Bifurcations of Some Planar Discrete Dynamical Systems with Applications

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane. The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species. The first Manuscript was on the study of the following equation: xn+1 = p+ xn xn−1 (1) Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation: xn+1 = p+ xn xn−1 The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the difference equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium. In my second Manuscript, I considered the difference equation: xn+1 = xn Cxn−1 +Dxn + F (2) where C,D and F are positive numbers and the initial conditions x−1 and x0 are non-negative numbers. Equation (2) which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium. The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts: Theorem 1 Let I be a compact interval of the real numbers and assume that f : I → I is a continuous function satisfying the following properties: 1. f(x, y, z) is non-decreasing in x and non-increasing in y and z 2. The system { f(M,m,m) = M f(m,M,M) = m has a unique solution M = m in I. Then the equation xn+1 = f(xn, xn−1, xn−2) has a unique equilibrium x̄ in I and every solution of it that enters I must converge to x̄. In addition, x̄ is globally asymptotically stable. As of my third manuscript, I focused on providing some possible scenarios for general discrete competitive dynamical systems in the plane. I applied the results achieved to a class of second order difference equations of the form: xn+1 = f(xn, xn−1), n = 0, 1, . . . where the function f(x, y) is decreasing in the variable x and increasing in the variable y. In my proofs I relied on a collection of well established theorems and results. Furthermore I illustrate my results with an application to equation: xn+1 = xn−1 cxn−1 + dxn + f , n = 0, 1, . . . (3) With initial conditions x−1 and x0 arbitrary nonnegative numbers and parameters c, d, f >0. Equation. (3) is a special case of: xn+1 = Cxn−1 +Dxn + F cxn−1 + dxn + f , n = 0, 1, . . . which of great interest to the field of difference equation and special cases of it were considered by different scholars. It also turns out to be a non-linear perturbation of the Sigmoid Beverton-Holt model. I characterize completely the global bifurcations and dynamics of equation.(3) with the basins of attraction of all its equilibria and periodic solutions. Moreover I provide techniques to investigates cases that are not covered by the established theorems in the theory of competitive maps. Finally in my fourth manuscript I considered extending some existing theorems and proving some new global stability results, namely for difference equations that are of the form xn+1 = f(xn, xn−1) where f(x, y) is either increasing in the first and decreasing in the second variable, or decreasing in both variables. In addition I illustrate my results with examples and applications. I also provide a new proof for Pielou’s equation (a mathematical model in population dynamics).

describes the growth or decay of single species.
The first Manuscript was on the study of the following equation: Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation: The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the difference equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium.
In my second Manuscript, I considered the difference equation: x n+1 = x n Cx 2 n−1 + Dx n + F (2) where C, D and F are positive numbers and the initial conditions x −1 and x 0 are non-negative numbers. Equation (2)  Then the equation x n+1 = f (x n , x n−1 , x n−2 ) has a unique equilibriumx in I and every solution of it that enters I must converge tox. In addition,x is globally asymptotically stable.
As of my third manuscript, I focused on providing some possible scenarios for general discrete competitive dynamical systems in the plane. I applied the results achieved to a class of second order difference equations of the form: x n+1 = f (x n , x n−1 ), n = 0, 1, . . .
where the function f (x, y) is decreasing in the variable x and increasing in the variable y. In my proofs I relied on a collection of well established theorems and results. Furthermore I illustrate my results with an application to equation: With initial conditions x −1 and x 0 arbitrary nonnegative numbers and parameters c, d, f >0. Equation.

Introduction and Preliminaries
In this paper I consider the difference equation where the parameter p is positive number and the initial conditions x −1 and x 0 are positive numbers. Notice that if x −1 , x 0 = 0 in equation (5) then x n > 0, n ≥ 1, and so without loss of generality we can assume that x −1 > 0, x 0 > 0. This implies that our results are global. Clearly equation (27) has the unique equilibrium point x = p + 1. Linear fractional version of equation (27) x n+1 = p + x n x n−1 , n = 0, 1, . . . , was considered in [3], where it was proved that the unique equilibriumx = p + 1 of equation (5) is globally asymptotically stable. Introduction of quadratic terms into equation (5) changes the local stability analysis and consequently the global dynamics as well. In particular, quadratic terms introduces the possibility of Naimark-Sacker bifurcation and the existence of locally stable periodic solution, see [6] for several similar examples.
The linearized equation of equation (5) at the equilibrium pointx = p + 1 is z n+1 = 2 p + 1 z n − 2 p + 1 z n−1 , n = 0, 1, . . . , with the characteristic equation and the characteristic roots Since |λ ± | = 2 p + 1 it is clear that that the equilibrium pointx = p + 1 is asymptotically stable if p > 1, non-hyperbolic if p = 1 and unstable if p < 1. In all cases the eigenvalues are complex conjugate numbers which indicates the presence of the Naimark-Sacker bifurcation at p = 1. We will prove that indeed the equilibrium pointx = p + 1 is globally asymptotically stable if p ≥ √ 2 and that the Naimark-Sacker bifurcation takes the place at p = 1. Our tool in proving global asymptotic stability of equation (5) is the result in [3,5]. We conjecture that the equilibrium pointx = p + 1 is globally asymptotically stable if a > 1. Furthermore, we give some numeric values of parameter a with corresponding periodic solutions. Our bifurcation diagram indicates a complicated behavior and possible chaos for the values p < 1.
Now, for the sake of completeness I give the basic facts about the Naimark-Sacker bifurcation.
The Hopf bifurcation is well known phenomenon for a system of ordinary differential equations in two or more dimension, whereby, when some parameter is varied, a pair of complex conjugate eigenvalues of the Jacobian matrix at a fixed point crosses the imaginary axis, so that the fixed point changes its behavior from stable to unstable and a limit cycle appears. In the discrete setting, the Naimark-Sacker bifurcation is the discrete analogue of the Hopf bifurcation.
The Naimark-Sacker bifurcation occurs for a discrete system depending on a parameter, λ say, with a fixed point whose Jacobian has a pair of complex conjugate µ(λ),μ(λ) which cross the unit circle transversally at λ = λ 0 .
Theorem 2 (Naimark-Sacker bifurcation) Let be a C 4 map depending on real parameter λ satisfying the following conditions: (i) F (λ, 0) = 0 for λ near some fixed λ 0 ; (ii) DF (λ, 0) has two non-real eigenvalues µ(λ) andμ(λ) for λ near λ 0 with |µ(λ 0 )| = 1; Then there is a smooth λ-dependent change of coordinate bringing F into the form and there are smooth function a(λ), b(λ), and ω(λ) so that in polar coordinates the function F(λ, x) is given by If a(λ 0 ) < 0, then there is a neighborhood U of the origin and a δ > 0 such that for |λ − λ 0 | < δ and x 0 ∈ U , then ω-limit set of x 0 is the origin if λ > λ 0 and belongs to a closed invariant C 1 curve Γ(λ) encircling the origin if λ < λ 0 . Furthermore, If a(λ 0 ) > 0, then there is a neighborhood U of the origin and a δ > 0 such that for |λ − λ 0 | < δ and x 0 ∈ U , then α-limit set of x 0 is the origin if λ < λ 0 and belongs to a closed invariant C 1 curve Γ(λ) encircling the origin if λ > λ 0 . Furthermore, Consider a general map F(λ 0 , x) that has a fixed point at the origin with where A is the Jacobian matrix of F evaluated at the fixed point (0, 0), and Here I denote µ(λ 0 ) = µ, A(λ 0 ) = A and G(λ 0 , x) = G(x). We let p and q be the eigenvectors of A associated with µ satisfying Aq = µq, pA = µp, pq = 1 and Φ = (q,q). Assume that and K 20 = (µ 2 I − A) −1 g 20 , Let The following result from [9] gives an approximate expression for the invariant curve from Theorem 5.
Corollary 1 Assume a(λ 0 ) = 0 and λ = λ 0 + η where η is a sufficiently small parameter. Ifx is a fixed point of F then the invariant curve Γ(λ) from Theorem 5 can be approximated by Here "Re" represents the real parts of the complex numbers.
The second section of the paper gives global asymptotic stability result for the values of parameter p ≥ √ 2 and the third section gives the reduction to the normal form and computation of the coefficients of the Naimark-Sacker bifurcation and the asymptotic approximation of the invariant curve. My computational method is based on the computational algorithm developed in [9] rather than more often used computational algorithm in [10]. The advantage of the computational algorithm of [9] lies in the fact that this algorithm computes also the approximate equation of the invariant curve in the Naimark-Sacker theorem, which is not provided by Wan's algorithm. Here I give numeric and visual evidence that the approximate equation of the invariant curve is accurate. See Figure 4.

