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PureMathematicsnØêÆ,2021,11(5),841-850
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115097
†*Ñéäko«•
Beddington-DeAngelis.
Ó .•)K•
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Á‡
©ïÄàgNeumann>.^‡e†*Ñéäko«•Beddington-DeAngelis.Ó
.•)K•,Ù¥†*ÑL« ø;Ô.ÄkA^‚5z•{©ÛšK~ê²ï)
-½5.ÙgA^•ŒŠn‰ Ñ)kO.•A^©ÜnØ?Ø•)•35.(
JL²,†*ÑkÏuÔ«•"
'…c
Ó .§Beddington-DeAngelis.õU‡A¼ê§o«•§†*ѧ•)
TheEffectofCross-Diffusionon
Beddington-DeAngelisType
Predator-PreyModelwith
ProtectionZone
KaiYan,LinaZhang
∗
,NanaLin
∗ÏÕŠö"
©ÙÚ^:Ap,ÜwA,AA.†*Ñéäko«•Beddington-DeAngelis.Ó .•)K•[J].n
ØêÆ,2021,11(5):841-850.DOI:10.12677/pm.2021.115097
Ap
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/
1.Úó
3Ó XÚ¥,õU‡A¼ê´4Ù-‡|¤Ü©.1975c, BeddingtonÚDeAngelis©
O3©[1]Ú[2]¥ÕáJÑBeddington-DeAngelis.õU‡A¼ê.TõU‡A¼êQ•Ä
HollingII.‡A¼ê™•ÄÓ öSÜƒpZ,q;•'Ç•6.õU‡A¼ê3$—
ÝžÛÉ1•,¤±Æö‚2•ïÄÚ?Ø.
DOI:10.12677/pm.2021.115097842nØêÆ
Ap
©?ØXe‘†*ÑÚo«•Beddington-DeAngelis.Ó .
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

















u
t
= ∆[(1+kρ(x)v)u]+u

λ−u−
b(x)v
1+mu+nv

,(x,t) ∈Ω×(0,∞),
v
t
= ∆v+v

µ−v+
cu
1+mu+nv

,(x,t) ∈Ω\Ω
0
×(0,∞),
∂u
∂ν
= 0,(x,t) ∈∂Ω×(0,∞),
∂v
∂ν
= 0,(x,t) ∈∂(Ω\Ω
0
)×(0,∞),
u(x,0) = u
0
(x),x∈Ω,v(x,0) = v
0
(x),x∈Ω\Ω
0
,
(1.1)
Ù¥Ω´R
N
¥>.1wk.«•,Ω
0
´Ω>.1wf8;ν´>.þü {•þ;
λ,c,k,m,n´~ê,µ´~ê;ρ(x),b(x)3Ω\Ω
0
S•~ê3Ω
0
S•";u,v©OL« 
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0
L«˜‡ Ugd?Ñ
o«•,Ó öØU?\Ω
0
•U3Ω
0
±Ó [3].
.(1.1)¥, k∆[ρ(x)vu]•†*Ñ‘,£ãÓ ö• —ÝOŒ••*Ñ,=Ó öJ
Å œ/.ù«†*Ñ‘•kdShigesada3©[4]¥é¿Lotka-Volterro.JÑ,
ƒÚ\ˆa¿.ÚÓ .?1ïÄ[5–8].,,Ú\Ó .¥†*Ñõê£ãÓ
ö• —Ý~••*Ñ,= -”1•u),X(1.1)¥•xÓ öJÅ †*Ñ
ïÄ.d,•C©z[9]é.(1.1)¥Ø¹†*Ñ(=k= 0)œ/?1ïÄ.
Ïd,©·‚ïÄ†* Ñéäko«•Beddington-DeAngelis.Ó .K•.-
Ω
1
= Ω\Ω
0
,U= (1+kρ(x)v)u,K(1.1)²ï¯K•

















∆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

= 0,x∈Ω,
∆v+v

µ−v+
cU
(1+nv)(1+kv)+mU

= 0,x∈Ω
1
,
∂
ν
U= 0,x∈∂Ω,
∂
ν
v= 0,x∈∂Ω
1
,
(1.2)
Ù¥,
ρ(x) =
(
1,x∈Ω\Ω
0
,
0,x∈Ω
0
,
b(x) =
(
β,x∈Ω\Ω
0
,
0,x∈Ω
0
,
(1.3)
Ù¥β´˜‡~ê.w,,(u,v)´XÚ(1.1)-)…=(U,v)´XÚ(1.2)
).Ïd,·‚3e¡Ü©•ïÄXÚ(1.2).
©1!©ÛXÚ(1.2)šK~ê²ï)-½5.1n!|^©ÜnØy²XÚ(1.2)l
Œ²…)-‚)©Ü)•35,|^•ŒŠnXÚ(1.2))kO.1o!ï
Ä†*ÑXêké•«•K•.
2.šK~ê²ï)-½5
Äk§·‚y²eãÚn.
DOI:10.12677/pm.2021.115097843nØêÆ
Ap
Ún2.1µ>0,é?¿‰½kÚΩ
0
,•3•˜λ
∗
(µ) ∈(0,βµ),¦
λ
N
1

b(x)µ−λ
∗
(1+nµ)
(1+nµ)(1+kρ(x)µ)
,Ω

= 0.
d,lim
µ→0
λ
∗
(µ) = 0.
y²Ï•AŠλ
N
1
(q,Ω)'uqëY…î‚üN4O,¼êq(x) =
b(x)µ−λ(1+nµ)
(1+nµ)(1+kρ(x)µ)
'u
λëY…î‚üN4~,Ïd¼ê
λ→λ
N
1

b(x)µ−λ(1+nµ)
(1+nµ)(1+kρ(x)µ)
,Ω

: (0,∞) →R
ëY…î‚üN4~.qduλ
N
1
(0,Ω) = 0,¤±
λ
N
1

b(x)µ
(1+nµ)(1+kρ(x)µ)
,Ω

>0.
5¿(1.3)¥b(x)Lˆª,k
λ
N
1

b(x)µ−βµ(1+nµ)
(1+nµ)(1+kρ(x)µ)
,Ω

<0.
ŠâëY¼ê0Š½n•,éu?¿µ>0•3•˜λ
∗
= λ
∗
(µ) ∈(0,βµ),¦
λ
N
1

b(x)µ−λ
∗
(1+nµ)
(1+nµ)(1+kρ(x)µ)
,Ω

= 0.
w,,lim
µ→0
λ
∗
(µ) = 0.
52.1λ
∗
(µ)'uµüN5Ã{ä.duBeddington−DeAngelis.õU‡A¼ê±9
†*ÑÑy,¦d?³¼ê´'uµg¼ê.¤±³¼ê'uµüN5Ã{(½.
ù†HollingII.õU‡A¼ê[3]!'Ç•6.õU‡A¼ê[10]±9XÚ(1.2)¥Ø¹†*
Ñ[9]œ/ÑØÓ.
w,,XÚ(1.2)kn‡šK~ê²ï):²…)(0,0),Œ²…)(λ,0)Ú(0,µ).ÏL‚ 5z
•{©Û,Œ±y²…)ÚŒ²…)ÛÜ-½5.
½n2.1(1)é¤kλ,(0,0)´Ø-½.
(2)-λ
∗
= −
µ
c+mµ
.eλ<λ
∗
,K(λ,0)´ÛÜìC-½;eλ>λ
∗
,K(λ,0)´Ø-½.
(3)eλ<λ
∗
,K(0,µ)´ÛÜìC-½;eλ>λ
∗
,K(0,µ)´Ø-½.
y²XÚ(1.2)3~ê)e
∗
= (U,v)?‚5zŽf•
L=
4+A(U,v)B(U,v)
C(U,v)4+D(U,v)
!
,
DOI:10.12677/pm.2021.115097844nØêÆ
Ap
ٽ•´X= {(φ,ψ) ∈H
2
(Ω)×H
2
(Ω
1
) : ∂
ν
φ= 0,x∈∂Ω,∂
ν
ψ= 0,x∈∂Ω
1
},Ù¥
A(U,v) =
λ
1+kρ(x)v
−
2U
(1+kρ(x)v)
2
−b(x)
v(1+nv)(1+kρ(x)v)
[(1+nv)(1+kρ(x)v)+mU]
2
,
B(U,v) = −
λkρ(x)U
(1+kρ(x)v)
2
+
2kρ(x)U
2
(1+kρ(x)v)
3
−b(x)
U(1+mU−nkρ(x)v
2
)
[(1+nv)(1+kρ(x)v)+mU]
2
,
C(U,v) =
cv(1+nv)(1+kv)
[(1+nv)(1+kv)+mU]
2
,D(U,v) = µ−2v+
cU(1+mU−nkv
2
)
[(1+nv)(1+kv)+mU]
2
.
(1)ee
∗
= (0,0),K
A(U,v)B(U,v)
C(U,v)D(U,v)
!




(U,v)=(0,0)
=
λ0
0µ
!
.
duλ>0´‚5zŽfL3:(0,0)?˜‡AŠ,¤±(0,0)´Ø-½.
(2)XÚ(1.2)éA‚5zXÚ3(λ,0)?AŠ¯K•











4f−λf+λ
2
kρ(x)g−b(x)
λ
1+mλ
g−ζf= 0,x∈Ω,
4g+µg+
cλ
1+mλ
g−ζg= 0,x∈Ω
1
,
∂
ν
f= 0,x∈∂Ω,∂
ν
g= 0,x∈∂Ω
1
,
§k˜¢AŠSζ
1
>ζ
2
>ζ
3
...>ζ
n
>...→−∞.ζ
1
<0ž,(λ,0)´ÛÜìC-½.
´„,ζ
1
=d'ug•§û½,…
−ζ
1
= λ
N
1

−µ−
cλ
1+mλ
,Ω
1

= −µ−
cλ
1+mλ
.
=ζ
1
= µ+
cλ
1+mλ
.Pλ
∗
= −
µ
c+mµ
,•yλ
∗
>0I•›µŠ‰Œ•µ∈(−
c
m
,0).Kλ=λ
∗
ž,ζ
1
=0;µ∈(−
c
m
,0)…λ<λ
∗
ž, ζ
1
<0,dž(λ,0)´ÛÜìC-½;µ≥0½ö
µ∈(−
c
m
,0)…λ>λ
∗
ž,ζ
1
>0,dž(λ,0)´Ø-½.
(3)XÚ(1.2)éA‚5zXÚ3(0,µ)?AŠ¯K•











4f+
λ(1+nµ)−b(x)µ
(1+nµ)(1+kρ(x)µ)
f−ηf= 0,x∈Ω,
4g−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ŠSη
1
>η
2
>η
3
...>η
n
>...→−∞.η
1
<0ž,(0,µ)´ÛÜìC-½.
η
1
=df•§(½,´„
−η
1
= λ
N
1

−λ(1+nµ)+b(x)µ
(1+nµ)(1+kρ(x)µ)
,Ω

.
(ÜÚn2.1•,eλ<λ
∗
,Kη
1
<0,l(0,µ)´ÛÜìC-½;eλ>λ
∗
,Kη
1
>0,l
(0,µ)´Ø-½.
DOI:10.12677/pm.2021.115097845nØêÆ
Ap
3.•)•35
e¡ÀλŠ•©Üëê,A^ÛÜ©ÜnØ[11],?ØXÚ(1.2)ü‡Œ²…)-‚:
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
).
½Â
φ
∗
= (−4+λ
∗
)
−1
Ω

λ
2
∗
kρ(x)−b(x)
λ
∗
1+mλ
∗

(3.1)
-φ
∗
•AŠ¯K
−4φ
∗
+
−λ
∗
(1+nµ)+b(x)µ
(1+nµ)(1+kρ(x)µ)
φ
∗
= 0, x∈Ω,∂
ν
φ
∗
= 0, x∈∂Ω(3.2)
ÌA¼ê,
ψ
∗
= (−4+µ)
−1
Ω
1

cµφ
∗
(1+nµ)(1+kµ)

.(3.3)
e¡·‚$^©ÜnØ?ØÛÜ©|)-‚•35,Œe(Ø.
½n3.1(1)(λ
∗
;λ
∗
,0)´Γ
U
þ•˜©Ü:,…3(λ
∗
;λ
∗
,0)•S•3XÚ(1.2)
)-‚Γ
∗
={(λ;U,v)=(λ(s);λ+ s(φ
∗
+
b
U(s)),s(1+ bv(s))):s∈(0,
b
δ)},Ù¥
b
δ>0¿©,
λ(s),
b
U(s),bv(s)•C
1
-‚,…÷vλ(0) = λ
∗
,
b
U(0) = bv(0) = 0,
R
Ω
1
bv(s)dx= 0.
(2)(λ
∗
;0,µ)´Γ
v
þ•˜©Ü:,…3(λ
∗
;0,µ)•S•3XÚ(1.2))-‚Γ
∗
=
{(λ;U,v)= (λ(s);s(φ
∗
+
e
U(s)),µ+s(ψ
∗
+ev(s))):s∈(0,
e
δ)},Ù¥
e
δ>0¿©,λ(s),
e
U(s),ev(s)
•C
1
-‚,…÷vλ(0) = λ
∗
,
e
U(0) = ev(0) = 0,
R
Ω
e
U(s)φ
∗
dx= 0.
y²(1)-z:= U−λ,½ÂNF: R×X
1
→X
2
:
F(λ;z,v) =


4z+
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)

4v+v

µ−v+
c(z+λ)
(1+nv)(1+kv)+m(z+λ)



.(3.4)
²{üOŽ´
F
(z,v)
(λ;0,0)[φ,ψ] =


4φ−λφ+λ
2
kρ(x)ψ−b(x)
λ
1+mλ
ψ
4ψ+(µ+
cλ
1+mλ
)ψ


.
dKrein-Rutman½n[12]•,…=λ= λ
∗
ž,F
(z,v)
(λ;0,0)[φ,ψ] = (0,0)k˜‡)ψ>0.¤
±,(λ
∗
;0,0)´Γ
U
þ•˜©|:…KerF
(z,v)
(λ
∗
;0,0)=span{(φ
∗
,1)}.Ù ¥φ
∗
3(3.1)¥‰Ñ.
DOI:10.12677/pm.2021.115097846nØêÆ
Ap
ÏddimKerF
(z,v)
(λ
∗
;0,0) = 1.ŠâFredholmJ˜½n[12]•,
RangeF
(z,v)
(λ
∗
;0,0) =

(φ,ψ)}∈X
2
:
Z
Ω
1
ψdx= 0

.
Ïd,codimRangeF
(z,v)
(λ
∗
;0,0) = 1.d,
F
λ(z,v)
(λ
∗
;0,0)[φ
∗
,1] =



−φ
∗
+2kρ(x)λ
∗
−
b(x)
(1+mλ
∗
)
2
c
(1+mλ
∗
)
2



*RangeF
(z,v)
(λ
∗
;0,0).
Ïd,dÛÜ©|½n[11],½n3.1(Ø(1).
(2)½ÂN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)
K²{üOŽŒ
G
(U,v)
(λ;0,µ)[φ,ψ] =



