×

zbMATH — the first resource for mathematics

Multiple attractors via CUSP bifurcation in periodically varying environments. (English) Zbl 1068.92038
Summary: Periodically forced (non-autonomous) single species population models support multiple attractors via tangent bifurcations, where the corresponding autonomous models support single attractors. S. Elaydi and R. J. Sacker [J. Differ. Equations 208, 258–273 (2005; Zbl 1067.39003)] obtained conditions for the existence of single attractors in periodically forced discrete-time models. In this paper, the Cusp Bifurcation Theorem is used to provide a general framework for the occurrence of multiple attractors in such periodic dynamical systems.

MSC:
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
39A11 Stability of difference equations (MSC2000)
Software:
Dynamics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1142/S0218127492000446 · Zbl 0870.58046 · doi:10.1142/S0218127492000446
[2] Begon M, Ecology: Individuals, Populations and Communities, Blackwell Science Ltd (1996)
[3] Brauer F, Texts in Applied Mathematics 40 (2001)
[4] DOI: 10.1006/bulm.1997.0017 · Zbl 0973.92034 · doi:10.1006/bulm.1997.0017
[5] DOI: 10.1137/0132006 · Zbl 0348.34031 · doi:10.1137/0132006
[6] Elaydi SN, Discrete Chaos (2000)
[7] Elaydi SN, Journal of Differential Equations
[8] Elaydi SN, Journal of Differential Equations
[9] DOI: 10.1080/10236190290027666 · Zbl 1048.39002 · doi:10.1080/10236190290027666
[10] DOI: 10.1016/S0022-247X(03)00417-7 · Zbl 1035.37020 · doi:10.1016/S0022-247X(03)00417-7
[11] DOI: 10.1006/jmaa.1996.0410 · Zbl 0864.92016 · doi:10.1006/jmaa.1996.0410
[12] DOI: 10.1016/0022-247X(92)90167-C · Zbl 0778.93012 · doi:10.1016/0022-247X(92)90167-C
[13] DOI: 10.1007/BF00160333 · Zbl 0735.92023 · doi:10.1007/BF00160333
[14] DOI: 10.1016/0375-9601(93)91119-P · doi:10.1016/0375-9601(93)91119-P
[15] DOI: 10.1111/j.1095-8312.1991.tb00549.x · doi:10.1111/j.1095-8312.1991.tb00549.x
[16] Hanski IA, Metapopulation Biology: Ecology, Genetics, and Evolution (1997)
[17] Hassell MP, Studies in Biol, The Camelot Press Ltd 72 (1976)
[18] DOI: 10.2307/3886 · doi:10.2307/3886
[19] DOI: 10.1016/S0167-2789(99)00231-6 · Zbl 0957.37018 · doi:10.1016/S0167-2789(99)00231-6
[20] DOI: 10.1006/bulm.1999.0136 · Zbl 1323.92169 · doi:10.1006/bulm.1999.0136
[21] S.M. Henson, R.F. Costantino, R.A. Desharnais, J.M. Cushing, B. Dennis, Basins of attraction: population dynamics with two stable 4-cycles (Preprint) · Zbl 0973.92034
[22] Henson SM, Journal of Mathematical Biology 34 pp 755– (1996)
[23] DOI: 10.1007/s002850050098 · Zbl 0890.92023 · doi:10.1007/s002850050098
[24] DOI: 10.1038/288699a0 · doi:10.1038/288699a0
[25] Kocic VL, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993)
[26] Kuznetsov YA, Elements of Applied Bifurcation Theory, Springer-Verlag (1995) · Zbl 0829.58029
[27] DOI: 10.1016/0025-5564(92)90012-L · Zbl 0746.92022 · doi:10.1016/0025-5564(92)90012-L
[28] DOI: 10.1086/283092 · doi:10.1086/283092
[29] DOI: 10.1038/261459a0 · Zbl 1369.37088 · doi:10.1038/261459a0
[30] May RM, Stability and Complexity in Model Ecosystems, Princeton University Press (1974)
[31] DOI: 10.2307/4142 · doi:10.2307/4142
[32] DOI: 10.2307/1934346 · doi:10.2307/1934346
[33] DOI: 10.1016/0040-5809(90)90040-3 · Zbl 0699.92020 · doi:10.1016/0040-5809(90)90040-3
[34] DOI: 10.1071/ZO9540001 · doi:10.1071/ZO9540001
[35] Nusse HE, Dynamics: Numerical Explorations (1997)
[36] DOI: 10.1139/f54-039 · doi:10.1139/f54-039
[37] DOI: 10.1016/S0167-2789(01)00324-4 · Zbl 1018.37047 · doi:10.1016/S0167-2789(01)00324-4
[38] DOI: 10.1080/1023619031000146887 · Zbl 1319.92053 · doi:10.1080/1023619031000146887
[39] Yodzis P, Introduction to Theoretical Ecology (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.