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Volumn 4, Issue 2, 2009, Pages 140-188

On nonlinear dynamics of predator-prey models with discrete delay

Author keywords

bifurcation; harvesting; predator prey model; stability; time delay

Indexed keywords

BOGDANOV-TAKENS BIFURCATIONS; DISCRETE DELAY; DISTRIBUTION OF ZEROS; FUNCTIONAL RESPONSE; NONMONOTONE; PREDATOR-PREY MODEL; PREDATOR-PREY MODELS;

EID: 77955469405     PISSN: 09735348     EISSN: 17606101     Source Type: Journal    
DOI: 10.1051/mmnp/20094207     Document Type: Article
Times cited : (157)

References (94)
  • 1
    • 84984082223 scopus 로고
    • A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates
    • J. F. Andrews. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng., 10 (1968), 707-723.
    • (1968) Biotechnol. Bioeng. , vol.10 , pp. 707-723
    • Andrews, J.F.1
  • 2
    • 0017362809 scopus 로고
    • The effect of a time-delay in a predator-prey model
    • R. Arditi, J.-M. Abillon, J. V. Da Silva. The effect of a time-delay in a predator-prey model. Math. Biosci., 33 (1977), 107-120.
    • (1977) Math. Biosci. , vol.33 , pp. 107-120
    • Arditi, R.1    Abillon, J.-M.2    Da Silva, J.V.3
  • 4
    • 0001813929 scopus 로고
    • On theoretical models for competitive and predatory biological systems
    • M. S. Bartlett. On theoretical models for competitive and predatory biological systems. Biometrika, 44 (1957), 27-42.
    • (1957) Biometrika , vol.44 , pp. 27-42
    • Bartlett, M.S.1
  • 5
    • 0020380018 scopus 로고
    • Harvesting from a prey-predator complex
    • J. R. Beddington, J. G. Cooke. Harvesting from a prey-predator complex. Ecol. Modelling, 14 (1982), 155-177.
    • (1982) Ecol. Modelling , vol.14 , pp. 155-177
    • Beddington, J.R.1    Cooke, J.G.2
  • 7
    • 0030589781 scopus 로고    scopus 로고
    • Convergence results in a well-known delayed predator-prey system
    • DOI 10.1006/jmaa.1996.0471
    • E. Beretta, Y. Kuang. Convergence results in a well-known delayed predator-prey system. J. Math. Anal. Appl., 204 (1996), 840-853. (Pubitemid 126167819)
    • (1996) Journal of Mathematical Analysis and Applications , vol.204 , Issue.3 , pp. 840-853
    • Beretta, E.1    Kuang, Y.2
  • 8
    • 0032058512 scopus 로고    scopus 로고
    • Global analyses in some delayed ratio-dependent predator-prey systems
    • PII S0362546X97004914
    • E. Beretta, Y. Kuang. Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal., 32 (1998), 381-408. (Pubitemid 128397593)
    • (1998) Nonlinear Analysis, Theory, Methods and Applications , vol.32 , Issue.3 , pp. 381-408
    • Beretta, E.1    Kuang, Y.2
  • 9
    • 0036373967 scopus 로고    scopus 로고
    • Geometric stability switch crteria in delay differential equations with delay dependent parameters
    • E. Beretta, Y. Kuang. Geometric stability switch crteria in delay differential equations with delay dependent parameters. SIAM J. Math. Anal., 33(2002), 1144-1165.
    • (2002) SIAM J. Math. Anal. , vol.33 , pp. 1144-1165
    • Beretta, E.1    Kuang, Y.2
  • 10
    • 21844482244 scopus 로고
    • Stability criteria for second-order dynamical systems involving several time delays
    • F. G. Boes. Stability criteria for second-order dynamical systems involving several time delays. SIAM J. Math. Anal., 26 (1995), 1306-1330.
