A prey-predator fractional order model with fear effect and group defense
In this work, a fractional-order prey-predator system with fear effect of predator on prey population and group defense has been proposed. The existence and uniqueness of the system have been studied. Non-negativity and boundedness are also theoretically demonstrated. Analysis of local stability wit...
Saved in:
| Published in | International journal of dynamics and control Vol. 9; no. 1; pp. 334 - 349 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.03.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | In this work, a fractional-order prey-predator system with fear effect of predator on prey population and group defense has been proposed. The existence and uniqueness of the system have been studied. Non-negativity and boundedness are also theoretically demonstrated. Analysis of local stability with examination of saddle-node and Hopf bifurcation at equilibrium points are performed by the help of numerical simulations along with analytical study. All the numerical simulations are performed using MATLAB and MAPLE. |
|---|---|
| AbstractList | In this work, a fractional-order prey-predator system with fear effect of predator on prey population and group defense has been proposed. The existence and uniqueness of the system have been studied. Non-negativity and boundedness are also theoretically demonstrated. Analysis of local stability with examination of saddle-node and Hopf bifurcation at equilibrium points are performed by the help of numerical simulations along with analytical study. All the numerical simulations are performed using MATLAB and MAPLE. |
| Author | Das, Meghadri Samanta, G. P. |
| Author_xml | – sequence: 1 givenname: Meghadri surname: Das fullname: Das, Meghadri organization: Department of Mathematics, Indian Institute of Engineering Science and Technology – sequence: 2 givenname: G. P. surname: Samanta fullname: Samanta, G. P. email: g_p_samanta@yahoo.co.uk, gpsamanta@math.iiests.ac.in organization: Department of Mathematics, Indian Institute of Engineering Science and Technology |
| BookMark | eNp9kE1PwzAMhiM0JMbYH-AUiXPBTdI2OU4TH5MmcQGJW-Q2yejUNSPpxPbv6SgCicMutg_vY9nPJRm1vrWEXKdwmwIUd1GA4FkCDBKAnOXJ_oyMWaqyhOVKjn5n-XZBpjGuAYClAphQY7KY0W2wh6QvBjsfqAtYdbVvsaE-GBvoxhvb0M-6e6fOYqDWOVt1FFtDV8HvttRYZ9tor8i5wyba6U-fkNeH-5f5U7J8flzMZ8uk4qnqEixzlSk0kEsjSpU6zpVTjoEorJIZSgaAeYGWIS9FnmWSSYVCgCmR84rzCbkZ9m6D_9jZ2Om134X-3qiZkIUQBZNZn5JDqgo-xmCdruoOj491AetGp6CP7vTgTvfu9Lc7ve9R9g_dhnqD4XAa4gMU-3C7suHvqhPUF3Sbgug |
| CitedBy_id | crossref_primary_10_1016_j_chaos_2023_114449 crossref_primary_10_1515_cmb_2020_0116 crossref_primary_10_1142_S0218127422500821 crossref_primary_10_3390_fractalfract5030084 crossref_primary_10_3390_axioms11100577 crossref_primary_10_3934_math_20221000 crossref_primary_10_1007_s11071_021_06435_x crossref_primary_10_1007_s40435_020_00646_7 crossref_primary_10_1016_j_matcom_2020_06_015 crossref_primary_10_1016_j_matcom_2021_08_005 crossref_primary_10_12677_AAM_2021_1011380 crossref_primary_10_1007_s12346_024_00981_6 crossref_primary_10_1142_S179300572450011X crossref_primary_10_3934_mbe_2022610 crossref_primary_10_1007_s40995_023_01540_5 crossref_primary_10_1155_2021_9963031 crossref_primary_10_3934_mbe_2023322 crossref_primary_10_1007_s00366_021_01383_x crossref_primary_10_1007_s11071_021_06719_2 crossref_primary_10_1016_j_nonrwa_2023_103888 crossref_primary_10_3390_fractalfract7100722 crossref_primary_10_3390_math11204367 crossref_primary_10_1007_s12190_022_01728_9 crossref_primary_10_1063_5_0130809 crossref_primary_10_1515_cmb_2020_0121 crossref_primary_10_3934_mbe_2024319 crossref_primary_10_3390_axioms14010053 crossref_primary_10_3934_math_2023091 crossref_primary_10_1016_j_csfx_2022_100073 crossref_primary_10_1155_2023_5030729 crossref_primary_10_3390_math10163020 crossref_primary_10_1007_s40435_021_00822_3 crossref_primary_10_1142_S021812742250002X crossref_primary_10_1155_2021_9358881 crossref_primary_10_1155_2021_6402459 crossref_primary_10_1007_s40808_023_01843_x crossref_primary_10_1007_s40324_021_00251_4 crossref_primary_10_1007_s12215_020_00570_x crossref_primary_10_1007_s40819_021_01224_x crossref_primary_10_1016_j_cnsns_2022_107043 crossref_primary_10_1007_s12591_021_00575_7 crossref_primary_10_1142_S0218127423500086 crossref_primary_10_1063_5_0241236 crossref_primary_10_1007_s12190_025_02382_7 crossref_primary_10_3390_math9070703 crossref_primary_10_11121_ijocta_2021_1123 crossref_primary_10_3390_math8081244 crossref_primary_10_2139_ssrn_4185431 |
| Cites_doi | 10.1007/s10336-010-0638-1 10.1155/2013/279681 10.1114/1.1574026 10.1016/j.amc.2019.03.034 10.1007/s11071-011-0157-5 10.1016/j.automatica.2009.04.003 10.1016/j.chaos.2017.03.034 10.1016/j.aml.2011.05.035 10.3934/dcdsb.2014.19.2267 10.1155/2014/631419 10.2193/0022-541X(2004)068[0519:VPRATA]2.0.CO;2 10.1080/10652469.2014.965704 10.1007/s10928-010-9153-5 10.1002/bit.260230909 10.1142/3779 10.1007/s00285-016-0989-1 10.1007/s00607-003-0033-3 10.1155/2011/298628 10.1002/bit.260100602 10.1126/science.1135918 10.1007/s11071-014-1439-5 10.1007/978-3-642-18101-6 10.1023/A:1016592219341 10.1111/j.1365-2656.2009.01552.x 10.1073/pnas.0902235106 10.1007/s12190-016-1017-8 10.1201/9781315367453 10.1142/S0218127407017732 10.1080/00207160802624331 10.1007/978-1-4020-6042-7 10.1016/j.jmaa.2006.01.087 10.1371/journal.pone.0002465 10.1016/j.tree.2007.12.004 10.1890/0012-9658(1997)078[1388:BMTCEO]2.0.CO;2 10.1086/661250 10.1890/090226 |
| ContentType | Journal Article |
| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature 2020 Springer-Verlag GmbH Germany, part of Springer Nature 2020. |
| Copyright_xml | – notice: Springer-Verlag GmbH Germany, part of Springer Nature 2020 – notice: Springer-Verlag GmbH Germany, part of Springer Nature 2020. |
| DBID | AAYXX CITATION |
| DOI | 10.1007/s40435-020-00626-x |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering |
| EISSN | 2195-2698 |
| EndPage | 349 |
| ExternalDocumentID | 10_1007_s40435_020_00626_x |
| GroupedDBID | -EM 0R~ 30V 4.4 406 96X AACDK AAHNG AAIAL AAJBT AAJKR AANZL AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAZMS ABAKF ABDZT ABECU ABFTV ABJNI ABJOX ABKCH ABMQK ABQBU ABTEG ABTKH ABTMW ABXPI ACAOD ACDTI ACGFS ACHSB ACIWK ACKNC ACMLO ACOKC ACPIV ACZOJ ADHHG ADHIR ADINQ ADKNI ADRFC ADURQ ADYFF ADZKW AEBTG AEFQL AEGNC AEJHL AEJRE AEMSY AEOHA AEPYU AESKC AETCA AEVLU AEXYK AFBBN AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGJBK AGMZJ AGQEE AGQMX AGRTI AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AILAN AITGF AJBLW AJRNO AJZVZ AKLTO ALFXC ALMA_UNASSIGNED_HOLDINGS AMKLP AMXSW AMYLF AMYQR ANMIH ASPBG AUKKA AVWKF AVXWI AXYYD AZFZN BGNMA DNIVK DPUIP EBLON EBS EIOEI EJD ESBYG FEDTE FERAY FIGPU FINBP FNLPD FRRFC FSGXE FYJPI GGCAI GGRSB GJIRD H13 HMJXF HRMNR HVGLF HZ~ I0C IKXTQ IWAJR J-C JBSCW JZLTJ KOV LLZTM M4Y NPVJJ NQJWS NU0 O9- O9J PT4 RLLFE ROL RSV SCL SHX SJYHP SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE TSG UG4 UOJIU UTJUX UZXMN VFIZW W48 ZMTXR AAYXX ABBRH ABDBE ABFSG ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION ABRTQ |
| ID | FETCH-LOGICAL-c319t-ab6959ad068d4b91f339f9f2047e985a8200a67ae2a3b46558289a440dba33c33 |
| IEDL.DBID | AGYKE |
| ISSN | 2195-268X |
| IngestDate | Wed Aug 13 06:10:18 EDT 2025 Tue Jul 01 02:44:15 EDT 2025 Thu Apr 24 22:57:04 EDT 2025 Fri Feb 21 02:48:40 EST 2025 |
| IsPeerReviewed | false |
| IsScholarly | true |
| Issue | 1 |
| Keywords | Bifurcation Caputo fractional differential equation Fear effect Predator-prey model Group defense Lyapunov exponent |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c319t-ab6959ad068d4b91f339f9f2047e985a8200a67ae2a3b46558289a440dba33c33 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 2487447285 |
| PQPubID | 2043900 |
| PageCount | 16 |
| ParticipantIDs | proquest_journals_2487447285 crossref_citationtrail_10_1007_s40435_020_00626_x crossref_primary_10_1007_s40435_020_00626_x springer_journals_10_1007_s40435_020_00626_x |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 2021-03-01 |
| PublicationDateYYYYMMDD | 2021-03-01 |
| PublicationDate_xml | – month: 03 year: 2021 text: 2021-03-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Berlin/Heidelberg |
| PublicationPlace_xml | – name: Berlin/Heidelberg – name: Heidelberg |
| PublicationTitle | International journal of dynamics and control |
