Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 295308, 35 pages
http://dx.doi.org/10.1155/2011/295308
Research Article

## Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

1Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina
2Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
3International University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina

Received 26 April 2011; Accepted 28 August 2011

Copyright Β© 2011 S. KalabuΕ‘iΔ 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 investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , are positive numbers, and initial conditions and are arbitrary nonnegative numbers such that . We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.

#### 1. Introduction

In this paper we consider the following rational system of difference equations where the parameters , , , , , are positive numbers, and initial conditions and are nonnegative numbers such that . System (1.1) was mentioned in [1] as one of three systems of open problem 3 which asked for the description of global dynamics of some rational systems of difference equations. In notation used to labels systems of linear fractional difference equations used in [1] system (1.1) is known as (3.19) and (4.1). In this paper, we provide the precise description of global dynamics of the system (1.1). We show that the system (1.1) may have between zero and three equilibrium points, which may have different local character. If the system (1.1) has one equilibrium point, then this point is either locally asymptotically stable or saddle point or nonhyperbolic equilibrium point. If the system (1.1) has two equilibrium points, then they are either locally asymptotically stable, and nonhyperbolic, or locally asymptotically stable and saddle point. If the system (1.1) has three equilibrium points then two of the equilibrium points are locally asymptotically stable and the third point, which is between these two points in South-East ordering defined below, is a saddle point. The major problem for global dynamics of the system (1.1) is determining the basins of attraction of different equilibrium points. The difficulty in analyzing the behavior of all solutions of the system (1.1) lies in the fact that there are many regions of parameters where this system possesses different equilibrium points with different local character and that in several cases the equilibrium point is nonhyperbolic. However, all these cases can be handled by using recent results in [2]. The dual of this system is the system where and replace their role, and it was labeled as system (4.1) and (3.19) in [1]. Dynamics of this system immediately follows from the results proven here, by simply replacing the roles of and .

System (1.1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [2, 3]. System (1.1) has a potential to be used as a mathematical model for competition. In fact, the first equation of (1.1) is of Leslie-Gower type, and the second equation can be considered to be of Leslie-Gower type with stocking (or immigration) represented with the term , see [4β7]. Here , are the inherent birth rates while and are related to the density-dependent effects on newborn recruitment. Finally, affects stocking for species with state variable .

In Section 2, we present some general results about competitive systems in the plane. In Section 3 contains some basic facts such as the nonexistence of period-two solution of system (1.1). In Section 4 analyzes local stability which is fairly complicated for this system. Finally, in Section 5 gives global dynamics for all values of parameters. This section finishes with an introduction of a new terminology for different type scenarios for competitive systems that can be used to give a simple classification of all possible global behavior for system (1.1). The interesting feature of this paper is that there are five regions of the parameters in which one of the equilibrium points is nonhyperbolic, and yet we are able to describe the global dynamics in all five cases. To achieve this goal, we use new method of proving stability of nonhyperbolic equilibrium points introduced in [2].

#### 2. Preliminaries

Consider a first-order system of difference equations of the form where are continuous functions on an interval , is nondecreasing in and non-increasing in , and is non-increasing in and nondecreasing in . Such system is called competitive. One may associate a competitive map to a competitive system (2.1) by setting and considering on .

We now present some basic notions about competitive maps in plane. Define a partial order βͺ― on so that the positive cone is the fourth quadrant, that is, if and only if and . For the order interval is the set of all such that . A set is said to be linearly ordered if βͺ― is a total order on . If a set is linearly ordered by βͺ―, then the infimum and supremum of exist in . If both and belong to , then the linearly ordered set is bounded, and conversely. We note that the ordering βͺ― may be extended to the extended plane in a natural way. For example, if or . If , we denote with , , the four quadrants in relative to , that is, , , and so on.

A map on a set is a continuous function . The map is smooth on if the interior of is nonempty and if is continuously differentiable on the interior of . A set is invariant for the map if . A point is a fixed point of if , and a minimal period-two point if and . A period-two point is either a fixed point or a minimal period-two point. The orbit of is the sequence . A minimal period two orbit is an orbit for which and . The basin of attraction of a fixed point is the set of all such that . A fixed point is a global attractor on a set if is a subset of the basin of attraction of . A fixed point is a saddle point if is differentiable at , and the eigenvalues of the Jacobian matrix of at are such that one of them lies in the interior of the unit circle in , while the other eigenvalue lies in the exterior of the unit circle. If is a map on , define the sets and . For and , define the distance from to as .

