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PureMathematicsnØêÆ,2023,13(5),1321-1332
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135135
˜a‘™êAHolling-Tanner.
Hopf©|
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Holling-Tanner.§™êA§²ï:§-½5§Hopf©|
HopfBifurcationofaHolling-Tanner
ModelwithFearEffect
QianZhao
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.18
th
,2023;accepted:May22
nd
,2023;published:May29
th
,2023
©ÙÚ^:ëÊ.˜a‘™êAHolling-Tanner.Hopf©|[J].nØêÆ,2023,13(5):1321-1332.
DOI:10.12677/pm.2023.135135
ëÊ
Abstract
Inthispaper,weinvestigatetheinfluenceofanti-predatorbehaviourduetothefear
of predators with a Holling-Tanner preydator-prey model incorporating a prey refuge.
First,thelocalasymptoticstabilityoftheequilibriumpointsisdiscussed,andthen
theconditionoftheexistenceofHopfbifurcationisgivenbytakingthefearlevelk
ofthepredatorasthebifurcationparameter.Finally,usingthecanonicaltheoryand
thecentralmanifoldtheorem,thedirectionofHopfbifurcationandthestabilityof
periodicsolutionofbifurcationareanalyzed.Throughcalculationandanalysis,itis
found that thefear effect can reducethe population densityof predator atthe positive
equilibrium.
Keywords
Holling-TannerModel,FearEffect,EquilibriumPoints,Stability,HopfBifurcation
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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x
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)−
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,
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dt
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x
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DOI:10.12677/pm.2023.1351351322nØêÆ
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K
.
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2.1.²ï:•35
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0
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∗
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∗
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∗
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∗
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f(y) := A
1
y
3
+A
2
y
2
+A
3
y+A
4
= 0.(2.1)
DOI:10.12677/pm.2023.1351351323nØêÆ
ëÊ
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A
1
= βb
2
kA+βb
3
k>0,
A
2
= βb
2
A+βb
3
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A
3
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2
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2
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0
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2
.
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r
β
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2
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3(0,+∞) þf
0
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ey=
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2
A+βb
3
+2bβmkA+2bβmk+abk)+
√
∆
3(βb
2
kA+βb
3
k)
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2
2
−3A
1
A
3
(ii) m=
r
β
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βm
2
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2
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r
β
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βm
2
(A+b)
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k<
b(rA+rb−2mβA−2bmβ−a)
βm
2
(A+b)
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Š;XJf(ey) <0,f(y) = 0kü‡Š.
nþ¤ã,eã(ؤá.
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1
):m<
r
β
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∗
1
=(x
∗
1
,y
∗
1
),Ù¥,
x
∗
1
= by
∗
1
+m.
(ii) XJm=
r
β
…k<
b(rA+rb−2mβA−2bmβ−a)
βm
2
(A+b)
,@oXÚ(1.3)k˜‡•˜²ï:,E
∗
2
=
(x
∗
2
,y
∗
2
),Ù¥,x
∗
2
= by
∗
2
+m,y
∗
2
=
−A
2
+
√
A
2
2
−4A
1
A
3
2A
1
.
(iii)XJm>
r
β
…k≥
b(rA+rb−2mβA−2bmβ−a)
βm
2
(A+b)
,XÚ(1.3)vkŠ.
(iv)XJm>
r
β
…k<
b(rA+rb−2mβA−2bmβ−a)
βm
2
(A+b)
.
DOI:10.12677/pm.2023.1351351324nØêÆ
ëÊ
(a)XJf(ey) <0,XÚ(1.3)kü‡²ï:E
∗
3
= (x
∗
3
,y
∗
3
)ÚE
∗
4
= (x
∗
4
,y
∗
4
),Ù¥,
y
∗
3
<ey<y
∗
4
,x
∗
i
= by
∗
i
+m(i= 3,4).
(b) XJf(ey)=0,@oE
∗
3
†E
∗
4
-Ü,XÚ(1.3)k˜‡•˜²ï:E
∗
5
=(x
∗
5
,y
∗
5
),Ù¥,
x
∗
5
= bey+m,y
∗
5
=ey.
(c)XJf(ey) >0,XÚ(1.3)vk²ï:.
Y©='5(H
1
) : m<
r
β
¤áœ/,Ù¦œ/Œaq?Ø.
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XÚ(1.3)3(x,y)?JacobiÝXe
J=
r
1+ky
−2βx−
aAy
2
(Ay+x−m)
2
−rkx
(1+ky)
2
−
a(x−m)
2
(Ay+x−m)
2
sby
2
(x−m)
2
s−
2sby
x−m
!
,(2.2)
e¡ÏLOŽXÚ(1.3)3z‡²ï:?JacobiÝAŠ,5(½ù²ï:-½5.
½n2(i)²…²ï:E
0
= (0,0)´Ã^‡Ø-½.
(ii)Œ²…²ï:E
1
= (
r
β
,0)´Ã^‡Ø-½.
(iii)b(H
1
)¤á,e÷v
(H
2
)2β(by
∗
+m) >
r
1+ky
∗
−
aA
(A+b)
2
,
²ï:E
∗
1
= (x
∗
1
,y
∗
1
)´ÛÜìC-½,‡ƒ´Ø-½.
y²(i)XÚ(1.3)3²ï:E
0
= (0,0)?JacobiÝ•
J
(E
0
)
=