    • (1995) SIAM J. Math. Anal. , vol.26 , pp. 1306-1330
    • Boes, F.G.1
  • 11
    • 0017394929 scopus 로고
    • Stability of some population models with delay
    • DOI 10.1016/0025-5564(77)90148-1
    • F. Brauer. Stability of some population models with delay. Math. Biosci., 33 (1977), 345-358. (Pubitemid 8102172)
    • (1977) Mathematical Biosciences , vol.33 , Issue.3-4 , pp. 345-358
    • Brauer, F.1
  • 12
    • 0018305697 scopus 로고
    • Characteristic return times for harvested population models with time lag
    • DOI 10.1016/0025-5564(79)90064-6
    • F. Brauer. Characteristic return times for harvested population models with time lag. Math. Biosci., 45 (1979), 295-311. (Pubitemid 9248090)
    • (1979) Mathematical Biosciences , vol.45 , Issue.3-4 , pp. 295-311
    • Brauer, F.1
  • 13
    • 38249035469 scopus 로고
    • Absolute stability in delay equations
    • F. Brauer. Absolute stability in delay equations. J. Differential Equations, 69 (1987), 185-191.
    • (1987) J. Differential Equations , vol.69 , pp. 185-191
    • Brauer, F.1
  • 14
    • 0018318922 scopus 로고
    • Stability regions and transition phenomena for harvested predator-prey systems
    • F. Brauer, A. C. Soudack. Stability regions and transition phenomena for harvested predatorprey systems. J. Math. Biol., 7 (1979), 319-337. (Pubitemid 9255726)
    • (1979) Journal of Mathematical Biology , vol.7 , Issue.4 , pp. 319-337
    • Brauer, F.1    Soudack, A.C.2
  • 15
    • 0018572296 scopus 로고
    • Stability regions in predator-prey systems with constant-rate prey harvesting
    • DOI 10.1007/BF00280586
    • F. Brauer, A. C. Soudack. Stability regions in predator-prey systems with constant-rate prey harvesting. J. Math. Biol., 8 (1979), 55-71. (Pubitemid 9249936)
    • (1979) Journal of Mathematical Biology , vol.8 , Issue.1 , pp. 55-71
    • Brauer, F.1    Soudack, A.C.2
  • 16
    • 0019862153 scopus 로고
    • Coexistence properties of some predator-prey systems under constant rate harvesting and stocking
    • DOI 10.1007/BF00275206
    • F. Brauer, A. C. Soudack. Coexistence properties of some predator-prey systems under constant rate harvesting and stocking. J. Math. Biol., 12 (1981), 101-114. (Pubitemid 11091190)
    • (1981) Journal of Mathematical Biology , vol.12 , Issue.1 , pp. 101-114
    • Brauer, F.1    Soudack, A.C.2
  • 17
    • 34347187625 scopus 로고
    • Sur le problème biologique héréditaiare de deux especès dévorante et dévorée
    • M. Brelot. Sur le problème biologique héréditaiare de deux especès dévorante et dévorée. Ann. Mat. Pura Appl., 9 (1931), 58-74.
    • (1931) Ann. Mat. Pura Appl. , vol.9 , pp. 58-74
    • Brelot, M.1
  • 18
    • 0017173061 scopus 로고
    • The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater
    • A. W. Bush, A. E. Cook. The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J. Theoret. Biol., 63 (1976), 385-395.
    • (1976) J. Theoret. Biol. , vol.63 , pp. 385-395
    • Bush, A.W.1    Cook, A.E.2
  • 19
    • 0000287120 scopus 로고    scopus 로고
    • Global attractivity in time-delayed predator-prey systems
    • Y. Cao, H. I. Freedman. Global attractivity in time-delayed predator-prey systems. J. Austral. Math. Soc. Ser. B, 38 (1996), 149-162.
    • (1996) J. Austral. Math. Soc. Ser. B , vol.38 , pp. 149-162
    • Cao, Y.1    Freedman, H.I.2
  • 20
    • 0001453657 scopus 로고
    • Time lag in population growth response of isochrysis galbana to a variable nitrate environment
    • J. Caperon. Time lag in population growth response of isochrysis galbana to a variable nitrate environment. Ecology, 50 (1969), 188-192.
    • (1969) Ecology , vol.50 , pp. 188-192
    • Caperon, J.1
  • 21
    • 0005689817 scopus 로고
    • Unconditional stability of systems with time-lags
    • Y.-S. Chin. Unconditional stability of systems with time-lags. Acta Math. Sinica, 1 (1960), 125-142.
