- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 463059, 7 pages

http://dx.doi.org/10.1155/2013/463059

## Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay

^{1}Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China^{2}Department of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, Jiangsu 210096, China^{3}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia^{4}Department 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.

#### Abstract

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.

#### 1. Introduction

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 [13], 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 [14], 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 [15] 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. [16] investigated the stability and bifurcation of a class of discrete-time Cohen-Grossberg neural networks with delays. Han and Liu [17] discussed the stability and bifurcation for a discrete-time model of Lotka-Volterra type with delay. He and Cao [18] 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 [19].

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. [20] 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 .

Set ; then there follows that By introducing a new variable , we can rewrite (6) in the form where and Clearly, the origin is a fixed point of (7) and the linear part of (7) is where

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 [20]), 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 .*

*Proof. *By Lemma 2(i), from (14) and (16), it follows that

*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

In this section, we will study the direction and the stability of the Neimark-Sacker bifurcation in model (4). The method we used is based on the techniques developed by Yuri [19].

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
Similarly,
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.

#### Acknowledgments

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.

#### References

- A. K. Laird, “Dynamics of tumour growth,”
*British Journal of Cancer*, vol. 13, pp. 490–502, 1964. View at Scopus - A. K. Laird, “Dynamics of tumour growth: comparison of growth rates and extrapolation of growth curve to one cell,”
*British Journal of Cancer*, vol. 19, pp. 278–291, 1965. View at Publisher · View at Google Scholar · View at Scopus - J. C. Panetta, “A logistic model of periodic chemotherapy,”
*Applied Mathematics Letters*, vol. 8, no. 4, pp. 83–86, 1995. View at Zentralblatt MATH · View at Scopus - J. C. Panetta, “A logistic model of periodic chemotherapy with drug resistance,”
*Applied Mathematics Letters*, vol. 10, no. 1, pp. 123–127, 1997. View at Zentralblatt MATH · View at Scopus - M. Bodnar and U. Forys, “Three types of simple DDE’s describing tumour growth,”
*Journal of Biological Systems*, vol. 15, no. 4, pp. 1–19, 2007. View at Publisher · View at Google Scholar - H. V. Jaina and H. M. Byrne, “Qualitative analysis of an integro-differential equation model of periodic chemotherapy,”
*Applied Mathematics Letters*, vol. 25, no. 12, pp. 2132–2136, 2012. View at Publisher · View at Google Scholar - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equtions of Population Dynamics*, Kluwer Academic, Boston, Mass, USA, 1992. - F. D. Chen, X. D. Xie, and X. X. Chen, “Permanence and global attractivity of a delayed periodic logistic equation,”
*Applied Mathematics and Computation*, vol. 177, no. 1, pp. 118–127, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Bodnar, U. Forys, et al., “Logistic type equations with discrete delay and quasi-periodic suppression rate,”
*Applied Mathematics Letters*, vol. 26, no. 6, pp. 607–611, 2013. View at Publisher · View at Google Scholar - S. Mohamad and K. Gopalsamy, “Extreme stability and almost periodicity in a discrete logistic equation,”
*Tohoku Mathematical Journal*, vol. 52, no. 1, pp. 107–125, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. Muroya, “Global attractivity for discrete models of nonautonomous logistic equations,”
*Computers and Mathematics with Applications*, vol. 53, no. 7, pp. 1059–1073, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Z. Li and F. D. Chen, “Almost periodic solutions of a discrete almost periodic logistic equation,”
*Mathematical and Computer Modelling*, vol. 50, no. 1-2, pp. 254–259, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Z. Zhou and X. Zou, “Stable periodic solutions in a discrete periodic logistic equation,”
*Applied Mathematics Letters*, vol. 16, no. 2, pp. 165–171, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. R. Sun and W. T. Li, “Qualitative analysis of a discrete logistic equation with several delays,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 515–525, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. M. Chen and Z. Zhou, “Global attractivity and oscillation in a nonlinear periodic delay difference equation,”
*Computers and Mathematics with Applications*, vol. 45, no. 6–9, pp. 943–950, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Q. Liu, R. Xu, and Z. Wang, “Stability and bifurcation of a class of discrete-time Cohen-Grossberg neural networks with delays,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 403873, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - W. Han and M. Liu, “Stability and bifurcation analysis for a discrete-time model of Lotka-Volterra type with delay,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5449–5457, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - W. He and J. Cao, “Stability and bifurcation of a class of discrete-time neural networks,”
*Applied Mathematical Modelling*, vol. 31, no. 10, pp. 2111–2122, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. K. Yuri,
*Elements of Applied Bifurcation Theory*, Springer, New York, NY, USA, 1995. - S. J. Guo, X. H. Tang, and L. H. Huang, “Stability and bifurcation in a discrete system of two neurons with delays,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 4, pp. 1323–1335, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus