GLOBAL DYNAMICS OF DISCRETE MONOTONE MAPS IN THE PLANE AND IN RN

This dissertation investigates the local and global behavior of some monotone systems of difference equations. In each study, general results are provided as well as specific examples. In Manuscript 2 it is shown that locally asymptotically equilibria of planar cooperative or competitive maps have basin of attraction B with relatively simple geometry. The boundary of each component of B consists of the union of two unordered curves, and the components of B are not comparable as sets. The curves are Lipschitz if the map is of class C. Further, if a periodic point is in ∂B, then ∂B is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given. In Manuscript 3 Sufficient conditions are given for planar cooperative maps to have the qualitative global dynamics determined solely on local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class C on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points, and also the minimal period-two points are ordered in a sense. An application is included. In Manuscript 4 we give a characterization of monotone discrete systems of equations in terms of associated signature matrix and give some properties of certain invariant surfaces of codimension 1, which often give the boundary of attraction of some fixed points. We present several examples that illustrate our results in the case of k dimensional systems where k ≥ 3.

iv Introduction

Difference Equation Basics
Difference equations describe the progression of a given quantity or population over discrete time intervals. If we consider the size of the population in the nth generation, which we denote x n and assume that the size of the population in the n+1st generation denoted x n+1 is a function of x n , then we get the following first order difference equation x n+1 = f (x n ) n = 0, 1, . . . (1) where f : R → R is a given function. We call (1) a one-dimensional dynamical system. Also, the function f is called the map associated with (1). If we are given an initial value for (1), say x 0 = d, then applying equation (1) to x 0 multiple times results in the sequence {x 0 , f (x 0 ), f (f (x 0 )), f (f (f (x 0 ))), . . .} which is called a solution of (1). Now the population of the n+1st generation can also be dependant on the size of several previous generations x n , x n−1 , x n−2 , . . .. When x n+1 is a function of x n and x n−1 we get the following second order difference equation x n+1 = f (x n , x n−1 ) n = 0, 1, . . .
where f : I × I → I is a given function. Similar to equation (1), we can be given initial conditions for equation (2) and find solutions.
In this thesis, we will be particularly interested in systems of difference equations which model two or more quantities or populations that depend on each other over discrete time intervals. A two-dimensional system of difference equations is of the form x n+1 = f (x n , y n ) y n+1 = g(x n , y n ), n = 0, 1, . . .
where f, g : D → R, D ∈ R 2 are given functions. Initial conditions of (3) are ordered pairs (x 0 , y 0 ) ∈ D. Systems of equations will be discussed throughout this thesis, in particular, monotone systems of equations.
In studying difference equations, the main goal is often to determine the global For each equilibrium point of (3) we call the basin of attraction of (x, y) denoted B(x, y) is defined as the set J that contains (x, y) such that if T is the map associated to (3) T n (x, y) → (x, y) as n → ∞ for all (x, y) ∈ J.
There is also the potential for periodic solutions of the equation. These periodic points are often important in determining the global dynamics of the equation.
A minimal period two point is a point (x, y) ∈ D such that T 2 (x, y) = (x, y) and T (x, y) = (x, y). The same definition can be extended to points of larger period.
When conducting the local analysis of difference equations we consider the behavior of the equation about the equilibrium points, and periodic points if they exist, in a process called local stability analysis. After determining the local be-havior, the global behavior the the equation is considered and characterized.

Local Stability Analysis
To determine the local dynamics of a difference equation we go through a process known as local stability analysis. In this process we consider each of the equilibrium points, and periodic points if they exist, of our difference equation and characterize it based on the behavior of points in a small neighborhood around the equilibrium point. We give the following definitions to characterize the different types of equilibrium points. The definitions and theorems for this section are found in [4]. For the following discussion, let T : D → D, D ∈ R 2 be the map associated to (3), and let f, g be continuously differentiable functions at (x, y).
Definition 1 1. An equilibrium point (x, y) of (3) is said to be stable if for any > 0 there is δ > 0 such that for every initial point (x 0 , y 0 ) for which ||(x 0 , y 0 ) − (x, y)|| < δ, the iterates (x n , y n ) of (x 0 , y 0 ) satisfy ||(x n , y n ) − (x, y)|| < for all n > 0. An equilibrium point (x, y) is said to be unstable if it is not stable.
2. An equilibrium point (x, y) of (3) is said to be locally asymptotically stable (LAS) if it is stable and if there exists r > 0 such that (x n , y n ) → (x, y) as n → ∞ for all (x 0 , y 0 ) that satisfy ||(x 0 , y 0 ) − (x, y)|| < r.

