Discrete Dynamics in Nature and Society

Volume 2015, Article ID 160672, 11 pages

http://dx.doi.org/10.1155/2015/160672

## Basin of Attraction through Invariant Curves and Dominant Functions

^{1}Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Alkhoud, 123 Muscat, Oman^{2}Sciences and Engineering, Paris-Sorbonne University Abu Dhabi, P.O. Box 38044, Abu Dhabi, UAE

Received 9 February 2015; Accepted 6 May 2015

Academic Editor: Garyfalos Papashinopoulos

Copyright © 2015 Ziyad AlSharawi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a second-order difference equation of the form , where both and are decreasing. We consider a set of invariant curves at and use it to characterize the behaviour of solutions when and when . The case is related to the Y2K problem. For , we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.

#### 1. Introduction

Second-order difference equations of the formwhere is a continuous function, have been widely used in applications [1–4]. Several models in mathematical biology take the form of (1) under the assumptions that is decreasing and is bounded and increasing [4, 5]. A well-known example is Pielou’s difference equation which has been suggested to model the growth of a single species with delayed-density dependence [4, 6]. In Pielou’s equation, takes the form . Adding or subtracting a constant from (1) can be interpreted mathematically as a perturbation of the model, or biologically as constant stocking or constant yield harvesting [7–9]. These meaningful applications motivate investigating the dynamics of the difference equation However, when the functions and are both decreasing, the corresponding difference equation becomes more abstract and so far little work has been done to investigate its dynamics. Although our ultimate goal is to reach a general theory for this type of difference equation, we find it interesting to consider as a prototype, and so we focus this work on the dynamics of the equation

Among the important aspects of solutions of a difference equation are boundedness and global stability. On several occasions, the question of boundedness of all solutions of a particular difference equation was settled by finding invariant curves. Invariant curves of a second-order difference equation are plane curves on which forward orbits that start on a curve remain on the curve. Finding invariant curves, studying their properties and their relation with Liapunov functions is an active area of research [10–13].

In recent years, several results that give sufficient conditions for global stability or global attractivity of equilibrium solutions of difference equations have been established. Most of the results rely on the monotonicity of the function defined by the difference equation under investigation; see, for instance, [1, 11, 14, 15] and the references therein. Another approach of establishing global attractivity is to relate solutions of the difference equation to the solutions of a one-dimensional map where (under some conditions) the global attractivity of the equilibrium of the map implies the global attractivity of the equilibrium of the difference equation. This technique, which is sometimes called enveloping or dominance, is well-known in first-order difference equations [16], and it has been extended recently to tackle some higher order difference equations [17, 18].

In this paper, we focus on (3), where and the initial conditions are restricted to assure the existence of positive solutions (persistent solutions). Throughout our work, solutions are meant to be well-defined solutions. We consider a set of invariant curves and verify certain inequalities they satisfy. The inequalities describe the movement of solutions in the phase plane, which are ultimately used to prove that the larger positive equilibrium of (3) is a global attractor for a certain range of the parameter . When , (3) is related (via a change of variables) to the Y2K difference equation where the global attractivity of its positive equilibrium was an open problem for several years till it was settled recently by Merino [19]. Also, the case is related (via a change of variables) to Lyness equation in which solutions remain on invariant curves. The case of (3) exhibits very interesting dynamics and is discussed in Section 3.

#### 2. **The Case **

In this section, we consider (3) and assume . Thus, throughout this section, we refer to the equationThe substitutions transform (4) toEquation (6) is known in the literature as the Y2K problem [18]. Observe that, for and , we have if and only if . Also, the restriction , on the -parameters is equivalent to the restriction on the -parameters. In (6), it is obvious that positive initial conditions give rise to positive solutions, and there exists a unique positive equilibrium. Proving that the positive equilibrium is a global attractor with respect to the positive quadrant was an open question for over a decade till it was settled by Merino [19]. Merino transformed (6) intowhere . Then he used the functionto prove the next crucial lemma, where is the positive equilibrium of (7), which is given by

Lemma 1 (see [19]). *Consider , and then either or .*

Merino used this crucial lemma to show that the positive equilibrium of (7) is a global attractor. However, the given proof of Lemma 1 depends heavily on Mathematica code. So here, we present an alternative proof of Lemma 1, which is more trackable. But in order to keep things within the context of our work, we translate the curves of (7) to the curves of (4), which are given bywhere and is the large equilibrium of (4). Observe that at , = constant is in fact an invariant curve for Lyness equation after a transformation.

For typographical reasons, we define , , and in (4). Without further mention, we consider to be in the region , which is in fact the positive quadrant of (6). We find where and . Because and , we have whenever while otherwise. It is worth mentioning here that if is located in the region between and , then the point is guaranteed to be out of the region. Now, we are in a position to give our trackable proof of Lemma 1 translated in terms of .

*Proof. *We consider and show that . Because then we need to show that We writeOur aim here is to show that the R.H.S. is larger than or equal to 1. We proceed by taking two cases, namely, the following.*Case 1.* Consider*Case 2.* Consider*Case 1*. Observe that the conditions we have on , , and force all factors in the numerator and denominator to be positive. Furthermore, we have and the fact that gives . Next, write (15) asand observe that each one of the first three factors in the R.H.S. is larger than 1. Thus, we obtainUse Figure 1(a) to conclude thatSo, the R.H.S. of Inq. (19) is larger than one which completes the proof of Case 1.*Case 2*. In this case, we have , and since , we obtain . To make all factors in both the numerator and denominator of (15) positives, we rewrite the equation asAgain we show that the R.H.S. is larger than 1. To achieve this task, we rewrite (21) asObserve that the first and last factors are larger than 1, and therefore, we obtainSince and (see Figure 1(b)), we obtainBecause the function is increasing on the interval as long as , then implies . Therefore,as required, which completes the proof.