| PublicationTitleAbbrev | Int. J. Dynam. Control |
| PublicationYear | 2021 |
| Publisher | Springer Berlin Heidelberg Springer Nature B.V |
| Publisher_xml | – name: Springer Berlin Heidelberg – name: Springer Nature B.V |
| References | AhmedEEl-SayedAEl-SakaHEquilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies modelsJ Math Anal Appl2007325542-1-7553227354410.1016/j.jmaa.2006.01.087 Hilfer R (ed) (2000) Applications of fractional calculus in physics. World Scientific Publishing Co., Inc, River Edge MainardiFOn some properties of the Mittag-Leffler function Eα,1(-ηtε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ E_{\alpha ,1}(-\eta t^{\varepsilon })$$\end{document}, completely monotone for t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 0$$\end{document} with 0<ε<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \varepsilon < 1$$\end{document}Discrete Contin Dyn Syst Ser B201419722672278325325710.3934/dcdsb.2014.19.22671303.26007 LiXWuRHopf bifurcation analysis of a new commensurate fractional-order hyper chaotic systemNonlinear Dyn201478127928810.1007/s11071-014-1439-5 LyapunovAMThe general problem of the stability of motion1892KharkovKharkov Mathematical Society CresswellWPredation in bird populationsJ Ornithol2011152125126310.1007/s10336-010-0638-1 PodlubnyIFractional differential equations1999San DiegoAcademic Press0924.34008 LeonovGAKuznetsovNVTime-Varying Linearization and the Perron effectsInt J Bifurc Chaos200717410791107232951610.1142/S0218127407017732 SchmitzOJBeckermanAPO’BrienKMBehaviorally mediated trophic cascades: effects of predation risk on food web interactionsEcology1997781388139910.1890/0012-9658(1997)078[1388:BMTCEO]2.0.CO;2 ElliottKHBetiniGSNorrisDRExperimental evidence for within- and cross-seasona effects of fear on survival and reproductionJ Anim Ecol2010201685507515 GarrappaROn linear stability of predictor-corrector algorithms for fractional differential equationsInt J Comput Math2010871022812290268014710.1080/00207160802624331 DeshpandeASDaftardar-GejjiVSukaleYVOn Hopf bifurcational dynamical systemsChaos Solut Fractals20179818919810.1016/j.chaos.2017.03.034 MooringMSFitzpatrickTANishihiraTTReisigDDVigilance, predation risk, and the Allee effect in desert bighorn sheepJ Wildl Manag20046851953210.2193/0022-541X(2004)068[0519:VPRATA]2.0.CO;2 DasMMaityASamantaGPStability analysis of a prey-predator fractional order model incorporating prey refugeEcol Genet Genom201873346 SimonTMittag-Leffler functions and complete monotonicityIntegral Transforms Spec Funct20152613650327544710.1080/10652469.2014.965704 AtanganaASecerAA note on fractional order derivatives and table of fractional derivatives of some special functionsAbstr Appl Anal201320138303916910.1155/2013/2796811276.26010 SvennugsenTOHolenOHLeimarOInducible defenses:continuous reaction norms or threshold traitsAM Nat2011178339741010.1086/661250 MillerKSRossBAn introduction to the fractional calculus and fractional differential equations1993New YorkWiley0789.26002 WangXZanetteLZouXMthe fear effect in predator-prey interactionsJ Math Biol20167311791204355536610.1007/s00285-016-0989-1 KexueLJigenPLaplace transform and fractional differential equationsAppl Math Lett2011241220192023282611810.1016/j.aml.2011.05.035 PetrasIFractional-order nonlinear systems: modeling aanlysis and simulation2011BeijingHigher Education Press10.1007/978-3-642-18101-6 DiethelmKFordNJFreedADA predictor-corrector approach for the numerical solution of fractional differential equationsNonlinear Dyn200229322192646610.1023/A:1016592219341 LiYChenYQPodlubnyIMittag-Leffler stability of fractional order non linear dynamic systemsAutomatica20094519651969287952510.1016/j.automatica.2009.04.003 CreelSChristiansonDRelationships between direct predation and risk effectsTrends Ecol Evol200823419420110.1016/j.tree.2007.12.004 IvlevVSExperimental ecology of the feeding of fishes1961New HavenYale University Press PreisserELBolnicDIThe many faces of fear:comparing the pathways and impacts on non consumptive predator effects on prey populationsPLoS ONE200836e246510.1371/journal.pone.0002465 AndrewsJFA mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesBiotechnol Bioeng19681070772310.1002/bit.260100602 DiethelmKEfficient solution of multi-term fractional differential equations using P(EC)mE methodsComputing2003714305319202721510.1007/s00607-003-0033-3 Stamova I, Stamov G (2017) Functional and impulsive differential equations of fractional order: qualitative analysis and applications SokolWHowellJAKinetics of phenol oxidation by washed cellsBiotechnol Bioeng1981232039204910.1002/bit.260230909 KilbasASrivastavaHTrujilloJTheory and application of fractional differential equations2006New YorkElsevier1092.45003 MalthusTRAn essay on the principle of population, and a summary view of the principle of populations1798PenguinHarmondsworth DasSFunctional fractional calculus for system identification and controls2007BerlinSpringer WirsingAJRippleWA comparison of shark and wolf research reveals similar behavioral responses by preyFront Ecol Environ2010933534110.1890/090226 KlimekMBłasikMExistence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional orderCentr Eur J Math2012101981199429831401260.26009 SabatierJAgrawalOPTenreiro MachadoJAAdvances in fractional calculus: theoretical developments and applications in physics and engineering2007BerlinSpringer10.1007/978-1-4020-6042-7 ZhangHFuSWangWImpact of the fear effect in a prey-predator model incorporating a prey refugeAppl Math Comput2019356328337393453710.1016/j.amc.2019.03.0341428.92099 LiHLZhangLHuCJiangYLTengZDynamical analysis of a fractional-order predator-prey model incorporating a prey refugeJ Appl Math Comput2016541–2435449364718510.1007/s12190-016-1017-81377.34062 CreelSWinnieJAChristiansonDGlucocorticoid stress hormones and the effect of predation risk on elk reproductionProc Natl Acad Sci USA2009106123881239310.1073/pnas.0902235106 ChoiSKKangBKooNStability for caputo fractional differential systemsAbstr Appl Anal2014316663510.1155/2014/63141907022779 SheriffMJKrebsCJBoonstraRThe sensitive hare: sublethal effects of predator stress on reproduction in snowshoe haresJ Anim Ecol2009781249125810.1111/j.1365-2656.2009.01552.x DjordjevicVDJaricJFabryBAnn Biomed Eng20033169210.1114/1.1574026 CreelSChristiansonDLilleySWinnieJAPredation risk effects reproductive physiology and demography of elkScience2007315581496096010.1126/science.1135918 LiangSWuRChenLLaplace transform of fractional order differential equationsElectron J Differ Equ2015139133585111346.34009 OdibatZShawagfehNGeneralized Taylors formulaAppl Math Comput2007186286-1-729323165141122.26006 Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math 298628. arXiv:0909.0230 [math.CA] DokoumetzidisAMaginRMacherasPA commentary on fractionalization of multi-compartmental modelsJ Pharmacokinet Pharmacodyn20103720320710.1007/s10928-010-9153-5discussion 217 TenerJSMuskoxen1965OttawaQueen’s Printer DelavariHBaleanuDSadatiJStability analysis of Caputo fractional-order non linear system revisitedNon linear Dyn2012672433243910.1007/s11071-011-0157-5 OJ Schmitz (626_CR8) 1997; 78 X Wang (626_CR24) 2016; 73 S Creel (626_CR5) 2007; 315 H Zhang (626_CR25) 2019; 356 X Li (626_CR20) 2014; 78 VS Ivlev (626_CR14) 1961 MS Mooring (626_CR12) 2004; 68 JS Tener (626_CR13) 1965 S Liang (626_CR29) 2015; 139 K Diethelm (626_CR37) 2002; 29 VD Djordjevic (626_CR17) 2003; 31 Y Li (626_CR31) 2009; 45 SK Choi (626_CR36) 2014 626_CR44 HL Li (626_CR23) 2016; 54 EL Preisser (626_CR4) 2008; 3 AJ Wirsing (626_CR7) 2010; 9 626_CR49 T Simon (626_CR48) 2015; 26 M Klimek (626_CR33) 2012; 10 W Sokol (626_CR15) 1981; 23 A Dokoumetzidis (626_CR26) 2010; 37 J Sabatier (626_CR47) 2007 TR Malthus (626_CR1) 1798 626_CR34 S Creel (626_CR11) 2009; 106 A Atangana (626_CR22) 2013; 2013 H Delavari (626_CR43) 2012; 67 W Cresswell (626_CR2) 2011; 152 KH Elliott (626_CR10) 2010; 2016 R Garrappa (626_CR39) 2010; 87 L Kexue (626_CR30) 2011; 24 S Das (626_CR42) 2007 K Diethelm (626_CR38) 2003; 71 M Das (626_CR21) 2018; 7 AM Lyapunov (626_CR32) 1892 JF Andrews (626_CR41) 1968; 10 I Podlubny (626_CR16) 1999 GA Leonov (626_CR40) 2007; 17 E Ahmed (626_CR18) 2007; 325 I Petras (626_CR27) 2011 TO Svennugsen (626_CR3) 2011; 178 AS Deshpande (626_CR19) 2017; 98 F Mainardi (626_CR35) 2014; 19 A Kilbas (626_CR45) 2006 KS Miller (626_CR46) 1993 S Creel (626_CR6) 2008; 23 Z Odibat (626_CR28) 2007; 186 MJ Sheriff (626_CR9) 2009; 78 |
| References_xml | – reference: DjordjevicVDJaricJFabryBAnn Biomed Eng20033169210.1114/1.1574026 – reference: AndrewsJFA mathematical model for the continuous culture of microorganisms utilizing inhibitory substratesBiotechnol Bioeng19681070772310.1002/bit.260100602 – reference: DiethelmKFordNJFreedADA predictor-corrector approach for the numerical solution of fractional differential equationsNonlinear Dyn200229322192646610.1023/A:1016592219341 – reference: PreisserELBolnicDIThe many faces of fear:comparing the pathways and impacts on non consumptive predator effects on prey populationsPLoS ONE200836e246510.1371/journal.pone.0002465 – reference: AhmedEEl-SayedAEl-SakaHEquilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies modelsJ Math Anal Appl2007325542-1-7553227354410.1016/j.jmaa.2006.01.087 – reference: DasMMaityASamantaGPStability analysis of a prey-predator fractional order model incorporating prey refugeEcol Genet Genom201873346 – reference: KexueLJigenPLaplace transform and fractional differential equationsAppl Math Lett2011241220192023282611810.1016/j.aml.2011.05.035 – reference: DelavariHBaleanuDSadatiJStability analysis of Caputo fractional-order non linear system revisitedNon linear Dyn2012672433243910.1007/s11071-011-0157-5 – reference: DokoumetzidisAMaginRMacherasPA commentary on fractionalization of multi-compartmental modelsJ Pharmacokinet Pharmacodyn20103720320710.1007/s10928-010-9153-5discussion 217 – reference: MainardiFOn some properties of the Mittag-Leffler function Eα,1(-ηtε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ E_{\alpha ,1}(-\eta t^{\varepsilon })$$\end{document}, completely monotone for t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t > 0$$\end{document} with 0<ε<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \varepsilon < 1$$\end{document}Discrete Contin Dyn Syst Ser B201419722672278325325710.3934/dcdsb.2014.19.22671303.26007 – reference: CreelSWinnieJAChristiansonDGlucocorticoid stress hormones and the effect of predation risk on elk reproductionProc Natl Acad Sci USA2009106123881239310.1073/pnas.0902235106 – reference: LiXWuRHopf bifurcation analysis of a new commensurate fractional-order hyper chaotic systemNonlinear Dyn201478127928810.1007/s11071-014-1439-5 – reference: CreelSChristiansonDRelationships between direct predation and risk effectsTrends Ecol Evol200823419420110.1016/j.tree.2007.12.004 – reference: SheriffMJKrebsCJBoonstraRThe sensitive hare: sublethal effects of predator stress on reproduction in snowshoe haresJ Anim Ecol2009781249125810.1111/j.1365-2656.2009.01552.x – reference: AtanganaASecerAA note on fractional order derivatives and table of fractional derivatives of some special functionsAbstr Appl Anal201320138303916910.1155/2013/2796811276.26010 – reference: KilbasASrivastavaHTrujilloJTheory and application of fractional differential equations2006New YorkElsevier1092.45003 – reference: LiYChenYQPodlubnyIMittag-Leffler stability of fractional order non linear dynamic systemsAutomatica20094519651969287952510.1016/j.automatica.2009.04.003 – reference: LiangSWuRChenLLaplace transform of fractional order differential equationsElectron J Differ Equ2015139133585111346.34009 – reference: KlimekMBłasikMExistence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional orderCentr Eur J Math2012101981199429831401260.26009 – reference: WangXZanetteLZouXMthe fear effect in predator-prey interactionsJ Math Biol20167311791204355536610.1007/s00285-016-0989-1 – reference: CreelSChristiansonDLilleySWinnieJAPredation risk effects reproductive physiology and demography of elkScience2007315581496096010.1126/science.1135918 – reference: MalthusTRAn essay on the principle of population, and a summary view of the principle of populations1798PenguinHarmondsworth – reference: WirsingAJRippleWA comparison of shark and wolf research reveals similar behavioral responses by preyFront Ecol Environ2010933534110.1890/090226 – reference: PodlubnyIFractional differential equations1999San DiegoAcademic Press0924.34008 – reference: MooringMSFitzpatrickTANishihiraTTReisigDDVigilance, predation risk, and the Allee effect in desert bighorn sheepJ Wildl Manag20046851953210.2193/0022-541X(2004)068[0519:VPRATA]2.0.CO;2 – reference: ZhangHFuSWangWImpact of the fear effect in a prey-predator model incorporating a prey refugeAppl Math Comput2019356328337393453710.1016/j.amc.2019.03.0341428.92099 – reference: PetrasIFractional-order nonlinear systems: modeling aanlysis and simulation2011BeijingHigher Education Press10.1007/978-3-642-18101-6 – reference: ElliottKHBetiniGSNorrisDRExperimental evidence for within- and cross-seasona effects of fear on survival and reproductionJ Anim Ecol2010201685507515 – reference: LyapunovAMThe general problem of the stability of motion1892KharkovKharkov Mathematical Society – reference: SvennugsenTOHolenOHLeimarOInducible defenses:continuous reaction norms or threshold traitsAM Nat2011178339741010.1086/661250 – reference: DasSFunctional fractional calculus for system identification and controls2007BerlinSpringer – reference: SimonTMittag-Leffler functions and complete monotonicityIntegral Transforms Spec Funct20152613650327544710.1080/10652469.2014.965704 – reference: Stamova I, Stamov G (2017) Functional and impulsive differential equations of fractional order: qualitative analysis and applications – reference: DiethelmKEfficient solution of multi-term fractional differential equations using P(EC)mE methodsComputing2003714305319202721510.1007/s00607-003-0033-3 – reference: GarrappaROn linear stability of predictor-corrector algorithms for fractional differential equationsInt J Comput Math2010871022812290268014710.1080/00207160802624331 – reference: TenerJSMuskoxen1965OttawaQueen’s Printer – reference: IvlevVSExperimental ecology of the feeding of fishes1961New HavenYale University Press – reference: ChoiSKKangBKooNStability for caputo fractional differential systemsAbstr Appl Anal2014316663510.1155/2014/63141907022779 – reference: LiHLZhangLHuCJiangYLTengZDynamical analysis of a fractional-order predator-prey model incorporating a prey refugeJ Appl Math Comput2016541–2435449364718510.1007/s12190-016-1017-81377.34062 – reference: CresswellWPredation in bird populationsJ Ornithol2011152125126310.1007/s10336-010-0638-1 – reference: SokolWHowellJAKinetics of phenol oxidation by washed cellsBiotechnol Bioeng1981232039204910.1002/bit.260230909 – reference: LeonovGAKuznetsovNVTime-Varying Linearization and the Perron effectsInt J Bifurc Chaos200717410791107232951610.1142/S0218127407017732 – reference: Hilfer R (ed) (2000) Applications of fractional calculus in physics. World Scientific Publishing Co., Inc, River Edge – reference: DeshpandeASDaftardar-GejjiVSukaleYVOn Hopf bifurcational dynamical systemsChaos Solut Fractals20179818919810.1016/j.chaos.2017.03.034 – reference: SchmitzOJBeckermanAPO’BrienKMBehaviorally mediated trophic cascades: effects of predation risk on food web interactionsEcology1997781388139910.1890/0012-9658(1997)078[1388:BMTCEO]2.0.CO;2 – reference: OdibatZShawagfehNGeneralized Taylors formulaAppl Math Comput2007186286-1-729323165141122.26006 – reference: SabatierJAgrawalOPTenreiro MachadoJAAdvances in fractional calculus: theoretical developments and applications in physics and engineering2007BerlinSpringer10.1007/978-1-4020-6042-7 – reference: Haubold HJ, Mathai AM, Saxena RK (2011) Mittag-Leffler functions and their applications. J Appl Math 298628. arXiv:0909.0230 [math.CA] – reference: MillerKSRossBAn introduction to the fractional calculus and fractional differential equations1993New YorkWiley0789.26002 – volume-title: Experimental ecology of the feeding of fishes year: 1961 ident: 626_CR14 – volume: 139 start-page: 1 year: 2015 ident: 626_CR29 publication-title: Electron J Differ Equ – volume: 7 start-page: 33 year: 2018 ident: 626_CR21 publication-title: Ecol Genet Genom – volume-title: Muskoxen year: 1965 ident: 626_CR13 – volume: 152 start-page: 251 issue: 1 year: 2011 ident: 626_CR2 publication-title: J Ornithol doi: 10.1007/s10336-010-0638-1 – volume: 2016 start-page: 507 issue: 85 year: 2010 ident: 626_CR10 publication-title: J Anim Ecol – volume: 2013 start-page: 8 year: 2013 ident: 626_CR22 publication-title: Abstr Appl Anal doi: 10.1155/2013/279681 – volume: 31 start-page: 692 year: 2003 ident: 626_CR17 publication-title: Ann Biomed Eng doi: 10.1114/1.1574026 – volume: 356 start-page: 328 year: 2019 ident: 626_CR25 publication-title: Appl Math Comput doi: 10.1016/j.amc.2019.03.034 – volume: 67 start-page: 2433 year: 2012 ident: 626_CR43 publication-title: Non linear Dyn doi: 10.1007/s11071-011-0157-5 – volume: 45 start-page: 1965 year: 2009 ident: 626_CR31 publication-title: Automatica doi: 10.1016/j.automatica.2009.04.003 – volume: 98 start-page: 189 year: 2017 ident: 626_CR19 publication-title: Chaos Solut Fractals doi: 10.1016/j.chaos.2017.03.034 – volume: 24 start-page: 2019 issue: 12 year: 2011 ident: 626_CR30 publication-title: Appl Math Lett doi: 10.1016/j.aml.2011.05.035 – volume-title: Fractional differential equations year: 1999 ident: 626_CR16 – volume: 19 start-page: 2267 issue: 7 year: 2014 ident: 626_CR35 publication-title: Discrete Contin Dyn Syst Ser B doi: 10.3934/dcdsb.2014.19.2267 – year: 2014 ident: 626_CR36 publication-title: Abstr Appl Anal doi: 10.1155/2014/631419 – volume: 68 start-page: 519 year: 2004 ident: 626_CR12 publication-title: J Wildl Manag doi: 10.2193/0022-541X(2004)068[0519:VPRATA]2.0.CO;2 – volume: 26 start-page: 36 issue: 1 year: 2015 ident: 626_CR48 publication-title: Integral Transforms Spec Funct doi: 10.1080/10652469.2014.965704 – volume: 186 start-page: 286-1-7293 year: 2007 ident: 626_CR28 publication-title: Appl Math Comput – volume: 37 start-page: 203 year: 2010 ident: 626_CR26 publication-title: J Pharmacokinet Pharmacodyn doi: 10.1007/s10928-010-9153-5 – volume: 23 start-page: 2039 year: 1981 ident: 626_CR15 publication-title: Biotechnol Bioeng doi: 10.1002/bit.260230909 – ident: 626_CR44 doi: 10.1142/3779 – volume: 73 start-page: 1179 year: 2016 ident: 626_CR24 publication-title: J Math Biol doi: 10.1007/s00285-016-0989-1 – volume-title: The general problem of the stability of motion year: 1892 ident: 626_CR32 – volume: 71 start-page: 305 issue: 4 year: 2003 ident: 626_CR38 publication-title: Computing doi: 10.1007/s00607-003-0033-3 – volume-title: An essay on the principle of population, and a summary view of the principle of populations year: 1798 ident: 626_CR1 – ident: 626_CR34 doi: 10.1155/2011/298628 – volume: 10 start-page: 707 year: 1968 ident: 626_CR41 publication-title: Biotechnol Bioeng doi: 10.1002/bit.260100602 – volume: 315 start-page: 960 issue: 5814 year: 2007 ident: 626_CR5 publication-title: Science doi: 10.1126/science.1135918 – volume: 78 start-page: 279 issue: 1 year: 2014 ident: 626_CR20 publication-title: Nonlinear Dyn doi: 10.1007/s11071-014-1439-5 – volume-title: Fractional-order nonlinear systems: modeling aanlysis and simulation year: 2011 ident: 626_CR27 doi: 10.1007/978-3-642-18101-6 – volume: 10 start-page: 1981 year: 2012 ident: 626_CR33 publication-title: Centr Eur J Math – volume-title: Theory and application of fractional differential equations year: 2006 ident: 626_CR45 – volume-title: An introduction to the fractional calculus and fractional differential equations year: 1993 ident: 626_CR46 – volume: 29 start-page: 3 year: 2002 ident: 626_CR37 publication-title: Nonlinear Dyn doi: 10.1023/A:1016592219341 – volume: 78 start-page: 1249 year: 2009 ident: 626_CR9 publication-title: J Anim Ecol doi: 10.1111/j.1365-2656.2009.01552.x – volume: 106 start-page: 12388 year: 2009 ident: 626_CR11 publication-title: Proc Natl Acad Sci USA doi: 10.1073/pnas.0902235106 – volume: 54 start-page: 435 issue: 1–2 year: 2016 ident: 626_CR23 publication-title: J Appl Math Comput doi: 10.1007/s12190-016-1017-8 – ident: 626_CR49 doi: 10.1201/9781315367453 – volume: 17 start-page: 1079 issue: 4 year: 2007 ident: 626_CR40 publication-title: Int J Bifurc Chaos doi: 10.1142/S0218127407017732 – volume: 87 start-page: 2281 issue: 10 year: 2010 ident: 626_CR39 publication-title: Int J Comput Math doi: 10.1080/00207160802624331 – volume-title: Advances in fractional calculus: theoretical developments and applications in physics and engineering year: 2007 ident: 626_CR47 doi: 10.1007/978-1-4020-6042-7 – volume: 325 start-page: 542-1-7553 year: 2007 