Global Asymptotic Stability
I use the method of embedding of equation (27) into higher order difference equation to which we apply one of global attractivity results [2] . By substituting in equation (27) we get: . Now by substituting for x n−1 in the term x n−1 of the last equation we we obtain From equation (10) we observe that p<x n <p + (1 + 1 p + 1 p 2 ) 2 for n ≥ 4. Also from (27) and (10) we have:

Consequently
x n x n−1 which implies: Replacing x n in (11) by p + x n−1 we obtain the equation Observe now that every solution of equation (27) is also a solution of equation (12), with initial values x −2 , x −1 and We also know that if U satisfies: It follows that given p > 1 such U exists and therefore I is invariant for f where . In the following we may assume p > 1 and U = p(p 2 +p+1) , so I is invariant for f .
Next, I prove that I is an attracting interval, that is every solution of equation (11) must enter the interval I. Observe that given the initial values x −2 , x −1 and x 0 for equation (11), we have x n > p for n ≥ 1.
Now if x 3 ≤ U then x n ∈ [p, U] for all n ≥ 3. Otherwise, from equation (12) given that x n−2 , x n−3 > p we have Thus by induction we can conclude that It is straightforward to check that when x 3 > U the right hand side of (13) is a decreasing sequence that converges to A . This limit is in fact U = p(p 2 +p+1) (p 2 −1) . It follows that there must exist k > 3 such that: p < x k < U. Otherwise x n must converge to U which is impossible.
Thus we have x k−1 , x k−2 > p and x k ≤ U, hence x k+1 ∈ [a, U]. Now it follows by induction that x n ∈ [p, U] for n ≥ k. Consequently every solution of equation (11) must enter the interval [p, U]. Now we check the conditions of Theorem A.0.5 [3], see also [5]: From the second equation we get On the other hand the system is equivalent to: Equations (14) and (15) yield which implies: This leads to the following quadratic equation: It is clear that when p > √ 2 there is no real solutions and when p = √ 2 there is one unique solution m = p + 1 = M . Consequently if p ≥ √ 2 the conditions of Theorem A.0.5 [3] or Theorem 1 [5] are fully satisfied and therefore every solution must converge to the unique equilibrium (p + 1). 2    Table 1) for numeric values.
Set u n = y n−1 and v n = y n for n = 0, 1, . . . and write equation (27) in the equivalent form: Let F be the corresponding map defined by: Then F has the unique fixed point (0, 0) and the Jacobian matrix of F at (0, 0) is given by It is easy to see that where The eigenvalues of Jac F (0, 0) are µ(p) and µ(p) where One can prove that for p = p 0 = 1 we obtain µ(p 0 ) = 1 and which implies that µ k (p 0 ) = 1 for k = 1, 2, 3, 4. Furthermore, we obtain d dp |µ(p)| = − 1 √ 2 The eigenvectors of Jac F (0, 0) corresponding to µ(p) and µ(p) are q(p) and where Hence, for p = p 0 system (37) is equivalent to Define the basis of R 2 by Φ = (q,q), where q = q(p 0 ), then we can represent (u, v) as By using K 20 , K 11 and K 02 we have that It is easy to see that pA = µp and pq = 1 where Thus I prove the following result: Theorem 4 There is a neighborhood U of the equilibrium pointx = p + 1 and a ρ > 0 such that for |p − 1| < ρ and x 0 , x −1 ∈ U , the ω-limit set of solution of equation (27), with initial condition x 0 , x −1 is the equilibrium pointx if p > 1 and belongs to a closed invariant C 1 curve Γ(p) encircling the equilibrium pointx if p < 1. Furthermore, Γ(1) = 0 and the invariant curve Γ(p) can be approximated by the parametric equation Proof. The proof follows from above discussion and Theorem 5 and Corollary 2.
2 Figure 1 shows convergence to the equilibrium for the values of p slightly larger than 1 which visually confirms Conjecture. Figure

