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Global dynamics of some systems of higher-order rational difference equations

Abdul Qadeer Khan, Muhammad Naeem Qureshi and Qamar Din*

Author Affiliations

Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan

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Advances in Difference Equations 2013, 2013:354  doi:10.1186/1687-1847-2013-354

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/354


Received:3 October 2013
Accepted:7 November 2013
Published:3 December 2013

© 2013 Khan et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the present work, we study the qualitative behavior of two systems of higher-order rational difference equations. More precisely, we study the local asymptotic stability, instability, global asymptotic stability of equilibrium points and rate of convergence of positive solutions of these systems. Our results considerably extend and improve some recent results in the literature. Some numerical examples are given to verify our theoretical results.

MSC: 39A10, 40A05.

Keywords:
system of difference equations; stability; global character; rate of convergence

1 Introduction

Recently, studying the qualitative behavior of difference equations and systems is a topic of great interest. Applications of discrete dynamical systems and difference equations have appeared recently in many areas such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, probability theory and resource management. Unfortunately, these are only considered as the discrete analogs of differential equations. It is a well-known fact that difference equations appeared much earlier than differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers where differential equations are solved by using their approximate difference equation formulations. The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. It is very interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the local asymptotic stability of their equilibrium points. Systems of rational difference equations have been studied by several authors. Especially there has been a great interest in the study of the attractivity of the solutions of such systems. For more results on the qualitative behavior of nonlinear difference equations, we refer the interested reader to [1-19].

Zhang et al.[20] studied the dynamics of a system of rational third-order difference equations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M1">View MathML</a>

Din et al.[11] investigated the dynamics of a system of fourth-order rational difference equations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M2">View MathML</a>

To be motivated by the above studies, our aim in this paper is to investigate the qualitative behavior of the following <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M3">View MathML</a>-order systems of rational difference equations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M4">View MathML</a>

(1)

where the parameters α, β, γ, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M7">View MathML</a> and initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M9">View MathML</a> are positive real numbers, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M10">View MathML</a>

(2)

where the parameters a, b, c, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M12">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M13">View MathML</a> and initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M9">View MathML</a> are positive real numbers. This paper is a natural extension of [11,20,21].

Let us consider <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M16">View MathML</a>-dimensional discrete dynamical system of the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M17">View MathML</a>

(3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M18">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M19">View MathML</a> are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M20">View MathML</a> of system (3) is uniquely determined by initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M21">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M22">View MathML</a>. Along with system (3), we consider the corresponding vector map <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M23">View MathML</a>. An equilibrium point of (3) is a point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> that satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M25">View MathML</a>

The point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is also called a fixed point of the vector map F.

Definition 1 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> be an equilibrium point of system (3).

(i) An equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is said to be stable if for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M29">View MathML</a> there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M30">View MathML</a> such that for every initial condition <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M31">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M22">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M33">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M34">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M36">View MathML</a> is the usual Euclidian norm in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M37">View MathML</a>.

(ii) An equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is said to be unstable if it is not stable.

(iii) An equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is said to be asymptotically stable if there exists <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M40">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M41">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M42">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43">View MathML</a>.

(iv) An equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is called a global attractor if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M42">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43">View MathML</a>.

(v) An equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> is called an asymptotic global attractor if it is a global attractor and stable.

Definition 2 Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> be an equilibrium point of the map

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M49">View MathML</a>

where f and g are continuously differentiable functions at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M50">View MathML</a>. The linearized system of (3) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M51">View MathML</a> is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M52">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M53">View MathML</a>

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M54">View MathML</a> is the Jacobian matrix of system (3) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M55">View MathML</a>.

Lemma 1[22]

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M57">View MathML</a> , is a system of difference equations and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>is the fixed point ofF. If all eigenvalues of the Jacobian matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M59">View MathML</a>about<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>lie inside an open unit disk<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M61">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>is locally asymptotically stable. If one of them has norm greater than one, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>is unstable.

Lemma 2[23]

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M56">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M65">View MathML</a> , is a system of difference equations and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>is the equilibrium point of this system. The characteristic polynomial of this system about the equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M58">View MathML</a>is<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M68">View MathML</a>, with real coefficients and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M69">View MathML</a>. Then all roots of the polynomial<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M70">View MathML</a>lie inside the open unit disk<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M71">View MathML</a>if and only if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M72">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M73">View MathML</a> , where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M74">View MathML</a>is the principal minor of orderkof the<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M75">View MathML</a>matrix

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M76">View MathML</a>

(4)

Let us consider a system of difference equations

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M77">View MathML</a>

(5)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78">View MathML</a> is an m-dimensional vector, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M79">View MathML</a> is a constant matrix, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M80">View MathML</a> is a matrix function satisfying

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M81">View MathML</a>

(6)

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M43">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M36">View MathML</a> denotes any matrix norm which is associated with the vector norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M84">View MathML</a>

Proposition 1 (Perron’s theorem)[24]

Suppose that condition (6) holds. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78">View MathML</a>is a solution of (5), then either<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M86">View MathML</a>for all largenor

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M87">View MathML</a>

(7)

exists and is equal to the modulus of one of the eigenvalues of matrixA.