4φ+
λ(1+nµ)−b(x)µ
(1+nµ)(1+kρ(x)µ)
φ
4ψ−µψ+
cµ
(1+nµ)(1+kµ)
φ



.
dKrein-Rutman½n[12]•,…=λ=λ
∗
ž, G
(U,v)
(λ;0,µ)[φ,ψ]=(0,0)k˜‡)φ>0.
¤±,(λ
∗
;0,µ)´Γ
v
þ•˜©|:…KerG
(U,v)
(λ
∗
;0,µ)=span{(φ
∗
,ψ
∗
)}.Ù¥φ
∗
, ψ
∗
d(3.2)
Ú(3.3)ª‰Ñ.dimKerG
(U,v)
(λ
∗
;0,µ) = 1.d,
RangeG
(U,v)
(λ
∗
;0,µ) =

(φ,ψ)}∈X
2
:
Z
Ω
φφ
∗
dx= 0

.
dþªŒ±íÑ
G
λ(U,v)
(λ
∗
;0,µ)[φ
∗
,ψ
∗
] =


φ
∗
1+kρ(x)µ
0


*RangeG
(U,v)
(λ
∗
;0,µ).
Ïd,dÛÜ©|½n[11]½n3.1(Ø(2).
e5,·‚ò‰ÑXÚ(1.2))˜‡kO,?©Û©|)-‚l©|:ž
Û©|(.
Ún3.1(U,v)´XÚ(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≥0ž,é?¿x∈Ω
1
kv(x) >µ.
DOI:10.12677/pm.2021.115097847nØêÆ
Ap
y²-U(x
0
) = max
x∈Ω
U(x),d•ŒŠ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+kmax
x∈Ω
1
v(x)).(3.6)
-v(x
1
) = max
x∈Ω
1
v(x),KXÚ(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.Ïd,é?¿x∈Ω
1
,kv(x) ≤µ+c/m.¤±é?¿
x∈Ω,k0 <U(x) ≤λ[1+k(µ+c/m)].,˜•¡,
−4v= v

µ−v+
cU
(1+nv)(1+kv)+mU

>v(µ−v)x∈Ω
1
, ∂
ν
v= 0x∈∂Ω
1
.
d'n,XJµ≥0,Kx∈Ω
1
ž,v(x) >µ.
aqu©[13]½n2.2y²L§,ÏLIOÛ©|?Ø,Œ±Xe(Ø.
½n3.2(1)eµ≥0,Kλ>λ
∗
(µ)ž,XÚ(1.2)–•3˜‡).
(2)eµ<0,Kλ>λ
∗
(µ)ž,XÚ(1.2)–•3˜‡).
4.†*Ñé•)K•
!ïÄ†*ÑXêké•«•K•.
½n4.1µ>0,Kλ
∗
(µ,k,Ω
0
)'ukî‚üN4~.
y²½µ>0ÚΩ
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)
db(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)
dλ
N
1
(q,Ω)'uqëY…î‚üN4O5Ÿ±9λ
N
1
(0,Ω) = 0,βµ−(1+nµ)λ
∗
(µ,k,Ω
0
) >0.
DOI:10.12677/pm.2021.115097848nØêÆ
Ap
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)
,
Ïd,
λ
N
1

b(x)µ−(1+nµ)λ
∗
(µ,k
1
,Ω
0
)
(1+nµ)(1+k
2
ρ(x)µ)
,Ω

<λ
N
1

b(x)µ−(1+nµ)λ
∗
(µ,k
1
,Ω
0
)
(1+nµ)(1+k
1
ρ(x)µ)
,Ω

= λ
N
1

b(x)µ−(1+nµ)λ
∗
(µ,k
2
,Ω
0
)
(1+nµ)(1+k
2
ρ(x)µ)
,Ω

.
Ïd,ek
2
>k
1
,Kλ
∗
(µ,k
1
,Ω
0
) >λ
∗
(µ,k
2
,Ω
0
).ùL²λ
∗
(µ,k,Ω
0
)'ukî‚üN4~.
54.1½n4.1(ØL²•«•‘X†*ÑXêkOŒ*Œ,Ï†*Ñk|u
Ô«•.
Ä7‘8
I[g,䮀7(11761063)"
ë•©z
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DOI:10.12677/pm.2021.115097850nØêÆ

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