Introduction and Preliminaries
In this paper I consider the difference equation where the parameters C, D and F are positive numbers and the initial conditions x −1 and x 0 are positive numbers.
Equation (27) can be considered as a nonlinear perturbation of the Beverton-Holt difference equation which is a major mathematical model in population dynamics see [1,13]. Furthermore, it is similar in appearance to the linear fractional equation of the form which was considered in [5]. Both equations (28) and (29) exhibit a global asymptotic stability of either zero or positive equilibrium solutions and exchange of stability bifurcation. As we will see in this paper the introduction of quadratic term will substantially change dynamics and will introduce the existence of a locally stable periodic solution and possibly chaos. I will show that local asymptotic stability of the zero equilibrium will also implies its global asymptotic stability. In the case of the positive equilibrium solution I will show that such statement is true in some subspace of the parametric region of local asymptotic stability and I pose the conjecture that the same property holds in the complete region of local asymptotic stability. Our tool in proving global asymptotic stability of the positive equilibrium solution consists of embedding considered equation into higher order equation and using global attractivity results for maps with invariant boxes, see [3,5,7]. Related rational difference equations which exhibit similar behavior were considered in [4,8].
Now, for the sake of completeness I give the basic facts about the Neimark-Sacker bifurcation.
The Hopf bifurcation is well known phenomenon for a system of ordinary differential equations in two or more dimension, whereby, when some parameter is varied, a pair of complex conjugate eigenvalues of the Jacobian matrix at a fixed point crosses the imaginary axis, so that the fixed point changes its behavior from stable to unstable and a limit cycle is generated.
In the discrete setting, the Neimark-Sacker bifurcation is the discrete analogue of the Hopf bifurcation. The Neimark-Sacker bifurcation occurs for a discrete system in the plane depending on a parameter, λ say, with a fixed point whose Jacobian matrix has a pair of complex conjugate eigenvalues µ(λ),μ(λ) which crosses the unit circle transversally at λ = λ 0 . In this case the periodic solution, which is in general, of unknown period appears and is locally stable. In this paper we use Murakami computational approach, see [12] to find an asymptotic formula for an invariant locally attracting curve in the phase plane, which represents a periodic solution.
Then there is a smooth λ-dependent change of coordinate bringing F into the form and there are smooth functions a(λ), b(λ), and ω(λ) so that in polar coordinates the function F(λ, x) is given by If a(λ 0 ) < 0, then there is a neighborhood U of the origin and a δ > 0 such that for |λ − λ 0 | < δ and x 0 ∈ U , then ω-limit set of x 0 is the origin if λ < λ 0 and belongs to a closed invariant C 1 curve Γ(λ) encircling the origin if λ > λ 0 . Furthermore, If a(λ 0 ) > 0, then there is a neighborhood U of the origin and a δ > 0 such that for |λ − λ 0 | < δ and x 0 ∈ U , then α-limit set of x 0 is the origin if λ > λ 0 and belongs to a closed invariant C 1 curve Γ(λ) encircling the origin if λ < λ 0 . Furthermore, Consider a general map F(λ 0 , x) that has a fixed point at the origin with where A is the Jacobian matrix of F evaluated at the fixed point (0, 0), and Here we denote µ(λ 0 ) = µ, A(λ 0 ) = A and G(λ 0 , x) = G(x). We let p and q be the eigenvectors of A associated with µ satisfying Aq = µq, pA = µp, pq = 1 and Φ = (q,q). Assume that and The next result of Murakami [12] gives an approximate formula for the periodic solution.
Corollary 2 Assume a(λ 0 ) = 0 and λ = λ 0 + η where η is a sufficient small parameter. Ifx is a fixed point of F then the invariant curve Γ(λ) from Theorem 5 can be approximated by Here "Re" represents the real parts of those complex numbers.

Local and global stability
The equilibrium solutions of Equation (27) is the positive solution of the

Now the following results hold:
Lemma 1 For the equilibrium pointx 0 the following holds: (i) If F > 1 the equilibrium pointx 0 is locally asymptotically stable.
(ii) If F < 1 the equilibrium pointx 0 is a saddle point.
The proof of part (iv) follows from the fact that every solution {x n } of Equation which shows that {x n } is non-increasing sequence and so convergent. Consequently lim n→∞ x n = 0. The proofs of parts (i) − (iii) are immediate.

Lemma 2
The positive equilibriumx satisfies the following: Proof. One can see that The rest of the proof follows from Theorem 2.13 [6]. 2 Now I give a global asymptotic stability result for the positive equilibrium solution. I will show that local asymptotic stability of the positive equilibrium will also imply its global asymptotic stability in substantial subregion of the parametric space.
Theorem 6 Assume that F < 1 and Then the positive equilibrium of Equation (27) is globally asymptotically stable.
Proof. Clearly we can consider solutions of Equation (27) which are positive, that is for which x 0 > 0. The substitution y n = D xn transforms Equation (27) into the equation One can easily show that Equation (35) has a unique equilibriumȳ = D x . I will show thatȳ is globally asymptotically stable when F <1 and C ≤ . Our major tool is global asymptotic stability result in [5], more precisely Theorem 1.4.5 [5]. Now I will check the assumptions of this theorem.
1. Clearly f (x, y) is non-decreasing in x and non-increasing in y.
2. There exists an interval I such that f : On the other hand for any U ≥ Therefore f (x, y) ∈ I, which shows that I is an invariant interval for f .
Next, consider the system of equations which is equivalent to : and show that M = m.
Subtracting the second equation from the first we get: Thus m satisfies the following quadratic equation: To show thatȳ is globally asymptotically stable, it is sufficient to show that every solution of Equation (35) must enter I. Observe that by Equation (35) and so by the result on difference inequalities, see [10] Since U can be chosen to be large there exists N ≥ 0 such that y n ≥ 1 1 − F for all n ≥ N . Furthermore as we can choose U ≥ Based on my simulation I state the following

Conjecture 2
The equilibrium pointx of equation (27) is globally asymptotically stable if it is locally asymptotically stable.

Reduction to the normal form
In this section I bring the system that corresponds to Equation (27)  Assume that 0 < F < 1 2 . If we make a change of variable y n = x n −x, then the transformed equation is given by Set u n = y n−1 and v n = y n for n = 0, 1, . . . and write Equation (27) in the equivalent form: Let F be the corresponding map defined by: Then F has the unique fixed point (0, 0) and the Jacobian matrix of F at (0, 0) is given by A straightforward calculation shows that where The eigenvalues of Jac F (0, 0) are µ(C) and µ(C) where One can prove that for One can see that µ k (C 0 ) = 1 for k = 1, 2, 3, 4 and Furthermore, we get and The eigenvectors corresponding to µ(C) and µ(C) are q(C) and q(C), where where Hence, for C = C 0 system (37) is equivalent to Define the basis of R 2 by Φ = (q,q). Let By using this, one can see that and By using K 20 , K 11 and K 02 we have that . (44) Next we have that pA = µp and pq = 1 where One can see that Theorem 7 Let 0 < F < 1 2 and Then there is a neighborhood U of the equilibrium pointx and a ρ > 0 such that Furthermore, Γ(C 0 ) = 0 and invariant curve Γ(C) can be approximated by Proof. The proof follows from above discussion and Theorem 5 and Corollary 2.
See Figure 6 for a graphical illustration.

