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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(3),1215-1223
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.123123
äkð½Nþ;J¤
FilippovÓ ö- .
ÛÄåÆ
öööëë맧§±±±
•ânóŒÆêƆÚOÆ§H•â
ÂvFϵ2023c221F¶¹^Fϵ2023c320F¶uÙFϵ2023c327F
Á‡
©ïÄ˜aäkð½Nþ;J¤FilippovÓ ö- ."A^Filippov•{ïÄ.
wÄåƧ¿?˜ÚïÄ.ÛÄåÆ"y²XÚ•3•˜²ï§…T²ï´
ÛìC-½"(JL²§ïá·Nþ;J¤k|u‘±Ó ö†Ó öƒm²ï"
'…c
Ó ö- XÚ§;J¤§²ï:§-½5
GlobalDynamicsofaFilippov
Predator-PreyModelwitha
Constant-CapacityRefuge
ShaJiang,PeiZhou
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Feb.21
st
,2023;accepted:Mar.20
th
,2023;published:Mar.27
th
,2023
©ÙÚ^:öë,±.äkð½Nþ;J¤ilippovÓ ö- .ÛÄåÆ[J].A^êÆ?Ð,2023,12(3):
1215-1223.DOI:10.12677/aam.2023.123123
ö맱
Abstract
Inthispaper,apredator-preymodelwithaconstant-capacityrefugeisstudied.We
applyFilippovmethodtostudytheslidingmodedynamicsofthemodel,andstudied
theglobaldynamicsfurtherly.Itisprovedthatthereexistsauniqueequilibriumand
theequilibriumisgloballyasymptoticallystable.Theresultsshowthattheestab-
lishmentofarefugewithappropriatecapacityisbeneficialtomaintainthebalance
betweenpredatorandprey.
Keywords
PredatorõõõPreyModel,Refuge,Equilibrium,Stability
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3g,.¥§)¹3Ó˜/•Ô«ƒm•)• ?1Xí¤Ì. Ôó´)XÚ
Ä(§´d «+•ÏuÐÚ?z/¤.~X§HÚ3Ó‚¸¥§‚¸¥ú„
jž§ Ô¿v§Ïdm©Œþ„ˆ.duêþO\§Hk¿v Ô§u´Hm©
Œþ„ˆ.HêþO\§oêþdu¯Åì~§ÏdH Ô~±–uH
êþ~.ù‡žÿ§S5Jp§¤±oêqm©O\§-Eù‡L§§†ˆ²ï.
éþãy–§¿Œ|Ͷ)ÔêÆ[VolterraïáXeêÆ.[1][2][3]:
(
˙x= rx−cxy
˙y= βxy−dy.
bx´ —ݧy´Ó ö—Ý.Ù¥r´ Ñ)ǧβL«Ó¼ =z•#
Ó öǧdL«Ó ökǧÓ öêþ:ìO\ž§é %•¬‘ƒO
\.•y «+Øàk§<am©Zý§• ïá;J¤´˜«k“Ž «
ý•{.Cc5§<‚‰ŒþóŠ5ïÄ ;J¤K•§X©z[3][4][5]§ŠöJ
Ñäkð½'~NþÔ;J¤Lotka-VolterraÓ ö..3ë•©z[6][7][8]¥§ïÄ
DOI:10.12677/aam.2023.1231231216A^êÆ?Ð
ö맱
HollingII.Úk;J¤Ó ö- XÚÄåÆ1•.
3©¥§·‚± —ÝŠ•KŠ5©Û±e.[4][9][10]:
(
˙x= rx

1−
x
K

−c(x−εR)y
˙y= β(x−εR)y−dy,
(1.1)
Ù¥K•Ù«1Uå§K>RÚK>σ§R´ ;J¤¤UNB•ŒNþ.
ε=
(
0,x>σ,
1,x<σ,
(1.2)
´››¼ê§σ>0L«KŠ. —Ýpuσž§ØI‡æ››„–.,§˜ —Ý$
uKŠσ§ÒATá=‹m;J¤5o .
©Ù{Ü©|„Xeµ31!§·‚‰ ÑFilippovXÚ˜½ÂÚ5Ÿ¶1n!©
Ûw•35ÚÄåÆ©Û¶31o!¥§y²²ï:Û-½5§¿‰ÑêŠ[
㶕§31ÊÜ©Ñ(Ø.
2..09ý•£
3ù˜!¥§·‚‰Ñ©¤I‡FilippovXÚ˜½ÂÚ5Ÿ.•õ[!žë
•[11][12][13][14].
(x,y) ∈R
2
+
§·‚òR
2
+
©¤±enÜ©µ
G
1
=

(x,y) ∈R
2
+
: x>σ

,
G
2
=

(x,y) ∈R
2
+
: x<σ

,
Ω =

(x,y) ∈R
2
+
: x= σ

,
·‚ÀJn= (1,0)Š•{•þ.w,§XÚ(1.1)dü‡fXÚ|¤



˙x= rx

1−
x
K

−cxy
˙y= βxy−dy
(2.1)



˙x= rx

1−
x
K

−cx−Ry
˙y= βx−Ry−dy
(2.2)
duXÚ(1.1)mýØëY5§©½ÂFilippov¿ÂFilie(1.1)).co[φ(I)]•φ(I)à4
•§@oco[φ(I)] = [φ(I
−
),φ(I
+
)]§Ù¥φ(I
−
)Úφ(I
+
)©OL«φ3I?†m4•.
e5§·‚©OïÄfXÚÛÄåÆ.
Äk3G
1
¥§w,XÚ(2.1)o´kü‡>.²ï:(0,0)Ú(K,0).d§(0,0)©ª´˜‡
Q:.XJβK<d§K(K,0)´˜‡-½(:§XJβK>d§K§´˜‡Q:.βK>dž§
DOI:10.12677/aam.2023.1231231217A^êÆ?Ð
ö맱
(2.1)k•˜²ï:E
1
= (x
∗
1
,y
∗
1
) =

d
β
,
rβK−rd
cβK

§§•3ž§ù´˜‡-½:½(:.
e53G
2
¥§w,XÚ(2.2)•o´kü‡>.²ï:(0,0)Ú(K,0).(0,0)©ª´˜‡
Q:.XJβK<d+βR§K(K,0)´˜‡-½(:§XJβK>d+βR§K§´˜‡Q:"
βK>d+βRž§(2.2)•k•˜²ï:E
2
= (x
∗
2
,y
∗
2
) =

βR+d
β
,
r[(βR+d)(βK−βR−d)]
βKcd

.§•
3ž§ù´˜‡-½:½(:.
·K2.1.b(x(t),y(t))•.(1.1)3«m[0,T),T∈(0,+∞)þ÷vЩ^‡x(0)=x
0
>
0Úy(0) = y
0
>0)§Ké¤kt∈[0,T)kx(t) >0…y(t) >0.
y².d‡y{y²y(t) >0.b•3t
0
¦y(t
0
) ≤0§@o•3t
∗
>0÷vy(t
∗
) =0¿…é
u¤kt∈[0,t
∗
)ky(t) >0.•Ä.(1.1)1‡•§§·‚é¤kt∈[0,t
∗
)k
dy
dt
= y[β(x−εR)−d],
¤±
y(t
∗
) = y(0)e
R
t
∗
0
[β(x(s)−εR)−d]ds
>0,
§Úy(t
∗
) = 0gñ.
e5d.(1.1)1˜‡•§§XJ3x=0?Œ
dx
dt
=εcRy≥0§Ï•®yy(t)>0§K
dx(0) >0Υx(t) >0.y..2
·K2.2.b(x(t),y(t))•.(1.1))§Ù¥é¤kt∈[0,+∞)kx(0) = x
0
>0Úy(0) =
y
0
>0.K)(x(t),y(t))3t∈[0,+∞)þk..
y².5¿
dx
dt


x=K
= −c(K−εR)y<0,
dx
dt


x>K
<0.
Ïdé(2.1)z‡)(x(t),y(t))§•3˜‡T>0§éut≥T¦x(t) <K.
Két≥T§·‚k
d

x(t)+
c
β
y(t)

dt
= rx

1−
x
K

−
cd
β
y
≤rK

1−
x
K

−
cd
β
y
= rK−

rx+
cd
β
y

≤rK−min{r,c}

x(t)+
d
β
y(t)

„Ìlim
t→∞
sup

x(t)+
c
β
y(t)

≤
rK
min{r,c}
§Ïd)3t∈[0,+∞) þk..2
·K2.3.XÚ(2.1)Ú(2.2)©Ok˜‡ÛìC-½²ï:E
1
,E
2
.
y².EDulac¼ê•B=
1
xy
§K
∂(Bf
i1
)
∂x
+
∂(Bf
i2
)
∂y
<0,
ŠâBendixson−Dulac{K•XÚ(2.1)Ú(2.2)Ø•34•‚"Ïd²ï:E
1
,E
2
ÛìC-
DOI:10.12677/aam.2023.1231231218A^êÆ?Ð
ö맱
½"2
3.w•359ÙÄåÆ
!·‚Äk?Øw•35§,ïÄÙÄåÆ1•.
dhn
1
,F
1
i= 0,hn
1
,F
2
i= 0.Œ-T
1
:=
r(K−σ)
cK
§T
2
:=
rσ(K−σ)
cK(σ−R)
§Ù¥K>σ.
σ>Rž§dhn
1
,F
1
i≤0hn
1
,F
2
i≥0§XÚ(2.1)3Ωþw«•µ
Σ
s
1
⊂Ω =

(x,y) ∈Ω;T
1
≤y≤T
2

.
σ≤Rž§XÚ(2.1)3Ωþw«•µ
Σ
s
2
⊂Ω =

(x,y) ∈Ω;T
1
≤y

.
ŠâFilippovà•{
x
0
(t)
y
0
(t)
!
= λF
1
+(1−λ)F
2
w•§•
f(y) =
βrσ(K−σ)
cK
−dy.(3.1)
XÚ•3•˜˜‡²ï)E
p
=

σ,y
∗
p

§Ù¥y
∗
p
=
βrσ(K−σ)
cKd
§σ>Rž§…=T
1
≤y
∗
p
≤
T
2
ž§3Σ
s
1
⊂Ωþ•3•˜–²ï:E
p
.σ≤Rž§…=T
1
≤y
∗
p
ž§3Σ
s
2
⊂Ωþ•3
•˜–²ï:E
p
.d
∂
∂y

βrσ(K−σ)
ck
−dy





E
p
= −d<0
ùL²)´áÚ§•Ò´`§–²ïE
p
•3ž§§´ÛìC-½.
4.ÛÄåÆ
Ún4.1.XÚ(2.1)Ø•3Œ7wΣ
s
4;.
y².e¡·‚æ^‡y{5y².
b•3˜‡•¹Σ
s
4•‚Γ.Xã1¤«§P4•‚†6/þe:©O•NÚM§4•
‚†x= σ+δþe:©O•N
1
= N−δÚM
1
= M+δ§Óž§P4•‚†x= σ−δþe:
©O•N
2
= N−δÚM
2
= M+δ§Ù¥δ>0¿©.d÷vlim
δ→0
M
i
= M,lim
δ→0
N
i
= N(i= 1,2).
-∂D
1
L«Γ
1
ÚN
1
M
1
¤Œ¤«•D
1
>.§∂D
2
L«Γ
2
ÚM
2
N
2
¤Œ¤«•D
2
>..À
JDulac¼ê•B=
1
xy
§
ZZ
D

∂(BF
1
)
∂x
+
∂(BF
2
)
∂y

dxdy
=
2
X
i=1
ZZ
D
i

∂(Bf
i1
)
∂x
+
∂(Bf
i2
)
∂y

dxdy<0.
(4.1)
DOI:10.12677/aam.2023.1231231219A^êÆ?Ð
ö맱
Figure1.Supposethatthereexists
alimitcycleΓcontainingΣ
s
ã1.b•3˜‡•¹Σ
s
4•‚Γ
3D
1
¥A^‚úªk
ZZ
D
1

