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AdvancesinAppliedMathematics
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,2023,12(3),1215-1223
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.123123
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5
GlobalDynamicsofaFilippov
Predator-PreyModelwitha
Constant-CapacityRefuge
ShaJiang,PeiZhou
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Feb.21
st
,2023;accepted:Mar.20
th
,2023;published:Mar.27
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,2023
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1215-1223.DOI:10.12677/aam.2023.123123
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Abstract
Inthispaper,apredator-preymodelwithaconstant-capacityrefugeisstudied.We
applyFilippovmethodtostudytheslidingmodedynamicsofthemodel,andstudied
theglobaldynamicsfurtherly.Itisprovedthatthereexistsauniqueequilibriumand
theequilibriumisgloballyasymptoticallystable.Theresultsshowthattheestab-
lishmentofarefugewithappropriatecapacityisbeneficialtomaintainthebalance
betweenpredatorandprey.
Keywords
Predator
õõõ
PreyModel,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/
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DOI:10.12677/aam.2023.1231231219
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DOI:10.12677/aam.2023.1231231220
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DOI:10.12677/aam.2023.1231231221
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[1]
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[M].
®
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,2004.
[2]
p
<
.
~
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©•
§
[M].
®
:
p
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,2009.
[3]
o
ï
,
Ü
H
.
é
ä
k
™
ê
Ú
2
o
¤
Ó
X
Ú
Ä
å
Æï
Ä
[J].
ê
Æ
¢
‚
†
@
£
,2021,
51(18):281-287.
[4]Gonz
S
lez-Olivares,E. andRamos-Jiliberto,R. (2003)Dynamic ConsequencesofPrey Refuges
inaSimpleModelSystem:MorePrey,FewerPredatorsandEnhancedStability.
Ecological
Modelling
,
166
,135-146.https://doi.org/10.1016/S0304-3800(03)00131-5
[5]Chen,F.,Ma,Z.andZhang,H.(2012)GlobalAsymptoticalStabilityofthePositiveEqui-
libriumofthe Lotka-VolterraPrey-Predator ModelIncorporatingaConstantNumberofPrey
Refuges.
NonlinearAnalysis:RealWorldApplications
,
13
,2790-2793.
https://doi.org/10.1016/j.nonrwa.2012.04.006
[6]Tang,S.andLiang,J.(2013)GlobalQualitativeAnalysisofaNon-SmoothGausePredator-
PreyModelwithaRefuge.
NonlinearAnalysis:Theory,MethodsandApplications
,
76
,165-
180.https://doi.org/10.1016/j.na.2012.08.013
[7]Wu,Y.,Chen,F.andDu,C.(2021)DynamicBehaviorsofaNonautonomousPredator-Prey
SystemwithHollingTypeIISchemesandaPreyRefuge.
AdvancesinDifferenceEquations
,
2021
,ArticleNo.62.https://doi.org/10.1186/s13662-021-03222-1
[8]Kar,T.K.(2005)StabilityAnalysisofaPrey-PredatorModelIncorporatingaPreyRefuge.
CommunicationsinNonlinearScienceandNumericalSimulation
,
10
,681-691.
https://doi.org/10.1016/j.cnsns.2003.08.006
[9]Li,W.,Chen,Y.andHuang,L.(2022)GlobalDynamicsofaFilippovPredator-PreyModel
withTwoThresholdsforIntegratedPestManagement.
Chaos,SolitonsandFractals
,
157
,
ArticleID:111881.https://doi.org/10.1016/j.chaos.2022.111881
[10]Chen, X. and Huang, L. (2015) AFilippovSystem Describing the Effectof Prey Refuge Use on
a Ratio-Dependent Predator-PreyModel.
JournalofMathematicalAnalysisandApplications
,
428
,817-837.https://doi.org/10.1016/j.jmaa.2015.03.045
[11]
ê
¦
w
,
‘
á
÷
,
Z
Ï
.
k
Z
ý
„
–
Filippov
+
ë
.
Û
Ä
å
Æ
[J].
²
L
ê
Æ
,2020,
37(3):208-213.
[12]Filippov,A.F.(1988)DifferentialEquationswithDiscontinuousRight-HandSides.Kluwer
AcademicPublishers,Dordrecht.
[13]Wang,A.and Xiao,Y.(2013) SlidingBifurcation andGlobal Dynamicsof aFilippov Epidem-
icModelwithVaccination.
InternationalJournalofBifurcationandChaos
,
23
,ArticleID:
1350144.https://doi.org/10.1142/S0218127413501447
[14]Zhu,J.andLiu,H.(2006)PermanenceoftheTwoInteractingPrey-PredatorwithRefuges.
JournalofNorthwestUniversityforNationalities
,
27
,1-3.
DOI:10.12677/aam.2023.1231231223
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