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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 482459, 14 pages
Chaos in a Discrete Delay Population Model
Department of Mathematics, Shandong Jianzhu University, Shandong, Jinan 250101, China
Received 2 August 2012; Accepted 21 August 2012
Academic Editor: Hua Su
Copyright © 2012 Zongcheng Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with chaos in a discrete delay population model. The map of the model is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Some computer simulations are provided to visualize the theoretical result.
Delay differential equations have been largely used to model phenomena in economics, biology, medicine, ecology, and other sciences. The studies on delay differential equations in population dynamics not only focus on the discussion of stability, attractivity, and persistence, but also involve many other dynamical behaviors such as periodic phenomenon, bifurcation, and chaos, see [1–5], and many references therein.
As we well know, the discrete time population models governed by difference equations are more appropriate than the continuous time population models governed by differential equations when the populations have nonoverlapping generations or the size of the population is rarely small. Moreover, some qualitative properties of the difference equations can also provide a lot of useful information for analyzing the properties of the original differential equations. In addition, discrete time models can also provide efficient computational models of continuous time models for numerical simulations. Therefore, many researchers studied the complex behaviors of the discrete population model, see, for example, [6–11].
Recently, some researchers used the Euler discretization to explore the complex dynamical behaviors of nonlinear differential systems, such as determining the bifurcation diagrams with Hopf bifurcation, observing stable or unstable orbits, and chaotic behavior, see [8–16], and so forth. However, some complicated behaviors such as chaos they observed were obtained only by numerical simulations, and have not been proved rigorously. It is noted that there could exist some false phenomena only by virtue of numerical simulations. Therefore, the existence of chaotic behavior of these systems needs to be studied rigorously.
In this paper, we study the chaotic behavior of the following discrete delay population model where , and are constants, is a positive integer, and is a parameter.
Equation (1.1) can be viewed as a discrete analogue of the following delay differential equation by using the forward Euler scheme when , which is the model of single species population with a quadratic per capita growth rate where , , and are constants, and is the delay. In , Gopalsamy studied the global attractivity of the equilibrium of (1.2). As in (1.2), Gopalsamy  discussed the existence conditions of Hopf bifurcation, and gave an approximate expression of the bifurcation periodic solution.
When in (1.2), it becomes which was studied by Gopalsamy and Ladas for the oscillation and asymptotic behavior in ; Rodrigues , Huang and Peng  studied the discretization of (1.3) by using the forward and backward Euler scheme, respectively. They obtained some results about oscillation and stability of the solutions. Peng  also studied the backward difference form of (1.3), and observed much rich dynamical behaviors, such as Neimark-Sacker bifurcation and chaotic behavior by using the computer-assisted method and computer simulations.
To the best of our knowledge, the research works on the chaotic behavior of (1.2) or its discrete analogue (1.1) with rigorously mathematical proof up to now are still few. The main purpose of this paper is to study the chaotic behavior of (1.1) by using the snap-back repeller theory.
The rest of the paper is organized as follows. In Section 2, some basic concepts and lemmas are introduced. The transformation of the chaos problem is given in Section 3. In Section 4, it is rigorously proved that there exists chaotic behavior in the delay population model by using the snap-back repeller theory. Finally, an illustrative example is provided with computer simulations.
In this section, some basic concepts and lemmas are introduced.
Since Li and Yorke  first introduced a precise mathematical definition of chaos, there appeared several different definitions of chaos, some are stronger and some are weaker, depending on the requirements in different problems. We refer to [17–23] for the some definitions of chaos and discussions of their relationships. For convenience, we present two definitions of chaos in the sense of Li-Yorke and Devaney.
Definition 2.1. Let be a metric space, be a map, and be a set of with at least two distinct points. Then is called a scrambled set of if for any two distinct points , (i); (ii). The map is said to be chaotic in the sense of Li-Yorke if there exists an uncountable scrambled set of .
There are three conditions in the original characterization of chaos in Li-Yorke's theorem . Besides the previous conditions (i) and (ii) mentioned in Definition 2.1, the third one is that for all and for all periodic point of , But conditions (i) and (ii) together (Definition 2.1) imply that the scrambled set contains at most one point that does not satisfy the above condition. So the third condition is not essential and can be removed.
Definition 2.2 (see ). Let be a metric space. A map is said to be chaotic on in the sense of Devaney if(i)the set of the periodic points of is dense in ;(ii) is topologically transitive in ;(iii) has sensitive dependence on initial conditions in .
In Definition 2.2, condition (i) implies that all systems with no periodic points are not chaotic; condition (ii) means that a chaotic system is indecomposable, that is, the system cannot be decomposed into the sum of two subsystems; condition (iii) says that the system is unpredictable, which means that a small change of initial conditions can cause an unavoidable error after many iterations. In 1992, Banks et al.  proved that conditions (i) and (ii) together imply condition (iii) if is continuous in . So, condition (iii) is redundant in the above definition. It has been proved that under some conditions, chaos in the sense of Devaney is stronger than that in the sense of Li-Yorke .
