Discrete Dynamics in Nature and Society

Volume 2017, Article ID 1295089, 8 pages

https://doi.org/10.1155/2017/1295089

## Global Dynamics of Rational Difference Equations and

^{1}School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061, China^{2}School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450045, China^{3}College of Business Administration, Huaqiao University, Quanzhou 362021, China

Correspondence should be addressed to Weizhou Zhong; nc.ude.utjx.liam@uohziew

Received 24 December 2016; Revised 9 March 2017; Accepted 15 March 2017; Published 3 May 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Keying Liu 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

Global dynamics of a system of nonlinear difference equations was investigated, which had five kinds of equilibria including isolated points and a continuum of nonhyperbolic equilibria along the coordinate axes. The local stability of these equilibria was analyzed which led to nine regions in the parameters space. The solution of the system converged to the equilibria or the boundary point or in each region depending on nonnegative initial conditions. These results completely described the behavior of the system.

#### 1. Introduction

In this paper, we focus on the global dynamics of the following system:where the parameters and are positive and the initial conditions and are nonnegative.

In [1], the stability of (1) was investigated. If and , the equilibrium of (1) is globally asymptotically stable. If and , the equilibria and of (1) are locally unstable. The global dynamics of (1) was considered only for the case and .

System (1) can be regarded as a generalization of the equationwith the parameter being positive and initial conditions , being nonnegative, which was studied in [2] on the stability of the equilibria, nonexistence of prime period-two solutions, and global dynamics of the equation. More accurate results were obtained in our forthcoming work that every positive solution of (2) converged to its equilibria, for and for .

The above equations and systems are also the special cases of a general equationwith nonnegative parameters and proper initial conditions. Several global asymptotic results for some special cases of (3) were obtained in [3–13].

As for the definition of stability and the method of linearized stability, see [1–21]. For other types of equations and systems, see [14–19, 22–39]. As for the definition of basin of attraction and the stable manifold and so on, see [35–39].

In this article, we try to determine a complete picture of the behavior of (1). First, we part completely the regions of parameters by equilibria. Second, by the theory of linearized stability, we describe the local stability of these equilibria for five cases and derive nine regions in plane. At last, we present the main results on global dynamics of (1) in these regions.

It is the first time that the parameters spaces are divided into nine regions and complex dynamics of (1) are derived according to the initial conditions for each region. It is also the first time that we determine the details that the equilibrium is nonhyperbolic or a saddle point if it is unstable. It is worth pointing out that the system has a continuum of nonhyperbolic equilibria along a vertical line or/and a horizontal line, which lead to interesting phenomena on the global dynamics.

#### 2. Existence of Equilibria

The study of dynamics of difference equations often requires that equilibria be calculated first, followed by a local stability analysis of the equilibria. This is then complemented by other considerations (existence of periodic points, etc.). If the analysis is applied to a class of equations dependent on one or more parameters, the task is complicated by the fact that a formula is not always available for equilibria, and even if it is, determination of stability of parameter-dependent equilibria may be a daunting task.

First of all, we obtain the existence of equilibria of (1).

As is known, the equilibrium of (1) satisfies from which it follows that one of five cases holds for the equilibrium points of (1):(1) if one of the following conditions holds:(i) and .(ii) and .(iii) and .(2) and if and .(3) and with arbitrary if and .(4) and with arbitrary if and .(5) and and with arbitrary and if and .

#### 3. Local Stability of Equilibria

Now, we consider the local stability of these equilibria of (1).

##### 3.1. Local Stability of

Theorem 1. *Suppose that is the equilibrium of (1). Then one of the following holds: *(1)* is locally asymptotically stable for and .*(2)* is nonhyperbolic of the stable type for one of the following three cases: Case 2.1: , . Case 2.2: , . Case 2.3: , .*(3)

*is nonhyperbolic of the unstable type for one of the following two cases:*(4)

*Case 3.1: , .**Case 3.2: , .**is a saddle point for one of the following three cases:*

*Case 4.1: and .**Case 4.2: and .**Case 4.3: and .**Proof. *The linearized system of (1) about isAs is shown in [1], we have its characteristic polynomialHere, we only focus on one of these two factors. Obviously, we have Thus, the distribution of solutions of is one of the following: (i)Two real roots in for (ii)One root being and the other being for (iii)One root in and the other in for By Theorem 1.2.1 in [21], we obtain the conclusions and complete the proof.

*3.2. Local Stability of *

*Now, we consider the local stability of the positive equilibrium of (1), which exists only for and .*

*Theorem 2. Assume that and and is the positive equilibrium of (1); then is a saddle point.*

*Proof. *The linearized equation of (1) about iswhere . It is obvious that for and .

As is shown in [1], its characteristic polynomial isWe have , , , and for . Thus, has one solution in and one in .

Now, we divide it into two cases to show the distribution of the other two solutions of .*Case 1* (). In view of , we conclude that has three solutions in and one in and thus is a saddle point by Theorem 1.2.1 in [21].*Case 2* (). In this case, we rewrite (9) as follows:where , , and satisfyingNext, we try to prove all roots of the equation to be inside the unit disk for , which is necessary and sufficient to prove by Theorem 1.2.2 in [21].

First, we try to show for .

From (14), it is obvious and thus is equivalent to . In view of for and , we try to prove . To this end, we try to determine the exact range of .

In view of , from and , we could obtain . Thus, we have and is proved.

Second, we show for . To this end, we only need to show for .

From (11), we obtain and thus as desired.

In fact, more precisely, in view of , we have that for and for .

Therefore, for , we have ; for and for . Hence, we obtain for and thus all roots of the equation lie inside the unit disk. By Theorem 1.2.1 in [21], of (1) is a saddle point for .

Thus, we conclude that of (1) is a saddle point if it exists and we complete the proof.

*3.3. Local Stability of *

*Now, we consider the local stability of the equilibria of (1), which exists only for and .*

*Theorem 3. Assume that and and are the equilibria of (1).If or and , then is nonhyperbolic of the stable type.If and , then is nonhyperbolic of the unstable type.*

*Proof. *The linearized equation of (1) about every isfrom which we have its characteristic polynomialIt is obvious that has two solutions and .

Similarly, if and , then has two solutions and .

If or and , then has two solutions in .

If and , then has one solution in and the other in .

By Theorem 1.2.1 in [21], we derive the conclusions and complete the proof.

*3.4. Local Stability of *

*Similar to the proof of Theorem 3, we have the following theorem.*

*Theorem 4. Assume that and and are the equilibria of (1).If or and , then is nonhyperbolic of the stable type.If and , then is nonhyperbolic of the unstable type.*

*3.5. Local Stability of and *

*In case of , the equilibria of (1) include , , and .*

*By Theorem 1, for , of (1) is nonhyperbolic of the stable type.*

*Similar to the proof of Theorem 3, the linearized equation of (1) about is (15) with and its characteristic polynomial is (16) with which has four roots in . It implies that every is nonhyperbolic of the stable type.*

*Similarly, every is nonhyperbolic of the stable type.*

*Theorem 5. Assume that , , , and ( and ) are the equilibria of (1); then they are nonhyperbolic of the stable type.*

*There are 9 cases in parametric space with distinct local stability of distinct equilibria. We list the above results in Table 1. For simplicity, if of (1) is locally asymptotically stable, we denote If is nonhyperbolic, we denote If is nonhyperbolic of the stable type or the unstable type, we denote or N.H.. If is a saddle point, we denote .*