Skip to content

Advertisement

  • Research
  • Open Access

Chaos in a single-species discrete population model with stage structure and birth pulses

Advances in Difference Equations20142014:175

https://doi.org/10.1186/1687-1847-2014-175

  • Received: 5 February 2014
  • Accepted: 4 June 2014
  • Published:

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.

Keywords

  • Chaotic Dynamic
  • Periodic Point
  • Chaotic Behavior
  • Stage Structure
  • Analytical Proof

1 Introduction

Many papers have been published on chaos in discrete models (see [111] 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 single-species discrete population model with stage structure and birth pulses:
{ u n + 1 = r u n + b e ( r + p ) u n q v n ( p u n + q v n ) , v n + 1 = p u n + q v n ,
(1.1)

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 [1217]). Recently, Liz and Ruiz-Herrera [12] 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 [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 , , 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 ψ : X X induces chaotic dynamics on two symbols if there exist two disjoint compact sets K 0 , K 1 X such that, for each two-sided sequence ( s i ) i Z { 0 , 1 } Z , there exists a corresponding sequence ( ω i ) i Z ( K 0 K 1 ) Z such that
ω i K s i and ω i + 1 = ψ ( ω i ) for all  i Z ,
(2.1)

and, whenever ( s i ) i Z is a k-periodic sequence (that is, s i + k = s i , i Z ) for some k 1 , there exists a k-periodic sequence ( ω i ) i Z ( K 0 K 1 ) Z satisfying (2.1).

The following basic facts are listed in [12]:
  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 semi-conjugate with the Bernoulli shift, the existence of periodic points of any period n N .

     
  2. 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 coin-tossing [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 Ruiz-Herrera.

We employ the usual maximum norm in R n ,
( x 1 , x 2 , , x n ) = max { | x i | : i = 1 , 2 , , n } ,

and use the notation J n = [ 1 , 1 ] n for the closed cube centered at 0 R n .

Definition 2.2 [12]

An h-set is a quadruple consisting of

  • a compact subset N of R n ,

  • a pair of numbers u = u ( N ) , s = s ( N ) { 0 , 1 , 2 , , } , with u + s = n ,

  • a homeomorphism c N : R n R n , such that c N ( N ) = J n .

In this setting, we employ the notation
N c = J u × J s , N c + = J u × J s .

Definition 2.3 [12]

Assume that N, M are h-sets, such that u ( N ) = u ( M ) = u and s ( N ) = s ( M ) = s . Let f : N R n be a continuous map, and define f c = c M f c N 1 : J n R n . We say that N f-covers M, and we write it as
N f M ,
if the following conditions are satisfied:
  1. 1.
    There exists a continuous homotopy H : [ 0 , 1 ] × J n R n , such that the following conditions hold true:
    H ( 0 , ) = f c ( ) , H ( [ 0 , 1 ] , N c ) J n = , H ( [ 0 , 1 ] , J n ) M c + = .
     
  2. 2.

    There exists a linear map A : R u R u , such that H ( 1 , ( p , q ) ) = ( A p , 0 ) for p J u and q J s , and A ( J u ) R u J u .

     

Lemma 2.4 [12]

Let F : D R n be a continuous map and assume that there exist two disjoint h-sets N 0 and N 1 such that
N i f N j

for all i , j = 0 , 1 . Then F induces chaotic dynamics on two symbols (with compact sets K 0 = N 0 and K 1 = N 1 ).

Definition 2.5 [12]

Let I be a real interval and g : I I a continuous map. We say that g is δ-strictly turbulent if there exist four constants α 0 < α 1 < β 0 < β 1 , and δ > 0 so that
g ( α 0 ) < α 0 δ < β 1 + δ < g ( α 1 ) , g ( β 1 ) < α 0 δ < β 1 + δ < g ( β 0 ) .

3 Chaotic dynamics in the model (1.1)

Associated to (1.1), we define the map in R 2
F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) ) = ( r x + b ( p x + q y ) e ( r + p ) x q y , p x + q y ) ,

where 0 < r < 1 , b > 0 , p > 0 , 0 < q < 1 .