r−a
0s




.(2.3)
Ý(2.3)AŠ•rÚs,Ïd,²ï:E
0
´Ø-½.
(ii)XÚ(1.3)3²ï:E
1
= (
r
β
,0)?JacobiÝ•
J
(E
1
)
=




−r−
kr
2
β
−a
0s




.(2.4)
Ý(2.4)AŠ•−rÚs,Ïd,²ï:E
1
´Ø-½.
DOI:10.12677/pm.2023.1351351325nØêÆ
ëÊ
(iii)XÚ(1.3)3²ï:E
∗
1
= (x
∗
1
,y
∗
1
)?JacobiÝ•
J
(E
∗
1
)
=




a
11
a
12
a
21
a
22




.(2.5)
Ù¥,
a
11
=
r
1+ky
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
,a
12
=
−rk(by
∗
1
+m)
(1+ky
∗
1
)
2
−
ab
2
(A+b)
2
,
a
21
=
s
b
,a
22
= −s.
Ý(2.5)A•§•
λ
2
−Tλ+D= 0.(2.6)
Ù¥,
D= det[J(E
∗
1
)] = −s[
r
1+ky
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
]+
s
b
[
rk(by
∗
1
+m)
(1+ky
∗
1
)
2
+
ab
2
(A+b)
2
],
T= tr[J(E
∗
1
)] =
r
1+ky
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
−s.
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∗
1
+m) >
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1+ky
∗
1
−
aA
(A+b)
2
ž,XÚ(1.3)
²ï:E
∗
1
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∗
1
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∗
1
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1
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∗
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∗
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r
1+k
∗
y
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
−s= 0.(3.1)
y²-
r
1+ky
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
−s= 0k
k=
r(A+b)
2
y
∗
1
[2β(by
∗
1
+m)(A+b)
2
+aA+s(A+b)
2
]
−
1
y
∗
1
.
-k
∗
= k,…(H
3
) : r>2β(by
∗
1
+m)+s+
aA
(A+b)
2
.Ïd,•3k
∗
>0,¦(3.1)ª¤á,…k
∗
>0
´•˜.
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1
),(H
3
)¤á,Kk= k
∗
ž,XÚ(1.3)3²ï:E
∗
1
?)Hopf©|.
DOI:10.12677/pm.2023.1351351326nØêÆ
ëÊ
y²bλ(k) = α(k)±ω(k)´A•§(2.6)Š,Ù¥
α(k) =
1
2
T=
1
2
[
r
1+ky
∗
1
−2β(by
∗
1
+m)−
aA
(A+b)
2
−s],
ω(k) =
1
2
√
4D−T
2
,
(3.2)
k= k
∗
ž,
α(k
∗
) = 0,α
0
(k
∗
) = −
ry
∗
1
(k
∗
)+rky
∗
0
1
(k
∗
)
2(1+ky
∗
1
(k
∗
))
2
−βby
∗
0
1
(k
∗
).
duy
∗
1
÷v(2.1),Ïd
y
∗
0
1
(k
∗
) =
(βb
2
A+βb
3
)y
∗3
1
+(2bβmA+2bβm+ab)y
∗2
1
+(βm
2
A+βm
2
b)y
∗
1
f
0
(y
∗
1
)
,
d2.1!?Ø¥•:e²ï:E
∗
1
•3,Kf
0
(y
∗
1
) >0,¤±,y
∗
0
1
(k
∗
) >0,?α
0
(k
∗
) <0.
nþ,e^‡(H
1
),(H
3
)¤á,k= k
∗
ž,
(i)[T(k)]
k=k
∗
= 0;
(ii)[D(k)]
k=k
∗
>0;
(iii)
d
dk
[T(k)]
k=k
∗
6= 0.Ïd,dPoinca´e-Andronov-Hopf©|½nŒ •,XÚ(1.3)3²ï:
E
∗
1
)Hopf©|.
4.Hopf©|••Ú-½5
!̇ïÄk=k
∗
ž,XÚ(1.3)3²ï:E
∗
1
NC)Hopf©|••9d©|
)±Ï)-½5.²£(x
∗
1
,y
∗
1
),-
ex= x−x
∗
1
,ey= y−y
∗
1
,
••B,C†E^x,y“Oex,eyKXÚ(1.3)C•









dx
dt
=
r(x+x
∗
1
)
1+k(y+y
∗
1
)
−β(x+x
∗
1
)
2
−
a(y+y
∗
1
)(x+x
∗
1
−m)
A(y+y
∗
1
)+(x+x
∗
1
)−m
,
dy
dt
= s(y+y
∗
1
)(1−
b(y+y
∗
1
)
(x+x
∗
1
)−m
),
(4.1)
ò(4.1)ª•Œ¤Xe/ª





dx
dt
dy
dt





= J(
E
∗
1
)




x
y




+




f(x,y,k)
g(x,y,k)




.(4.2)
DOI:10.12677/pm.2023.1351351327nØêÆ
ëÊ
Ù¥
f(x,y,k) = a
1
x
2
+a
2
xy+a
3
y
2
+a
4
x
3
+a
5
x
2
y+a
6
xy
2
+a
7
y
3
+···,
g(x,y,k) = b
1
x
2
+b
2
xy+b
3
y
2
+b
4
x
3
+b
5
x
2
y+b
6
xy
2
+b
7
y
3
+···.
±9
a
1
= −β+
aAy
∗2
1
(Ay
∗
1
+x
∗
1
−m)
3
,a
2
=
−rk
(1+ky
∗
1
)
2
−
2aAy
∗
1
(x
∗
1
−m)
(Ay
∗
1
+x
∗
1
−m)
3
,
a
3
=
rk
2
x
∗
1
(1+ky
∗
1
)
3
+
aA(x
∗
1
−m)
2
(Ay
∗
1
+x
∗
1
−m)
3
,a
4
= −
aAy
∗2
1
(Ay
∗
1
+x
∗
1
−m)
4
,
a
5
=
2aAy
∗
1
(Ay
∗
1
+x
∗
1
−m)−3aA
2
y
∗
1
(Ay
∗
1
+x
∗
1
−m)
4
,
a
6
=
rk
2
(1+ky
∗
1
)
3
−
aA(x
∗
1
−m)(Ay
∗
1
+x
∗
1
−m)−3aA
2
y
∗
1
(x
∗
1
−m)
(Ay
∗
1
+x
∗
1
−m)
4
,
a
7
=
−rk
3
x
∗
1
(1+ky
∗
1
)
4
−
aA
2
(x
∗
1
−m)
2
(Ay
∗
1
+x
∗
1
−m)
4
,b
1
= −
sby
∗2
1
(x
∗
1
−m)
3
,
b
2
=
2sby
∗
1
(x
∗
1
−m)
2
,b
3
= −
sb
x
∗
1
−m
,
b
4
=
sby
∗2
1
(x
∗
1
−m)
4
,b
5
= −
2sby
∗
1
(x
∗
1
−m)
3
,
b
6
=
sb
(x
∗
1
−m)
2
,b
7
= 0.
½ÂÝ
R=




10
MN




.
Ù¥,M=
−a
11
a
12
,N= −
ω(k)
a
12
,Kk
R
−1
J
(E
∗
1
)
R=




α(k)−ω(k)
ω(k)α(k)




.
@o
R
−1
=




10
−
M
N
1
N




.
DOI:10.12677/pm.2023.1351351328nØêÆ
ëÊ
k= k
∗
ž
M
∗
:= M|
k=k
∗
,N
∗
:= N|
k=k
∗
,ω
∗
:= ω|
k=k
∗
.
ŠC†(x,y)
T
= R(u,v)
T
,XÚ(4.2)Œ•





du
dt
dv
dt





= R
−1
J
(E
∗
1
)
R




u
v




+R
−1




fR(u,v,k)
gR(u,v,k)




,
=





du
dt
dv
dt





=




α(k)−ω(k)
ω(k)α(k)








u
v




=




f
1
(u,v,k)
g
1
(u,v,k)




.(4.3)
Ù¥
f
1
(u,v,k) = f(u,Mu+Nv,k)
= (a
1
+a
2
M+a
3
M
2
)u
2
+(a
2
+2a
3
M)Nuv+a
3
N
2
v
2
+(a
4
+a
5
M+a
6
M
2
+a
7
M
3
)u
3
+(a
5
+2a
6
M+3a
7
M
2
)Nu
2
v
+(a
6
+3a
7
M)N
2
uv
2
+a
7
N
3
v
3
+···,
g
1
(u,v,k) = −
M
N
f(u,Mu+Nv,k)+
1
N
g(u,Mu+Nv,k)
=
1
N
(−a
1
M+b
1
−a
2
M
2
+b
2
M−a
3
M
3
+b
3
M
2
)u
2
+(−a
2
M+b
2
−2a
3
M
2
+2b
3
M)uv+(−a
3
M+b
3
)Nv
2
+
1
N
(−a
4
M+b
4
−a
5
M
2
+b
5
M−a
6
M
3
+b
6
M
2
−a
7
M
4
)u
3
+(−a
5
M+b
5
−2a
6
M
2
+2b
6
M−3a
7
M
3
)u
2
v+(−a
6
M+b
6
−3a
7
M
2
)Nuv
2
−a
7
MN
2
v
3
+···.
e¡?14‹IC†,XÚ(4.3)du







˙τ= α(k)τ+p(k)τ
3
+···,
˙
θ= ω(k)+q(k)τ
2
+···,
(4.4)
DOI:10.12677/pm.2023.1351351329nØêÆ
ëÊ
3k= k
∗
??1TaylorÐm