    • (1960) Acta Math. Sinica , vol.1 , pp. 125-142
    • Chin, Y.-S.1
  • 22
    • 0000209631 scopus 로고
    • Discrete delay, distributed delay and stability switches
    • K. L. Cooke, Z. Grossman. Discrete delay, distributed delay and stability switches. J. Math. Anal. Appl., 86 (1982), 592-627.
    • (1982) J. Math. Anal. Appl. , vol.86 , pp. 592-627
    • Cooke, K.L.1    Grossman, Z.2
  • 25
    • 0002419555 scopus 로고
    • Stability and maturation periods in age structured populations
    • S. Busenberg and K. L. Cooke (Eds.) Academic Press, New York
    • J. M. Cushing. Stability and maturation periods in age structured populations. In "Differential Equations and Applications in Ecology, Epidemics, and Population Problems", S. Busenberg and K. L. Cooke (Eds.), Academic Press, New York, 1981, pp. 163-182.
    • (1981) Differential Equations and Applications in Ecology, Epidemics, and Population Problems , pp. 163-182
    • Cushing, J.M.1
  • 26
    • 0019993839 scopus 로고
    • A predator prey model with age structure
    • DOI 10.1007/BF01832847
    • J. M. Cushing, M. Saleem. A predator prey model with age structure. J. Math. Biol., 14 (1982), 231-250. Erratum: 16 (1983), 305. (Pubitemid 12043653)
    • (1982) Journal of Mathematical Biology , vol.14 , Issue.2 , pp. 231-250
    • Cushing, J.M.1    Saleem, M.2
  • 27
    • 0032002567 scopus 로고    scopus 로고
    • Coexistence region and global dynamics of a harvested predator-prey system
    • PII S0036139994275799
    • G. Dai, M. Tang. Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math., 58 (1998), 193-210. (Pubitemid 128587708)
    • (1998) SIAM Journal on Applied Mathematics , vol.58 , Issue.1 , pp. 193-210
    • Dai, G.1    Tang, M.2
  • 28
    • 0019473512 scopus 로고
    • Nonconstant periodic solutions in predator-prey systems with continuous time delay
    • DOI 10.1016/0025-5564(81)90044-4
    • L. S. Dai. Nonconstant periodic solutions in predator-prey systems with continuous time delay. Math. Biosci., 53 (1981), 149-157. (Pubitemid 11129085)
    • (1981) Mathematical Biosciences , vol.53 , Issue.1-2 , pp. 149-157
    • Dai, L.S.1
  • 29
    • 0000226236 scopus 로고
    • A procedure for determination of the exponential stability of certain differential difference equations
    • R. Datko. A procedure for determination of the exponential stability of certain differential difference equations. Quart. Appl. Math., 36 (1978), 279-292.
    • (1978) Quart. Appl. Math. , vol.36 , pp. 279-292
    • Datko, R.1
  • 32
    • 0035866130 scopus 로고    scopus 로고
    • Stability and Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion
    • DOI 10.1006/jmaa.2000.7182, PII S0022247X00971828
    • T. Faria. Stability and bifurcation for a delayed predator-prey model and the effect of diffusion. J. Math. Anal. Appl., 254 (2001), 433-463. (Pubitemid 33380901)
    • (2001) Journal of Mathematical Analysis and Applications , vol.254 , Issue.2 , pp. 433-463
    • Faria, T.1
  • 33
    • 0000840165 scopus 로고
    • Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity
    • T. Faria, L. T. Magalhães. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differential Equations, 122 (1995), 201-224.
    • (1995) J. Differential Equations , vol.122 , pp. 201-224
    • Faria, T.1    Magalhães, L.T.2
  • 34
    • 0000840164 scopus 로고
    • Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations
    • T. Faria, L. T. Magalhães. Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations. J. Differential Equations, 122 (1995), 181-200.
    • (1995) J. Differential Equations , vol.122 , pp. 181-200
    • Faria, T.1    Magalhães, L.T.2
  • 35
    • 0005729247 scopus 로고
    • Multiparameter bifurcation diagrams in predator-prey models with time lag
    • A. Farkas, M. Farkas, G. Szabó. Multiparameter bifurcation diagrams in predator-prey models with time lag. J. Math. Biol., 26 (1988), 93-103.