3.
A periodic point (x p , y p ) of period m is stable (respectively unstable or asymptotically stable) if (x p , y p ) is stable (respectively unstable or asymptotically stable) fixed point of T m .
To determine the local stability of the equilibrium points as defined above, we first find the Jacobian matrix of the map T at each (x, y) and use a theorem to characterize the points.
Definition 2 Let (x, y) be a fixed point of the map T , the Jacobian matrix of T at (x, y) is the matrix J T (x, y) = ∂f ∂x (x, y) ∂f ∂y (x, y) ∂g ∂x (x, y) ∂g ∂y (x, y) .
The characteristic equation associated with the Jacobian matrix is The following two theorems will provide criteria to easily characterize the equilibrium points of (3).
Theorem 1 Let T = (f, g) be a continuously differentiable function defined on an open set W in R 2 , and let (x, y) in W be a fixed point of T .
1. If all the eigenvalues of the Jacobian matrix J T (x, y) have modulus less than one, then the equilibrium point (x, y) is locally asymptotically stable.
2. If at least one of the eigenvalues of the Jacobian matrix J T (x, y) has modulus greater than one, then the equilibrium point (x, y) is unstable.
Definition 3 1. If both eigenvalues of the Jacobian matrix J T (x, y) have modulus bigger than one, such a fixed point is called a source or repeller.
2. If one eigenvalue of the Jacobian matrix J T (x, y) has modulus less than one and the other eigenvalue has modulus bigger than one, such a fixed point is called a saddle.
Theorem 2 1. An equilibrium point (x, y) of (3) is locally asymptotically stable if every solution of the characteristic equation lies inside the unit circle, that is, if |trJ T (x, y)| < 1 + detJ T (x, y) < 2.
3. An equilibrium point (x, y) of (3) is locally a saddle point if the characteristic equation has one root that lies inside the unit circle and one root that lies outside the unit circle, that is, if |trJ T (x, y)| > |1 + detJ T (x, y)| By using these theorems we can characterize each of the equilibrium points of (3) which gives the local dynamics of (3).

Monotone Systems of Difference Equations
Since this thesis will be discussing monotone systems of difference equations throughout, we also provide basic definitions and theorems for monotone systems.
Consider (3) with T the map associated with (3). Then we have the following definitions and theorems about monotone systems.
if and only if x ≤ w and y ≤ z. Also set (x, y) and (x, y) = (w, z), and (x, y) < < N E (w, z) if and only if x < w and y < z. The South-east partial order ≤ SE is defined by (x, y) ≤ SE (w, z) if and only if x ≤ w and y ≥ z. The symbols < SE and < < SE are similarly defined to < N E and < < N E .
This definition is similar for a South-east order interval.
The following theorem and corollary is a theorem of Dancer and Hess in [2].
Let the map T : X → X be monotone and T (X) have compact closure in X. Then, at least one of the following holds: 1. There is a fixed point c such that a < c < b.
2. There exists an entire orbit from a to b that is increasing, and strictly increasing if T is strictly monotone.
3. There exists an entire orbit from b to a that is decreasing, and strictly decreasing if T is strictly monotone. These definitions and theorem are essential for understanding the analysis of monotone systems. The focus of this thesis is to provide some general results for the basins of attraction of fixed points in some planar cooperative maps.

Introduction and Preliminaries
Fixed points and periodic points of planar maps often have basins of attraction that have very complex boundary. This is the case even if the map is smooth. An which has two repelling fixed points and a single minimal period-two pair, namely {(−1, 0), (0, 0)}. The basin of attraction of the minimal period-two pair has fractal boundary [13,14]. See [12,14,15,16,17] for further properties of the basins of attraction for general maps in the plane or in higher dimension.
In this paper we consider maps T (x, y) = (f (x, y), g(x, y)), where f and g are continuous functions defined on some subset of R 2 with non-empty interior, such that f and g are non-decreasing in all of its arguments. Such maps are said to be cooperative. It is shown in this paper that, in stark contrast to the general case of planar maps, basins of attraction B of fixed points and periodic points of cooperative maps have simple geometry. In particular, when B contains a neighborhood of the periodic orbit, it is then bounded by unordered curves (in the sense of north-east order), which is to say that they are the graphs of decreasing functions. Moreover, at any fixed or periodic points on ∂B, the latter is tangential to a line with direction of an eigenvector associated with a characteristic value of the map at the point in question. If B has more than one connected component, then any two components are non-comparable, and if the map is of class C 1 , then the curves bounding B are Lipschitz.
As a motivating example consider the difference equation from [1] x n+1 = x 3 n + x 3 which has associated map The fixed points of the map are (0, 0), ( 1 , where the origin is locally asymptotically stable other two fixed points are saddle points. There are no periodic points. By using results from [9], it is shown in [1] that the basin of attraction B of (0, 0) is unbounded, and it consists of the union of the stable manifolds of the two nonzero fixed points, see Fig. 2.1(a). Notice that F is cooperative and F 2 is strongly cooperative.
A variation on (4) is the difference equation whose associated map has three fixed points: the point (0, 0) which is LAS, and the points  The basin of attraction B of the zero fixed point of the map T (x, y) = (y 3 , x 3 + y 3 ). The set B is bounded, and ∂B contains two fixed points p 1 and p 2 (saddles) and a repelling minimal period-two pair q 1 and q 2 . The union of the stable manifolds of p 1 and p 2 gives ∂B.
The previous examples suggest the question of whether the geometry of the basin of locally asymptotically stable fixed or periodic points of planar monotone maps is particularly simple and amenable to a "nice" characterization.
We note that the maps in (5) and (7) are (locally) invertible, and that in each of both cases the boundary of the basin B of the origin contains two saddle points.
This allows, by using the results from [9] for example, the characterization of ∂B as the union of stable manifolds of the saddle points. However, local invertibility of a cooperative or competitive map is not always true. Also, there is the question of the components of the basin of attraction in other cases, in addition to the possible presence of other fixed points (perhaps nonhyperbolic) on the boundary of the basin.
In general, the basin of attraction B(E) of locally asymptotically stable fixed point E of a map T satisfies where B 0 (E) is a largest connected invariant set containing E, and T 0 is the identity function. The problem of characterization of B(E) is finding the properties of The basin of attraction of a fixed pointx of a map T , denoted as B(x), is defined as the set of all initial points x 0 for which the sequence of iterates T n (x 0 ) converges tox. The map T is monotone if x y implies T (x) T (y) for all x, y ∈ R, and it is strongly monotone on R if x ≺ y implies that T (x) T (y) for all x, y ∈ R.
The map is strictly monotone on R if x ≺ y implies that T (x) ≺ T (y) for all x, y ∈ R. Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x 1 , y 1 ) ne (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 and the South-East (SE) ordering defined as (x 1 , y 1 ) se (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≥ y 2 . A map T on a nonempty set R ⊂ R 2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive. If T is continuously differentiable on an open set, a sufficient condition for T to be strongly cooperative (respectively, strongly competitive) is that at every point of the set, the jacobian matrix has positive entries (resp. positive diagonal entries and negative off-diagonal entries). For x ∈ R 2 , define Q i (x) for i = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction, for example, Q 1 (x) = {y ∈ R 2 : x ne y}. A set A is said to be order convex if for every x, y ∈ A, the order interval [x, y] is a subset of A. A general reference for difference equations and maps is [2]. For some basic notions about monotone discrete systems in the plane, see [1,5,6,7,8,9,10,18].

