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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 463059, 7 pages
Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay
1Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China
2Department of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, Jiangsu 210096, China
3Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
4Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Received 18 August 2013; Accepted 30 October 2013
Academic Editor: Zhan Zhou
Copyright © 2013 Daiyong Wu 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.
We consider a nonlinear discrete logistic model with delay. The characteristic equation of the linearized system at the positive equilibrium is a polynomial equation involving high order terms. We obtain the conditions ensuring the asymptotic stability of the positive equilibrium and the existence of Neimark-Sacker bifurcation, with respect to the parameter of the model. Based on the bifurcation theory, we discuss Neimark-Sacker bifurcation direction and the stability of bifurcated solutions. Finally, some numerical simulations are performed to illustrate the theoretical results.
Logistic type models are often used to model a single species dynamics, for example, the underlying dynamics of tumour cells [1–4]. Moreover, time delay is sometimes necessary to better reflect the description of real processes [5, 6]. The continuous single population model described by the ordinary differential equations has been studied very deeply [7–9]. There are also very many results concerning the discrete logistic model (see [10–15] and references therein).
In , Zhou and Zou considered a discrete logistic equation: where and are positive -periodic sequences. Sufficient conditions were obtained for the existence of a positive and globally asymptotically stable -periodic solution.
In , Sun and Li considered the qualitative analysis of the following discrete logistic equation with several delays: where for , are positive integers, and . They obtained sufficient conditions for the global attractivity of all positive solutions about the positive equilibrium of model (2). Moreover, the oscillation about the positive equilibrium of model (2) was also discussed.
Chen and Zhou  discussed the following nonlinear periodic delay difference equation: where , , and are periodic sequences with a common period , , , , are positive integers, and , are positive constants with . Some sufficient conditions for the global attractivity and oscillation about the periodic solution of (3) were obtained.
Recently, Liu et al.  investigated the stability and bifurcation of a class of discrete-time Cohen-Grossberg neural networks with delays. Han and Liu  discussed the stability and bifurcation for a discrete-time model of Lotka-Volterra type with delay. He and Cao  studied the explicit formula for determining the direction of Neimark-Sacker bifurcation and the stability of periodic solution by using the normal form method and the center manifold theory.
To the best of our knowledge, no similar results have been reported regarding the nonlinear discrete logistic model with delay. In this paper, we are interested in the bifurcation analysis and the direction and the stability of the Neimark-Sacker bifurcations for the following nonlinear discrete logistic model: where , , and are positive constants, together with the initial condition
Our works focus on the stability and bifurcation analysis and the direction analysis of the Neimark-Sacker bifurcations by applying the center manifold theorem and the normal form theory. The method of the paper is similar to the work of Yuri .
The paper is organized as follows. In Section 2, we analyze the distribution of the characteristic equation associated with the model (4) and obtain the existence of the Neimark-Sacker bifurcation. In Section 3, the direction and stability of closed invariant curve from the Neimark-Sacker bifurcation of the model (4) are determined. In Section 4, some numerical simulations are performed to illustrate the theoretical results. A brief discussion is given in Section 5.
2. Stability Analysis
In this section, we will employ the techniques of Guo et al.  to study the distribution of the characteristic roots of model (4). Then we will obtain the stability of the positive equilibrium and the existence of local Neimark-Sacker bifurcations. Clearly, model (4) has a unique positive equilibrium .
The characteristic equation of is given by
Lemma 1. There exists such that for all roots of (11) have modulus less than one.
Proof. When , (11) becomes
The equation has, at , an -fold root and a simple root .
Consider the root such that . This root depends continuously on and is a differential function of . From (11), we have Consequently, for all sufficient small . Thus, all roots of (11) are inside the unit circle for sufficient small positive , and the existence of the maximal follows.
In the sequel, we define a parametric curve with Let . It is easy to see that for all such that . Therefore, as increases from 0 to , the corresponding point on the curve moves from origin and anticlockwise around origin. Moreover, it follows from that the part of the curve with parameter is simple; that is, it cannot intersect itself. Let be the zeros of in the interval and . Obviously, we have Moreover, from the anticlockwise property of the curve , we further have that From , it follows that . This means that the curve is contained in the region
Accordingly, we set where is the greatest integer function.
By Lemma 1 and the idea of Guo et al. (see Lemma 1 in ), we can obtain the distribution of the roots of the characteristic equation (11).
