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Chaos in a singlespecies discrete population model with stage structure and birth pulses
 Hui Fang^{1}Email author
https://doi.org/10.1186/168718472014175
© Fang; licensee Springer. 2014
 Received: 5 February 2014
 Accepted: 4 June 2014
 Published: 18 July 2014
Abstract
This paper gives an analytical proof of the existence of chaotic dynamics for a singlespecies discrete population model with stage structure and birth pulses. The approach is based on a general existence criterion for chaotic dynamics of ndimensional maps and inequality techniques. An example is given to illustrate the effectiveness of the result.
Keywords
 Chaotic Dynamic
 Periodic Point
 Chaotic Behavior
 Stage Structure
 Analytical Proof
1 Introduction
where $0<r<1$, $b>0$, $p>0$, $0<q<1$. 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 RuizHerrera [12] established a general existence criterion for chaotic dynamics of ndimensional maps under a new definition of chaos, and they applied it to prove analytically the existence of chaotic dynamics in some classical discretetime agestructured population models. This novel analytical approach is very effective in detecting chaos of discretetime 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 RuizHerrera [12].
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.
2 Preliminaries
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 [12].
In this paper, we denote by ℕ, ℤ, ℝ the set of all positive integers, integers, and real numbers, respectively.
Definition 2.1 [12]
and, whenever ${({s}_{i})}_{i\in \mathbb{Z}}$ is a kperiodic sequence (that is, ${s}_{i+k}={s}_{i}$, $\mathrm{\forall}i\in \mathbb{Z}$) for some $k\ge 1$, there exists a kperiodic sequence ${({\omega}_{i})}_{i\in \mathbb{Z}}\in {({K}_{0}\cup {K}_{1})}^{\mathbb{Z}}$ satisfying (2.1).
 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 semiconjugate with the Bernoulli shift, the existence of periodic points of any period $n\in \mathbb{N}$.
 2.
A map that is chaotic according to Definition 2.1 is also chaotic in the sense of Block and Coppel [18] and in the sense of cointossing [19, 20].
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 RuizHerrera.
and use the notation ${J}_{n}={[1,1]}^{n}$ for the closed cube centered at $0\in {\mathbb{R}}^{n}$.
Definition 2.2 [12]
An hset is a quadruple consisting of

a compact subset N of ${\mathbb{R}}^{n}$,

a pair of numbers $u=u(N),s=s(N)\in \{0,1,2,,\dots \}$, with $u+s=n$,

a homeomorphism ${c}_{N}:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$, such that ${c}_{N}(N)={J}_{n}$.
Definition 2.3 [12]
 1.There exists a continuous homotopy $H:[0,1]\times {J}_{n}\to {\mathbb{R}}^{n}$, such that the following conditions hold true:$\begin{array}{c}H(0,\cdot )={f}_{c}(\cdot ),\hfill \\ H([0,1],{N}_{c}^{})\cap {J}_{n}=\mathrm{\varnothing},\hfill \\ H([0,1],{J}_{n})\cap {M}_{c}^{+}=\mathrm{\varnothing}.\hfill \end{array}$
 2.
There exists a linear map $A:{\mathbb{R}}^{u}\to {\mathbb{R}}^{u}$, such that $H(1,(p,q))=(Ap,0)$ for $p\in {J}_{u}$ and $q\in {J}_{s}$, and $A(\partial {J}_{u})\subset {\mathbb{R}}^{u}\mathrm{\setminus}{J}_{u}$.
Lemma 2.4 [12]
for all $i,j=0,1$. Then F induces chaotic dynamics on two symbols (with compact sets ${\mathcal{K}}_{0}={N}_{0}$ and ${\mathcal{K}}_{1}={N}_{1}$).
Definition 2.5 [12]
3 Chaotic dynamics in the model (1.1)
where $0<r<1$, $b>0$, $p>0$, $0<q<1$.
First, we provide a technical lemma, which will play a key role in the proof of the existence of chaotic dynamics.
 (a)
${F}_{1}^{2}(x,y)>{r}^{2}x+{f}^{2}(x)\cdot {e}^{[(r+p)r+pq]x}\cdot {e}^{[{q}^{2}+(r+p)bq+q]y}$;
 (b)
${F}_{1}^{2}(x,y)\le [{r}^{2}+rbp+bp(r+q)]x+[rqb+b{q}^{2}+pq{b}^{2}]y+{f}^{2}(x){[{e}^{\frac{bp}{e}}]}^{[{e}^{qy}1]}$;
 (c)
$0<{F}_{2}^{2}(x,y)<\frac{p{F}_{1}^{2}(x,y)}{r}+pqx+{q}^{2}y$.
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 [12] with appropriate modifications.
for $i,j=0,1$.
where $A(x,y)=(2x,0)$.
taking the linear map $A(x,y)=(2x,0)$. The proof is complete. □
Now we apply Theorem 3.2 in a particular example.
4 Conclusions
This paper rigorously proves the existence of chaotic dynamics for a singlespecies discrete population model with stage structure and birth pulses. The result shows that the second composition map of a twodimensional map associated to this model is chaotic in the sense of Liz and RuizHerrera.
Author’s contributions
The author contributed to the manuscript and read and approved the final manuscript.
Declarations
Acknowledgements
This research is supported by the National Natural Science Foundation of China (Grant No. 10971085). The author also thanks the reviewers for helpful comments.
Authors’ Affiliations
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