Denote
F 2 ( x , y ) = F ( F 1 ( x , y ) , F 2 ( x , y ) ) = ( F 1 2 ( x , y ) , F 2 2 ( x , y ) ) .
Set f ( x ) = b p x e ( r + p ) x , then one has
f 2 ( x ) = f ( f ( x ) ) = b 2 p 2 x e ( r + p ) x e [ ( r + p ) b p x e ( r + p ) x ] .

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 0 ,
  1. (a)

    F 1 2 ( x , y ) > r 2 x + f 2 ( x ) e [ ( r + p ) r + p q ] x e [ q 2 + ( r + p ) b q + q ] y ;

     
  2. (b)

    F 1 2 ( x , y ) [ r 2 + r b p + b p ( r + q ) ] x + [ r q b + b q 2 + p q b 2 ] y + f 2 ( x ) [ e b p e ] [ e q y 1 ] ;

     
  3. (c)

    0 < F 2 2 ( x , y ) < p F 1 2 ( x , y ) r + p q x + q 2 y .

     
Proof The first component of F 2 has the following expression:
F 1 2 ( x , y ) = r F 1 ( x , y ) + b ( p F 1 ( x , y ) + q F 2 ( x , y ) ) e [ ( r + p ) F 1 ( x , y ) q F 2 ( x , y ) ] = r 2 x + r b ( p x + q y ) e ( r + p ) x q y + b [ p r x + p b ( p x + q y ) e ( r + p ) x q y + p q x + q 2 y ] e [ ( r + p ) [ r x + b ( p x + q y ) e ( r + p ) x q y ] q ( p x + q y ) ] .
We easily deduce that, for x > 0 , y 0 ,
F 1 2 ( x , y ) > r 2 x + [ b 2 p 2 x e ( r + p ) x q y ] e [ ( r + p ) [ r x + b ( p x + q y ) e ( r + p ) x q y ] q ( p x + q y ) ] = r 2 x + [ b 2 p 2 x e ( r + p ) x q y ] e [ ( r + p ) r x q ( p x + q y ) ] e [ ( r + p ) b p x e ( r + p ) x q y ] e [ ( r + p ) b q y e ( r + p ) x q y ] r 2 x + [ b 2 p 2 x e ( r + p ) x q y ] e [ ( r + p ) r x q ( p x + q y ) ] e [ ( r + p ) b p x e ( r + p ) x ] e [ ( r + p ) b q y ] = r 2 x + e [ ( r + p ) r x q ( p x + q y ) ] [ b 2 p 2 x e ( r + p ) x e [ ( r + p ) b p x e ( r + p ) x ] ] e [ ( r + p ) b q y q y ] = r 2 x + f 2 ( x ) e [ ( r + p ) r + p q ] x e [ q 2 + ( r + p ) b q + q ] y ,

which implies assertion (a) holds.

On the other hand, for x > 0 and y 0 ,
F 1 2 ( x , y ) r 2 x + r b p x + r b q y + b e [ ( r + p ) b p x e ( r + p ) x q y ] [ p ( r + q ) x + q 2 y + p 2 b x e ( r + p ) x + p b q y ] [ r 2 + r b p + b p ( r + q ) ] x + [ r q b + b q 2 + p q b 2 ] y + b 2 p 2 x e ( r + p ) x e [ ( r + p ) b p x e ( r + p ) x e q y ] = [ r 2 + r b p + b p ( r + q ) ] x + [ r q b + b q 2 + p q b 2 ] y + f 2 ( x ) [ e ( r + p ) b p x e ( r + p ) x ] [ e q y 1 ] .
Now using
f ( x ) = b p x e ( r + p ) x b p ( r + p ) e ,
we arrive at
F 1 2 ( x , y ) [ r 2 + r b p + b p ( r + q ) ] x + [ r q b + b q 2 + p q b 2 ] y + f 2 ( x ) [ e b p e ] [ e q y 1 ] ,

which implies assertion (b) holds.