Lemma 2. Consider the following.(i) has zeros on the unit circle if and only if for some . Moreover, if for some , then all the zeros of of modulus 1 are , which are simple.(ii) has a simple zero if and only if is even and .(iii)For a fixed , there exist and a -mapping such that and is a zero of for all .(iv)All zeros of are inside the unit circle if and only if Now we verify that the transversality condition is satisfied.
Lemma 3. Along (11), one obtains where .
Remark 4. Since , for ; that is, the nondegeneracy condition is satisfied.
Theorem 5. Consider the following.(i)If then the fixed point of model (4) is asymptotically stable.(ii)If then the fixed point of model (4) is unstable.(iii)If then the model (4) undergoes a Neimark-Sacker bifurcation at the positive fixed point ; that is, a unique closed invariant curve bifurcates form . Moreover, if is even, the model (4) undergoes a flip bifurcation when
3. Direction and Stability of the Neimark-Sacker Bifurcation
Without loss of generality, denote by and the critical value by at which map (7) undergoes a Neimark-Sacker bifurcation at the origin.
For map (7), we obtain where Let be a complex eigenvector corresponding to , We also introduce an adjoint eigenvector having the properties and satisfying the normalization , where .
Lemma 6. Let be the eigenvector of corresponding to eigenvalue and let be the eigenvector of corresponding to eigenvalue . Then where
Proof. Let be an eigenvector of corresponding to eigenvalue . Then
For satisfying the normalization , we choose
Let denote a real eigenspace corresponding to , which is two-dimensional and is spanned by , and let be a real eigenspace corresponding to all eigenvalues of other than being dimensional.
For any , we have its decomposition where , , and . The complex variable can be viewed as a new coordinate on , and In this coordinate, the map at has the form
Using Taylor expansions, we obtain where , , , and
Now, we seek the center manifold which has the representation where . Substituting (40) into (38), we have When restricting (38) to the center manifold, up to cubic term, then we have
Define Substituting into (43), we can obtain .
From the above argument, we have the following result.
Theorem 7. The direction and stability of Neimark-Sacker bifurcation of model (4) can be determined by the sign of ; that is, the Neimark-Sacker bifurcation of model (4) at is supercritical (resp., subcritical) and unique closed invariant curve bifurcation from is asymptotically stable (resp., unstable) when (resp., .
4. Numerical Simulations
In this section we will give an example to illustrate the analytic results. Here we take model (4) with , , , and , respectively. From , , and , we have and . By the simple calculation, we obtain . Similarly, when , we have , , and = .
From Theorem 5, we conclude that, with , , and , the positive equilibrium of model (4) is asymptotically stable when , and model (4) undergoes the Neimark-Sacker bifurcation at when . Furthermore, from Theorem 7 and the Neimark-Sacker bifurcation of model (4) at is supercritical and unique closed invariant curve bifurcating from is asymptotically stable. These are shown by Figures 1 and 2.
Similarly, with , , and , the positive equilibrium of model (4) is asymptotically stable when , and, from Theorem 7 and , the Neimark-Sacker bifurcation of model (4) at is supercritical and unique closed invariant curve bifurcating from is asymptotically stable. These are shown by Figures 3 and 4.
5. Conclusion and Discussion
There has been a large body of work discussing the stability and bifurcation in logistic model, but most of them dealt with only the continuous logistic models, or only the discrete logistic model without delays. In this paper, we discuss the dynamical behaviors of nonlinear discrete logistic model with delay. The characteristic equation (11) is a polynomial equation involving high order terms, which make it difficult to find all parameters such that the characteristic roots have modulus 1 or less. By analyzing the characteristic equation, some sufficient and necessary conditions are derived to ensure that all the characteristic roots have modulus less than 1. Moreover, we obtain that when the parameter varies, the positive equilibrium of the model (4) exchanges its stability and Neimark-Sacker bifurcation occurs. Furthermore, the direction and stability of closed invariant curve from the Neimark-Sacker bifurcation of the model (4) are determined. Numerical simulations also show the occurrence of the stable bifurcate periodic solutions when passes the critical .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the Associate Editor, Professor Zhan Zhou, and the anonymous reviewers for their valuable time and professional comments. This paper is jointly supported by the National Natural Science Foundation of China under Grant nos. 11072059 and 61272530, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20110092110017, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186. This work is also funded by the Deanship of Scientific Research, King Abdulaziz University (KAU), under Grant no. 3-130/1434/HiCi. Therefore, the authors acknowledge technical and financial support of KAU.
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