α
3
+
cZ
)
YZ
−
d
3
Z
.
(2.1)
Œ
±
U
¤
U
t
−
D
∆
U
=
J
(
U
)
,x
∈
Ω
,t>
0
,
∂
U
∂ν
= 0
,x
∈
∂
Ω
,t>
0
,
U
(
x,
0)
≥
0
,x
∈
Ω
.
(2.2)
DOI:10.12677/aam.2022.1174594326
A^
ê
Æ
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¾
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§
Š
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l
3.
.
-
½
5
ù
Ü
©
¥
,
Ì
‡
•
Ä
X
Ú
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é
A
ODE
X
Ú
²
ï
:
•
3
5
!
-
½
5
!
Hopf
©
|
•
3
5
Ú
5
Ÿ
.ODE
.
X
e
:
d
U
d
t
=
J
(
U
)
.
(3.1)
X
Ú
(3.1)
k
±
e
š
K
~
ê
)
:
(1)
²
…
)
E
0
(0
,
0
,
0);
(2)
+
S
Ó
ö
«
+
Ø
•
3
ž
,
X
J
^
‡
(
H
1
):
r
−
d
1
>
0
¤
á
,
k
>
.
²
ï
:
E
1
(
X
1
,Y
1
,
0),
X
1
=
d
2
β
1
,Y
1
=
(
r
−
d
1
)
α
1
;
(3)
+
S
«
+
Ø
•
3
ž
,
X
J
^
‡
(
H
2
):
r
−
d
1
>
0
¤
á
,
k
>
.
²
ï
:
E
2
(
X
2
,
0
,Z
2
),
X
2
=
d
3
β
2
,Z
2
=
(
r
−
d
1
)
α
2
;
(4)
•
²
ï
:
E
∗
(
X
∗
,Y
∗
,Z
∗
),
Ù
¥
X
∗
=
c
(
Z
∗
)
2
+
α
3
Z
∗
+
d
2
β
1
,Y
∗
=
d
3
−
β
2
X
∗
β
3
(
α
3
+
cZ
∗
)
,
Z
∗
´
e
˜
g
•
§
Š
:
A
1
Z
2
+
A
2
Z
+
A
3
= 0
,
(3.2)
Ù
¥
A
1
=
c
(
α
1
β
2
−
α
2
β
1
β
3
)
,
A
2
=
cβ
1
β
3
(
r
−
d
1
)+
α
3
(
α
1
β
2
−
α
2
β
1
β
3
)
,
A
3
=
α
3
β
1
β
3
(
r
−
d
1
)+
α
1
(
d
2
β
2
−
d
3
β
1
)
.
,
Š
â
(
k
Î
Ò
O
K
,
Œ
±
w
Ñ
X
Ú
(3.1)
¥
•
²
ï
:
•
3
5
´
š
~
E
,
.
Ï
d
,
©
•
?
Ø
Ù
¥
˜
«
œ
¹
,
=
X
Ú
(3.1)
•
k
˜
‡
•
²
ï
:
^
‡
•
(
H
3
) :
d
3
−
β
2
X
∗
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1
>
0
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2
>
0
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3
<
0.
e
5
,
ò
|
^
I
O
‚
5
z
•{
5
ï
Ä
z
‡
²
ï
:
-
½
5
.
.
(3.1)
Jacobian
Ý
•
J
U
=
J
11
J
12
J
13
J
21
J
22
J
23
J
31
J
32
J
33
,
(3.3)
Ù
¥
J
11
=
r
−
d
1
−
α
1
Y
−
α
2
Z,J
12
=
−
α
1
X,
J
13
=
−
α
2
X,J
21
=
β
1
Y,
J
22
=
β
1
X
−
(
α
3
+
cZ
)
Z
−
d
2
,J
23
=
−
α
3
Y
−
2
cYZ,
J
31
=
β
2
Z,J
32
=
β
3
(
α
3
+
cZ
)
Z,
J
33
=
β
2
X
+
β
3
(
α
3
Y
+2
cYZ
)
−
d
3
.
Ú
n
3.1.
X
Ú
(3.1)
²
…
²
ï
:
E
0
(0
,
0
,
0)
o
´
Ø
-
½
.
y
²
.
X
Ú
(3.1)
3
²
ï
:
E
0
?
A
•
§
•
[
λ
−
(
r
−
d
1
)](
λ
+
d
2
)(
λ
+
d
3
) = 0
.
Œ
±
é
A
A
Š
•
λ
1
=
r
−
d
1
>
0
,λ
2
=
−
d
2
<
0
,λ
3
=
−
d
3
<
0
.
DOI:10.12677/aam.2022.1174594327
A^
ê
Æ
?
Ð
¾
ÿ
§
Š
#
l
¤
±
X
Ú
(3.1)
3
²
ï
:
E
0
?
´
Ø
-
½
.
