- Open Access
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.
- Chaotic Dynamic
- Periodic Point
- Chaotic Behavior
- Stage Structure
- Analytical Proof
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 
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.
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