Introduction
Consider the second-order quadratic fractional difference equation where the initial conditions x −1 and x 0 are arbitrary nonnegative numbers and the parameters satisfy that c, d, f >0. Notice that Equation (45) is a special case of the equation where the initial conditions x −1 and x 0 are arbitrary nonnegative numbers and the Both Equations (45) and (46) are special cases of the general second-order quadratic fractional difference equation where all parameters are nonnegative numbers and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers such that the solution is well-defined. Many special cases of Equation (47) have been studied in [1,2,11,13] etc.
The first systematic study of global dynamics of a special quadratic fractional case of Equation (47) where Another special case of Equation (47) was given in [9] and uses the theory of monotone maps in the plane. Indeed, in [9] the unique coexistence of a unique locally asymptotically stable equilibrium point and a locally asymptotically stable minimal period-two solution was obtained.
Equation (45), on the other hand, can have as many as three fixed points and up to three period-two solutions and its dynamics is similar to the dynamics of investigated in [18]. The possible dynamic scenarios for Equation (45) will be our motivation for getting the corresponding results for the general second order difference equation in section 3.3.
Many other interesting special cases of Equation (47) are studied in [11,13,14,19,20], which reveal the potential for rich dynamical behaviors that include the Allee effect, period-doubling bifurcation, Neimark-Sacker bifurcation, and chaos.
Equation (45) has an interesting special case when d = 0; the well-known sigmoid Beverton-Holt equation whose interesting dynmaics is given in [18]. Thus Equation (45) can be considered as a perturbation of Equation (48).

Preliminaries
In this section I provide some basic facts about competitive maps and systems of difference equations in the plane.
Definition 1 Let R be a subset of R 2 with nonempty interior, and let T : R → R be a map (i.e., a continuous function). Set T (x, y) = (f (x, y), g(x, y)). The map T is competitive if f (x, y) is non-decreasing in x and non-increasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. If both f and g are nondecreasing in x and y, we say that T is cooperative. If T is competitive (cooperative), the associated system of difference equations is said to be competitive (cooperative). The map T and associated difference equations system are said to be strongly competitive (strongly cooperative) if the adjectives non-decreasing and non-increasing are replaced by increasing and decreasing.
First I provide some theorems from [16,17] used in [9] that will be of particular importance in my investigation of the global dynamics of Equation (45).
x is not the NW or SE vertex of R), and T is strongly competitive on ∆.
Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian J T (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T .
We shall see in Theorem 10 that the situation where the endpoints of C are boundary points of R is of interest. The following result gives a sufficient condition for this case.

Theorem 9
For the curve C of Theorem 8 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied.
i. The map T has no fixed points nor periodic points of minimal period two in ∆.
ii. The map T has no fixed points in ∆, det J T (x) > 0, and T (x) = x has no solutions x ∈ ∆.
iii. The map T has no points of minimal period-two in ∆, det J T (x) < 0, and For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 8 reduces just to |λ| < 1. This follows from a change of variables that allows the Perron-Frobenius Theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis. The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 10 (A) Assume the hypotheses of Theorem 8, and let C be the curve whose existence is guaranteed by Theorem 8. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W − := {x ∈ R\C : ∃y ∈ C with x se y} and W + := {x ∈ R\C : ∃y ∈ C with y se x} , such that the following statements are true.
(B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q 1 (x) ∪ Q 3 (x) except for x, and the following statements are true.
(iii) For every x ∈ W − there exists n 0 ∈ N such that T n (x) ∈ int Q 2 (x) for n ≥ n 0 .
(iv) For every x ∈ W + there exists n 0 ∈ N such that T n (x) ∈ int Q 4 (x) for n ≥ n 0 .
If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set When T is non-invertible, the set W s (x) may not be connected, can consist of infinitely many curves, or W u (x) 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 R, the sets W s (x) and W u (x) are the stable and unstable manifolds of x.
Theorem 11 In addition to the hypotheses of part (B) of Theorem 10, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis.
If the curve C of Theorem 8 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T .
The following result is for strictly order preserving maps [23]. The result is stated for a partial order in R n , but it also holds in Banach spaces. where A ⊂ R n , and let I = u 1 , u 2 ⊂ A. Then at least one of the following holds.
(a) T has a fixed point in I distinct from u 1 and u 2 .
(b) There exists an entire orbit {x n } n∈Z of T in I joining u 1 to u 2 and satisfying x n x n+1 .
(c) There exists an entire orbit {x n } n∈Z of T in I joining u 2 to u 1 and satisfying x n+1 x n . The following result is a direct consequence of Theorem 29. The following theorem from [5] applies to Equation (45): Theorem 13 Let I be a set of real numbers and f : I × I → I be a function which is non-increasing in the first variable and non-decreasing in the second variable.
Then, for every solution {x n } ∞ n=−1 of the equation the subsequences {x 2n } ∞ n=0 and {x 2n−1 } ∞ n=0 of even and odd terms of the solution do exactly one of the following: (i) Eventually they are both monotonically increasing.
(ii) Eventually they are both monotonically decreasing.
(iii) One of them is monotonically increasing and the other is monotonically decreasing.
The consequence of Theorem 28 is that every bounded solution of Equation (50) converges to either an equilibrium, a period-two solution, or to the point on the boundary, so we try to determine the basins of attraction of these solutions.

Remark 1
We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D 1 f negative and first partial derivative D 2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Equation (50) follows from the fact that if f is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to Equation (50) is a strictly competitive map on I × I (see [17]). Set x n−1 = u n and x n = v n in Equation(50) to obtain the equivalent system f (v, u)). The second iterate T 2 is given by and it is strictly competitive on I × I, see [17].
has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decreasing in first and increasing in second variable. Thus the applicability of Theorems 8-11 depends on the nonexistence of a minimal period-two solution.

