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Chaos in a singlespecies discrete population model with stage structure and birth pulses
Advances in Difference Equations volume 2014, Article number: 175 (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.
1 Introduction
Many papers have been published on chaos in discrete models (see [1–11] 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 2005, Gao and Chen [10] proposed a singlespecies discrete population model with stage structure and birth pulses:
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]
Consider $(X,d)$ a metric space. We say that a continuous map $\psi :X\to X$ induces chaotic dynamics on two symbols if there exist two disjoint compact sets ${K}_{0},{K}_{1}\subset X$ such that, for each twosided sequence ${({s}_{i})}_{i\in \mathbb{Z}}\in {\{0,1\}}^{\mathbb{Z}}$, there exists a corresponding sequence ${({\omega}_{i})}_{i\in \mathbb{Z}}\in {({K}_{0}\cup {K}_{1})}^{\mathbb{Z}}$ such that
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).
The following basic facts are listed in [12]:

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.
We employ the usual maximum norm in ${\mathbb{R}}^{n}$,
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 ${c}_{N}(N)={J}_{n}$.
In this setting, we employ the notation
Definition 2.3 [12]
Assume that N, M are hsets, such that $u(N)=u(M)=u$ and $s(N)=s(M)=s$. Let $f:N\to {\mathbb{R}}^{n}$ be a continuous map, and define ${f}_{c}={c}_{M}\circ f\circ {c}_{N}^{1}:{J}_{n}\to {\mathbb{R}}^{n}$. We say that N fcovers M, and we write it as
if the following conditions are satisfied:

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]
Let $F:D\to {\mathbb{R}}^{n}$ be a continuous map and assume that there exist two disjoint hsets ${N}_{0}$ and ${N}_{1}$ such that
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]
Let I be a real interval and $g:I\to I$ a continuous map. We say that g is δstrictly turbulent if there exist four constants ${\alpha}_{0}<{\alpha}_{1}<{\beta}_{0}<{\beta}_{1}$, and $\delta >0$ so that
3 Chaotic dynamics in the model (1.1)
Associated to (1.1), we define the map in ${\mathbb{R}}^{2}$
where $0<r<1$, $b>0$, $p>0$, $0<q<1$.
Denote
Set $f(x)=bpx{e}^{(r+p)x}$, then one has
First, we provide a technical lemma, which will play a key role in the proof of the existence of chaotic dynamics.
Lemma 3.1 The first component ${F}_{1}^{2}(x,y)$ and the second component ${F}_{2}^{2}(x,y)$ of ${F}^{2}(x,y)$ satisfy the following inequalities: for $x>0$, $y\ge 0$,

(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)
$0<{F}_{2}^{2}(x,y)<\frac{p{F}_{1}^{2}(x,y)}{r}+pqx+{q}^{2}y$.
Proof The first component of ${F}^{2}$ has the following expression:
We easily deduce that, for $x>0$, $y\ge 0$,
which implies assertion (a) holds.
On the other hand, for $x>0$ and $y\ge 0$,
Now using
we arrive at
which implies assertion (b) holds.
For the second component ${F}_{2}^{2}(x,y)$, noticing that $ax{e}^{cx}\le \frac{a}{ce}$ and
we obtain, for $x>0$, $y\ge 0$,
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.
Theorem 3.2 Assume that $f(x)=bpx{e}^{(r+p)x}$ satisfies the requirement that ${f}^{2}$ is δstrictly turbulent with parameters $0<{\alpha}_{0}<{\alpha}_{1}<{\beta}_{0}<{\beta}_{1}$ and $\delta >0$. Suppose that $r>q$, and the following inequalities are fulfilled:
Then ${F}^{2}$ induces chaotic dynamics on two symbols relative to
Proof Set
Then we have
From this, it is easy to check that ${N}_{0}$ and ${N}_{1}$ are hsets, with
and
where ${t}_{v}$ and ${t}_{w}$ are the translations according to the vectors
respectively, and
In order to apply Lemma 2.4, it suffices to demonstrate that
for $i,j=0,1$.
We give the proof only for the case $i=0$. Indeed, consider the homotopy
where $A(x,y)=(2x,0)$.
Define $f(x)=bpx{e}^{(r+p)x}$. Then it follows from (a) and (c) of Lemma 3.1 that, for all $(x,y)\in {N}_{i}$ ($i=0,1$),
As ${f}^{2}({\alpha}_{0})<{\alpha}_{0}\delta $, from (b) of Lemma 3.1, we obtain, for all $y\in [0,\frac{p{\beta}_{1}}{rq}]$,
By using (3.1) together with the inequality ${\alpha}_{0}<{\beta}_{0}$, we get from (3.4), for all $y\in [0,\frac{p{\beta}_{1}}{rq}]$,
Analogously, as ${f}^{2}({\alpha}_{1})>{\beta}_{1}+\delta $, from (a) of Lemma 3.1, we obtain, for all $y\in [0,\frac{p{\beta}_{1}}{rq}]$,
By using (3.2) together with the inequality ${\beta}_{1}>{\alpha}_{1}$, we get from (3.6), for all $y\in [0,\frac{p{\beta}_{1}}{rq}]$,
From inequalities (3.3), (3.5), and (3.7), it follows that, for $j=0,1$,
These properties, together with the expression of A, lead to the conclusion
This leads to the covering relations
It can be similarly verified for the covering relations
taking the linear map $A(x,y)=(2x,0)$. The proof is complete. □
Now we apply Theorem 3.2 in a particular example.
Example 3.3 Take $f(x)=bpx{e}^{(r+p)x}$ with $bp=exp(3.2)$ and $r+p=0.6$, then ${f}^{2}$ is 0.545strictly turbulent with parameters $2.5<7.5<8.5<13.9$. Straightforward computations show that conditions (3.1), (3.2) in Theorem 3.2 hold for
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.
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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.
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Keywords
 Chaotic Dynamic
 Periodic Point
 Chaotic Behavior
 Stage Structure
 Analytical Proof