    • (1988) J. Math. Biol. , vol.26 , pp. 93-103
    • Farkas, A.1    Farkas, M.2    Szabó, G.3
  • 37
    • 0002455609 scopus 로고
    • Nonoccurence of stability switching in systems with discrete delays
    • H. I. Freedman, K. Gopalsamy. Nonoccurence of stability switching in systems with discrete delays. Canad. Math. Bull., 31 (1988), 52-58.
    • (1988) Canad. Math. Bull. , vol.31 , pp. 52-58
    • Freedman, H.I.1    Gopalsamy, K.2
  • 38
    • 0001352326 scopus 로고
    • The tradeoff between mutual interference and time lags in predator-prey systems
    • H. I. Freedman, V. S. H. Rao. The tradeoff between mutual interference and time lags in predator-prey systems. Bull. Math. Biol., 45 (1983), 991-1004.
    • (1983) Bull. Math. Biol. , vol.45 , pp. 991-1004
    • Freedman, H.I.1    Rao, V.S.H.2
  • 39
    • 0022768009 scopus 로고
    • Stability criteria for a system involving two time delays
    • H. I. Freedman, V. S. H. Rao. Stability criteria for a system involving two time delays. SIAM J. Appl. Anal., 46 (1986), 552-560. (Pubitemid 16607820)
    • (1986) SIAM Journal on Applied Mathematics , vol.46 , Issue.4 , pp. 552-560
    • Freedman, H.I.1    Rao, V.S.H.2
  • 40
    • 0022849138 scopus 로고
    • Predator-prey systems with group defence: The paradox of enrichment revisited
    • H. I. Freedman, G. S. K.Wolkowicz. Predator-prey systems with group defence: The paradox of enrichment revisited. Bull. Math. Biol., 48 (1986), 493-508.
    • (1986) Bull. Math. Biol. , vol.48 , pp. 493-508
    • Freedman, H.I.1    Wolkowicz, G.S.K.2
  • 41
    • 35949030757 scopus 로고
    • On the Volterra and other nonlinear models of interacting populations
    • N. S. Goel, S. C. Maitra, E. W. Montroll. On the Volterra and other nonlinear models of interacting populations. Rev. Modern Phys., 43 (1971), 231-276.
    • (1971) Rev. Modern Phys. , vol.43 , pp. 231-276
    • Goel, N.S.1    Maitra, S.C.2    Montroll, E.W.3
  • 42
    • 0000429668 scopus 로고
    • Harmless delay in model systems
    • K. Gopalsamy. Harmless delay in model systems. Bull. Math. Biol., 45 (1983), 295-309.
    • (1983) Bull. Math. Biol. , vol.45 , pp. 295-309
    • Gopalsamy, K.1
  • 43
    • 0005686511 scopus 로고
    • Delayed responses and stability in two-species systems
    • K. Gopalsamy. Delayed responses and stability in two-species systems. J. Austral. Math. Soc. Ser. B, 25 (1984), 473-500.
    • (1984) J. Austral. Math. Soc. Ser. B , vol.25 , pp. 473-500
    • Gopalsamy, K.1
  • 45
    • 15844390604 scopus 로고    scopus 로고
    • A stage structured predator-prey model and its dependence on maturation delay and death rate
    • DOI 10.1007/s00285-004-0278-2
    • S. Gourley, Y. Kuang. A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol., 49 (2004), 188-200. (Pubitemid 40876393)
    • (2004) Journal of Mathematical Biology , vol.49 , Issue.2 , pp. 188-200
    • Gourley, S.A.1    Kuang, Y.2
  • 49
    • 0000083221 scopus 로고
    • Age-dependent predation is not a simple process: I. continuous time models
    • A. Hastings. Age-dependent predation is not a simple process: I. continuous time models. Theoret. Pop. Biol., 23 (1983), 347-362.