˙τ= α
0
(k
∗
)(k−k
∗
)τ+p(k
∗
)τ
3
+o((k−k
∗
)
2
τ,(k−k
∗
)τ
3
,τ
5
),
˙
θ= ω(k
∗
)+ω
0
(k
∗
)(k−k
∗
)+q(k
∗
)τ
2
+o((k−k
∗
)
2
,(k−k
∗
)τ
2
,τ
4
).
(4.5)
•äHopf©|••±9d©|)±Ï)-½5,IOŽp(k
∗
)ÎÒ,=
p(k
∗
) :=
1
16
(f
1
uuu
+f
1
uvv
+g
1
uuv
+g
1
vvv
)
+
1
16ω(k
∗
)
[f
1
uv
(f
1
uu
+f
1
vv
)−g
1
uv
(g
1
uu
+g
1
vv
)−f
1
uu
g
1
uu
+f
1
vv
g
1
vv
],
Ù¥¤k êÑŠu©|:(u,v,k) = (0,0,k
∗
)…
f
1
uuu
(0,0,k
∗
) = 6(a
4
+a
5
M
0
+a
6
M
2
0
+a
7
M
3
0
),f
1
uvv
(0,0,k
∗
) = 2(a
6
+3a
7
M
0
)N
2
0
,
g
1
uuv
(0,0,k
∗
) = 2(−a
5
M
0
+b
5
−2a
6
M
2
0
+2b
6
M
0
−3a
7
M
3
0
),g
1
vvv
(0,0,k
∗
) = −6a
7
M
0
N
2
0
,
f
1
uu
(0,0,k
∗
) = 2(a
1
+a
2
M
0
+a
3
M
2
0
),f
1
uv
(0,0,k
∗
) = (a
2
+2a
3
M
0
)N
0
,
f
1
vv
(0,0,k
∗
) = 2a
3
N
2
0
,g
1
uu
(0,0,k
∗
) =
2
N
0
(−a
1
M
0
+b
1
−a
2
M
2
0
+b
2
M
0
−a
3
M
3
0
+b
3
M
2
0
),
g
1
uv
(0,0,k
∗
) = −a
2
M
0
+b
2
−2a
3
M
2
0
+2b
3
M
0
,g
1
vv
(0,0,k
∗
) = 2(−a
3
M
0
+b
3
)N
0
.
½Â˜LyapunovXêOŽúªXe
µ
2
= −
p(k
∗
)
α
0
(k
∗
)
.
Ï•α
0
(k
∗
) <0,dPoinca´e-Andronov-Hopf©|½nŒ.
½n5 b(H
1
)-(H
3
)¤á,K3k= k
∗
ž,XÚ(1.3)3E
∗
1
?)Hopf©|.
(i)p(k
∗
) <0,Hopf©|±Ï)´ìC-½…©|••´æ..
(ii)XJp(k
∗
) >0,Hopf©|±Ï)´Ø-½…©|••´‡..
5.Hopf™êAé ÚÓ öK•
duԗ݆™êY²kÃ',¤±™êAéÔ—ÝvkK•.Ïd,·‚•?Ø™ê
AéÓ ö—ÝK•.
DOI:10.12677/pm.2023.1351351330nØêÆ
ëÊ
òy
∗
wŠ´'ukëY¼ê,'uk¦:
dy
∗
dk
= −
(βb
2
A+βb
3
)y
∗3
+(2bβmA+2bβm+ab)y
∗2
+(βm
2
A+βm
2
b)y
∗
f
0
(y
∗
)
éu²ï:E
∗
1
XJ•3,@of
0
(y
∗
1
)>0,¤±þªmàu0,=
dy
∗
1
dk
<0¤á.=Ó ö«+
²ï—Ýy
∗
‘kOŒÅì~,ù´Ï•™êY²Œž,Œ<;Ó 5õ,Ó ö
Œ¼ 5,KÓ ö²ï—Ý‘ƒ~.
6.(Ø
©ïÄ˜a‘k™êAHolling-Tanner..Äk?Ø²ï:ÛÜìC-½
5,,±Ó ö™êY²k•©|ëê,‰ÑHopf©|•3^‡.•,|^5‰.nØ,
Poinca´e-Andronov-Hopf©|½nÚ¥%6/½n©ÛHopf©|••9©|±Ï)-½5.
²©ÛOŽuy™êAŒ±ü$Ó ö3²ïGe«+—Ý.
ë•©z
[1]Liang, Z.and Pan, H.(2007) QualitativeAnalysis ofa Ratio-DependentHolling-TannerModel.
JournalofMathematicalAnalysisandApplications,334,954-964.
https://doi.org/10.1016/j.jmaa.2006.12.079
[2]Wang,X.L. andWang,W.D. (2011)HopfBifurcationAnalysis ofa Ratio-Dependent Holling-
TannerPredator-PreyModel.JournalofSouthwestUniversity(NaturalScienceEdition),33,
1-4.
[3]Rebaza,J.(2012)DynamicsofPreyThresholdHarvestingandRefuge.JournalofComputa-
tionalandAppliedMathematics,236,1743-1752.https://doi.org/10.1016/j.cam.2011.10.005
[4]Ma,Z., Li,W.,Zhao,Y.,Wang,W., Zhang,H.andLi, Z.(2009)Effectsof PreyRefuges ona
Predator-PreyModel withaClass ofFunctionalResponses:TheRoleof Refuges.Mathematical
Biosciences,218,73-79.https://doi.org/10.1016/j.mbs.2008.12.008