For the second component F 2 2 ( x , y ) , noticing that a x e c x a c e and
F 1 2 ( x , y ) > r 2 x + r b ( p x + q y ) e ( r + p ) x q y ,
we obtain, for x > 0 , y 0 ,
0 < F 2 2 ( x , y ) = p r x + p b ( p x + q y ) e ( r + p ) x q y + p q x + q 2 y < p F 1 2 ( x , y ) r + p q x + q 2 y ,

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 ) = b p x e ( r + p ) x satisfies the requirement that f 2 is δ-strictly turbulent with parameters 0 < α 0 < α 1 < β 0 < β 1 and δ > 0 . Suppose that r > q , and the following inequalities are fulfilled:
b p e [ e p q β 1 r q 1 ] < ln ( α 0 [ r 2 + 2 r 2 b q + p 2 q b 2 r q ] β 1 α 0 δ ) ,
(3.1)
[ ( r + p ) r + p q r + ( r + p ) b p q + p q r q ] β 1 < ln ( β 1 + δ β 1 r 2 α 1 ) .
(3.2)
Then F 2 induces chaotic dynamics on two symbols relative to
N 0 = { ( x , y ) : α 0 x α 1 , 0 y p r q x } , N 1 = { ( x , y ) : β 0 x β 1 , 0 y p r q x } .
Proof Set
g 0 ( x , y ) = ( x , α 1 y x ) , g 1 ( x , y ) = ( x , β 1 y x ) .
Then we have
g 0 ( N 0 ) = { ( x , y ) : α 0 x α 1 , 0 y p α 1 r q } , g 1 ( N 1 ) = { ( x , y ) : β 0 x β 1 , 0 y p β 1 r q } .
From this, it is easy to check that N 0 and N 1 are h-sets, with
u ( N 0 ) = u ( N 1 ) = 1 ( x -direction ) , s ( N 0 ) = s ( N 1 ) = 1 ( y -direction )
and
c N 0 = h 0 t v g 0 , c N 1 = h 1 t w g 1 ,
where t v and t w are the translations according to the vectors
v = ( α 0 + α 1 2 , α 1 p 2 ( r q ) ) , w = ( β 0 + β 1 2 , β 1 p 2 ( r q ) ) ,
respectively, and
h 0 ( x , y ) = ( 2 x α 1 α 0 , 2 ( r q ) y α 1 p ) , h 1 ( x , y ) = ( 2 x β 1 β 0 , 2 ( r q ) y β 1 p ) .
In order to apply Lemma 2.4, it suffices to demonstrate that
N i F 2 N j

for i , j = 0 , 1 .

We give the proof only for the case i = 0 . Indeed, consider the homotopy
H j ( t , ( x , y ) ) = ( 1 t ) ( c N j F 2 c N 0 1 ) ( x , y ) + t A ( x , y ) ( j = 0 , 1 ) ,

where A ( x , y ) = ( 2 x , 0 ) .

