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AdvancesinAppliedMathematicsA^êÆ?Ð,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
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,2022
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©ÙÚ^:¾ÿ,Š#l.äkÓ ÜŠ*Ñ +Ó .[J].A^êÆ?Ð,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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,0)?´ÛÜ
ìC-½.
y².ŠâÝ(3.3),.(3.1)3²ï: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
.
Ïd,λ
1
= B
33
,λ
2
Úλ
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©ÛŒ±Ñλ
2
<0Úλ
3
<0,B
33
<0ž,XÚ(3.1)3>.²ï:E
1
?´ÛÜ
ìC-½.
Ún3.3.3^‡(H
2
)÷vž,C
22
<0,G
3
≥0ž,X Ú(3.1)3>.²ï:E
2
(X
2
,0,Z
2
)
?´ÛÜìC-½.
y²..(3.1)3²ï: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
.Ïdλ
1
= C
22
,λ
2
Úλ
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Ñλ
1
<0,λ
2
<0,λ
3
<0…=C
22
<0,G
3
≥0,ÏdXÚ(3.1)3>.²ï:E
2
?´ÛÜìC-½.
e5,©Û•²ï:E
∗
(X
∗
,Y
∗
,Z
∗
)-½5.dÝ(3.3)Œ±Ñ.(3.1)3²ï
:E
∗
?A•§Xe
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
∗
−2cY
∗
Z
∗
,
D
31
= β
2
Z
∗
,D
32
= β
3
(α
3
+cZ
∗
)Z
∗
,D
33
= cβ
3
Y
∗
Z
∗
.
ΥG
5
<0,G
6
>0.ÄuRouth-Hurwitz-½5⌱±e½n.
½n3.1.b^‡(H
3
)÷v,kG
5
<0,XÚ(3.1)3•²ï:E
∗
(X
∗
,Y
∗
,Z
∗
)?´Ø-
½.
DOI:10.12677/aam.2022.1174594328A^êÆ?Ð
¾ÿ§Š#l
4.‡A*Ñ.ÄåÆ
-0=µ
1
<µ
2
<µ
3
<···´Žf−∆3àgNeumann>.^‡eAŠ,E(µ
i
)´
3H
1
(Ω)þµ
i
éAA˜m.-{φ
ij
:j=1,2,...,dim(E(µ
i
))}´E(µ
i
)IOÄ,…
X= [H
1
(Ω)]
3
,X
ij
= {cφ
ij
: c∈R
3
},K
X=
+∞
M
i=1
X
i
,X
i
=
dimE(µ
i
)
M
j=1
X
ij
.(4.1)
b^‡(H
3
)¤á,-L=D∆+J
U
(U
∗
),X Ú(2.2)3U
∗
?‚5zXÚ•U
t
=LU.
éz‡i≥1,X
i
3ŽfLe´ØC,…λ´L˜‡AŠ…=éi≥1,XJλ´Ý
−µ
i
D+J
U
(U
∗
)˜‡AŠ,K3ŽfX
i
¥•3˜‡A•þ.−µ
i
D+J
U
(U
∗
)Aõ‘
ª´
ϕ
i
(λ) = λ
3
+H
1i
λ
2
+H
2i
λ+H
3i
,
Ù¥
H
1i
= (d
11
+d
22
+d
33
)µ
i
+G
5
,
H
2i
= (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
3i
= 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
ÚG
i
3ªf(3.4)¥‰Ñ.XJd
33
≥d
11
,KkH
1i
,H
2i
,H
3i
>0.ÏLOŽŒ
H
1i
H
2i
−H
3i
= 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
+2d
11
d
22
+2d
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
.
XJd
22
≥d
11
,d
33
,KkG
13
>0.
½n4.1.b^‡(H
3
),±9d
22
≥d
33
≥d
11
÷v,duG
5
<0,G
7
>0,G
5
G
6
−G
7
<0,¤
±XÚ(2.2)•²ï:E
∗
(X
∗
,Y
∗
,Z
∗
)´Ø-½.
5.êŠ[
3ù˜Ü©,òÏLêŠ[5y²cA!©Û(J.XÚ(2.1)Њb•X(0)=
0.41,Y(0)=0.21,Z(0) = 0.2.Ù¦ëêŠÀJXe:r= 0.7,d
1
= 0.12,α
1
= 1.9,α
2
= 2.5,β
1
=
2.2,α
3
= 2.57,c= 0.5,d
2
= 0.2,β
2
= 1.1,β
3
= 2,d
3
= 0.5.
Äk,y²~‡©.(3.1)-½5.Šâ½n3.1,.(3.1)k²ï:E
∗
=
(0.3430,0.0237,0.2140)…´Ø-½(„ã1Úã2).
e5,½ëêr= 0.7,d
1
= 0.12,α
1
= 1.9,α
2
= 2.5,β
1
= 2.2,α
3
= 2.57,d
2
= 0.2,β
2
=
1.1,β
3
= 2,d
3
= 0.5,XÚ(3.1)vkÓ ööÜŠAž,3²ï:E
∗
?ÄåÆ1•„ã
3.