    • (1983) Theoret. Pop. Biol. , vol.23 , pp. 347-362
    • Hastings, A.1
  • 50
    • 0021742869 scopus 로고
    • Delays in recruitment at different trophic levels: Effects on stability
    • A. Hastings. Delays in recruitment at different trophic levels: effects on stability. J. Math. Biol., 21 (1984), 35-44. (Pubitemid 15180123)
    • (1984) Journal of Mathematical Biology , vol.21 , Issue.1 , pp. 35-44
    • Hastings, A.1
  • 51
    • 0030098369 scopus 로고    scopus 로고
    • Stability and delays in a predator-prey system
    • DOI 10.1006/jmaa.1996.0087
    • X.-Z. He. Stability and delays in a predator-prey system. J. Math. Anal. Appl., 198 (1996), 355-370. (Pubitemid 126161466)
    • (1996) Journal of Mathematical Analysis and Applications , vol.198 , Issue.2 , pp. 355-370
    • He, X.-Z.1
  • 52
    • 0032138492 scopus 로고    scopus 로고
    • The Lyapunov functionals for delay Lotka-Volterra-type models
    • PII S0036139995295116
    • X.-Z. He. The Lyapunov functionals for delay Lotka-Volterra-type models. SIAM J. Appl. Math., 58 (1998), 1222-1236. (Pubitemid 128689431)
    • (1998) SIAM Journal on Applied Mathematics , vol.58 , Issue.4 , pp. 1222-1236
    • He, X.-Z.1
  • 53
  • 54
    • 4544266198 scopus 로고
    • Algebraic criteria on the stability of the zero solutions of the second order delay differential equations
    • W. Huang. Algebraic criteria on the stability of the zero solutions of the second order delay differential equations. J. Anhui University, (1985), 1-7.
    • (1985) J. Anhui University , pp. 1-7
    • Huang, W.1
  • 55
    • 0028160322 scopus 로고
    • What can be learned from the collapse of a renewable resource? Atlantic code, Gadus morhua, of Newfoundland and Labrador
    • J. A. Hutchings, R. A. Myers. What can be learned from the collapse of a renewable resource? Atlantic code, Gadus morhua, of Newfoundland and Labrador. Can. J. Fish. Aquat. Sci., 51 (1994), 2126-2146.
    • (1994) Can. J. Fish. Aquat. Sci. , vol.51 , pp. 2126-2146
    • Hutchings, J.A.1    Myers, R.A.2
  • 57
    • 0003236870 scopus 로고
    • Elements of applied bifurcation theory
    • Springer-Verlag, New York
    • Y. A. Kuznetsov. Elements of applied bifurcation theory. Applied Mathematical Sciences 112, Springer-Verlag, New York, 1995.
    • (1995) Applied Mathematical Sciences , vol.112
    • Kuznetsov, Y.A.1
  • 58
    • 0037032855 scopus 로고    scopus 로고
    • Recent progress on stage-structured population dynamics
    • PII S0895717702002790
    • S. Liu, L. Chen, R. Agarwal. Recent progress on stage-structured population dynamics. Math. Computer Model.,36 (2002), 1319-1360. (Pubitemid 35474120)
    • (2002) Mathematical and Computer Modelling , vol.36 , Issue.11-13 , pp. 1319-1360
    • Liu, S.1    Chen, L.2    Agarwal, R.3
  • 59
    • 3442890192 scopus 로고    scopus 로고
    • Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response
    • DOI 10.1016/j.jmaa.2004.04.051, PII S0022247X04003191
    • Z. Liu, R. Yuan. Stability and bifurcation in a delayed predator-prey system with Beddinton- DeAngelis functional response. J. Math. Anal. Appl., 296 (2004), 521-537. (Pubitemid 39007480)
    • (2004) Journal of Mathematical Analysis and Applications , vol.296 , Issue.2 , pp. 521-537
    • Liu, Z.1    Yuan, R.2
  • 60
    • 0031108505 scopus 로고    scopus 로고
    • Global stability for two-species Lotka-Volterra systems with delay
    • Z. Lu,W.Wang. Global stability for two-species Lotka-Volterra systems with delay. J. Math. Anal. Appl., 208 (1997), 277-280.
    • (1997) J. Math. Anal. Appl. , vol.208 , pp. 277-280
    • Lu, Z.1    Wang, W.2
  • 61
    • 0000236923 scopus 로고
    • Stability of predation models with time delay
    • Z. Ma. Stability of predation models with time delay. Applicable Anal., 22 (1986), 169-192.