ident: 626_CR18 publication-title: J Math Anal Appl doi: 10.1016/j.jmaa.2006.01.087 – volume-title: Functional fractional calculus for system identification and controls year: 2007 ident: 626_CR42 – volume: 3 start-page: e2465 issue: 6 year: 2008 ident: 626_CR4 publication-title: PLoS ONE doi: 10.1371/journal.pone.0002465 – volume: 23 start-page: 194 issue: 4 year: 2008 ident: 626_CR6 publication-title: Trends Ecol Evol doi: 10.1016/j.tree.2007.12.004 – volume: 78 start-page: 1388 year: 1997 ident: 626_CR8 publication-title: Ecology doi: 10.1890/0012-9658(1997)078[1388:BMTCEO]2.0.CO;2 – volume: 178 start-page: 397 issue: 3 year: 2011 ident: 626_CR3 publication-title: AM Nat doi: 10.1086/661250 – volume: 9 start-page: 335 year: 2010 ident: 626_CR7 publication-title: Front Ecol Environ doi: 10.1890/090226 |
| SSID | ssj0002140249 ssib031263557 |
| Score | 2.4047394 |
| Snippet | In this work, a fractional-order prey-predator system with fear effect of predator on prey population and group defense has been proposed. The existence and... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 334 |
| SubjectTerms | Complexity Control Control and Systems Theory Dynamical Systems Engineering Fear Hopf bifurcation Mathematical models Predator-prey simulation Predators Stability analysis Vibration |
| Title | A prey-predator fractional order model with fear effect and group defense |
| URI | https://link.springer.com/article/10.1007/s40435-020-00626-x https://www.proquest.com/docview/2487447285 |
| Volume | 9 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV07T8MwELZQu8DAG1EoyAMbuErsPMcKtRQQTFQqU2TH9gIKVZtKiF_PnZO0UAESS5bYjnM-29_Zd98RcqFjrT2hciZUFIKBElqmjI2ZzRXg0RCDwvFA_-ExGo2Du0k4qYPC5o23e3Ml6VbqZbAb8sBgNDFGQgMMZ4Ac24A_vKBF2v2b5_tBo0fCR4aV-jYPV2QOVgR3SJhjakIeJZM6fubnhr_vUSvguXZX6rag4Q4ZN52vPE9eeotS9fKPNV7H__7dLtmuMSntV0q0RzZMsU-2vjAVHpDbPp3CoDN4aDTTqZ1VERFQz7F3UpdSh-KxLrUwe2jlKEJloamLHKHaWDCZzSEZDwdP1yNWZ2FgOUzPkkkVpWEqtRclOlCpb4VIbWq5F8QmTUIJEMKTUSwNl0IhGxvacDIIPK2kELkQR6RVvBXmmNA0h09ZZLg3JpCKp0ieZkEhZOhjboUO8Ru5Z3lNUY6ZMl6zJbmyE1MGYsqcmLL3Drlc1plWBB1_lu42w5nVk3We8QBzAMQ8gQ5cNaOzev17ayf_K35KNjl6xDgPti5plbOFOQNIU6rzWoM_AZaW6KU |
| linkProvider | Springer Nature |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV07T8MwELZQOwADb0ShgAc2SJXaeTRjhVpa-phaqUyRHdsLKFR9SIhfz52TtFABUpcssR3nfLa_s---I-ROhUq5XCYOl4EPBopvHKlN6JhEAh71MSgcD_QHw6Az9p4n_iQPCpsX3u7FlaRdqVfBbsgDg9HEGAkNMNwB5Fj2wAYHvS43n156rUKPeB0ZVvLbPFyRGVgRzCJhhqkJWdCY5PEzvzf8c49aA8-Nu1K7BbUPybjofOZ58lpbLmQt-dzgddz2747IQY5JaTNTomOyo9MTsv-NqfCUdJt0CoPuwEOhmU7NLIuIgHqWvZPalDoUj3WpgdlDM0cRKlJFbeQIVdqAyazPyLjdGj12nDwLg5PA9Fw4QgaRHwnlBg3lyahuOI9MZJjrhTpq-AIghCuCUGgmuEQ2NrThhOe5SgrOE87PSSl9T_UFoVECnzLIcK-1JySLkDzNgEIIv465FSqkXsg9TnKKcsyU8RavyJWtmGIQU2zFFH9UyP2qzjQj6Pi3dLUYzjifrPOYeZgDIGQN6MBDMTrr13-3drld8Vuy2xkN-nG_O-xdkT2G3jHWm61KSovZUl8DvFnIm1ybvwCtEOuU |
| linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV07T8MwELZQkRAMvBHl6YENUlI7j3qsoOVdMVCpTJEd2wsoVCWVEL-eOydpoQIkxJIlsRPb5_g7-77vCDnSsdY-V6nHVRSCgxJaTxkbezZVgEdDJIXjhv5dL7rsB9eDcPCJxe-i3asjyYLTgCpNWX461PZ0QnxDTRhkFiMrGiC5ByhyPkBtuxqZb1883nQqm-JNVFspT_bw78zAo2AOFTNMU8ii1qDk0nxf8df1agpCZ85N3XLUXSGyakgRhfLUGOeqkb7PaDz-p6WrZLnEqrRdGNcamTPZOln6pGC4Qa7adAjG4MFFo_tO7ahgSkA5p-pJXaoditu91MKsokUACZWZpo5RQrWx4EqbTdLvdh7OLr0yO4OXwrTNPakiEQqp_ailAyWalnNhhWV-EBvRCiVAC19GsTRMcoUqbejbySDwtZKcp5xvkVr2kpltQkUKr7KofG9MIBUTKKpmwVBk2MScC3XSrMYgSUvpcsyg8ZxMRJddNyXQTYnrpuStTo4nZYaFcMevT-9VQ5uUk_g1YQHmBohZCz7gpBqp6e2fa9v52-OHZOH-vJvcXvVudskiw6AZF-S2R2r5aGz2AfXk6qA07A9kLfR4 |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+prey-predator+fractional+order+model+with+fear+effect+and+group+defense&rft.jtitle=International+journal+of+dynamics+and+control&rft.au=Das%2C+Meghadri&rft.au=Samanta%2C+G.+P.&rft.date=2021-03-01&rft.pub=Springer+Berlin+Heidelberg&rft.issn=2195-268X&rft.eissn=2195-2698&rft.volume=9&rft.issue=1&rft.spage=334&rft.epage=349&rft_id=info:doi/10.1007%2Fs40435-020-00626-x&rft.externalDocID=10_1007_s40435_020_00626_x |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2195-268X&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2195-268X&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2195-268X&client=summon |