∂(Bf
11
)
∂x
+
∂(Bf
12
)
∂y

dxdy=
I
∂D
1
(Bf
11
)dy−
I
∂D
1
(Bf
12
)dx
=
Z
Γ
1
Bf
11
dy+
Z
−−−−→
N
1
M
1
Bf
11
dy−

Z
Γ
1
Bf
12
dx+
Z
−−−−→
N
1
M
1
Bf
12
dx

=
Z
Γ
1
(Bf
11
·f
12
−Bf
12
·f
11
)dt+
Z
−−−−→
N
1
M
1
Bf
11
dy
=
Z
−−−−→
N
1
M
1
Bf
11
dy.
Ù¥
dx
dt
= f
11
,
dy
dt
= f
12
§
R
−−−−→
N
1
M
1
Bf
12
dx= 0.
Ó3D
2
¥A^‚úªk
ZZ
D
2

∂(Bf
21
)
∂x
+
∂(Bf
22
)
∂y

dxdy=
Z
−−−−→
M
2
N
2
Bf
21
dy.
duN>M§lim
δ→0
M
i
= M,lim
δ→0
N
i
= N(i= 1,2)§Œ•
lim
δ→0

Z
−−−−→
N
1
M
1
Bf
11
dy+
Z
−−−−→
M
2
N
2
Bf
21
dy

=lim
δ→0

Z
M
1
N
1

r
y
−
rx
Ky
−c

dy+
Z
N
2
M2

r
y
−
rx
Ky
−c+
cR
x

dy

=

rlny−cy−
rx
K
lny




M
N
+

rlny−
rx
K
lny−cy+
cRy
x





N
M
=
cR
x
(N−M) >0.
ù†ª(4.1)gñ§=XÚ(2.1)Ø•3Œ7wΣ
s
4;.2
Ún4.2.XÚ(2.1)Ø•3•¹Ü©wΣ
s
4;.
y².3ù«œ¹e§æ^‡y{5y²§ Ø”˜„5§bE
1
•¢²ï:§E
2
•J²ï:§
XÚ(2.1)•3•¹Ü©Σ
s
4;Γ§KΓ˜½lƒ:T
1
Ñu¿…ˆΣ
s
§Xã2¤«§dž4;
DOI:10.12677/aam.2023.1231231220A^êÆ?Ð
ö맱
Figure2.Thepossibleclosed
orbitcontainingapartof
¯
Σ
s
ã2.ŒU•3•¹Ü©w
Σ
s
4;
¡);‚ØU?\4;Sܧù†E
1
3«•G
1
Û-½5´gñ§ÏdlT
1
Ñu;‚Ø
¬ˆΣ
s
.2
Ún4.3.XJXÚ(2.1)Œ;T
+
´k.§@o§4•8Ω(T)‡o•¹˜‡²ï:§‡
o•¹˜‡4;§„[12].
½n1.²ï:E
1
,E
2
Ñ•3ž§e¡(ؤáµ
£1¤σ<x
∗
1
ž§¢²ï:E
1
´ÛìC-½¶
£2¤σ>x
∗
2
ž§¢²ï:E
2
´ÛìC-½¶
£3¤x
∗
1
<σ<x
∗
2
ž§–²ï:E
p
´ÛìC-½.
y².Äky²(Ø£1¤.σ<x
∗
1
ž§dw•§(3.1)5ŸŒ•§f(T
1
)=
r(βr−d)(K−σ)
cK
<
0,f(T
2
)=
rσ(βσ−βR−d)(K−σ)
cK
<0.=3«m(T
1
,T
2
)þf(y)<0§ù`²–²ï:Ø•3§… w
þ;, dT
2
T
1
.qÏ•)k.…ŠâÚnŒ•Ø•3Ù§4;§KdÚn(4.3)ŒÑ•˜
¢²ï:E
1
´ÛìC-½§Xã3¤«.aq /•ŒÑ(Ø£2¤µσ>x
∗
2
ž§¢²ï:E
2
´ÛìC-½§Xã4¤«.
e5y²(Ø£3¤.x
∗
1
<σ<x
∗
2
ž§f(T
1
) =
r(βr−d)(K−σ)
cK
>0,f(T
2
) =
rσ(βσ−βR−d)(K−σ)
cK
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