Main Results
In this section I present some global dynamics scenarios for competitive system (49) which will be applied to Equation (45). If the initial point (x 0 , y 0 ) is above W s (E 2 ) ∪ W s (E 7 ) one can find the point (x l , y l ) on the y-axis and a point ( that (x l , y l ) se (x 0 , y 0 ) se (x u , y u ). This will imply that T n ((x l , y l )) se T n ((x 0 , y 0 )) se T n ((x u , y u )), and so T n ((x 0 , y 0 )) ∈ int E 1 , E 7 eventually. Now, in view of Corollary 7 T n ((x 0 , y 0 )) → E 1 as n → ∞.
In a similar way the case when the initial point (x 0 , y 0 ) is below W s (E 4 ) ∪ W s (E 7 ) can be handled.
(b) The existence of the global stable and unstable manifolds of the saddle fixed points is guaranteed by Theorems 8 -11. Both global stable manifolds W s (E 2 ) and W s (E 4 ) have an end point at E 6 . The existence of curves C 1 and C 2 follows from Theorem 8. The proof that the region between the stable manifolds W s (E 2 ) and W s (E 4 ) eventually enters int E 2 , E 4 and so it converges to E 3 is the same as in part (a).
In a similar way as in the proof of part (a) we can show that if the initial point (x 0 , y 0 ) is above W s (E 2 ) ∪ C 2 it will eventually enter int E 1 , E 6 and so it will converge to E 1 . In a similar way we can show that if the initial point (x 0 , y 0 ) is below W s (E 4 ) ∪ C 1 it will eventually enter int E 6 , E 5 and so it will converge to E 5 .
Finally, if the initial point (x 0 , y 0 ) is between C 1 and C 2 then one can find the point (x l , y l ) ∈ C 2 and a point (x u , y u ) ∈ C 1 , such that (x l , y l ) se (x 0 , y 0 ) se (x u , y u ). This will imply that T n ((x l , y l )) se T n ((x 0 , y 0 )) se T n ((x u , y u )), and so T n ((x 0 , y 0 )) → E 6 as T n ((x u , y u )) → E 6 , T n ((x l , y l )) → E 6 .
(c) The proof that the region between the stable manifolds W s (E 2 ) and W s (E 4 ) is the basin of attraction of E 3 is same as in part (a) and will be omitted.
The proof that all solutions which start above W s (E 2 ) ∪ W s (E 8 ) converges to E 1 and all solutions which start below W s (E 4 ) ∪ W s (E 9 ) converge to E 5 is same as in part (a) and so will be ommitted.
If the initial point (x 0 , y 0 ) is between W s (E 8 ) and W s (E 9 ) then one can find the point (x l , y l ) ∈ W s (E 8 ) and a point (x u , y u ) ∈ W s (E 9 ), such that (x l , y l ) se (x 0 , y 0 ) se (x u , y u ). This will imply that T n ((x l , y l )) se T n ((x 0 , y 0 )) se T n ((x u , y u )), and so T n ((x 0 , y 0 )) ∈ int E 8 , E 9 as T n ((x u , y u )) → E 9 , T n ((x l , y l )) → E 8 as n → ∞. Now in view of Corollary 8 T n ((x 0 , y 0 )) → E 7 as n → ∞.

2
In the case of Equation (50)  and C 2 converges to E.
(c) Assume that Equation (50) has three equilibrium points where E 0 and E + are locally asymptotically stable and E − is repeller. Furthermore assume that Equation (50) has three minimal period-two solutions If (1) has the following linearized equation: Proof.
1. Since p = q = 0 forx = 0 then E 0 corresponds to the unique eigenvalue λ = 0, thus E 0 is locally asymptotically stable for all values of c ,d and f . clearly that E * is non-hyperbolic of unstable type.
3. The roots of the characteristic equation are: On the other hand one can use the fact thatx − < 1+d 2c to show that λ − < − 1.
As of E + : Sincex + > 1−d 2c one can similarly show that λ + <1. Moreover a simple algebraic verification shows the following: Consequently we conclude that: E − is a repeller whenever it exists while: i E + is locally asymptotically stable: ii E + is non-hyperbolic of stable type: iii E + is a saddle point:

Local stability analysis of minimal period two solutions
Here I present the results about the existence and the stability of minimal period two solutions of Eq.(1)
2. If 4f c = 1 then Eq.(1) has a minimal period two solution: (1) has two minimal period two solutions: and P 2 x (1) has three minimal period two solutions: x , P 1 y , P 2 x , P 2 y and P i ∓, P i ± where: ) and: Proof. For the sake of obtaining minimal period two solutions we must seek the ordered pairs (φ, ψ) that satisfy the following system of equations: If φ = 0 and ψ = 0 then the system is equivalent to: Now given that φ = ψ we get φ = 1+d c − ψ which implies: while equation (**) has one unique solution which is an equilibria φ = ψ =x + . Therefore Eq.(1) has two minimal period two solutions: and P 2 x , 0 , P 2 y 0, 1+ while equation(**) has the solutions: Therefore Eq.(1) has three minimal period two solutions: together with: ) .

2
Now consider the following substitution: u n = x n−1 and v n = x n then the behavior of the solutions of Eq.(1) can be investigated by the following two dimensional system: which corresponds to the following map: (53) The second iteration of the map T is given by: where: clearly the map T 2 is competitive and its Jacobian matrix is given by: Where: The following theorem describes the local stability of minimal period two solutions of Eq.(1) whenever they exist. Proof.
1. The minimal period solutions {P x , P y } exist when 4f c = 1, thus the Jacobian matrix of the second iterate of the map T at P x and P y is the following: both correspond to the eigenvalues λ 1 = 0 and λ 2 = 1 therefore {P x , P y } are non-hyperbolic of stable type.

(a) For {P 1
x , P 1 y }; the Jacobian matrix is of the form: Both with eigenvalues λ 1 = 0 and are saddle points.

Now at the interior period two solutions
Observe that: Consequently; the interior period two solutions {P i ∓ , P i ± } are saddle points whenever they exist.
2 Remark 3 Observe that the interior period two solutions {P i ∓ , P i ± } exist if and only if:   If 4f c>1, then the equilibrium E 0 is globally asymptotically stable. see Figure 10, Proof. First observe that every solution of Eq.(1) is bounded; as for x n−1 = 0 2. If 4f c = 1 then: 3. If 4f c<1 then: Proof.