[5]Wang,L.andChen,W.(2008)QualitativeAnalysisofaPredator-PreyModelwithHolling
TypeIIFunctionalResponse.JournalofSoutheastUniversity(NaturalScienceEdition),38,
No.2.
[6]Sih,A.(1987)PreyRefugesandPredator-PreyStability.TheoreticalPopulationBiology,31,
1-12.https://doi.org/10.1016/0040-5809(87)90019-0
[7]Huang,Y.,Chen,F.andZhong,L.(2006)StabilityAnalysisofaPrey-PredatorModelwith
HollingTypeIIIResponseFunctionIncorporatingaPreyRefuge.AppliedMathematicsand
Computation,182,672-683.https://doi.org/10.1016/j.amc.2006.04.030
DOI:10.12677/pm.2023.1351351331nØêÆ
ëÊ
[8]Uttam, D., Kar,T.K.and Pahari, U.K.(2013)Global Dynamicsof anExploited Prey-Predator
ModelwithConstantPreyRefuge.ISRNBiomathematics,2013,ArticleID:637640.
https://doi.org/10.1155/2013/637640
[9]ûr,@.Äu'Ç…•¹ ;JHolling-Tanner.©Û[J].ìÀŒÆÆ:nƇ,
2009,44(3):56-60.
[10]Xia,L.andXing,Y.(2013)BifurcationsofaRatio-DependentHolling-TannerSystemwith
RefugeandConstantHarvesting.AbstractandAppliedAnalysis,2013,ArticleID:478315.
https://doi.org/10.1155/2013/478315
[11]Wang,X.Y.,Zanette,L.andZou,X.F.(2016)ModellingtheFearEffectinPredator-Prey
Interactions.JournalofMathematicalBiology,73,1179-1204.
https://doi.org/10.1007/s00285-016-0989-1
[12]Zhang,H.,Cai,Y.,Fu,S. andWang,W.(2019)ImpactoftheFearEffectinaPrey-Predator
ModelIncorporatingaPreyRefuge.AppliedMathematicsandComputation,356,328-337.
https://doi.org/10.1016/j.amc.2019.03.034
[13]½©ï.‘™êÏfIVlev.Ó ö- †*Ñ.ž˜€ã[D]:[a¬Æ Ø©].=²:
Ü“‰ŒÆ,2022.
[14]Mondal,S.(2020)DynamicsofaDelayedPredator-PreyInteractionIncorporatingNonlinear
PreyRefugeundertheInfluenceofFearEffectandAdditionalFood.JournalofPhysicsA:
MathematicalandTheoretical,53, Article295601.https://doi.org/10.1088/1751-8121/ab81d8
[15]Xie,B.F.andZhang,N.(2022)InfluenceofFearEffectonaHollingTypeIIIPrey-Predator
SystemwiththePreyRefuge.AIMSMathematics,7,1811-1830.
https://doi.org/10.3934/math.2022104
[16]Wang,J.,Cai,Y.L.,Fu,S.M.andWang,W.M.(2019)TheEffectoftheFearFactoronthe
Dynamics of a Predator-Prey Model Incorporating the Prey Refuge.Chaos(Woodbury,N.Y.),
29,Article083109.
[17]Shi,H.B.andRuan,S.(2015)Spatial,TemporalandSpatiotemporalPatternsofDiffusive
PredatorPreyModelswithMutualInterference.IMAJournalofAppliedMathematics,80,
1534-1568.
[18]Kuznetsov, Y.A.(2013) Elements ofApplied BifurcationTheory.Springer Science& Business
Media,Berlin.
DOI:10.12677/pm.2023.1351351332nØêÆ

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