Remark 2.3. Some researchers consider that condition (i) in Definition 2.2 is not essential in the chaotic behavior. In 1990, Wiggins  gave another definition of chaos, that is, is said to be chaotic on in the sense of Wiggins if it satisfies conditions (ii) and (iii) in Definition 2.2. It is evident that chaos in the sense of Devaney is stronger than that in the sense of Wiggins.
For convenience, we present some definitions in .
Definition 2.4 ([25, Definitions 2.1–2.4]). Let be a metric space and be a map. (i)A point is called an expanding fixed point (or a repeller) of in for some constant , if and there exists a constant such that where is the closed ball centered at . The constant is called an expanding coefficient of in . Furthermore, is called a regular expanding fixed point of in if is an interior point of , where is the open ball centered at . Otherwise, is called a singular expanding fixed point of in . (ii)Assume that is an expanding fixed point of in for some . Then is said to be a snap-back repeller of if there exists a point with and for some positive integer . Furthermore, is said to be a nondegenerate snap-back repeller of if there exist positive constants and such that and is called a regular snap-back repeller of if is open and there exists a positive constant such that and for each positive constant , is an interior point of . Otherwise, is called a singular snap-back repeller of .
Remark 2.5. In 1978, Marotto  introduced the concept of snap-back repeller for maps in the Euclidean space . It is obvious that Definition 2.4 extended the concept of snap-back repeller to maps in metric spaces. According to the above classifications of snap-back repellers for maps in metric spaces, the snap-back repeller in the Marotto paper  is regular and nondegenerate.
We now present two lemmas which will be used to study chaos in the delay population model.
Lemma 2.6. Let be a continuously differentiable map. Assume that , then for a sufficiently small neighborhood of and any bounded interval of , there exists a positive constant such that the equation has a solution for any and .
Proof. Since is continuously differentiable on , and , , for any sufficiently small neighborhood of , there exist two neighborhoods and of such that , and is a homeomorphism by [27, Theorem 10.39]. In addition, for any bounded interval , there exists a positive constant such that Take , then for any . Hence, for any and , the equation , that is, has a solution . This completes the proof.
Lemma 2.7 ([28, Theorem 4.4]). Let be a map with a fixed point . Assume that (1) is continuously differentiable in a neighborhood of and all the eigenvalues of have absolute values larger than 1, which implies that there exist a positive constant and a norm in such that is expanding in in , where is the closed ball of radius centered at in ; (2) is a snap-back repeller of with , , for some and some positive integer , where is the open ball of radius centered at in . Furthermore, is continuously differentiable in some neighborhoods of , respectively, and for , where for . Then for each neighborhood of , there exist a positive integer and a Cantor set such that is topologically conjugate to the symbolic system . Consequently, is chaotic on in the sense of Devaney and is chaotic in the sense of Li-Yorke. Further, there exists a compact and perfect invariant set , containing the Cantor set , such that is chaotic on in the sense of Devaney.
Remark 2.8. The conclusions of Lemma 2.7 is slightly different from the original Theorem 4.4 in . From [28, Theorem 4.4], we get that is chaotic on in the sense of Devaney. By [29, Lemma 2.4], we obtain that is chaotic in the sense of Li-Yorke. Consequently is chaotic in the sense of Li-Yorke. The last conclusion of Lemma 2.7 can be conferred to [30, Theorem 4.2]. Under the conditions of Lemma 2.7, is a regular and nondegenerate snap-back repeller. Therefore, Lemma 2.7 can be briefly stated as the following: “a regular and nondegenerate snap-back repeller in implies chaos in the sense of both Devaney and Li-York.” We refer to [25, 31] for details.
3. Transformation of the Chaos Problem
In this section, we will transform the delay population model (1.1) into a -dimensional discrete dynamical system, and give some definitions about chaos of the two systems.
Let for , then system (1.1) can be transformed into the following -dimensional discrete system where .
The map is said to be induced by , and system (3.1) is said to be induced by system (1.1). It is evident that a solution of system (1.1) with an initial condition corresponds to a solution of system (3.1) with an initial condition . We call the solution of (3.1) is induced by the solution of (1.1). Therefore, the dynamical behavior of system (1.1) is the same as that of its induced system (3.1) in . So, we introduce some relative concepts for system (1.1), which are motivated from some works in [31, Definitions 5.1 and 5.2].
Definition 3.1. (i) A point is called an -periodic point of system (1.1) if is an -periodic point of its induced system (3.1), that is, and for . In the special case of , is called a fixed point or steady state of system (1.1).
(ii) The concept of snap-back repeller and its classifications of system (1.1) are defined similarly to those for its induced system (3.1) in .
(iii) The concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and the invariant set for system (1.1) are defined similarly to those for its induced system (3.1) in .