First recall that every solution of Eq.(1) is bounded and by theorem
6 every solution must either converge to an equilibrium or a minimal period two solution. It follows that every solution generated by T 2 must converge to an equilibrium. Now consider {s n } ∞ n=1 the solution with initial n +f , 0) and observe that: ii. If x n < 1 2c then s n = (x n , 0) → (0, 0) , +∞) and monotone increasing in ) thus: , +∞) then s n = (x n , 0) → ( 1+ ) then s n = (x n , 0) → (0, 0).
The remaining part of the proof follows similarly by considering solutions of the form: 2
There exist two invariant curves C 1 and C 2 which are graphs of strictly increasing continuous functions of the first coordinate on an interval with endpoints in P x and P y respectively. Basins of attraction of the minimal period-two solutions are while the basin of attraction of the equilibrium point E 0 is the region between curves See Figure 11.  Proof. The Jacobian matrix of the second iterate of the map at P x is given by: with two eigenvalues λ 1 = 0 associated with the eigenvector d 1 and λ 2 = 1 which corresponds the the eigenvector 1 0 .
Observe that the eigenvector associated with λ 1 is not parallel to the x-axis and the map T 2 is strongly competitive. It follows by theorems 8-11 and 15; that there exists an invariant curve C 1 through the point P x which a subset of W s (P x ).
Moreover C 1 is the graph of a strictly increasing continuous that separates the first quadrant into two connected subregions: an upper one W − (C 1 ) and a lower one . As of the Jacobian matrix of P y : J T 2 (P y ) = 0 0 0 1 , it has two eigenvectors that are parallel to the coordinate axis; 1 0 , 0 1 corresponding to λ 1 = 0 and λ 2 = 1 respectively. By Hartman Grobman theorem [22], we know that there exist a C 1 curve C through P y that is tangential at P y to the eigenspace associated with λ = 0 such that T 2 (C) ⊂ C.

Claim 1
The stable manifold at P y is a linearly strongly ordered curve in the northeast ordering, where it is given for δ positive and small enough as : W s loc (P y ) = {(t, φ(t)) : Since T 2 is strongly competitive we have: T 2 (P y ) se T 2 u 0 , 1 2c and that implies: T 2 u 0 , 1 2c ∈ int (Q 4 (P y )) . So, there exists a ball B ε T 2 u 0 , 1 2c such that: ⊂ int (Q 4 (P y )) . Since the map T 2 is continuous on R 2 + , there exists a ball B δ 1 u 0 , 1 2c such that: It follows that: W s loc (P y ) ∩ int (Q 4 (P y )) = ∅ Now observe that φ (0) = 0 as its curve must be tangential to the horizontal eigenspace. Moreover φ ≥ 0 in a small neighborhood of t = 0 otherwise: Therefore for sufficiently small δ 1 : W s loc (P y ) = {(t, φ(t)) : 0 ≤ t ≤ δ 1 } is linearly ordered in the northeast ordering and as T 2 is competitive: W s loc (P y ) ∩ R 2 + can be extended to an unbounded curve (global stable manifold) C 2 , see [16,17].

2
Hence the curve C 2 separates the region into two connected components an upper subregion W − (C 2 ) and a lower subregion W + (C 2 ).
Clearly the basin of attraction of P y is B (P y ) = C 2 ∪ W − (C 2 ), and finally the basin of attraction of the zero equilibrium E 0 is:
• Two minimal period two solutions P 1 x , P 1 y which are saddle points and P 2 x , P 2 y which are Locally asymptotically stable.
There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are basins of attractions of P 1 x , P 1 y and the unstable manifolds have the following form: The basin of attraction of the equilibrium point E 0 = (0, 0) is the region between the global stable sets Basin of attraction of the minimal period-two solutions P 2 x , P 2 y is given with the following B P 2 x = W + P 1 x , B P 2 y = W − P 1 y .
Proof. Recall that: associated with the eigenvector 1 0 . Thus there exists a local stable manifold at P 1 x that is linearly strongly ordered in the north east ordering with P 1 x as an endpoint. As T 2 is competitive the local stable manifold can be extended to a curve W s (P 1 x ) which separates the region into two connected components W + (P 1 x ) and . on the other hand with eigenvalues λ 1 = 0 with eigenvector 1 0 and λ 2 = √ 1 − 4fc + 1 associated with the eigenvector 0 1 .
By Theorem 19 we know that (0, 1− ) ⊂ B(E 0 ), thus we know that the local stable manifold at P 1 y is tangential the horizontal Eigenspace but cannot enter the box (0, 1− ). I conclude that the local stable manifold at P 1 y is a linearly strongly ordered curve (in the northeast ordering) with P 1 y as an endpoint. Similarly we conclude its extension to a global stable manifold W s P 1 y which separates the region into two connected components W + P 1 y and Finally by the uniqueness of the stable manifold of the saddle point P 1 x we know that no solution in W + (P 1 x ) will converge to P 1 x , on the other hand all solutions are bounded and we know that by monotonicity of the map T every solution must converge to an equilibrium. It follows that B(P 2 x ) = W + (P 1 x ) and analogously B(P 2 y ) = W + (P 1 y ) 2 Figure 12. One equilibrium, two P-2 Figure 13. Two equilibriums, two P-2

Theorem 22
If d<1 and (d − 1) 2 = 4f c<1, the Eq.(1) has: • E 0 is locally asymptotically stable, is a non-hyperbolic point of unstable type, and two minimal period-two solutions: are saddle points, are locally asymptotically stable.
• There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are the basins of attraction of the periodic solutions {P 1 x , P 1 y } and which are tangential at the equilibrium point E * .
• There exists a global stable manifold W s (E * ) contained in Q 1 (E * ) which is the basin of attraction of the equilibrium E * .
The Basin of attraction of equilibrium point E 0 is the region between those stable manifolds i.e.
The basins of attraction of P 2 x and P 2 y are given by: See Figure 13.