    • (1986) Applicable Anal. , vol.22 , pp. 169-192
    • Ma, Z.1
  • 62
    • 0020155008 scopus 로고
    • A test for stability of linear differential delay equations
    • J. M. Mahaffy. A test for stability of linear differential delay equations. Quart. Appl. Math., 40 (1982), 193-202.
    • (1982) Quart. Appl. Math. , vol.40 , pp. 193-202
    • Mahaffy, J.M.1
  • 63
    • 0035460603 scopus 로고    scopus 로고
    • Predator-prey models with delay and prey harvesting
    • A. Martin, S. Ruan. Predator-prey models with delay and prey harvesting. J. Mathematical Biology, 43 (2001), 247-267. (Pubitemid 33773344)
    • (2001) Journal of Mathematical Biology , vol.43 , Issue.3 , pp. 247-267
    • Martin, A.1    Ruan, S.2
  • 64
    • 0001249569 scopus 로고
    • Time delay versus stability in population models with two and three trophic levels
    • R. M. May. Time delay versus stability in population models with two and three trophic levels. Ecology, 4 (1973), 315-325.
    • (1973) Ecology , vol.4 , pp. 315-325
    • May, R.M.1
  • 66
    • 0030616815 scopus 로고    scopus 로고
    • Why do fish stocks collapse? the example of cod in Atlantic Canada
    • R. A. Myers, J. A. Hutchings, N. J. Barrowman. Why do fish stocks collapse? The example of cod in Atlantic Canada. Ecol. Appl., 7 (1997), 91-106.
    • (1997) Ecol. Appl. , vol.7 , pp. 91-106
    • Myers, R.A.1    Hutchings, J.A.2    Barrowman, N.J.3
  • 67
    • 0038688384 scopus 로고    scopus 로고
    • Rapid worldwide depletion of predatory fish communities
    • DOI 10.1038/nature01610
    • R. A. Myers, B. Worm. Rapid worldwide depletion of large predatory fish communities. Nature, 423 (2003), 280-283. (Pubitemid 40852696)
    • (2003) Nature , vol.423 , Issue.6937 , pp. 280-283
    • Myers, R.A.1    Worm, B.2
  • 68
    • 0000360599 scopus 로고
    • An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking
    • M. R. Myerscough, B. F. Gray, W. L. Hogarth, J. Norbury. An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking. J. Math. Biol., 30 (1992), 389-411.
    • (1992) J. Math. Biol. , vol.30 , pp. 389-411
    • Myerscough, M.R.1    Gray, B.F.2    Hogarth, W.L.3    Norbury, J.4
  • 69
    • 33846872987 scopus 로고    scopus 로고
    • Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system
    • S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Math. Biosci. Engineer., 3 (2006), 173-187. (Pubitemid 46227039)
    • (2006) Mathematical Biosciences and Engineering , vol.3 , Issue.1 , pp. 173-187
    • Nakaoka, S.1    Saito, Y.2    Takeuchi, Y.3
  • 70
    • 0022042327 scopus 로고
    • The effect of long time delays in predator-prey systems
    • L. Nunney. The effect of long time delays in predator-prey systems. Theoret. Pop. Biol., 27 (1985), 202-221.
    • (1985) Theoret. Pop. Biol. , vol.27 , pp. 202-221
    • Nunney, L.1
  • 71
    • 0005749085 scopus 로고
    • Absolute stability in predator-prey models
    • L. Nunney. Absolute stability in predator-prey models. Theoret. Pop. Biol., 28 (1985), 209-232.
    • (1985) Theoret. Pop. Biol. , vol.28 , pp. 209-232
    • Nunney, L.1
  • 72
    • 34248370155 scopus 로고    scopus 로고
    • Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure
    • DOI 10.1007/s11071-006-9133-x
    • Y. Qu, J. Wei. Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure. Nonlinear Dynamics, 49 (2007), 285-294. (Pubitemid 46739715)
    • (2007) Nonlinear Dynamics , vol.49 , Issue.1-2 , pp. 285-294
    • Qu, Y.1    Wei, J.2
  • 73
    • 0015449004 scopus 로고
    • A difference-differential model in population dynamics
    • G. G. Ross. A difference-differential model in population dynamics. J. Theoret. Biol., 37 (1972), 477-492.