4. Chaos in the Model
It is obvious that system (3.1) has three fixed points where , . It is noticed that the Jacobian matrix of at the fixed-point always has a -multiple eigenvalue . So the fixed-point cannot be a snap-back repeller. However, the fixed-points and can be regular and nondegenerate snap-back repellers of system (3.1) when satisfies some conditions. We only show the fixed-point can be a regular and nondegenerate snap-back repeller of system (3.1) under some conditions, since the situation for the fixed point is similar.
Theorem 4.1. There exists a positive constant such that for any , the fixed-point is a regular and nondegenerate snap-back repeller of system (3.1). Then system (3.1) and consequently, system (1.1), is chaotic in the sense of both Devaney and Li-Yorke.
Proof. The idea in the proof is motivated by the proof of [32, Theorem 3.2]. We will apply Lemma 2.7 to prove this theorem. So, it suffices to show that all the assumptions in Lemma 2.7 are satisfied.
For convenience, we translate the fixed-point to the origin . Let . Then system (3.1) becomes the following where satisfying and . Therefore, is a fixed point of the map , and we only need to prove that is a regular and nondegenerate snap-back repeller of system (4.2) under some conditions.
First, we show that there exists a positive constant such that is an expanding fixed point of the map in some norm in for any . In fact, the map is continuously differentiable in , and the Jacobian matrix of at is Its eigenvalues are determined by the following: Let . From (4.4), we get that all the eigenvalues of have absolute values larger than 1 for any . Otherwise, suppose that there exists an eigenvalue of with , then we get the following inequality which is a contradiction. Hence, it follows from the first condition of Lemma 2.7, there exist a positive constant and a norm in such that is an expanding fixed point of in in the norm , that is, where is an expanding coefficient of in , and is the closed ball centered at of radius with respect to the norm .
Next, we show that is a snap-back repeller of in the norm . Suppose that is an arbitrary neighborhood of in , then there exists a small interval containing such that . In the following, we will show that there exists a positive constant such that for any , there exists a point with satisfying which implies that is a snap-back repeller of .
For convenience, Let , and . It is clear that is continuously differentiable on and satisfies
For . From Lemma 2.6, it follows that there exists a positive constant such that for any , there exist two points satisfying the following equations which can be written as follows Set , then we get that with for any . It follows from (4.10) that , , .
For . It also follows from Lemma 2.6 that there exists a positive constant such that for any , there exist two points satisfying the following equations which can also be written as follows Set , then we get that with for any . It follows from (4.12) that , for , and .
Take . Then for any , there exists a point with satisfying , in the two cases. Let . Then we get that is a snap-back repeller of for .
Now, it is clear that is continuously differentiable in , we shall show that for , where for . It is obvious that is continuously differentiable on and satisfies Hence, from the second conclusion of (4.14), it follows that there exists a sufficiently small neighborhood containing such that for all . We can take sufficiently large such that , obtained in the above, also lie in for and satisfy (4.10) or (4.12). Consequently, we have A direct calculation shows that for any ,
For , we get that , , . It follows from the third conclusion of (4.14), (4.15), and (4.16) that for ,
For , we get that , , and for . Hence, from the last two conclusions of (4.14), (4.15) and (4.16), we get that for ,
Therefore, all the assumptions in Lemma 2.7 are satisfied and is a regular and nondegenerate snap-back repeller of system (4.2). Consequently, is a regular and nondegenerate snap-back repeller of system (3.1). Hence, system (3.1), that is, system (1.1), is chaotic in the sense of both Devaney and Li-Yorke. The proof is complete.
Remark 4.2. From the proof of Theorem 4.1, we see there exists some positive constant such that for any , system (1.1) is chaotic in the sense of both Devaney and Li-Yorke. However, it is very difficult to determine the concrete value since the concrete expanding area of a fixed point is not easy to obtain. This will be left for our further research.
In order to help better visualize the theoretical result, six computer simulations are done, which exhibit complicated dynamical behaviors of the induced system (3.1), that is, system (1.1). We take , , and as a bifurcation parameter for computer simulations. It is clear that , and . Then, it follows from the proof of Theorem 4.1 that is an expanding fixed point of system (4.2) when . Furthermore, there exists some positive constant , such that for , is a regular and nondegenerate snap-back repeller of system (4.2). Consequently, is a regular and nondegenerate snap-back repeller of system (3.1). Some simulation results are shown in Figures 1, 2, 3, 4, 5, and 6 for , which show the complicated dynamical behaviors of system (3.1), that is, system (1.1).
In this paper, we rigorously prove the existence of chaos in a discrete delay population model. The map of the system is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Computer simulations confirm the theoretical analysis. The system (3.1) consists of a -dimensional linear subsystem and one-dimensional nonlinear subsystem. That is, the folding and stretching only occur in the variable , and all the other variables are taken placed by for . So system (3.1) can be viewed as one of the simplest discrete systems that can show higher-dimensional chaos. Consequently, system (1.1) can be viewed as one of the simplest delay difference systems that show chaos.
This work was supported by the National Natural Science Foundation of China (Grant nos. 11101246 and 11101247).
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