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PureMathematics
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PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115097
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TheEffectofCross-Diffusionon
Beddington-DeAngelisType
Predator-PreyModelwith
ProtectionZone
KaiYan,LinaZhang
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CollegeofMathematicsandStatistics,NorthwestNormalUniversity,Lanzhou730070,China
Received:Apr.12
th
,2021;accepted:May17
th
,2021;published:May24
th
,2021
Abstract
Theeffectof cross-diffusionon Beddington-DeAngelistypepredator-preymodelwith
protectionzoneandthehomogeneousNeumannboundaryconditionsisconsidered,
wherethecross-diffusionrepresentsthetendencyofpreytokeepawayfromits
predator.Firstly,thestabilityofthenonnegativeconstantsteadystatesolutionsis
analyzedbythelinearizationmethod.Secondly,aprioriestimatesofpositivesteady
statesolutionsisgivenbyapplyingmaximumprinciple.Finally,theexistenceof
coexistencesolutionsisdiscussedbyusingbifurcationtheory.Asaresult,itisshown
thatthecross-diffusionisbeneficialforspeciescoexistence.
Keywords
Predator-PreyModel,Beddington-DeAngelisFunctionalResponse,ProtectionZone,
Cross-Diffusion,CoexistenceSolutions
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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∗
>
0
I
•
›
µ
Š
‰
Υ
µ
∈
(
−
c
m
,
0).
K
λ
=
λ
∗
ž
,
ζ
1
=0;
µ
∈
(
−
c
m
,
0)
…
λ<λ
∗
ž
,
ζ
1
<
0,
d
ž
(
λ,
0)
´
Û
Ü
ì
C
-
½
;
µ
≥
0
½
ö
µ
∈
(
−
c
m
,
0)
…
λ>λ
∗
ž
,
ζ
1
>
0,
d
ž
(
λ,
0)
´
Ø
-
½
.
(3)
X
Ú
(1
.
2)
é
A
‚
5
z
X
Ú
3
(0
,µ
)
?
A
Š
¯
K
•
4
f
+
λ
(1+
nµ
)
−
b
(
x
)
µ
(1+
nµ
)(1+
kρ
(
x
)
µ
)
f
−
ηf
= 0
,x
∈
Ω
,
4
g
−
µg
+
cµ
(1+
nµ
)(1+
kµ
)
f
−
ηg
= 0
,x
∈
Ω
1
,
∂
ν
f
= 0
,x
∈
∂
Ω
,∂
ν
g
= 0
,x
∈
∂
Ω
1
,
§
k
˜
¢
A
Š
S
η
1
>η
2
>η
3
...>η
n
>...
→−∞
.
η
1
<
0
ž
,(0
,µ
)
´
Û
Ü
ì
C
-
½
.
η
1
=
d
f
•
§
(
½
,
´
„
−
η
1
=
λ
N
1
−
λ
(1+
nµ
)+
b
(
x
)
µ
(1+
nµ
)(1+
kρ
(
x
)
µ
)
,
Ω
.
(
Ü
Ú
n
2.1
•
,
e
λ<λ
∗
,
K
η
1
<
0,
l
(0
,µ
)
´
Û
Ü
ì
C
-
½
;
e
λ>λ
∗
,
K
η
1
>
0,
l
(0
,µ
)
´
Ø
-
½
.
DOI:10.12677/pm.2021.115097845
n
Ø
ê
Æ
A
p
3.
•
)
•
3
5
e
¡
À
λ
Š
•
©
Ü
ë
ê
,
A^
Û
Ü
©
Ü
n
Ø
[11],
?
Ø
X
Ú
(1
.
2)
ü
‡
Œ
²
…
)
-
‚
:
Γ
U
=
{
(
λ
;
U,v
) = (
λ
;
λ,
0) :
λ>
0
}
,
Γ
v
=
{
(
λ
;
U,v
) = (
λ
;0
,µ
) :
λ>
0
}
(
µ>
0)
.
é
?
¿
p>N
,
P
X
1
=
(
φ,ψ
) :
φ
∈
W
2
,p
(Ω)
,ψ
∈
W
2
,p
(Ω
1
)
,∂
ν
φ
|
∂
Ω
=
∂
ν
ψ
|
∂
Ω
1
= 0
,X
2
=
L
p
(Ω)
×
L
p
(Ω
1
)
.
½
Â
φ
∗
= (
−4
+
λ
∗
)
−
1
Ω
λ
2
∗
kρ
(
x
)
−
b
(
x
)
λ
∗
1+
mλ
∗
(3.1)
-
φ
∗
•
A
Š
¯
K
−4
φ
∗
+
−
λ
∗
(1+
nµ
)+
b
(
x
)
µ
(1+
nµ
)(1+
kρ
(
x
)
µ
)
φ
∗
= 0
, x
∈
Ω
,∂
ν
φ
∗
= 0
, x
∈
∂
Ω(3.2)
Ì
A
¼
ê
,
ψ
∗
= (
−4
+
µ
)
−
1
Ω
1
cµφ
∗
(1+
nµ
)(1+
kµ
)
.
(3.3)
e
¡
·
‚
$
^
©
Ü
n
Ø
?
Ø
Û
Ü
©
|
)
-
‚
•
3
5
,
Œ
e
(
Ø
.
½
n
3.1
(1)(
λ
∗
;
λ
∗
,
0)
´
Γ
U
þ
•
˜
©
Ü
:
,
…
3
(
λ
∗
;
λ
∗
,
0)
•
S
•
3
X
Ú
(1.2)
)
-
‚
Γ
∗
=
{
(
λ
;
U,v
)=(
λ
(
s
);
λ
+
s
(
φ
∗
+
b
U
(
s
))
,s
(1+
b
v
(
s
))):
s
∈
(0
,
b
δ
)
}
,
Ù
¥
b
δ>
0
¿
©
,
λ
(
s
)
,
b
U
(
s
)
,
b
v
(
s
)
•
C
1
-
‚
,
…
÷
v
λ
(0) =
λ
∗
,
b
U
(0) =
b
v
(0) = 0,
R
Ω
1
b
v
(
s
)
dx
= 0.
(2)(
λ
∗
;0
,µ
)
´
Γ
v
þ
•
˜
©
Ü
:
,
…
3
(
λ
∗
;0
,µ
)
•
S
•
3
X
Ú
(1.2)
)
-
‚
Γ
∗
=
{
(
λ
;
U,v
)= (
λ
(
s
);
s
(
φ
∗
+
e
U
(
s
))
,µ
+
s
(
ψ
∗
+
e
v
(
s
))):
s
∈
(0
,
e
δ
)
}
,
Ù
¥
e
δ>
0
¿
©
,
λ
(
s
)
,
e
U
(
s
)
,
e
v
(
s
)
•
C
1
-
‚
,
…
÷
v
λ
(0) =
λ
∗
,
e
U
(0) =
e
v
(0) = 0,
R
Ω
e
U
(
s
)
φ
∗
dx
= 0.
y
²
(1)
-
z
:=
U
−
λ
,
½
Â
N
F
:
R
×
X
1
→
X
2
:
F
(
λ
;
z,v
) =
4
z
+
z
+
λ
1+
kρ
(
x
)
v
λ
−
z
+
λ
1+
kρ
(
x
)
v
−
b
(
x
)
v
(1+
kρ
(
x
)
v
)
(1+
nv
)(1+
kρ
(
x
)
v
)+
m
(
z
+
λ
)
4
v
+
v
µ
−
v
+
c
(
z
+
λ
)
(1+
nv
)(1+
kv
)+
m
(
z
+
λ
)
.
(3.4)
²
{
ü
O
Ž
´
F
(
z,v
)
(
λ
;0
,
0)[
φ,ψ
] =
4
φ
−
λφ
+
λ
2
kρ
(
x
)
ψ
−
b
(
x
)
λ
1+
mλ
ψ
4
ψ
+(
µ
+
cλ
1+
mλ
)
ψ
.
d
Krein-Rutman
½
n
[12]
•
,
…
=
λ
=
λ
∗
ž
,
F
(
z,v
)
(
λ
;0
,
0)[
φ,ψ
] = (0
,
0)
k
˜
‡
)
ψ>
0.
¤
±
,(
λ
∗
;0
,
0)
´
Γ
U
þ
•
˜
©
|
:
…
Ker
F
(
z,v
)
(
λ
∗
;0
,
0)=span
{
(
φ
∗
,
1)
}
.
Ù
¥
φ
∗
3
(3
.
1)
¥
‰
Ñ
.
DOI:10.12677/pm.2021.115097846
n
Ø
ê
Æ
A
p
Ï
d
dimKer
F
(
z,v
)
(
λ
∗
;0
,
0) = 1
.