Proposition 2[24]

Suppose that condition (6) holds. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M78">View MathML</a>is a solution of (5), then either<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M86">View MathML</a>for all largenor

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M90">View MathML</a>

(8)

exists and is equal to the modulus of one of the eigenvalues of matrixA.

2 On the system <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M91">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M92">View MathML</a>

In this section, we shall investigate the qualitative behavior of system (1). Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> be an equilibrium point of system (1), then for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95">View MathML</a>, system (1) has two positive equilibrium points <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M96">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M97">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M98">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M99">View MathML</a>.

To construct the corresponding linearized form of system (1), we consider the following transformation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M100">View MathML</a>

(9)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M101">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M103">View MathML</a>, … , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M104">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M105">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M106">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M107">View MathML</a>, … , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M108">View MathML</a>. The Jacobian matrix about the fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> under the transformation (9) is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M110">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M111">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M112">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M113">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M114">View MathML</a>.

Theorem 1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116">View MathML</a>, then every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M117">View MathML</a>of system (1) is bounded.

Proof It is easy to verify that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M118">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M119">View MathML</a>

Take <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M120">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M121">View MathML</a>. Then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M122">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M123">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124">View MathML</a> . □

Theorem 2The equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125">View MathML</a>of system (1) is locally asymptotically stable.

Proof The linearized system of (1) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M127">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M128">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M129">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M130">View MathML</a> denote the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M131">View MathML</a> eigenvalues of matrix E. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M132">View MathML</a> be a diagonal matrix, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M133">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M134">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M135">View MathML</a>, and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M136">View MathML</a>

Clearly, D is invertible. Computing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M137">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M138">View MathML</a>

We obtain the following two inequalities:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M139">View MathML</a>

which implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M140">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M141">View MathML</a>

Furthermore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M142">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M143">View MathML</a>

It is a well-known fact that E has the same eigenvalues as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M137">View MathML</a>. Hence, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M145">View MathML</a>

Hence, the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125">View MathML</a> of system (1) is locally asymptotically stable. □

Theorem 3The positive equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M147">View MathML</a>of system (1) is unstable.

Proof The linearized system of (1) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M147">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M149">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M150">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M151">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M152">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M153">View MathML</a>. The characteristic polynomial of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M154">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M155">View MathML</a>

(10)

From (10), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M156">View MathML</a>

(11)

It is clear that not all of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M157">View MathML</a>. Therefore, by Lemma 1, the unique positive equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M158">View MathML</a> is unstable. □

Theorem 4Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95">View MathML</a>, and let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161">View MathML</a>be a solution of system (1). Then, for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M162">View MathML</a>, the following statements are true:

(i) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M163">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M164">View MathML</a>

(ii) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M165">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M166">View MathML</a>

Proof It follows from induction. □

Theorem 5Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116">View MathML</a>, then the equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125">View MathML</a>of system (1) is globally asymptotically stable.

Proof For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M115">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M116">View MathML</a>, from Theorem 2, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125">View MathML</a> is locally asymptotically stable. From Theorem 1, every positive solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M173">View MathML</a> is bounded, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M174">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M175">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124">View MathML</a> , where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M177">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M178">View MathML</a>. So, it is sufficient to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161">View MathML</a> is decreasing. From system (1), one has

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M180">View MathML</a>

This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M181">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M182">View MathML</a>. Hence, the subsequences

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M183">View MathML</a>

are decreasing, i.e., the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M184">View MathML</a> is decreasing. Also,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M185">View MathML</a>

This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M186">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M187">View MathML</a>. Hence, the subsequences

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M188">View MathML</a>

are decreasing, i.e., the sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M189">View MathML</a> is decreasing. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M190">View MathML</a>. □

Theorem 6Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M94">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M95">View MathML</a>. Then, for a solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M117">View MathML</a>of system (1), the following statements are true:

(i) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M194">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M195">View MathML</a>.

(ii) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M196">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M197">View MathML</a>.

2.1 Rate of convergence

We investigate the rate of convergence of a solution that converges to the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M125">View MathML</a> of system (1).

Assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M200">View MathML</a>. First we will find a system of limiting equations for the map F. The error terms are given as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M201">View MathML</a>

Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M202">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M203">View MathML</a>, one has

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M204">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M205">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M207">View MathML</a>

Taking the limits, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M208">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M210">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M211">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M213">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M215">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M217">View MathML</a>. Hence, the limiting system of error terms at <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218">View MathML</a> can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M219">View MathML</a>

(12)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M220">View MathML</a>

which is similar to the linearized system of (1) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M221">View MathML</a>. Using proposition (1), one has the following result.

Theorem 7Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161">View MathML</a>is a positive solution of system (1) such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M224">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218">View MathML</a>. Then the error vector<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M226">View MathML</a>of every solution of (1) satisfies both of the following asymptotic relations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M227">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M228">View MathML</a>are the characteristic roots of the Jacobian matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M229">View MathML</a>about<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a>.

3 On the system <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M231">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M232">View MathML</a>

In this section, we shall investigate the qualitative behavior of system (2). Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> be an equilibrium point of system (2), then system (2) has a unique equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a>. To construct the corresponding linearized form of system (2), we consider the following transformation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M235">View MathML</a>

(13)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M236">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M102">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M103">View MathML</a>, …, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M104">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M240">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M106">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M107">View MathML</a>, … , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M108">View MathML</a>. The Jacobian matrix about the fixed point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M24">View MathML</a> under the transformation (13) is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M245">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M246">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M247">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M248">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M249">View MathML</a>.

Theorem 8Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161">View MathML</a>be a positive solution of system (2), then for every<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M251">View MathML</a>, the following results hold.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M252">View MathML</a>

Lemma 3Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M253">View MathML</a>, then every solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M254">View MathML</a>of system (2) is bounded.

Proof Assume that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M255">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M256">View MathML</a>

Then from Theorem 8 one can easily see that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M257">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M258">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M259">View MathML</a> . □

Theorem 9The equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a>of equation (2) is locally asymptotically stable.

Proof The linearized system of (2) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M262">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M263">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M264">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M265">View MathML</a> denote the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M131">View MathML</a> eigenvalues of matrix E. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M267">View MathML</a> be a diagonal matrix, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M133">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M134">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M135">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M271">View MathML</a>

Clearly, D is invertible. Computing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M272">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M273">View MathML</a>

Next, we have the following two inequalities:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M274">View MathML</a>

which implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M275">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M276">View MathML</a>

Furthermore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M277">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M278">View MathML</a>

Now H has the same eigenvalues as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M272">View MathML</a>, we obtain that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M280">View MathML</a>

Hence, the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> of system (2) is locally asymptotically stable. □

Theorem 10Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M282">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M283">View MathML</a>, then the equilibrium point<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a>of system (2) is globally asymptotically stable.

Proof Assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M282">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M283">View MathML</a>. Then from Theorem 9 the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> of system (2) is locally asymptotically stable. Moreover, from Lemma 3 every positive solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M173">View MathML</a> is bounded, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M174">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M175">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M124">View MathML</a> , where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M292">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M293">View MathML</a>. Now, it is sufficient to prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M294">View MathML</a> is decreasing. From system (2) one has

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M295">View MathML</a>

This implies that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M296">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M297">View MathML</a>.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M298">View MathML</a>

This implies that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M299">View MathML</a>

Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M300">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M301">View MathML</a>. Hence, the subsequences

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M302">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M303">View MathML</a>

are decreasing. Therefore the sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M184">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M189">View MathML</a> are decreasing. Hence, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M190">View MathML</a>. □

3.1 Rate of convergence

Assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M308">View MathML</a>. First we will find a system of limiting equations for system (2). The error terms are given as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M309">View MathML</a>

Set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M202">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M203">View MathML</a>, then one has

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M312">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M313">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M314">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M316">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M317">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M206">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M319">View MathML</a>

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M320">View MathML</a>. Taking the limits, we obtain <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M321">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M323">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M324">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M325">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M326">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M324">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M328">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M329">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M212">View MathML</a>, So, the limiting system of error terms can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M331">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M332">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M333">View MathML</a>

which is similar to the linearized system of (2) about the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M221">View MathML</a>. Using proposition (1), one has the following result.

Theorem 11Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M161">View MathML</a>is a positive solution of system (2) such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M199">View MathML</a>, and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M224">View MathML</a>, where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218">View MathML</a>. Then the error vector<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M226">View MathML</a>of every solution of (2) satisfies both of the following asymptotic relations:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M340">View MathML</a>

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M228">View MathML</a>are the characteristic roots of the Jacobian matrix<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M229">View MathML</a>about<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M218">View MathML</a>.

4 Examples

In order to verify our theoretical results, we consider some interesting numerical examples in this section. These examples show that the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> of both systems (1) and (2) is globally asymptotically stable.

Example 1 Consider system (1) with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M345">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M346">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M347">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M350">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M352">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M353">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M354">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M355">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M356">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M357">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M358">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M360">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M361">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M362">View MathML</a>. Moreover, choose the parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M363">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M364">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M365">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M366">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M367">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M368">View MathML</a>. Then system (1) can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M369">View MathML</a>

(14)

with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M345">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M346">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M347">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M348">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M350">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M352">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M353">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M354">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M355">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M356">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M357">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M358">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M385">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M361">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M362">View MathML</a>. Moreover, in Figure 1, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388">View MathML</a> is shown in Figure 1a, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389">View MathML</a> is shown in Figure 1b, and an attractor of system (14) is shown in Figure 1c.

thumbnailFigure 1. Plots for system (14).

Example 2 Consider system (1) with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M390">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M391">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M392">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M393">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M394">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M395">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M396">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M397">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M398">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M399">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M400">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M402">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M403">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M404">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M405">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M406">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M407">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M408">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M409">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M411">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M412">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M413">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M414">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M415">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M416">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M417">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M418">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M420">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M421">View MathML</a>. Moreover, choose the parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M422">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M423">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M424">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M425">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M426">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M427">View MathML</a>. Then system (1) can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M428">View MathML</a>

(15)

with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M390">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M391">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M392">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M393">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M394">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M395">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M396">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M397">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M398">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M399">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M400">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M349">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M402">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M403">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M404">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M405">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M406">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M407">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M408">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M409">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M411">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M412">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M413">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M414">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M415">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M416">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M417">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M418">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M420">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M421">View MathML</a>. Moreover, in Figure 2, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388">View MathML</a> is shown in Figure 2a, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389">View MathML</a> is shown in Figure 2b, and an attractor of system (15) is shown in Figure 2c.

thumbnailFigure 2. Plots for system (15).

Example 3 Consider system (2) with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M463">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M464">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M465">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M466">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M468">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M469">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M471">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M472">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M475">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M476">View MathML</a>. Moreover, choose the parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M477">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M478">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M479">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M480">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M481">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M482">View MathML</a>. Then system (2) can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M483">View MathML</a>

(16)

with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M463">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M464">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M465">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M466">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M351">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M468">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M469">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M471">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M472">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M419">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M475">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M476">View MathML</a>. Moreover, in Figure 3, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388">View MathML</a> is shown in Figure 3a, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389">View MathML</a> is shown in Figure 3b, and an attractor of system (16) is shown in Figure 3c.

thumbnailFigure 3. Plots for system (16).

Example 4 Consider system (2) with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M500">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M501">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M502">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M503">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M504">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M505">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M506">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M507">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M508">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M509">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M510">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M511">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M512">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M513">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M515">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M516">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M517">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M518">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M520">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M521">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M523">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M524">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M525">View MathML</a>. Moreover, choose the parameters <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M526">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M527">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M528">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M529">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M530">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M531">View MathML</a>. Then system (2) can be written as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M532">View MathML</a>

(17)

with initial conditions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M500">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M501">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M502">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M503">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M504">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M505">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M506">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M507">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M508">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M509">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M510">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M511">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M512">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M513">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M410">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M515">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M516">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M517">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M518">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M470">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M520">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M521">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M359">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M523">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M524">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M525">View MathML</a>. Moreover, in Figure 4, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M388">View MathML</a> is shown in Figure 4a, the plot of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M389">View MathML</a> is shown in Figure 4b, and an attractor of system (17) is shown in Figure 4c.

thumbnailFigure 4. Plots for system (17).

Conclusion

This work is a natural extension of [11,20,21]. In the paper, we have investigated the qualitative behavior of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M16">View MathML</a>-dimensional discrete dynamical systems. Each system has only one equilibrium point which is stable under some restriction to parameters. The linearization method is used to show that equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> is locally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. An approach to this problem consists of determining the possible global behaviors of the system and determining which initial conditions lead to these long-term behaviors. In case of higher-order dynamical systems, it is crucial to discuss global behavior of the system. Some powerful tools such as semiconjugacy and weak contraction cannot be used to analyze global behavior of systems (1) and (2). In the paper, we prove the global asymptotic stability of equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> by using simple techniques. We have carried out a systematical local and global stability analysis of both systems. The most important finding here is that the unique equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> can be a global asymptotic attractor for systems (1) and (2). Moreover, we have determined the rate of convergence of a solution that converges to the equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/354/mathml/M126">View MathML</a> of systems (1) and (2). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.

Competing interests

The authors have no competing interests.

Authors’ contributions

All authors contributed equally in drafting this manuscript and giving the main proofs.

Acknowledgements

This work was supported by the Higher Education Commission of Pakistan.

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