DOI:10.12677/aam.2022.1174594329A^êÆ?Ð
¾ÿ§Š#l
Figure1.The co-existenceequilibriumE
∗
of model(3.1)isunstable.(a)Sharedresource;(b)IG prey;(c)IG
predator
ã1..(3.1)•²ï:E
∗
´Ø-½"(a)•]¶(b) +S ¶(c) +SÓ ö
Figure2.Thedynamicbehaviorofthemodel(3.1).(a)Unstablebehaviorofpopulation;
(b)phaseportrait
ã2..(3.1)ÄåÆ1•"(a)«+Ø-½1•¶(b)ƒã
d,Ú\Ó ööÜŠAž,XÚ(3.1)3E
∗
?ÄåÆ1•„ã4.TãL²Ó
öÜŠAo´¦XÚØ-½.
•,·‚ÀJ˜Xëê5[XÚ(2.1):Ω=[0,10π],r=0.7,d
1
= 0.12,α
1
= 1.9,α
2
=
2.5,β
1
=2.2,α
3
=2.57,c=0.3,d
2
=0.2,β
2
=1.1,β
3
=2,d
3
=0.5.Šâ½n4.1,
d
11
= 10,d
22
= 20,d
33
= 15ž,XÚ(2.1)•²ï:E
∗
´Ø-½(„ã5).
6.o(†Ð"
©ïÄ˜‡äkÓ ööÜŠ*Ñ +Ó ..Ù¥,Ó öö܊σ•Ä
 +SÓ öé +S HollingI.õU‡A¥.Äk,3vk*Ñœ¹e,?Ø
.²ï:•35Ú-½5.ÏLêŠ[,uyÚ\Ó ööÜŠëêcž,o¬¦XÚ
(3.1)CØ-½.d,é*ÑXÚ(2.1),3˜½^‡e,XÚo´Ø-½.
3©ïĤJÄ:þ,²Ù¦ƒ'©zuy„kNõÙ¦óŠk–ïÄ.1˜,„Œ±
•ÄÙ§©|,X:Bogdanov-Taken©|!Ó‰©|;Ùg,vk•ÄÚ\Bž¢¦.•
DOI:10.12677/aam.2022.1174594330A^êÆ?Ð
¾ÿ§Š#l
Figure3.Thedynamicbehaviorofthesystemat
thepositiveequilibriumE
∗
withouttheeffectof
huntingcooperation
ã3.XÚvkÓ ööÜŠAž§3²ï:
E
∗
?ÄåÆ1•
Figure4.ThedynamicbehaviorofthesystematthepositiveequilibriumE
∗
withtheeffectofhunting
cooperation.(a)Sharedresource;(b)IGprey;(c)IGpredator
ã4.•kÓ ööÜŠAž§XÚ3²ï:E
∗
?ÄåÆ1•"(a)•]¶(b) +S ¶(c)
 +SÓ ö
\ý¢.Ïd,3.(2.1)Ä:þ,•Ä•]«+Bž¢,KŒïıe.:































∂X(x,t)
∂t
= d
11
∆X+rX−d
1
X−α
1
XY−α
2
XZ,x∈Ω,t>0,
∂Y(x,t)
∂t
= d
22
∆Y+β
1
X(t−τ)Y−(α
3
+cZ)YZ−d
2
Y,x∈Ω,t>0,
∂Z(x,t)
∂t
= d
33
∆Z+β
2
X(t−τ)Z+β
3
(α
3
+cZ)YZ−d
3
Z,x∈Ω,t>0,
∂X(x,t)
∂ν
=
∂Y(x,t)
∂ν
=
∂Z(x,t)
∂ν
= 0,x∈∂Ω,t>0,
X(x,0) = X
0
(x) ≥0,Y(x,0) = Y
0
(x) ≥0,Z(x,0) = Z
0
(x) ≥0,x∈Ω,
٥Щ^‡!>.^‡†.(2.1)˜,τL«•]«+Bž¢.
DOI:10.12677/aam.2022.1174594331A^êÆ?Ð
¾ÿ§Š#l
Figure5.Unstablebehaviorforsystem(2.1)withd
11
= 1,d
22
= 0.05,d
33
= 0.01,m
1
= 0.8
ã5.d
11
= 1,d
22
= 0.05,d
33
= 0.01m
1
= 0.8§XÚ(2.1)Ø-½51•
Ä7‘8
I[g,‰ÆÄ7‘8(12161054911661050)¶[‹Žg,‰ÆÄ7‘8(20JR10RA156)"
ë•©z
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