Chaos in a single-species discrete population model with stage structure and birth pulses
© Fang; licensee Springer. 2014
Received: 5 February 2014
Accepted: 4 June 2014
Published: 18 July 2014
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.
where , , , . System (1.1) describes the numbers of immature population and mature population at a pulse in terms of values at the previous pulse. They proved numerically that system (1.1) can be chaotic.
Since numerical simulations may lead to erroneous conclusions, numerical evidence of the existence of chaotic behaviors still needs to be confirmed analytically. Some researchers proved analytically the existence of chaotic behavior of discrete systems under different definitions of chaos (for example, see [12–17]). Recently, Liz and Ruiz-Herrera  established a general existence criterion for chaotic dynamics of n-dimensional maps under a new definition of chaos, and they applied it to prove analytically the existence of chaotic dynamics in some classical discrete-time age-structured population models. This novel analytical approach is very effective in detecting chaos of discrete-time dynamical systems.
The main purpose of this paper is to give an analytical proof of the existence of chaotic dynamics of (1.1). To this end, we use the analytical approach for detecting chaos developed by Liz and Ruiz-Herrera .
The rest of the paper is organized as follows. In Section 2, some basic definitions and tools are introduced. In Section 3, it is rigorously proved that there exists chaotic behavior in the discrete population model (1.1). Finally, our conclusions are presented in Section 4.
For the reader’s convenience, we give a brief introduction to the main tools and definitions that we use in this paper. For more details, we refer the reader to .
In this paper, we denote by ℕ, ℤ, ℝ the set of all positive integers, integers, and real numbers, respectively.
Definition 2.1 
and, whenever is a k-periodic sequence (that is, , ) for some , there exists a k-periodic sequence satisfying (2.1).
Definition 2.1 guarantees natural properties of complex dynamics, such as sensitive dependence on the initial conditions or the presence of an invariant set Λ being transitive and semi-conjugate with the Bernoulli shift, the existence of periodic points of any period .
We understand chaos in the sense of Definition 2.1. A map that is chaotic according to Definition 2.1 is called chaotic in the sense of Liz and Ruiz-Herrera.
and use the notation for the closed cube centered at .
Definition 2.2 
An h-set is a quadruple consisting of
a compact subset N of ,
a pair of numbers , with ,
a homeomorphism , such that .
Definition 2.3 
- 1.There exists a continuous homotopy , such that the following conditions hold true:
There exists a linear map , such that for and , and .
Lemma 2.4 
for all . Then F induces chaotic dynamics on two symbols (with compact sets and ).
Definition 2.5 
3 Chaotic dynamics in the model (1.1)
where , , , .
First, we provide a technical lemma, which will play a key role in the proof of the existence of chaotic dynamics.
which implies assertion (a) holds.
which implies assertion (b) holds.
which implies assertion (c) holds. The proof is complete. □
Next, we prove the following result by following the idea of the proof of Theorem 5.1 in  with appropriate modifications.
taking the linear map . The proof is complete. □
Now we apply Theorem 3.2 in a particular example.
This paper rigorously proves the existence of chaotic dynamics for a single-species discrete population model with stage structure and birth pulses. The result shows that the second composition map of a two-dimensional map associated to this model is chaotic in the sense of Liz and Ruiz-Herrera.
The author contributed to the manuscript and read and approved the final manuscript.
This research is supported by the National Natural Science Foundation of China (Grant No. 10971085). The author also thanks the reviewers for helpful comments.
- 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.View ArticleMathSciNetGoogle Scholar
- 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.View ArticleGoogle Scholar
- 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. 10.1016/j.chaos.2003.12.044View ArticleMATHGoogle Scholar
- Peng MS, Yuan Y: Stability, symmetry-breaking bifurcation and chaos in discrete delayed models. Int. J. Bifurc. Chaos 2008, 18: 1477–1501. 10.1142/S0218127408021117MathSciNetView ArticleMATHGoogle Scholar
- He ZM, Lai X: Bifurcation and chaotic behavior of a discrete-time predator-prey system. Nonlinear Anal., Real World Appl. 2011, 12: 403–417. 10.1016/j.nonrwa.2010.06.026MathSciNetView ArticleMATHGoogle Scholar
- Fan DJ, Wei JJ: Bifurcation analysis of discrete survival red blood cells model. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 3358–3368. 10.1016/j.cnsns.2009.01.015MathSciNetView ArticleMATHGoogle Scholar
- Tuzinkevich AV: Bifurcations and chaos in a time-discrete integral model of population dynamics. Math. Biosci. 1992, 109: 99–126. 10.1016/0025-5564(92)90041-TMathSciNetView ArticleMATHGoogle Scholar
- Çelik C, Duman O: Allee effect in a discrete-time predator-prey system. Chaos Solitons Fractals 2009, 40: 1956–1962. 10.1016/j.chaos.2007.09.077MathSciNetView ArticleMATHGoogle Scholar
- Sun GQ, Zhang G, Jin Z: Dynamic behavior of a discrete modified Ricker & Beverton-Holt model. Comput. Math. Appl. 2009, 57: 1400–1412. 10.1016/j.camwa.2009.01.004MathSciNetView ArticleMATHGoogle Scholar
- 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. 10.1016/j.chaos.2004.09.091MathSciNetView ArticleMATHGoogle Scholar
- 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. 10.1016/j.chaos.2009.03.027View ArticleGoogle Scholar
- Liz E, Ruiz-Herrera A: Chaos in discrete structured population models. SIAM J. Appl. Dyn. Syst. 2012, 11: 1200–1214. 10.1137/120868980MathSciNetView ArticleMATHGoogle Scholar
- Li ZC, Zhao QL, Liang D: Chaos in a discrete delay population model. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 482459Google Scholar
- Thunberg H: Periodicity versus chaos in one-dimensional dynamics. SIAM Rev. 2001, 43: 3–30. 10.1137/S0036144500376649MathSciNetView ArticleMATHGoogle Scholar
- Ugarcovici I, Weiss H: Chaotic attractors and physical measures for some density dependent Leslie population models. Nonlinearity 2007, 20: 2897–2906. 10.1088/0951-7715/20/12/008MathSciNetView ArticleMATHGoogle Scholar
- Shi YM, Chen GR: Chaos of discrete dynamical systems in complete metric spaces. Chaos Solitons Fractals 2004, 22: 555–571. 10.1016/j.chaos.2004.02.015MathSciNetView ArticleMATHGoogle Scholar
- Shi YM, Yu P: Study on chaos induced by turbulent maps in noncompact sets. Chaos Solitons Fractals 2006, 28: 1165–1180. 10.1016/j.chaos.2005.08.162MathSciNetView ArticleMATHGoogle Scholar
- Block LS, Coppel WA Lectures Notes in Mathematics. In Dynamics in One Dimension. Springer, Berlin; 1992.Google Scholar
- Aulbach B, Kieninger B: On three definitions of chaos. Nonlinear Dyn. Syst. Theory 2001, 1: 23–37.MathSciNetMATHGoogle Scholar
- Kirchgraber U, Stoffer D: On the definition of chaos. Z. Angew. Math. Mech. 1989, 69: 175–185. 10.1002/zamm.19890690703MathSciNetView ArticleMATHGoogle Scholar
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