Chaos in a single-species discrete population model with stage structure and birth pulses
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RESEARCH
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Chaos in a single-species discrete population model with stage structure and birth pulses Hui Fang* *
Correspondence: [email protected] Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
Abstract 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.
1 Introduction Many papers have been published on chaos in discrete models (see [–] and references cited therein). However, in most cases, chaotic behaviors they observed were obtained only by numerical simulations and have not been proved rigorously. In , Gao and Chen [] proposed a single-species discrete population model with stage structure and birth pulses:
un+ = run + be–(r+p)un –qvn (pun + qvn ), vn+ = pun + qvn ,
(.)
where < r < , b > , p > , < q < . System (.) 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 (.) 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 [–]). 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 (.). 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 , some basic definitions and tools are introduced. In Section , it is rigorously proved that there exists chaotic behavior in the discrete population model (.). Finally, our conclusions are presented in Section . © 2014 Fang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fang Advances in Difference Equations 2014, 2014:175 http://www.advancesindifferenceequations.com/content/2014/1/175
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2 Preliminaries For the reader’s convenience, we give a brief introduction to the main tools and definitions that we use in
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