Š
â
Fredholm
J
˜
½
n
[12]
•
,
Range
F
(
z,v
)
(
λ
∗
;0
,
0) =
(
φ,ψ
)
}∈
X
2
:
Z
Ω
1
ψdx
= 0
.
Ï
d
,codimRange
F
(
z,v
)
(
λ
∗
;0
,
0) = 1
.
d
,
F
λ
(
z,v
)
(
λ
∗
;0
,
0)[
φ
∗
,
1] =
−
φ
∗
+2
kρ
(
x
)
λ
∗
−
b
(
x
)
(1+
mλ
∗
)
2
c
(1+
mλ
∗
)
2
*
Range
F
(
z,v
)
(
λ
∗
;0
,
0)
.
Ï
d
,
d
Û
Ü
©
|
½
n
[11],
½
n
3.1
(
Ø
(1).
(2)
½
Â
N
G
:
R
×
X
1
→
X
2
:
G
(
λ
;
U,v
) =
∆
U
+
U
1+
kρ
(
x
)
v
λ
−
U
1+
kρ
(
x
)
v
−
b
(
x
)
v
(1+
kρ
(
x
)
v
)
(1+
nv
)(1+
kρ
(
x
)
v
)+
mU
∆
v
+
v
µ
−
v
+
cU
(1+
nv
)(1+
kv
)+
mU
.
(3.5)
K
²
{
ü
O
Ž
Œ
G
(
U,v
)
(
λ
;0
,µ
)[
φ,ψ
] =
4
φ
+
λ
(1+
nµ
)
−
b
(
x
)
µ
(1+
nµ
)(1+
kρ
(
x
)
µ
)
φ
4
ψ
−
µψ
+
cµ
(1+
nµ
)(1+
kµ
)
φ
.
d
Krein-Rutman
½
n
[12]
•
,
…
=
λ
=
λ
∗
ž
,
G
(
U,v
)
(
λ
;0
,µ
)[
φ,ψ
]=(0
,
0)
k
˜
‡
)
φ>
0.
¤
±
,(
λ
∗
;0
,µ
)
´
Γ
v
þ
•
˜
©
|
:
…
Ker
G
(
U,v
)
(
λ
∗
;0
,µ
)=span
{
(
φ
∗
,ψ
∗
)
}
.
Ù
¥
φ
∗
,
ψ
∗
d
(3
.
2)
Ú
(3
.
3)
ª
‰
Ñ
.
dimKer
G
(
U,v
)
(
λ
∗
;0
,µ
) = 1
.
d
,
Range
G
(
U,v
)
(
λ
∗
;0
,µ
) =
(
φ,ψ
)
}∈
X
2
:
Z
Ω
φφ
∗
dx
= 0
.
d
þ
ª
Œ
±
í
Ñ
G
λ
(
U,v
)
(
λ
∗
;0
,µ
)[
φ
∗
,ψ
∗
] =
φ
∗
1+
kρ
(
x
)
µ
0
*
Range
G
(
U,v
)
(
λ
∗
;0
,µ
)
.
Ï
d
,
d
Û
Ü
©
|
½
n
[11]
½
n
3.1
(
Ø
(2).
e
5
,
·
‚
ò
‰
Ñ
X
Ú
(1
.
2)
)
˜
‡
k
O
,
?
©
Û
©
|
)
-
‚
l
©
|
:
ž
Û
©
|
(
.
Ú
n
3.1
(
U,v
)
´
X
Ú
(1
.
2)
?
¿
)
,
K
0
<U
(
x
)
≤
λ
h
1+
k
µ
+
c
m
i
x
∈
Ω
,
0
<v
(
x
)
≤
µ
+
c
m
x
∈
Ω
1
.
d
,
µ
≥
0
ž
,
é
?
¿
x
∈
Ω
1
k
v
(
x
)
>µ
.
DOI:10.12677/pm.2021.115097847
n
Ø
ê
Æ
A
p
y
²
-
U
(
x
0
) = max
x
∈
Ω
U
(
x
),
d
•
Œ
Š
n
U
(
x
0
)
1+
kρ
(
x
0
)
v
(
x
0
)
λ
−
U
(
x
0
)
1+
kρ
(
x
0
)
v
(
x
0
)
−
b
(
x
0
)
v
(
x
0
)(1+
kρ
(
x
0
)
v
(
x
0
))
(1+
nv
(
x
0
))(1+
kρ
(
x
0
)
v
(
x
0
))+
mU
(
x
0
)
≥
0
,
=
U
(
x
0
)
≤
λ
(1+
kρ
(
x
0
)
v
(
x
0
))
≤
λ
(1+
k
max
x
∈
Ω
1
v
(
x
))
.