Main Results
The main result applies to cooperative maps on an order interval whose k-th power (for some k ≥ 1) is strongly cooperative. Smoothness of the map is not assumed, but it is considered later in Theorems 5 and 6. Unbounded domains are discussed in Remark 2, competitive maps in Remark 3, and periodic points in Theorem 4 Let R be an order interval in R 2 with nonempty interior, and let T : int(R) → int(R) be a cooperative map whose k-th power (for some k ≥ 1) Points on a boundary curve that are interior to R are non-comparable. The boundary curves C − and C + have common endpoints, and these are their only common points.     ii. If C − and C + intersect at a hyperbolic periodic point p ∈ int(R), then p is a source.
Remark 3 A version of Theorems 4, 5 and 6 and corollaries 2 and 3 are valid for maps T that are competitive (instead of cooperative). To obtain these results, replace the word cooperative by the word competitive, and replace the north-east partial order by the south-east partial order and vice-versa. With these modifications, the proofs carry over word for word, so those will be omitted. See Example 2 in Section 2.3.

Remark 4
If the boundary of the set B * in Theorem 4 has a fixed or periodic saddle point, the local stable manifold can be extended to a global stable manifold by using topological arguments or results such as those in [9]. In these cases it is possible to obtain a description of ∂B * . But often the sufficient conditions for global stable manifold are difficult to verify or are not applicable at all. In these cases, Theorems 4, 5, 6 and corollaries give the existence of invariant Lipschitz curves where other methods fail.

Remark 5
The results of this section are applicable to locally asymptotically stable minimal period k points p of a map T . To do this, consider the iterates p, T (p),. . . , T k−1 (p) as a fixed points of T k . The basin of the orbit of p is then the union of the basins of points of the orbit as fixed points of T k .

Examples
In this section we provide two applications. Example 1 is a discussion on the global dynamics of a strongly cooperative map whose domain is R 2 . We show that the origin is LAS, with basin of attraction that has more than one component.
Admittedly the example is somewhat contrived, but it is the only example of co-  V is defined as a specific perturbation of U , so that the origin has bounded basin of attraction consisting of three components.
Consider the map This is a strongly cooperative map for which the origin is LAS, as can be easily We now consider a perturbation of U of the form We shall choose ∆ so that V is a strongly cooperative map with the origin being a LAS fixed point with basin of attraction having more than one component. One way to accomplish this is by further specializing ∆ to have the form where φ is a smooth real valued odd function of a real variable to be chosen later.
Since φ is an odd function we have, Since U (x, −x) = (0, 0), the dynamics of V (x, y) on the line x + y = 0 are exactly the dynamics of φ on the real line.
We shall require that φ(0) = 0, which is necessary for the origin to be a fixed point of V (x, y). Also desirable is a small value of |φ (0)| so the origin retains local stability after perturbing the original map. The function φ must give a cooperative V , which can be ensured by choosing φ with suitable growth restrictions. Consider the function (see Figure 4) With φ as in (13)  φ has locally asymptotically stable fixed points 0, b = 2.06, and a repelling fixed point a = 0.95. The real numbers c = 6.03 and d = 12.80 are pre-images of a. The basin of attraction of 0 on the semi-axis consists of the intervals 0 ≤ t < a and c < t < d. All decimal numbers have been rounded to two decimals. Example 2 Consider maps of the form The map T is competitive on its domain and strongly competitive on its interior, the open positive quadrant, as can be concluded from the jacobian matrix The origin o is a singular point, and there are three fixed points, namely a(α, 0), d(0, δ) and b(1, 1). A straightforward calculation gives that a and d are s The following is a complete characterization of the global dynamics of map (14) for all allowed values of the parameters. See Figure 7.
Proposition 1 Let T be as in (14). For all values of α and δ in (0, 1), the set