    • (1972) J. Theoret. Biol. , vol.37 , pp. 477-492
    • Ross, G.G.1
  • 74
    • 0035281981 scopus 로고    scopus 로고
    • Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays
    • S. Ruan. Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart. Appl. Math., 59 (2001), 159-173. (Pubitemid 32248363)
    • (2001) Quarterly of Applied Mathematics , vol.59 , Issue.1 , pp. 159-173
    • Ruan, S.1
  • 75
    • 34248517099 scopus 로고    scopus 로고
    • Delay differential equations in single species dynamics
    • "Delay Differential Equations with Applications," O. Arino, M. Hbid and E. Ait Dads (Eds.), Springer, Berlin
    • S. Ruan. Delay differential equations in single species dynamics. In "Delay Differential Equations with Applications," O. Arino, M. Hbid and E. Ait Dads (Eds.), NATO Science Series II: Mathematics, Physics and Chemistry, Vol. 205, Springer, Berlin, 2006, pp. 477-517.
    • (2006) NATO Science Series II: Mathematics, Physics and Chemistry , vol.205 , pp. 477-517
    • Ruan, S.1
  • 77
    • 0034839054 scopus 로고    scopus 로고
    • Global analysis in a predator-prey system with nonmonotonic functional response
    • DOI 10.1137/S0036139999361896, PII S0036139999361896
    • S. Ruan, D. Xiao. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J, Appl. Math., 61 (2001), 1445-1472. (Pubitemid 32825187)
    • (2001) SIAM Journal on Applied Mathematics , vol.61 , Issue.4 , pp. 1445-1472
    • Ruan, S.1    Xiao, D.2
  • 78
    • 84942111823 scopus 로고
    • Kinetics of phenol oxidation by washed cells
    • W. Sokol, J. A. Howell. Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng., 23 (1980), 2039-2049.
    • (1980) Biotechnol. Bioeng. , vol.23 , pp. 2039-2049
    • Sokol, W.1    Howell, J.A.2
  • 79
    • 34548261392 scopus 로고    scopus 로고
    • Bifurcations for a predator-prey system with two delays
    • Y. Song, Y. Peng, J. Wei. Bifurcations for a predator-prey system with two delays. J. Math. Anal. Appl., 337 (2008), 466-479.
    • (2008) J. Math. Anal. Appl. , vol.337 , pp. 466-479
    • Song, Y.1    Peng, Y.2    Wei, J.3
  • 80
    • 10144242643 scopus 로고    scopus 로고
    • Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system
    • DOI 10.1016/j.jmaa.2004.06.056, PII S0022247X04005700
    • Y. Song, J.Wei. Local Hopf bifurcation and global periodic solutions in a delayed predatorprey system. J. Math. Anal. Appl., 301 (2005), 1-21. (Pubitemid 39615204)
    • (2005) Journal of Mathematical Analysis and Applications , vol.301 , Issue.1 , pp. 1-21
    • Song, Y.1    Wei, J.2
  • 81
    • 0001513451 scopus 로고
    • Great delay in a predator-prey model
    • G. Stépán. Great delay in a predator-prey model. Nonlinear Anal., 10 (1986), 913-929.
    • (1986) Nonlinear Anal. , vol.10 , pp. 913-929
    • Stépán, G.1
  • 82
    • 0002885593 scopus 로고
    • Periodic solutions of a planar delay equation
    • P. Táboas. Periodic solutions of a planar delay equation. Proc. Roy. Soc. Edinburgh, 116A (1990), 85-101.
    • (1990) Proc. Roy. Soc. Edinburgh , vol.116 A , pp. 85-101
    • Táboas, P.1
  • 83
    • 0002604954 scopus 로고
    • Variazionie fluttuazioni del numbero d'individui in specie animali conviventi
    • V. Volterra. Variazionie fluttuazioni del numbero d'individui in specie animali conviventi. Mem. Acad. Lincei., 2 (1926), 31-113.
    • (1926) Mem. Acad. Lincei. , vol.2 , pp. 31-113
    • Volterra, V.1
  • 85
    • 0031124171 scopus 로고    scopus 로고
    • A predator-prey system with stage-structure for predators
    • W. Wang, L. Chen. A predator-prey system with stage-structure for predators. Computers Math. Appl., 33 (1997), No. 8, 83-91.
