Chaos in a single-species discrete population model with stage structure and birth pulses. (English) Zbl 1344.92132
Summary: This paper gives an analytical proof of the existence of chaotic dynamics for a single-species discrete population model with stage structure and birth pulses. The approach is based on a general existence criterion for chaotic dynamics of \(n\)-dimensional maps and inequality techniques. An example is given to illustrate the effectiveness of the result.
MSC:
| 92D25 | Population dynamics (general) |
| 60J85 | Applications of branching processes |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
Keywords:
single-species discrete population model; stage structure; birth pulses; chaotic dynamics; \(n\)-dimensional mapsReferences:
| [1] | Moghtadaei M, Hashemi Golpayegani MR, Malekzade R: Periodic and chaotic dynamics in a map-based model of tumor-immune interaction.J. Theor. Biol. 2013, 334:130-140. · Zbl 1397.92350 · doi:10.1016/j.jtbi.2013.05.031 |
| [2] | Mazrooei-Sebdani R, Farjami S: Bifurcations and chaos in a discrete-time-delayed Hopfield neural network with ring structures and different internal decays.Neurocomputing 2013, 99:154-162. · doi:10.1016/j.neucom.2012.06.007 |
| [3] | Peng MS, Uçar A: The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximations of delay differential equations.Chaos Solitons Fractals 2004, 21:883-891. · Zbl 1054.65126 · doi:10.1016/j.chaos.2003.12.044 |
| [4] | Peng MS, Yuan Y: Stability, symmetry-breaking bifurcation and chaos in discrete delayed models.Int. J. Bifurc. Chaos 2008, 18:1477-1501. · Zbl 1147.39301 · doi:10.1142/S0218127408021117 |
| [5] | He ZM, Lai X: Bifurcation and chaotic behavior of a discrete-time predator-prey system.Nonlinear Anal., Real World Appl. 2011, 12:403-417. · Zbl 1202.93038 · doi:10.1016/j.nonrwa.2010.06.026 |
| [6] | Fan DJ, Wei JJ: Bifurcation analysis of discrete survival red blood cells model.Commun. Nonlinear Sci. Numer. Simul. 2009, 14:3358-3368. · Zbl 1221.37181 · doi:10.1016/j.cnsns.2009.01.015 |
| [7] | Tuzinkevich AV: Bifurcations and chaos in a time-discrete integral model of population dynamics.Math. Biosci. 1992, 109:99-126. · Zbl 0761.92046 · doi:10.1016/0025-5564(92)90041-T |
| [8] | Çelik C, Duman O: Allee effect in a discrete-time predator-prey system.Chaos Solitons Fractals 2009, 40:1956-1962. · Zbl 1198.34084 · doi:10.1016/j.chaos.2007.09.077 |
| [9] | Sun GQ, Zhang G, Jin Z: Dynamic behavior of a discrete modified Ricker & Beverton-Holt model.Comput. Math. Appl. 2009, 57:1400-1412. · Zbl 1186.34074 · doi:10.1016/j.camwa.2009.01.004 |
| [10] | Gao SJ, Chen LS: The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses.Chaos Solitons Fractals 2005, 24:1013-1023. · Zbl 1061.92059 · doi:10.1016/j.chaos.2004.09.091 |
| [11] | Zhao M, Yu HG, Zhu J: Effects of a population floor on the persistence of chaos in a mutual interference host-parasitoid model.Chaos Solitons Fractals 2009, 42:1245-1250. · doi:10.1016/j.chaos.2009.03.027 |
| [12] | Liz E, Ruiz-Herrera A: Chaos in discrete structured population models.SIAM J. Appl. Dyn. Syst. 2012, 11:1200-1214. · Zbl 1260.37021 · doi:10.1137/120868980 |
| [13] | Li, ZC; Zhao, QL; Liang, D., Chaos in a discrete delay population model, No. 2012 (2012) · Zbl 1253.37085 |
| [14] | Thunberg H: Periodicity versus chaos in one-dimensional dynamics.SIAM Rev. 2001, 43:3-30. · Zbl 1049.37027 · doi:10.1137/S0036144500376649 |
| [15] | Ugarcovici I, Weiss H: Chaotic attractors and physical measures for some density dependent Leslie population models.Nonlinearity 2007, 20:2897-2906. · Zbl 1134.37039 · doi:10.1088/0951-7715/20/12/008 |
| [16] | Shi YM, Chen GR: Chaos of discrete dynamical systems in complete metric spaces.Chaos Solitons Fractals 2004, 22:555-571. · Zbl 1067.37047 · doi:10.1016/j.chaos.2004.02.015 |
| [17] | Shi YM, Yu P: Study on chaos induced by turbulent maps in noncompact sets.Chaos Solitons Fractals 2006, 28:1165-1180. · Zbl 1106.37008 · doi:10.1016/j.chaos.2005.08.162 |
| [18] | Block, LS; Coppel, WA, Lectures Notes in Mathematics (1992), Berlin · Zbl 0746.58007 |
| [19] | Aulbach B, Kieninger B: On three definitions of chaos.Nonlinear Dyn. Syst. Theory 2001, 1:23-37. · Zbl 0991.37010 |
| [20] | Kirchgraber U, Stoffer D: On the definition of chaos.Z. Angew. Math. Mech. 1989, 69:175-185. · Zbl 0713.58035 · doi:10.1002/zamm.19890690703 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