Define f ( x ) = b p x e ( r + p ) x . Then it follows from (a) and (c) of Lemma 3.1 that, for all ( x , y ) N i ( i = 0 , 1 ),
0 < F 2 2 ( x , y ) < p F 1 2 ( x , y ) r + p q x + q 2 y p F 1 2 ( x , y ) r + p q x + p q 2 r q x = p F 1 2 ( x , y ) r + r p q r q x p F 1 2 ( x , y ) r + p q F 1 2 ( x , y ) r ( r q ) = p r q F 1 2 ( x , y ) .
(3.3)
As f 2 ( α 0 ) < α 0 δ , from (b) of Lemma 3.1, we obtain, for all y [ 0 , p β 1 r q ] ,
F 1 2 ( α 0 , y ) [ r 2 + r b p + b p ( r + q ) ] α 0 + [ r q b + b q 2 + p q b 2 ] p β 1 r q + ( α 0 δ ) [ e b p e ] e [ p q β 1 r q ] 1 < [ r 2 + r b p + b p ( r + q ) ] β 1 + [ r q b + b q 2 + p q b 2 ] p β 1 r q + ( α 0 δ ) [ e b p e ] e [ p q β 1 r q ] 1 = [ r 2 + 2 r 2 b q + p 2 q b 2 r q ] β 1 + ( α 0 δ ) [ e b p e ] e [ p q β 1 r q ] 1 .
(3.4)
By using (3.1) together with the inequality α 0 < β 0 , we get from (3.4), for all y [ 0 , p β 1 r q ] ,
F 1 2 ( α 0 , y ) < α 0 < β 0 .
(3.5)
Analogously, as f 2 ( α 1 ) > β 1 + δ , from (a) of Lemma 3.1, we obtain, for all y [ 0 , p β 1 r q ] ,
F 1 2 ( α 1 , y ) > r 2 α 1 + f 2 ( α 1 ) e [ ( r + p ) r + p q ] α 1 e [ q 2 + ( r + p ) b q + q ] p β 1 r q > r 2 α 1 + ( β 1 + δ ) e [ ( r + p ) r + p q r + ( r + p ) b p q + p q r q ] β 1 .
(3.6)
By using (3.2) together with the inequality β 1 > α 1 , we get from (3.6), for all y [ 0 , p β 1 r q ] ,
F 1 2 ( α 1 , y ) > β 1 > α 1 .
(3.7)
From inequalities (3.3), (3.5), and (3.7), it follows that, for j = 0 , 1 ,
c N j F 2 c N 0 1 ( { 1 } × [ 1 , 1 ] ) { ( x , y ) : x < 1 } , c N j F 2 c N 0 1 ( { 1 } × [ 1 , 1 ] ) { ( x , y ) : x > 1 } , c N j F 2 c N 0 1 ( [ 1 , 1 ] 2 ) { ( x , y ) : 1 < y < 1 } .
These properties, together with the expression of A, lead to the conclusion
H j ( [ 0 , 1 ] , { 1 , 1 } × [ 1 , 1 ] ) [ 1 , 1 ] 2 = ( j = 0 , 1 ) , H j ( [ 0 , 1 ] , [ 1 , 1 ] 2 ) ( [ 1 , 1 ] × { 1 , 1 } ) = ( j = 0 , 1 ) .
This leads to the covering relations
N 0 F 2 N j ( j = 0 , 1 ) .
It can be similarly verified for the covering relations
N 1 F 2 N j ( j = 0 , 1 ) ,

taking the linear map A ( x , y ) = ( 2 x , 0 ) . The proof is complete. □

Now we apply Theorem 3.2 in a particular example.

Example 3.3 Take f ( x ) = b p x e ( r + p ) x with b p = exp ( 3.2 ) and r + p = 0.6 , then f 2 is 0.545-strictly 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
r = 0.003 , p = 0.597 , q = 1.5 × 10 7 , b = exp ( 3.2 ) / p .

4 Conclusions

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.

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

(1)
Department of Mathematics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. Li ZC, Zhao QL, Liang D: Chaos in a discrete delay population model. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 482459Google Scholar
  14. Thunberg H: Periodicity versus chaos in one-dimensional dynamics. SIAM Rev. 2001, 43: 3–30. 10.1137/S0036144500376649MathSciNetView ArticleMATHGoogle Scholar
  15. 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
  16. 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
  17. 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
  18. Block LS, Coppel WA Lectures Notes in Mathematics. In Dynamics in One Dimension. Springer, Berlin; 1992.Google Scholar
  19. Aulbach B, Kieninger B: On three definitions of chaos. Nonlinear Dyn. Syst. Theory 2001, 1: 23–37.MathSciNetMATHGoogle Scholar
  20. Kirchgraber U, Stoffer D: On the definition of chaos. Z. Angew. Math. Mech. 1989, 69: 175–185. 10.1002/zamm.19890690703MathSciNetView ArticleMATHGoogle Scholar

Copyright

Advertisement