Ú
n
3.2.
3
^
‡
(
H
1
)
¤
á
ž
,
X
J
B
33
<
0
,
X
Ú
(3.1)
3
>
.
²
ï
:
E
1
(
X
1
,Y
1
,
0)
?
´
Û
Ü
ì
C
-
½
.
y
²
.
Š
âÝ
(3.3),
.
(3.1)
3
²
ï
:
E
1
?
A
•
§
•
(
λ
−
B
33
)
λ
2
−
(
B
11
+
B
22
)
λ
+
B
11
B
22
−
B
12
B
21
= 0
,
Ù
¥
B
11
=
r
−
d
1
−
α
1
Y
1
,B
12
=
−
α
1
X
1
,B
21
=
β
1
Y
1
,B
22
=
β
1
X
1
−
d
2
,B
33
=
β
2
X
1
+
α
3
β
3
Y
1
−
d
3
.
Ï
d
,
λ
1
=
B
33
,
λ
2
Ú
λ
3
´
e
•
§
Š
:
λ
2
+
G
1
λ
+
G
2
= 0
,
Ù
¥
G
1
=
−
(
B
11
+
B
22
)
>
0
,G
2
=
B
11
B
22
−
B
12
B
21
>
0
.
l
±
þ
©
Û
Œ
±
Ñ
λ
2
<
0
Ú
λ
3
<
0,
B
33
<
0
ž
,
X
Ú
(3.1)
3
>
.
²
ï
:
E
1
?
´
Û
Ü
ì
C
-
½
.
Ú
n
3.3.
3
^
‡
(
H
2
)
÷
v
ž
,
C
22
<
0
,G
3
≥
0
ž
,
X
Ú
(3.1)
3
>
.
²
ï
:
E
2
(
X
2
,
0
,Z
2
)
?
´
Û
Ü
ì
C
-
½
.
y
²
.
.
(3.1)
3
²
ï
:
E
2
?
A
•
§
•
(
λ
−
C
22
)
λ
2
−
(
C
11
+
C
33
)
λ
+
C
11
C
33
−
C
13
C
31
= 0
,
Ù
¥
C
11
=
r
−
d
1
−
α
2
Z
2
,C
13
=
−
α
2
X
2
,C
22
=
β
1
X
2
−
(
α
3
+
cZ
2
)
Z
2
−
d
2
,C
31
=
β
2
Z
2
,C
33
=
β
2
X
2
−
d
3
.
Ï
d
λ
1
=
C
22
,
λ
2
Ú
λ
3
´
e
•
§
Š
:
λ
2
+
G
3
λ
+
G
4
= 0
,
Ù
¥
G
3
=
−
(
C
11
+
C
33
)
,G
4
=
C
11
C
33
−
C
13
C
31
>
0.
l
Ñ
λ
1
<
0
,λ
2
<
0
,λ
3
<
0
…
=
C
22
<
0
,G
3
≥
0,
Ï
d
X
Ú
(3.1)
3
>
.
²
ï
:
E
2
?
´
Û
Ü
ì
C
-
½
.
e
5
,
©
Û
•
²
ï
:
E
∗
(
X
∗
,Y
∗
,Z
∗
)
-
½
5
.
d
Ý
(3.3)
Œ
±
Ñ
.
(3.1)
3
²
ï
:
E
∗
?
A
•
§
X
e
λ
3
+
G
5
λ
2
+
G
6
λ
+
G
7
= 0
,
(3.4)
Ù
¥
G
5
=
−
(
D
11
+
D
33
)
,G
6
=
D
11
D
33
−
D
13
D
31
−
D
12
D
21
−
D
23
D
32
,
G
7
=
D
11
D
23
D
32
+
D
12
D
21
D
33
−
D
21
D
13
D
32
−
D
12
D
23
D
31
,
D
11
= 0
,D
12
=
−
α
1
X
∗
,D
13
=
−
α
2
X
∗
,
D
21
=
β
1
Y
∗
,D
22
= 0
,D
23
=
−
α
3
Y
∗
−
2
cY
∗
Z
∗
,
D
31
=
β
2
Z
∗
,D
32
=
β
3
(
α
3
+
cZ
∗
)
Z
∗
,D
33
=
cβ
3
Y
∗
Z
∗
.
Œ
•
G
5
<
0
,G
6
>
0.
Ä
u
Routh-Hurwitz
-
½
5
â
Œ
±
±
e
½
n
.
½
n
3.1.
b
^
‡
(
H
3
)
÷
v
,
k
G
5
<
0
,
X
Ú
(3.1)
3
•
²
ï
:
E
∗
(
X
∗
,Y
∗
,Z
∗
)
?
´
Ø
-
½
.
DOI:10.12677/aam.2022.1174594328
A^
ê
Æ
?
Ð
¾
ÿ
§
Š
#
l
4.
‡
A
*
Ñ
.
Ä
å
Æ
-
0=
µ
1
<µ
2
<µ
3
<
···
´
Ž
f
−
∆
3
à
g
Neumann
>
.
^
‡
e
A
Š
,
E
(
µ
i
)
´
3
H
1
(Ω)
þ
µ
i
é
A
A
˜
m
.
-
{
φ
ij
:
j
=1
,
2
,...,
dim(
E
(
µ
i
))
}
´
E
(
µ
i
)
I
O
Ä
,
…
X
= [
H
1
(Ω)]
3
,
X
ij
=
{
c
φ
ij
:
c
∈
R
3
}
,
K
X
=
+
∞
M
i
=1
X
i
,
X
i
=
dim
E
(
µ
i
)
M
j
=1
X
ij
.
(4.1)
b
^
‡
(
H
3
)
¤
á
,
-
L
=
D
∆+
J
U
(
U
∗
),
X
Ú
(2.2)
3
U
∗
?
‚
5
z
X
Ú
•
U
t
=
L
U
.
é
z
‡
i
≥
1
,
X
i
3
Ž
f
L
e
´
ØC
,
…
λ
´
L
˜
‡
A
Š
…
=
é
i
≥
1,
X
J
λ
´
Ý
−
µ
i
D
+
J
U
(
U
∗
)
˜
‡
A
Š
,
K3
Ž
f
X
i
¥
•
3
˜
‡
A
•
þ
.
−
µ
i
D
+
J
U
(
U
∗
)
A
õ
‘
ª´
ϕ
i
(
λ
) =
λ
3
+
H
1
i
λ
2
+
H
2
i
λ
+
H
3
i
,
Ù
¥
H
1
i
= (
d
11
+
d
22
+
d
33
)
µ
i
+
G
5
,
H
2
i
= (
d
11
d
22
+
d
11
d
33
+
d
22
d
33
)
µ
2
i
−
[
D
33
d
11
+(
D
11
+
D
33
)
d
22
+
D
11
d
33
]
µ
i
+
G
6
,
H
3
i
=
d
11
d
22
d
33
µ
3
i
−
(
D
33
d
11
+
D
11
d
33
)
d
22
µ
2
i
−
[
D
23
D
32
d
11
+
D
12
D
21
d
33
+(
D
13
D
31
−
D
11
D
33
)
d
22
]
µ
i
+
G
7
,
D
ij
Ú
G
i
3
ª
f
(3.4)
¥
‰
Ñ
.
X
J
d
33
≥
d
11
,
K
k
H
1
i
,H
2
i
,H
3
i
>
0.
Ï
L
O
Ž
Œ
H
1
i
H
2
i
−
H
3
i
=
G
11
µ
3
i
+
G
12
µ
2
i
+
G
13
µ
i
+
G
5
G
6
−
G
7
,
(4.2)
G
11
= (
d
11
+
d
22
+
d
33
)(
d
11
d
22
+
d
11
d
33
+
d
22
d
33
)
−
d
11
d
22
d
33
>
0
,
G
12
= (
d
11
d
33
+
d
2
22
+2
d
11
d
22
+2
d
22
d
33
)
G
5
−
(
d
11
+
d
33
)(
D
33
d
11
+
D
11
d
33
)
>
0
,
G
13
= (
D
11
+
D
33
)[
D
33
d
11
+
D
11
d
33
+(
D
11
+
D
33
)
d
22
]+
G
6
(
d
11
+
d
33
)+
D
23
D
32
d
11
+
D
12
D
21
d
33
−
(
D
12
D
21
+
D
23
D
32
)
d
22
.
X
J
d
22
≥
d
11
,d
33
,
K
k
G
13
>
0.
½
n
4.1.
b
^
‡
(
H
3
)
,
±
9
d
22
≥
d
33
≥
d
11
÷
v
,
du
G
5
<
0
,G
7
>
0
,G
5
G
6
−
G
7
<
0
,
¤
±
X
Ú
(2.2)
•
²
ï
:
E
∗
(
X
∗
,Y
∗
,Z
∗
)
´
Ø
-
½
.
5.
ê
Š
[
3ù
˜
Ü
©
,
ò
Ï
L
ê
Š
[
5
y
²
c
A
!
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DOI:10.12677/aam.2022.1174594329
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Figure1.
The co-existenceequilibrium
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of model(3.1)isunstable.(a)Sharedresource;(b)IG prey;(c)IG
predator
ã
1.
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(3.1)
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Figure2.
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(b)phaseportrait
ã
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(3.1)
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DOI:10.12677/aam.2022.1174594330
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thepositiveequilibrium
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huntingcooperation
ã
3.
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withtheeffectofhunting
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4.
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DOI:10.12677/aam.2022.1174594331
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Figure5.
Unstablebehaviorforsystem(2.1)with
d
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05
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