Predator–prey models with component Allee effect for predator reproduction

Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Terry, Alan J. [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2015

Schlagwörter:	Predator–prey model Allee effect Predator birth rate Trapping region Prey resurgence

Anmerkung:	© Springer-Verlag Berlin Heidelberg 2015

Übergeordnetes Werk:	Enthalten in: Journal of mathematical biology - Berlin : Springer, 1974, 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352
Übergeordnetes Werk:	volume:71 ; year:2015 ; number:6-7 ; day:20 ; month:02 ; pages:1325-1352

Links:	Volltext

DOI / URN:	10.1007/s00285-015-0856-5

Katalog-ID:	SPR003703967

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520			\|a Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program.
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matchkey_str	article:14321416:2015----::rdtrryoesihopnnalefetope
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publishDate	2015
allfields	10.1007/s00285-015-0856-5 doi (DE-627)SPR003703967 (SPR)s00285-015-0856-5-e DE-627 ger DE-627 rakwb eng Terry, Alan J. verfasserin aut Predator–prey models with component Allee effect for predator reproduction 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213 Enthalten in Journal of mathematical biology Berlin : Springer, 1974 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 (DE-627)242065082 (DE-600)1421292-4 1432-1416 nnns volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352 https://dx.doi.org/10.1007/s00285-015-0856-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 71 2015 6-7 20 02 1325-1352
spelling	10.1007/s00285-015-0856-5 doi (DE-627)SPR003703967 (SPR)s00285-015-0856-5-e DE-627 ger DE-627 rakwb eng Terry, Alan J. verfasserin aut Predator–prey models with component Allee effect for predator reproduction 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213 Enthalten in Journal of mathematical biology Berlin : Springer, 1974 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 (DE-627)242065082 (DE-600)1421292-4 1432-1416 nnns volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352 https://dx.doi.org/10.1007/s00285-015-0856-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 71 2015 6-7 20 02 1325-1352
allfields_unstemmed	10.1007/s00285-015-0856-5 doi (DE-627)SPR003703967 (SPR)s00285-015-0856-5-e DE-627 ger DE-627 rakwb eng Terry, Alan J. verfasserin aut Predator–prey models with component Allee effect for predator reproduction 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213 Enthalten in Journal of mathematical biology Berlin : Springer, 1974 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 (DE-627)242065082 (DE-600)1421292-4 1432-1416 nnns volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352 https://dx.doi.org/10.1007/s00285-015-0856-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 71 2015 6-7 20 02 1325-1352
allfieldsGer	10.1007/s00285-015-0856-5 doi (DE-627)SPR003703967 (SPR)s00285-015-0856-5-e DE-627 ger DE-627 rakwb eng Terry, Alan J. verfasserin aut Predator–prey models with component Allee effect for predator reproduction 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213 Enthalten in Journal of mathematical biology Berlin : Springer, 1974 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 (DE-627)242065082 (DE-600)1421292-4 1432-1416 nnns volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352 https://dx.doi.org/10.1007/s00285-015-0856-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 71 2015 6-7 20 02 1325-1352
allfieldsSound	10.1007/s00285-015-0856-5 doi (DE-627)SPR003703967 (SPR)s00285-015-0856-5-e DE-627 ger DE-627 rakwb eng Terry, Alan J. verfasserin aut Predator–prey models with component Allee effect for predator reproduction 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag Berlin Heidelberg 2015 Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213 Enthalten in Journal of mathematical biology Berlin : Springer, 1974 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 (DE-627)242065082 (DE-600)1421292-4 1432-1416 nnns volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352 https://dx.doi.org/10.1007/s00285-015-0856-5 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 71 2015 6-7 20 02 1325-1352
language	English
source	Enthalten in Journal of mathematical biology 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352
sourceStr	Enthalten in Journal of mathematical biology 71(2015), 6-7 vom: 20. Feb., Seite 1325-1352 volume:71 year:2015 number:6-7 day:20 month:02 pages:1325-1352
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Predator–prey model Allee effect Predator birth rate Trapping region Prey resurgence
isfreeaccess_bool	false
container_title	Journal of mathematical biology
authorswithroles_txt_mv	Terry, Alan J. @@aut@@
publishDateDaySort_date	2015-02-20T00:00:00Z
hierarchy_top_id	242065082
id	SPR003703967
language_de	englisch
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author	Terry, Alan J.
spellingShingle	Terry, Alan J. misc Predator–prey model misc Allee effect misc Predator birth rate misc Trapping region misc Prey resurgence Predator–prey models with component Allee effect for predator reproduction
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topic_title	Predator–prey models with component Allee effect for predator reproduction Predator–prey model (dpeaa)DE-He213 Allee effect (dpeaa)DE-He213 Predator birth rate (dpeaa)DE-He213 Trapping region (dpeaa)DE-He213 Prey resurgence (dpeaa)DE-He213
topic	misc Predator–prey model misc Allee effect misc Predator birth rate misc Trapping region misc Prey resurgence
topic_unstemmed	misc Predator–prey model misc Allee effect misc Predator birth rate misc Trapping region misc Prey resurgence
topic_browse	misc Predator–prey model misc Allee effect misc Predator birth rate misc Trapping region misc Prey resurgence
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title	Predator–prey models with component Allee effect for predator reproduction
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title_full	Predator–prey models with component Allee effect for predator reproduction
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title_sort	predator–prey models with component allee effect for predator reproduction
title_auth	Predator–prey models with component Allee effect for predator reproduction
abstract	Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. © Springer-Verlag Berlin Heidelberg 2015
abstractGer	Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. © Springer-Verlag Berlin Heidelberg 2015
abstract_unstemmed	Abstract We present four predator–prey models with component Allee effect for predator reproduction. Using numerical simulation results for our models, we describe how the customary definitions of component and demographic Allee effects, which work well for single species models, can be extended to predators in predator–prey models by assuming that the prey population is held fixed. We also find that when the prey population is not held fixed, then these customary definitions may lead to conceptual problems. After this discussion of definitions, we explore our four models, analytically and numerically. Each of our models has a fixed point that represents predator extinction, which is always locally stable. We prove that the predator will always die out either if the initial predator population is sufficiently small or if the initial prey population is sufficiently small. Through numerical simulations, we explore co-existence fixed points. In addition, we demonstrate, by simulation, the existence of a stable limit cycle in one of our models. Finally, we derive analytical conditions for a co-existence trapping region in three of our models, and show that the fourth model cannot possess a particular kind of co-existence trapping region. We punctuate our results with comments on their real-world implications; in particular, we mention the possibility of prey resurgence from mortality events, and the possibility of failure in a biological pest control program. © Springer-Verlag Berlin Heidelberg 2015
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title_short	Predator–prey models with component Allee effect for predator reproduction
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Predator–prey models with component Allee effect for predator reproduction

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