    • (1997) Computers Math. Appl. , vol.33 , Issue.8 , pp. 83-91
    • Wang, W.1    Chen, L.2
  • 86
    • 0001519022 scopus 로고
    • Time lag in prey-predator population models
    • P. J. Wangersky, W. J. Cunningham. Time lag in prey-predator population models. Ecology, 38 (1957), 136-139.
    • (1957) Ecology , vol.38 , pp. 136-139
    • Wangersky, P.J.1    Cunningham, W.J.2
  • 87
    • 0001555614 scopus 로고
    • Bifurcation analysis of a predator-prey system involving group defence
    • G. S. K.Wolkowicz. Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math., 48 (1988), 592-606.
    • (1988) SIAM J. Appl. Math. , vol.48 , pp. 592-606
    • Wolkowicz, G.S.K.1
  • 88
    • 33646864830 scopus 로고    scopus 로고
    • Symmetric functional differential equations and neural networks with memory
    • DOI 10.1090/S0002-9947-98-02083-2
    • J.Wu. Symmetric functional differential equations and neural networks with memory. Trans. Amer. Math. Soc., 350 (1998), 4799-4838. (Pubitemid 128166065)
    • (1998) Transactions of the American Mathematical Society , vol.350 , Issue.12 , pp. 4799-4838
    • Wu, J.1
  • 89
    • 79960804111 scopus 로고    scopus 로고
    • The effects of harvesting and time delay on predator-prey systems with Holling type II functional response
    • revised
    • J. Xia, Z. Liu, R. Yuan, S. Ruan. The effects of harvesting and time delay on predator-prey systems with Holling type II functional response. SIAM J. Appl. Math. (revised).
    • SIAM J. Appl. Math.
    • Xia, J.1    Liu, Z.2    Yuan, R.3    Ruan, S.4
  • 90
    • 0037299742 scopus 로고    scopus 로고
    • Stability and bifurcation in a delayed ratio-dependent predator-prey system
    • DOI 10.1017/S0013091500001140
    • D. Xiao, W. Li. Stability and bifurcation in a delayed ratio-dependent predator-prey system. Proc. Edinburgh Math. Soc., 46A (2003), 205-220. (Pubitemid 36357338)
    • (2003) Proceedings of the Edinburgh Mathematical Society , vol.46 , Issue.1 , pp. 205-220
    • Xiao, D.1    Wenxia, L.I.2
  • 91
    • 0000052872 scopus 로고    scopus 로고
    • Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting
    • D. Xiao, S. Ruan. Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. Fields Institute Communications, 21 (1999), 493-506.
    • (1999) Fields Institute Communications , vol.21 , pp. 493-506
    • Xiao, D.1    Ruan, S.2
  • 92
    • 0035507108 scopus 로고    scopus 로고
    • Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response
    • DOI 10.1006/jdeq.2000.3982
    • D. Xiao, S. Ruan. Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response. J. Differential Equations, 176 (2001), 494-510. (Pubitemid 33082434)
    • (2001) Journal of Differential Equations , vol.176 , Issue.2 , pp. 494-510
    • Xiao, D.1    Ruan, S.2
  • 93
    • 33744545227 scopus 로고    scopus 로고
    • Hopf bifurcation and global periodic solutions in a delayed predator-prey system
    • DOI 10.1016/j.amc.2005.11.020, PII S0096300305009227
    • X.-P. Yan, W.-T. Li. Hopf bifurcation and global periodic solutions in a delayed predatorprey system. Appl. Math. Computat., 177 (2006), 427-445. (Pubitemid 43815543)
    • (2006) Applied Mathematics and Computation , vol.177 , Issue.1 , pp. 427-445
    • Yan, X.-P.1    Li, W.-T.2
  • 94
    • 0031126691 scopus 로고    scopus 로고
    • Global existence of periodic solutions in a class of delayed gause-type predator-prey systems
    • T. Zhao, Y. Kuang, H. L. Smith. Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems. Nonlinear Anal., 28 (1997), 1373-1394. (Pubitemid 127425178)
    • (1997) Nonlinear Analysis, Theory, Methods and Applications , vol.28 , Issue.8 , pp. 1373-1394
    • Zhao, T.1    Kuang, Y.2    Smith, H.L.3


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