bounded by north-east ordered Lipschitz curves
C + and C − , which have endpoints a, b and d, b respectively. Also, C + and C − are tangential to the line y = x at the point b. Proof. We begin by verifying that the origin has a relative neighborhood that is a subset of B. This can be seen as follows. The relations T (x, 0) se (x, 0) for 0 < x < α, and (0, y) se (0, y) for 0 < y < δ imply that for (u, v) with 0 < u < x and 0 < v < y, T n (0, y) se T n (u, v) se T n (x, 0). Since T n (x, 0) → (0, 0) and x < 1} consists of points (x, x) such that T n (x, x) → (0, 0). Also by (16) the ray In particular, it follows that B has only one component.
Let C − and C + be as in Theorem 4. Since no points outside of the unit square belong to B, it follows that b is an endpoint of both C + and C − . Also a is an endpoint of C − and d is an endpoint of C + , due to the fact that the axes are unstable manifolds of a and d. The rest of the proposition follows from Theorems 4, 5 and 6, and their corollaries.
Without loss of generality we assume for the rest of this section that T is a strongly monotonic map (k = 1).
We prove several claims first. The first two claims are about certain properties of B and its boundary set. Proof. Arguing by contradiction, suppose ∂B contains a ne linearly ordered line for some open neighborhoods V of T (x) and W of T (z). Now both V and W contain points in B, say v and w. In particular, v < < ne T (y) < < ne w. Since B is order-convex, it follows that T (y) ∈ B, which contradicts invariance of ∂B. 2 We now proceed to define functions φ ± of a real variable that are key to establishing further properties of the boundary of B. Denote with π 1 the projection operator on R 2 given by in R, and it has a finite or countable number of connected components (intervals).
Note that the definition of φ ± implies graph(φ ± ) ⊂ ∂B and Properties of φ ± are investigated in Claims 3-8 below.

Claim 3
The functions φ ± are non-increasing on I.
Thus φ + is non-increasing on I. The proof of the corresponding statement for φ − is similar. 2 For each q ∈ Q the restriction of the function φ − (resp. φ + ) to I q is nonincreasing, hence it has a natural extension to the closure of I q in the extended real line given by choosing the value at each endpoint of I q as the one-sided limit of φ − (resp. φ + ). We denote such extensions with φ q − and φ q + . It is a consequence of Claim 3 that for q ∈ Q, the functions φ q − and φ q + are non-increasing, and their graphs are contained in ∂B.
Proof. Suppose φ + is not continuous at some t 0 in clos(I). By the monotonic character of φ, the discontinuity is of the "jump" variety. More specifically, assume that φ + is defined on an interval t 0 < t < t 0 + δ for some δ > 0, and y 0 > y + , where y 0 := φ + (t 0 ) and y + := lim t→t + 0 φ + (t). In this case, (t 0 , y + ) ∈ B, which is not possible. an interval such that I q = π 1 (B ) for some q ∈ Q. Define the curves C ± by cases as follows (see Figure 8).
(I) If π 1 (R) and π 1 (B ) have no common endpoints, C ± is the curve given by the graph of φ q ± .
(II) If π 1 (R) and π 1 (B ) have β and only β as common endpoint, C − is the curve given by the graph of φ q − and C + is the curve given by the graph of φ q + together with the line segment joining (β, φ q + (β)) to (β, φ q − (β).
(III) If π 1 (R) and π 1 (B ) have α and only α as common endpoint, C + is the curve given by the graph of φ q + and C − is the curve given by the graph of φ q − together with the line segment joining (α, φ q − (α)) to (α, φ q + (α).
(IV) If π 1 (R) and π 1 (B ) have common endpoints α and β, C + is the curve given by and C − is the curve given by the graph of φ q − together with the line segment The different cases are illustrated in Lemma 1 Let J be a 2 × 2 matrix with positive entries. Let v be an eigenvector of J that is associated with the eigenvalue of J that has the smallest modulus. Let C be a closed convex double cone in R 2 with vertex at the origin such that v ∈ C.
Then there exists an integer m such that J m (C) ⊂ Q 1 ∪ Q 3 .
Proof. Let λ 1 and λ 2 be eigenvalues of J, with associated eigenvectors v 1 , v 2 . Assume Note that v 2 has both coordinates with the same sign, by Perron-Frobenious The- (19) that for m large enough, J m z has both coordinates with sign equal to the sign of α 2 . Hence J m z ∈ Q 1 ∪ Q 3 and Proof of Theorem 5. We present here a proof for the case where the point p is a fixed point of T . The case where p is a minimal period-m point may be treated by considering the map T m , for which p is a fixed point, and it is not given here.
By Theorem 4, B = C − ∪ C + . Without loss of generality we assume p ∈ C + . If the conclusion is not true, then there exists a double cone C containing \ {p} in its interior and a sequence {x m } on the curve C + such that x m → p and x m ∈ C.
Choose an integer m as in Lemma 1, and set J m := ãb cd . Let {(t, φ(t)) : t ∈ I} be a parametrization of C + near p, such that for some t * in the real interval I, From Lemma 1, L n ∈ Q 1 ∪ Q 3 . Also, L n = (0, 0), since the null space ofJ, if nontrivial, is contained in the cone C. Then, for each n ≥ 0, L n has nonzero coordinates with the same sign.
Let K be a compact interval containing t * in its interior and define, for t ∈ K and The functions Φ 1 and Φ 2 are uniformly continuous on K × [−1, 1], and by (21), Φ 1 (t, 0) and Φ 2 (t, 0) are nonzero and have the same sign for all n. By uniform continuity, there exists δ > 0 such that Φ 1 (t, 1 ) and Φ 2 (t, 2 ) are nonzero and have the same sign for all t ∈ K and | 1 | < δ, | 2 | < δ. Without loss of generality we may assume that Let J andJ be the jacobian matrices of of T and T m at p respectively. Since p is a fixed point, the chain rule givesJ = J m . Note the entries ofJ are positive. Set Since T is continuously differentiable, Rearranging terms in (24) and using (20), (23) and (25) we have By (21), (23) and (25) and the assumption thatã,b,c andd are positive, both coordinates in the right-hand side of (26) have the same sign for large n, and therefore either T (t n , φ(t n )) T (t * , φ(t * )) or T (t * , φ(t * )) T (t n , φ(t n )). But this contradicts (i) of Theorem 4, which requires points on C + to be non-comparable.

2
Proof of Theorem 6. (i) Let p ∈ C + , and let {(t, φ(t)) : t ∈ I} be a parametrization of C + near p, such that for some t * ∈ I, (t * , φ(t * )) = p. Here I ⊂ R is an interval. The function φ is decreasing. If φ is not Lipschitz at t * , then there exists a sequence {t n } in I such that t n → t * and Without loss of generality we may assume that t n > t * and φ(t n ) < φ(t * ) for all n, that is, Let ( a b c d ) be the jacobian matrix of T at p. For each n ∈ N define o (1) n and o Since T is countinuously differentiable, Rearranging terms in (29) and using (28) and (30) we have By (28) and (30) and the assumption that a, b, c and d are positive, both coordinates in the right-hand side of (31) are negative for large n, and therefore T (t n , φ(t n )) T (t * , φ(t * )). But this contradicts (i) of Theorem 4, which requires points on C + to be non-comparable. Thus φ is Lipschitz.
(ii) We present the proof for the case when p is a fixed point of T . Note p is necessarily unstable since p ∈ ∂B. Since it is hyperbolic, it is either a saddle point or a source. If p is a saddle point, then it has a local stable manifold M s , which is tangential to v with v not comparable to the origin by the Krein-Rutman theorem.
There exist points x in B * that are arbitrarily close to p and which belong to to the union of quadrants Q 2 (p) and Q 4 (p). Furthermore, such points x may be chosen to be comparable to points on M s , which would prevent the iterates of such points from converging to p, thus contradicting the definition of stable manifold. to have a finite number of strongly ordered fixed points, and also the minimal period-two points are ordered in a sense. An application is included.

Introduction
In this paper we consider a cooperative system of the form where the transition functions f, g are non-decreasing in all its arguments and its corresponding map F = (f, g). Sufficient conditions are given for such planar cooperative map to have the qualitative global dynamics determined by local stability information obtained from fixed and minimal period-two points. The results are given for a class of strongly cooperative planar maps of class C 1 defined on an order interval. The maps are assumed to have a finite number of strongly ordered fixed points and minimal period-two points. Our results holds in hyperbolic case as well as in some non-hyperbolic cases as well. Our results are motivated by global dynamic results for the systems and see [3,4,5,17]. System (33) was considered in [3] and it was proved that all The system (34), studied in [4] exhibited the appearance of period-two solutions, which played an important role in the global dynamics of this system. Cases where provided for such system which had 1, 2 or 3 period-two solutions and in the last case one of these period-two solutions had substantial basin of attraction [4]. Papers [3,4] extensively used the algebraic techniques to find the regions of existence and stability of equilibrium solutions and period-two solutions. The results in this paper will be proven through the geometric analysis of the equilibrium curves and by using some major results about the global stable and unstable manifolds of cooperative systems in the plane [21]- [24]. The results of this paper are applicable to systems (33) and (34). Our results have immediate extension to the competitive systems of difference equations in the plane.
The paper is organized as follows. In Section 3.2 we list some basic results that are relevant to this paper, see [21]- [24]. See also [1,2,6,9,10,11,12,15,16,19,20,25,26,27,28] for some related competitive systems. In Section 3.3 we present the three main theorems. In section 3.4 we apply the theorems from Section 3.3 to a parametrized cooperative system whose transition functions are of the Holling's type [17]. Finally the proofs of the results in Section 3.3 are presented in Section 3.5.

Preliminaries
Let be a partial order on R n with nonnegative cone P . For x, y ∈ R n the order interval x, y is the set of all z such that x z y. We say x ≺ y if x y and x = y, and x y if y − x ∈ int(P ). A map T on a subset of R n is order whenever x ≺ y, and strongly order preserving if T (x) T (y) whenever x ≺ y.
Let T : R → R be a map with a fixed point x and let R be an invariant subset of R that contains x. We say that x is stable (asymptotically stable) relative to in V convergent to u. The notions of order stable from above and strongly order stable from above are defined similarly. A (strongly) order stable fixed point has the respective stability property from above and below [7].
Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x 1 , y 1 ) ne (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 and the South-East (SE) ordering defined as (x 1 , y 1 ) se (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≥ y 2 .
A map T on a nonempty set R ⊂ R 2 which is monotone with respect to the North-East (NE) ordering is called cooperative and a map monotone with respect to the South-East (SE) ordering is called competitive. A map T on a nonempty set R ⊂ R 2 which second iterate T 2 is monotone with respect to the North-East (resp. South-East) ordering is called anti-cooperative (resp. anti-competitive).
If T is differentiable map on a nonempty set R, a sufficient condition for T to be strongly monotone with respect to the NE ordering is that the Jacobian matrix at all points x has the sign configuration sign (J T (x)) = + + + + , provided that R is open and convex.
The next result in [24] is stated for order-preserving maps on R n . See [14] for a more general version valid in ordered Banach spaces.
Theorem 7 For a nonempty set R ⊂ R n and a partial order on R n , let T : R → R be an order preserving map, and let a, b ∈ R be such that a ≺ b and a, b ⊂ R. ii.) If T is strongly order preserving, then there exists a fixed point in a, b which is stable relative to a, b .
iii.) If there is only one fixed point in a, b , then it is a global attractor in a, b and therefore asymptotically stable relative to a, b .
The following result is a direct consequence of the Trichotomy Theorem, see [14,24], and is helpful for determining the basins of attraction of the equilibrium points.
Corollary 4 If the nonnegative cone of a partial ordering 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 .
Next result is a simple and useful geometric test for checking when the fixed point of the cooperative map is non-hyperbolic.
Lemma 2 Let (x,ȳ) be an interior fixed point of a cooperative map R(x, y) = (f (x, y), g(x, y)), and let r be the spectral radius of the Jacobian matrix J R (x,ȳ).
Suppose the tangent lines to f (x, y) = x and g(x, y) = y at (x,ȳ) are not parallel to one of the axes. Denote with m 1 and m 2 respectively the slopes of the tangent lines. The following statements are true: (i) If 0 < m 2 < m 1 , then r < 1.

Main Results
In this section we present three theorems.  Then the following statements are true.  and y can be chosen in C + (a) and C − (b) respectively so that x ≺ y. Then for all n ≥ 1, T n (x) ≺ T n (y). But for n large enough, T n (x) and T n (y) both belong to C, which is strongly ordered in the southeast order. This contradiction completes the argument.

Global dynamics of a cooperative system
The purpose of this section is to illustrate the application of the results in this paper. Consider the following parametrized system of difference equations of Holling type: where a, b, c, d, δ 1 , δ 2 > 0, x 0 , y 0 ≥ 0. Let T : R 2 + → R 2 + be the map associated to (37), that is Proof. The jacobian matrix of T at (x, y) is Since J T (x, y) has positive entries, T is strongly monotonic on R 2 + . The increasing character of the coordinate entries of T with respect to each variable gives that T . By direct substitution in (38) we have the origin e 0 is a fixed point for all values of the parameters. Fixed points of T are common points of the equilibrium curves It is obvious that the origin is a fixed point. Since (40) where A := Numerical searches performed by the authors of this article suggest that four is the maximum number of fixed points for system (37), and that three is the maximum number of minimal period-two orbits.
In the rest of this section we illustrate the application of the results of this paper for different parameter choices given in Table 1. The different cases are presented in Figures 9 -14. For specific values of the map T , fixed points and minimal period-two points can be easily found with a computer algebra system (CAS). Also a CAS can be used to determine that a specific fixed point has only one pre-image. CAS do not work to investigate existence of minimal period-four points algebraically due to the complexity of the equations involved. In this case one can use other means such as the approach mentioned in Figure 15.   Table 1. There exists a unique interior fixed point e 1 , which is non-hyperbolic and stable from above. The origin e 0 is stable from above. There are no minimal period-two points. (a) shows the equilibrium curves, which have a tangential contact point at e 1 , as implied by Lemma 2. (b) By Theorem 9 applied to the restriction of the map to the invariant order interval [e 0 , u], there exists a southeast ordered curve C through e 1 , such that points below C are attracted to e 0 and points on or above C are attracted to e 1 . Since there are no period-two points outside [e 0 , u], the curve C has an extension to a southeast ordered curve on R 2 + with endpoints on the boundary which separates the basins of attraction of e 0 and e 1 . has only two fixed points, so by the Trichotomy Theorem one of the fixed points is stable and the other is unstable. Thus either all even indexed fixed points are stable, or all odd indexed fixed points are stable. By boundedness of T , interior unstable fixed points belong to an order interval in R determined by two stable fixed points. Supposex andx +2 are stable, andx +1 is unstable. By Theorem 4 and Theorem 6 in [18] there exist south-east ordered Lipschitz curves C + , C − , such that ∂B(x ) = C + ∪ C − , and whose only possible common points are endpoints, and in this case such points are either fixed points or minimal period-two points.

Proofs of Theorems
Neither of those two possibilities is allowed by the hypotheses of the Theorem, hence endpoints do not coincide. Note thatx +1 ∈ C + . Thus the dynamics of T  Table 1. There exist interior fixed points e 1 (unstable) and e 2 (stable). The origin e 0 is stable. There are no minimal period-two points. (a) shows the equilibrium curves, which have a nontangential contact points at e 1 and e 2 . Either a calculation or Lemma 2 may be used to determine local stability of e 1 and e 2 . (b) By Theorem 9 applied to the restriction of the map to the invariant order interval [e 0 , e 2 ], there exists a southeast ordered curve C through e 1 , such that points below C are attracted to e 0 points above C are attracted to e 2 , and points on C are attracted to e 1 . Since there are no period-two points outside [e 0 , e 2 ], the curve C has an extension to a southeast ordered curve on R 2 + with endpoints on the boundary which separates the basins of attraction of e 0 and e 2 .
on the curve C + and C − is one-dimensional, bounded, with only one fixed point, namelyx +1 and no minimal period-two points. By Theorem C.3 in [8], the iterates of each point on C + must converge to a fixed point. Such point can only bex +1 .
We conclude thatx +1 is a saddle point. A similar argument can be made with the pointx +1 and the curve C +2 − . Thus C + and C +2 − coincide with a section of the local stable manifold W s loc of T atx +1 . We claim that C + = C +2 − . To prove this, assume the contrary statement. Then there exist points x ∈ C + and y ∈ C +2 − such that x ≺ y. Hence T n (x) ≺ T n (y) for all n ≥ 1. Now for n large enough, both T n (x) and T n (y) enter W s loc , which is strongly ordered in the south-east order, so in particular T n (x) and T n (y) are not comparable in the north-east order. It follows Proof of Theorem 9 Assume both a and c are order stable from above. By Theorem 4 in [18] the boundary of the basin of attraction of a is a strongly south-east  Table 1. There exist hyperbolic interior fixed points e 1 (stable), e 2 (unstable) and e 3 (stable). The origin e 0 is unstable. There are no minimal period-two points. (a) shows the equilibrium curves, which have a nontangential contact points at e , = 1, 2, 3. Either a calculation or Lemma 2 may be used to determine local stability of interior fixed points. (b) By Theorem 9 applied to the restriction of the map to the invariant order interval [e 0 , e 3 ], there exists a southeast ordered curve C through e e , such that non-zero points below C are attracted to e 1 , points above C are attracted to e 2 , and points on C are attracted to e 2 . Since there are no period-two points outside [e 0 , e 3 ], the curve C has an extension to a southeast ordered curve on R 2 + with endpoints on the boundary which separates the basins of attraction of e 1 and e 3 .  Table 1. Theorem 10 guarantees the existence of curves C 1 and C 2 in [e 1 , e 3 ] and through the period-two points and the point e 2 . These curves separate [e 1 , e 3 ] in regions attracted to e 1 , e 3 , and {p 2 , T (p 2 )}. Theorem 4 and Corollary 2 in [18] implies that C 1 and C 2 can be extended to invariant curves C 1 and C 2 that bound the basin of attraction of e 1 and e 3 in R 2 + respectively. C 1 and C 2 are south-east ordered and extend to the boundary of R 2 + . The restriction of the map T 2 to each of the curves C 1 and C 2 exhibits one-dimensional dynamics of a bounded map on the real line that has two fixed points and no minimal periodtwo points. By Theorem C.3 in [8], iterates of points on C 1 and C 2 must converge to a fixed point of T 2 . The point e 2 is a repeller with only itself as pre-image. Consequently for = 1, 2, for every point x in C \ {e 2 } T n (x) is attracted to {p , T (p )}. In particular, iterates of points x ∈ (C 1 ∪ C 2 ) \ [e 1 , e 3 ] must enter [e 1 , e 3 ] after a finite number of iterations. Since for every z in region between the curves there exist x ∈ C 1 and y ∈ C 2 such that x ≺ y ≺ z, Then T n (z) must enter [e 1 , e 3 ], and it is attracted to the nonhyperbolic minimal period two orbit. The curves C 1 and C 2 separate R 2 + into regions attracted to e 1 , e 3 , and {p 2 , T (p 2 )}. Claim: Suppose z is a fixed point or minimal period-two point of T in C, and let τ and ρ be the characteristic values of T 2 at z, where |τ | < ρ. If τ < 1, then in every neighborhood V of y there exist x, y ∈ C ∩ V such that x < < se T 4 (x) ≤ se z ≤ se T 4 (y) < < se y, and if τ > 1, then in every neighborhood V of z there exist x, y ∈ C ∩ V such that T 4 (x) < < se x ≤ se z ≤ se y < < se T 4 (y). By Theorem 3 in [18], C is tangential at z to the eigenspace associated to the characteristic value τ .