Proof.
The existence and orientation of the global stable manifold at P 1 x can be determined as in theorem 21. However in general the orientation can be determined by studying the curvature of the local curves given by: and δ 2 small enough, where if f (x, y) and g(x, y) are the coordinate functions of This is useful when the local curve is tangent parallel to the axis at the fixed point which is the case here for P 1 y .
By differentiating both sides of the equation above we get: which confirms the argument used in theorem 21. I conclude the existence of curves C 1 and C 2 (global stable manifolds) which are linearly ordered in the northeast ordering. Furthermore the curves cannot intersect the interior of the sets Q 2 (E * ) ∩ R 2 and Q 4 (E * )∩R 2 , as the monotonicity of T 2 forces the latter sets to be invariant.
Thus T −2n (P ) → E * for all P ∈ C l , l = 1, 2, therefore C 1 and C 2 are also center manifolds of E * .
On the other hand by letting T 2 (f (x, y), g(x, y) the center manifold φ(x) must satisfy: By using a Taylor expansion substitution we can approximate the center manifold by: The dynamics on the center manifold are given by the reduced difference equation u n+1 = f (u n , φ(u n )) which has the following asymptotic representation: Clearlyū = 1−d 2c is a semi-stable fixed point for the latter scalar difference equation, it follows that E * is a semi=stable fixed point for T 2 , furthermore the coefficient of the lowest nonlinear term in the reduced map is negative thus by [21] the local basin of attraction of the equilibrium E * is a one dimensional curve. I conclude that there is a unique center manifold curve U which satisfies T 2 (U) ⊂ U. Moreover U is tangential to the eigenspace associated with λ = 1 namely Span 1 1 .
It follows that U is contained in Q 1 (E * ) and is linearly ordered in the northeast ordering and therefore can be extended to an unbounded curve C (The global stable manifold).
Now for all point q ∈ W − (P 1 x ) ∩ W + P 1 y there exist q x ∈ W s (P 1 x ) and q y ∈ W s P 1 y such that: q y se q se q x which implies that: T 2n (q y ) se T 2n (q) se T 2n (q x ), but we know that: T 2n (q y ) → P 1 y and T 2n (q x ) → P 1 x Consequently there exist N such that: It follows by theorem 19 that q ∈ B(E 0 ). As of the basins of attractions of P 2 x and P 2 y The proof is analogous to the one given in theorem 21. 2
• There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are the basins of attraction of the periodic solutions {P 1 x , P 1 y } and which are tangential at the equilibrium point E − .
• There exists a global stable manifold W s (E + ) which is an unbounded curve of an increasing function contained in Q 1 (E + ) ∪ Q 3 (E + ) with an endpoint at E − . W s (E + ) is the basin of attraction of the equilibrium E + .
• There exist a global unstable manifold W u (E + ) which is a curve of a decreasing function contained in Q 2 (E + ) ∪ Q 4 (E + ) with endpoints P 2 x and P 2 y .
• The Basin of attraction of equilibrium point E 0 is the region between those stable manifolds i.e.
• The basins of attraction of P 2 x and P 2 y are given by: See Figure 14. Proof.
The existence of W s (P 1 x ) and W s (P 1 y ) as well as the basin of attraction of E 0 were discussed in theorem 22.
The existence of the stable W s (E + ) and the unstable manifold W u (E + ) follows from theorem8-11 and 15. The basins of attraction of P 2 x and P 2 y were discussed in theorem 21. 2 Figure 14. Three equilibriums, two P-2 Figure 15. Three equilibriums, two P-2
• There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are the basins of attraction of the periodic solutions {P 1 x , P 1 y } and which are tangential at the equilibrium point E − .
• There exists a global stable manifold W s (E + ) which is an unbounded curve of an increasing function contained in Q 1 (E + ) ∪ Q 3 (E + ) with an endpoint at is the basin of attraction of the equilibrium E + .
• The Basin of attraction of equilibrium point E 0 is the region between those stable manifolds i.e.
• The basins of attraction of P 2 x and P 2 y are given by: See Figure 15. Proof.
The existence and orientation of the global stable manifold W s (E + ) follows from Three equilibrium points: • E 0 is locally asymptotically stable, • E − is repeller, • E + is locally asymptotically stable, and three minimal period-two solutions: • P 1 x , P 1 y are saddle points, • P 2 x , P 2 y are locally asymptotically stable, • P i ∓ , P i ± are saddle points.
• There exist global stable manifolds W s (P 1 x ) and W s P 1 y which are the basins of attraction of the periodic solutions {P 1 x , P 1 y } and which are tangential at the equilibrium point E − .
• The Basin of attraction of equilibrium point E 0 is the region between those stable manifolds i.e.
• There exists a global stable manifold W s P i ∓ which is an unbounded curve of an increasing function contained in Q 1 ((P i ∓ ) ∪ Q 3 ((P i ∓ ) with an endpoint at E − . W s P i ∓ is the basin of attraction of the equilibrium P i ∓ .
• There exists a global stable manifold W s P i ± which is an unbounded curve of an increasing function contained in Q 1 ((P i ± ) ∪ Q 3 ((P i ± ) with an endpoint at E − . W s P i ± is the basin of attraction of the equilibrium P i ± .
• The Basin of attraction of equilibrium point E + is the region between those stable manifolds i.e.
• There exist a global unstable manifold W u P i ∓ which is a curve of a decreasing function contained in Q 2 (P i ∓ ) ∪ Q 4 (P i ∓ ) with endpoints P 2 y and E + .
• There exist a global unstable manifold W u P i ± which is a curve of a decreasing function contained in Q 2 (P i ± ) ∪ Q 4 (P i ± ) with endpoints P 2 x and E + .
• The basins of attraction of P 2 x and P 2 y are given by: See Figure 16. Proof.
The existence and orientation of the stable manifold W s P i ∓ follows from theo- On the other hand W u P i ∓ ∩ (P i ∓ , E + = ∅ and (P i ∓ , E + is invariant. Thus W u P i ∓ cannot leave the latter set and must end at E + . Analogous arguments and conclusions also hold for W s P i ± and W u P i ± . In addition we know that for p ∓ ∈ W s P i ∓ and p ± ∈ W s P i ± : Furthermore for all p ∈ W − P i ± ∩ W + P i ∓ . there exist p ∓ ∈ W s P i ∓ and p ± ∈ W s P i ± such that: . It follows that there exists N >0 such that T 2N (p) = q ∈ P i ∓ , P i ± . Thus there exist q ∓ ∈ W u P i ∓ and q ± ∈ W u P i ± such that: where T 2n (q ∓ ) → E + and T 2n (q ± ) → E + which implies that T 2n (q) → E + ⇒ T 2n (p) → E + I conclude that: As of W s (P 1 x ), W s (P 1 y ), B(E 0 ) , B(P 2 x ) and B(P 2 y ) the proof is analogous to the discussion in theorems 21 and 22.