A map is competitive if whenever , and is strongly competitive if implies that . If is differentiable, a sufficient condition for to be strongly competitive is that the Jacobian matrix of at any has the sign configuration For additional definitions and results (e.g., repeller, hyperbolic fixed points, stability, asymptotic stability, stable and unstable manifolds) see [8, 9] for competitive maps, and [10, 11] for difference equations.

If is any subset of , we shall use the notation to denote the closure of in , and to denote the interior of .

The next results are stated for order-preserving maps on and are known but given here for completeness. See [12] for a more general version valid in ordered Banach spaces.

Theorem 2.1. For a nonempty set and βͺ― a partial order on , let be an order-preserving map, and let be such that and . If and , then is invariant and(i)there exists a fixed point of in , (ii)if is strongly order preserving, then there exists a fixed point in which is stable relative to , (iii)if there is only one fixed point in , then it is a global attractor in and therefore asymptotically stable relative to .

Corollary 2.2. If the nonnegative cone of is a generalized quadrant in , and if has no fixed points in other than and , then the interior of is either a subset of the basin of attraction of or a subset of the basin of attraction of .

Define a rectangular region in to be the cartesian product of two intervals in .

Remark 2.3. It follows from the Perron-Frobenius theorem and a change of variables [9] that, at each point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that if the map is strongly competitive then no eigenvector is aligned with a coordinate axis.

Theorem 2.4. Let be a competitive map on a rectangular region . Let be a fixed point of such that is nonempty (i.e., is not the NW or SE vertex of ), and is strongly competitive on . Suppose that the following statements are true. (a)The map has a extension to a neighborhood of . (b)The Jacobian matrix of at has real eigenvalues , such that , where , and the eigenspace associated with is not a coordinate axes. Then there exists a curve through that is invariant and a subset of the basin of attraction of , such that is tangential to the eigenspace at , and is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of in the interior of are either fixed points or minimal period-two points. In the latter case, the set of endpoints of is a minimal period-two orbit of .

We shall see in Theorem 2.7 and in the examples in [2] that the situation where the endpoints of are boundary points of is of interest. The following result gives a sufficient condition for this case.

Theorem 2.5. For the curve of Theorem 2.4 to have endpoints in , it is sufficient that at least one of the following conditions is satisfied. (i)The map has no fixed points nor periodic points of minimal period two in . (ii)The map has no fixed points in , , and has no solutions . (iii)The map has no points of minimal period two in , , and has no solutions .

In many cases, one can expect the curve to be smooth.

Theorem 2.6. Under the hypotheses of Theorem 2.4, suppose that there exists a neighborhood of in such that is of class on for some , and that the Jacobian matrix of at each is invertible. Then, the curve in the conclusion of Theorem 2.4 is of class .

In applications, it is common to have rectangular domains for competitive maps. If a competitive map has several fixed points, often the domain of the map may be split into rectangular invariant subsets such that Theorem 2.4 could be applied to the restriction of the map to one or more subsets. For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2.4 reduces just to . This follows from a change of variables [9] that allows the Perron-Frobenius theorem to be applied to give that at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axes.

Smith performed a systematic study of competitive and cooperative maps in [9, 13, 14] and in particular introduced invariant manifolds techniques in his analysis [13β15] with some results valid for maps on -dimensional space. Smith restricted attention mostly to competitive maps that satisfy additional constraints. In particular, is required to be a diffeomorphism of a neighborhood of that satisfies certain conditions (this is the case if is orientation preserving or orientation reversing), and that the coordinate semiaxes are invariant under . For such class of maps (as well as for cooperative maps satisfying similar hypotheses), Smith obtained results on invariant manifolds passing through hyperbolic fixed points and a fairly complete description of the phase-portrait when , especially for those cases having a unique fixed point on each of the open positive semiaxes. In our results, presented here, we removed all these constraints and added the precise analysis of invariant manifolds of nonhyperbolic equilibrium points. The invariance of coordinate semiaxes seems to be serious restriction in the case of competitive models with constant stocking or harvesting, see [16] for stocking.

The next result is useful for determining basins of attraction of fixed points of competitive maps. Compare to Theoremβ4.4 in [13], where hyperbolicity of the fixed point is assumed, in addition to other hypotheses.

Theorem 2.7. Assume the hypotheses of Theorem 2.4, and let be the curve whose existence is guaranteed by Theorem 2.4. If the endpoints of belong to , then separates into two connected components, namely, such that the following statements are true: (i) is invariant, and as for every . (ii) is invariant, and as for every . If, in addition to the hypotheses of part (A), is an interior point of , and is and strongly competitive in a neighborhood of , then has no periodic points in the boundary of except for , and the following statements are true. (iii) For every there exists such that for . (iv)For every there exists such that for .

Basins of attraction of period-two solutions or period-two orbits of certain systems or maps can be effectively treated with Theorems 2.4 and 2.7. See [2, 6, 11] for the hyperbolic case; for the nonhyperbolic case, see examples in [2, 17].

If is a map on a set and if is a fixed point of , the stable set of is the set , and unstable set of is the set When is noninvertible, the set may not be connected and made up of infinitely many curves, or may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on , the sets and are the stable and unstable manifolds of .

Theorem 2.8. In addition to the hypotheses of part (B) of Theorem 2.7, suppose that and that the eigenspace associated with is not a coordinate axes. If the curve of Theorem 2.4 has endpoints in , then is the stable set of , and the unstable set of is a curve in that is tangential to at and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of in are fixed points of .

The following result gives information on local dynamics near a fixed point of a map when there exists a characteristic vector whose coordinates have negative product and such that the associated eigenvalue is hyperbolic. This is a well-known result, valid in much more general setting which we include it here for completeness. A point is a subsolution if , and is a supersolution if . An order interval is the cartesian product of the two compact intervals and .

Theorem 2.9. Let be a competitive map on a rectangular set with an isolated fixed point such that . Suppose that has a extension to a neighborhood of . Let be an eigenvector of the Jacobian matrix of at , with associated eigenvalue . If , then there exists an order interval which is also a relative neighborhood of such that, for every relative neighborhood of , the following statements are true. (i)If , then contains a subsolution, and contains a supersolution. In this case, for every , there exists such that for . (ii)If , then contains a supersolution and contains a subsolution. In this case, for every .

In the nonhyperbolic case, we have the following result.

Theorem 2.10. Assume that the hypotheses of Theorem 2.9 hold, that is real analytic at , and that . Let , , be defined by the Taylor series Suppose that there exists an index such that and for . If either (a), or , is affine in t, or , is affine in , then there exists an order interval which is also a relative neighborhood of such that, for every relative neighborhood of , the following statements are true. (i)If is odd and , then contains a supersolution, and contains a subsolution. In this case, for every , there exists such that for . (ii)If is odd and , then contains a subsolution and contains a supersolution. In this case, for every . (iii)If is even and , then contains a subsolution and contains a subsolution. In this case, for every , and for every , there exists such that for .(iv)If is even and , then contains a supersolution and contains a supersolution. In this case, for every , and, for every there exists, such that for .

#### 3. Some Basic Facts

In this section, we give some basic facts about the nonexistence of period-two solutions, local injectivity of map at the equilibrium point.

##### 3.1. Equilibrium Points

The equilibrium points of the system (1.1) satisfy Solutions of System (3.1) are(i), when , that is, (ii) If , then using System (3.1), we obtain

Solutions of System (3.3) are where , which gives a pair of the equilibrium points and .

Geometrically, the equilibrium points are the intersections of two equilibrium curves: and . Depending on the values of parameters, may have between 0 and 3 intersection points with two lines which constitutes .

The algebraic criteria for the existence of the equilibrium points are summarized in Table 1.

Table 1

Where

Remark 3.1. Observe the following: If the system (1.1) has two or three equilibrium points , , and then, . Indeed, consider the critical curve . Observe that , , and . It is obvious that the following holds . Since, the critical curve decreases, we have , that is, .

Lemma 3.2. Assume that , . Then the following statements are true for solutions of the system (1.1). (i)If , then , for all , and , .(ii)If , then , for all , and , .(iii) If , then , and , for all , , .
Assume that and . Then, the following statements are true for all :(iv). (v) andβ(a), , where is arbitrarily small positive number.β(b), , , where is arbitrarily small positive number.

Proof. Since (i)β(iv) are immediate consequences of the system (1.1), we will prove only (v).
Take and . Then, we have for all , and Solution of (3.6), when is which immediately implies (i) and (ii). Statement (iii) follows from (3.6).
Equation implies that Using the last inequality, we have which by difference inequality theorem [18] implies the following Furthermore, second equation in (1.1) implies that which, by the difference inequalities argument, see [18], implies that , where satisfies (3.6). In view of (3.7) we obtain our conclusion.

##### 3.2. Period-Two Solution

In this section, we prove that System (1.1) has no minimal period-two solution which will be essential for application of Theorems 2.5β2.7. The map associated to System (1.1) is given by

Lemma 3.3. System (1.1) has no minimal period-two solution.

Proof. We have Period-two solution satisfies We show that this system has no other positive solutions except the equilibrium points.
Equation (3.15) is equivalent to
If then we obtain the fixed point . So assume that . Then, using (3.17), we have Equation (3.19) implies that Substituting (3.20) into (3.18), we have or Equation (3.21) implies that, and (3.23) implies that Replacing (3.23) into (3.19), we get
Solutions of (3.24) are the equilibrium points.
Consider (3.25). Discriminant of this equation is given by Now, implies that where Using (3.23), we obtain where We prove the following claims.
Claim 3.4. For all values of parameters.Proof. If , then . Now, we assume that . Then, Equation (3.32) implies that Since if and only if This implies that (3.33) is true. That is .Claim 3.5. Assume that . Then that .Proof. Assume that . This is equivalent to Now which is equivalent to or
Also,
Since,
This inequality and (3.36) imply (3.40).
Since
Last inequality, (3.36) and (3.39) imply that . So we prove, if , then .
Assume that .
We have
Since, we have that if and only if which is true, because
Replacing with in the formula for , we obtain that .
Hence, there does not exist period-two solution.

#### 4. Linearized Stability Analysis

The Jacobian matrix of the map , given by (3.13), has the form The determinant of (4.1) is given by The value of the Jacobian matrix of at the equilibrium point , is The determinant of (4.3) is given by and the trace of (4.3) is The characteristic equation has the form

Theorem 4.1. Assume that . Then there exists the equilibrium point and (i) is locally asymptotically stable if ,(ii) is a saddle point if . The corresponding eigenvalues are (iii) is nonhyperbolic if . The corresponding eigenvalues are and the corresponding eigenvectors are and , respectively.

Proof. The Jacobian matrix (4.1) at the equilibrium point ,
Note that the Jacobian matrix (4.9) implies that the map is not strongly competitive at the equilibrium point .
The determinant of (4.9) is given by Note that, under the hypothesis of Theorem, the determinant is greater than zero.
The trace of (4.9) is
An equilibrium point is locally asymptotically stable if the following conditions are satisfied
Now, these two conditions become
Condition implies that
If , then this condition is satisfied.
Condition is equivalent to It is easy to see that the condition is satisfied if .
Next, we prove (ii).
An equilibrium point is a saddle if and only if the following conditions are satisfied
The first condition is equivalent to which is satisfied if . The second condition is equivalent to
Finally, we prove (iii).
An equilibrium point is nonhyperbolic if the following conditions are satisfied
The first condition is equivalent to which is satisfied if .
The second condition becomes establishing part (iii).

We now perform a similar analysis for the other cases in Table 1.

Theorem 4.2. Assume that Then , , and exist and(i)the equilibrium point is locally asymptotically stable,(ii)the equilibrium point is a saddle point. Furthermore, if and , then the smaller eigenvalue belongs to the interval , and the larger eigenvalue belongs to . In all other cases, the smaller eigenvalues is in .That is is given by and the corresponding eigenvector is Eigenvalue , where , is given by (iii) The equilibrium point is locally asymptotically stable.

Proof. By Theorem 4.1, (i) holds.
Evaluating the Jacobian matrix (4.3) at the equilibrium point , we obtain Note that the Jacobian matrix (4.27) implies that the map is strongly competitive.
The determinant of (4.27) is given by and the trace of (4.27) is given by The equilibrium point is a saddle if and only if (4.17) is satisfied. The first condition is equivalent to which is equivalent to
In light of (3.3) , and by using (3.4), . Now, we have This implies that which is true.
Condition is equivalent to which is true.
To prove the second part of the statement (ii), we use the characteristic equation (4.6) of System (1.1) at the equilibrium point. Now, we have
Since the map is strongly competitive, the Jacobian matrix (4.27) has two real and distinct eigenvalues, the larger one in absolute value being positive.
The first equation implies that either both eigenvalues are positive, or the smaller one is negative. First, we show that, under hypothesis (ii) of theorem, the product of these two eigenvalues is less than zero. In order to prove that, it is enough to prove that .
We have Now if and only if which holds if
Also, we have
In all other cases .
This proves that the smaller eigenvalue is negative. Since the equilibrium point is a saddle point, it has to belong to . The larger one belongs to . The proof of second statement is similar.
Now, we prove that is locally asymptotically stable.
Notice that implies that which is true.

Theorem 4.3. Assume that Then exist and:(i)The equilibrium point is locally asymptotically stable. (ii)The equilibrium point is nonhyperbolic. The eigenvalues of the Jacobian matrix evaluated at are and the corresponding eigenvectors, respectively, are Furthermore, if , then . If , then .

Proof. By Theorem 4.1, is a locally asymptotically stable.
Now, we prove that is nonhyperbolic.
The Jacobian matrix (4.3) at the equilibrium point is
The eigenvalues of (4.44) satisfy