(3.6)
-
v
(
x
1
) = max
x
∈
Ω
1
v
(
x
),
K
X
Ú
(1
.
2)
1
‡
•
§
÷
v
v
(
x
1
)
µ
−
v
(
x
1
)+
cU
(
x
1
)
(1+
nv
(
x
1
))(1+
kv
(
x
1
))+
mU
(
x
1
)
≥
0
.
l
,
v
(
x
1
)
≤
µ
+
cU
(
x
1
)
(1+
nv
(
x
1
))(1+
kv
(
x
1
))+
mU
(
x
1
)
.
(3.7)
r
(3.6)
“
\
(3.7)
v
(
x
1
)
≤
µ
+
c/m
.
Ï
d
,
é
?
¿
x
∈
Ω
1
,
k
v
(
x
)
≤
µ
+
c/m
.
¤
±
é
?
¿
x
∈
Ω,
k
0
<U
(
x
)
≤
λ
[1+
k
(
µ
+
c/m
)].
,
˜
•
¡
,
−4
v
=
v
µ
−
v
+
cU
(1+
nv
)(1+
kv
)+
mU
>v
(
µ
−
v
)
x
∈
Ω
1
, ∂
ν
v
= 0
x
∈
∂
Ω
1
.
d
'
n
,
X
J
µ
≥
0,
K
x
∈
Ω
1
ž
,
v
(
x
)
>µ
.
a
q
u
©
[13]
½
n
2.2
y
²
L
§
,
Ï
L
I
O
Û
©
|
?
Ø
,
Œ
±
X
e
(
Ø
.
½
n
3.2
(1)
e
µ
≥
0,
K
λ>λ
∗
(
µ
)
ž
,
X
Ú
(1
.
2)
–
•
3
˜
‡
)
.
(2)
e
µ<
0,
K
λ>λ
∗
(
µ
)
ž
,
X
Ú
(1
.
2)
–
•
3
˜
‡
)
.
4.
†
*
Ñ
é
•
)
K
•
!
ï
Ä
†
*
Ñ
X
ê
k
é
•
«
•
K
•
.
½
n
4.1
µ>
0
,
K
λ
∗
(
µ,k,
Ω
0
)
'
u
k
î
‚
ü
N4
~
.
y
²
½
µ>
0
Ú
Ω
0
,
5
¿
λ
∗
(
µ,k,
Ω
0
)
÷
v
λ
N
1
b
(
x
)
µ
−
(1+
nµ
)
λ
∗
(
µ,k,
Ω
0
)
(1+
nµ
)(1+
kρ
(
x
)
µ
)
,
Ω
= 0
.
(4.1)
d
b
(1.3)
•
,
b
(
x
)
µ
−
(1+
nµ
)
λ
∗
(
µ,k,
Ω
0
)
(1+
nµ
)(1+
kρ
(
x
)
µ
)
=
βµ
−
(1+
nµ
)
λ
∗
(
µ,k,
Ω
0
)
(1+
nµ
)(1+
kµ
)
,x
∈
Ω
\
Ω
0
,
−
λ
∗
(
µ,k,
Ω
0
)
,x
∈
Ω
0
.
(4.2)
d
λ
N
1
(
q,
Ω)
'
u
q
ë
Y
…
î
‚
ü
N4
O
5
Ÿ
±
9
λ
N
1
(0
,
Ω) = 0,
βµ
−
(1+
nµ
)
λ
∗
(
µ,k,
Ω
0
)
>
0
.
DOI:10.12677/pm.2021.115097848
n
Ø
ê
Æ
A
p
é
?
¿
÷
v
(4.1)
k
1
,k
2
,
Ø
”
k
2
>k
1
,
k
βµ
−
(1+
nµ
)
λ
∗
(
µ,k
1
,
Ω
0
)
(1+
nµ
)(1+
k
2
µ
)
<
βµ
−
(1+
nµ
)
λ
∗
(
µ,k
1
,
Ω
0
)
(1+
nµ
)(1+
k
1
µ
)
,
Ï
d
,
λ
N
1
b
(
x
)
µ
−
(1+
nµ
)
λ
∗
(
µ,k
1
,
Ω
0
)
(1+
nµ
)(1+
k
2
ρ
(
x
)
µ
)
,
Ω
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