Introduction and Preliminaries
The following results were obtained first in [13,14] and were extended to the case of higher order difference equations and systems in [14,17,21,23,24].
Theorem 26 Let [a, b] be a compact interval of real numbers and assume that is a continuous function satisfying the following properties: Similar results has been proved for other two cases of coordinate-wise monotone function f , see [14]. These results have been very useful in proving attractivity results for equilibrium or periodic solutions of Eq.(55) as well as for higher order difference equations and systems of difference equations, see [6,9,14,15]. Theorems 26 and 27 have attracted considerable attention of the leading specialists in difference equations and discrete dynamical systems and have been generalized and extended to the case of maps in R n , see [17], and maps in Banach space with the cone, see [21] and [23,24], as well as in the case of monotone mappings in partially ordered complete metric spaces, see [4,2].
The global behavior of solutions of Equation (55) in the case where f is either increasing in both variables or decreasing in the first and increasing in the second variable is well described by the following result from [1,5]. The consequence of Theorem 28 is that every bounded solution of (55) converges to either an equilibrium or a period-two solution or to the point on the boundary where equation is not defined, see [3,10]. Thus the most important question becomes determining the basins of attraction of these solutions. The answer to this question follows from an application of theory of monotone maps in the plane, which was developed in [18,19,23].
The global behavior of solutions of Equation (55) in the case where f is either decreasing in both variables or increasing in the first and decreasing in the second variable is much more complicated and it can range from global asymptotic stability of the unique equilibrium as in the cases of difference equations x n x n−1 , n = 0, 1, . . . , a > 0, see [14], [20], and [14] to the conservative chaos as in the case of Lyness' difference equation or the following difference equation see [8]. Also such equations may exhibit Neimark-Sacker bifurcation such as see [12] and global convergence to singular zero solution The proofs for Equations (57) and (59)  In this paper we show that we can use the theory of monotone maps to improve results of Theorems 26 and 27 in the case where f is either increasing in first and decreasing in second argument or is decreasing in both arguments. In fact, we will give an interesting special result which shows that under certain mild condition the local stability implies global asymptotic stability. Our method will be based on embedding Equation (55) into related monotone two dimensional system of difference equations to which we will apply the global attractivity theorems of monotone systems. This will imply the global asymptotic stability of an equilibrium of the corresponding monotone two dimensional system as well as global asymptotic stability of the equilibrium of Equation (55).
In the following I provide an application of theorem (54) for higher order difference equations.

Example 1
A new proof for Pielou's second order difference equation: Pielou's equation is a mathematical model in population biology that was introduced by Pielou as a discrete analogue of the logistic equation with delay, see [6,14] and was investigates by [ references] using various method. in the following I will provide an alternative proof of the global stability of the positive equilibrium using The M-m theorem.
Pielou's equation is given by: Let x, y, z ∈ I and f (x, y, z) = 1 Next is to solve the system: Set x n−1 = u n and x n = v n in Equation (55) to obtain the equivalent system , n = 0, 1, . . . . u)). Then F maps I 2 into itself and the second iterate T := F 2 is given by , u), v)) .

Now a map associated to Equation
and it is clearly strictly cooperative on I 2 , when f is increasing in both arguments and competitive on I 2 , when f is decreasing in first and increasing in second variable. Unfortunately, there is no such a result in the case when f is either decreasing in both variables or increasing in the first and decreasing in the second variable.

Remark 5 The characteristic equation of Equation (55) at an equilibrium point
where p = f x (x,x) and q = f y (x,x). If f is increasing in both arguments then Equation (66) has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ.
Here D i f, i = 1, 2 denotes the partial derivative with respect to the i-th variable.
The following result is for strictly order preserving maps [7]. The result is stated for a partial order in R n , but it also holds in Banach spaces.
Theorem 29 (Order Interval Trichotomy of Dancer and Hess, [7]) Let u 1 u 2 be distinct fixed points of a strictly order preserving map T : where A ⊂ R n , and let I = u 1 , u 2 ⊂ A. Then at least one of the following holds.
(a) T has a fixed point in I distinct from u 1 and u 2 .
(b) There exists an entire orbit {x n } n∈Z of T in I joining u 1 to u 2 and satisfying x n x n+1 .
(c) There exists an entire orbit {x n } n∈Z of T in I joining u 2 to u 1 and satisfying x n+1 x n . The following result is a direct consequence of Theorem 29.
Corollary 7 If the nonnegative cone of is a generalized quadrant in R n , and if T has no fixed points in u 1 , u 2 other than u 1 and u 2 , then the interior of u 1 , u 2 is either a subset of the basin of attraction of u 1 or a subset of the basin of attraction of u 2 .
A simple consequence of this result is the following

Main Results
The next result is an extension and improvement of Theorem 26.
Theorem 30 Let (b) System (54) has at most three solutions; (c) p − q < 1, where p and q are defined in Remark 5.
Then x is globally asymptotically stable in [a, b].
Proof. Set with eigenvalues λ ± = p ± q. In view of the fact that p > 0, q < 0 the condition |λ ± | < 1 becomes equivalent to the condition p − q < 1. The well known condition for local asymptotic stability of the equilibriumx, under the restrictions p > 0, q < 0 is that Clearly, condition (69) implies (70), which means whenever E is local attractor for system (67) thenx is local attractor for Equation (55), but converse is not true.

2
The next result is an extension and improvement of Theorem 26. (a) f (x, y) is non-increasing in both variables; (b) System (56) has at most three solutions; (c) q > −1, q + p > −1, where p and q are defined in Remark 5.
Then x is globally asymptotically stable in [a, b] Proof. Set where is an increasing function. If the equilibriumx is global attractor for Equation (72) then it is also global attractor for Equation (55).
The Jacobian matrix of system (71) evaluated at E is with eigenvalues λ ± = q. The well known condition for local asymptotic stability of the equilibrium E(x,x) of system (71), under the restrictions p, q < 0 is that q > −1 and q + p > −1, which shows that E is locally stable. Since system (71) is anti-cooperative, using the result for global attractivity of such systems [11] we conclude that E attracts the interior of the box (m, m), (M, M ) , which completes the proof.
2 Remark 6 Theorems 26 and 27 were originally applied to the difference equation x n+1 = a + bx n + cx n−1 A + Bx n + Cx n−1 , n = 0, 1, . . . , where all parameters and the initial conditions x −1 , x 0 are non-negative and such that A + Bx n + Cx n−1 > 0 for every n. It is interesting to observe that in the case

Examples
In this section we give some examples of difference equations where Theorems 30 and 31 apply.

Example 2 Equation
x n+1 = 1 Bx n + Cx n−1 , n = 0, 1, . . . , where B, C > 0, x −1 , x 0 ≥ 0, x −1 + x 0 > 0 was considered in [14], where we proved that every solution of this equation converges to the unique equilibriumx = 1 B+C . We used the method of limiting sequences in [14]. Here we use Remark 6 to prove this result. Indeed, we need to find an invariant and attracting interval [L, U ]. Since (56) is automatically satisfied every solution of Equation (74) converges to the unique equilibrium, which is globally asymptotically stable.
Theorem 32 The unique equilibrium of Equation (76) is globally asymptotically stable.

Example 5
Consider the equation: x n+1 = f (x n , x n−1 ) = a + x n A + Bx n + x n−1 Consider the function G(x, y) = a+x A+(B+1)y . which is non-decreasing in x and non-increasing in y.