The claim follows from this fact.
The set C := C ∩ Q 4 (c) is an unordered closed curve with c at one endpoint Investigating existence of minimal period-four points through graphical means. Boundedness of the map implies that any minimal periodic point is in the order interval given by the smallest and largest fixed points. This determines the initial domain for a contour plot of T 4 (x) − x . This first plot shows approximate locations of period four points, see the plot on the right. Then contour plots of T 2 (x) − x and T 4 (x) − x are produced with domains near these locations. If these plots show the same locations for values near zero, then this suggests that period-four points are actually period-two points, implying that there are no minimal period-four points.  Note that C is invariant for T 2 , and that both c and p are the only fixed points of T 2 in C , and by hypothesis T 2 has no minimal period-two points in C .
We now introduce a parametrization of C . With c = (c 1 , c 2 ) and d = ( y is such that (x, y) ∈ C . The function φ is well defined due to the strongly monotonic character of C . It is straightforward to verify that φ is one-to-one and onto, continuous, and satisfies φ(0) = c, φ(1) = d. Define f : [0, 1] → [0, 1] by i.e., the following diagram commutes: Thus f has exactly two fixed points, namely 0 and a fixed point t * ∈ (0, 1) where φ(t * ) = p. Note by the hypothesis on the point c the relation f (t) = 0 is only satisfied by t = 0. To see that p is not a repeller, assume it is. By the Claim above, the function f (t) satisfies f (t) > t for t ∈ (t * , d 1 ], which is not possible. Thus p is locally asymptotically stable or non-hyperbolic of stable type. In particular, t * is locally asymptotically stable for f (t). We now prove that c is a repeller. Assume the contrary, i.e., c is a saddle point. By the Claim and the fact that f (t) has only two fixed points, it follows that f (t) < t for t ∈ [c 1 , t * ). But this contradicts the Claim's conclusion of p being locally asymptotically stable. Thus c is a repeller. The following result is a corollary to P. Hartman's Lemma 5.1 and Corollary 5.1 in [13].
Lemma 4 Let c ∈ R 2 be a fixed point of a planar map F which is of class C 1 in a neighborhood of c. Suppose that the characteristic values of F at c are real numbers τ and ρ such that |τ | < min(1, ρ). Then there exists a C 1 curve C * through c that is locally invariant under F which is tangential to the eigenspace V associated with τ , such that for any x, if x ∈ C * then T n (x) → c, and if x ∈ C * and F n (x) → c tangentially to V, then there exists n 0 ∈ N such that F n (x) ∈ C * for n ≥ n 0 .
Proof. There is no loss of generality in assuming c = (0, 0) and that the map F has the form where f 1 , f 2 and their first partial derivatives are all zero at (0, 0). By Hartman's Lemma 5.1 there exists a function y = φ(x) of class C 1 for small |x| satisfying φ(0) = φ (0) = 0, and such that the graph of φ is locally invariant under F . By the same Lemma it may be assumed φ(x) = 0 and f 2 (x, 0) = 0 for small |x|, by performing a C 1 change of variables if necessary. The curve C * is now taken to consist of points x = (x, 0) with small |x|. Choose a real number θ 0 so that . If x = (x, 0) ∈ C * , then F (x, 0) = (τ x + f 1 (x, 0), 0), and by the proof of Corollary 5.1 in page 238 of [13], |F (x, 0)| < (τ + θ 0 )|x| < 1+τ 2 |x|. Hence F n (x) → (0, 0). Now consider x such that (x n , y n ) := F n (x) satisfies x n = 0 for all n ≥ 0, (x n , y n ) → (0, 0) and y n /x n → 0. To complete the proof it must be shown that y n = 0 for all large enough n. If for some m the point (x m , y m ) satisfies y m = 0 and |x m | is small enough, then y m+k = 0 for k = 0, 1, 2, . . . and there is nothing else to prove. Now assume y n = 0 for all n ≥ 0. The proof of Corollary 5.1 in page 238 of [13] gives the inequality |y n+1 | ≥ (ρ − 2θ 0 ) |y n |, which by the definition of θ 0 implies |y n+1 | ≥ 1 2 (ρ + τ ) |y n |.
Since f 1 (x, y) and its derivatives are zero at the origin, we have The assumption y n /x n → 0 and (46) imply that there exists n 0 ∈ N such that f 1 (x n , y n ) x n ≤ 1 2 (ρ − τ ) , n = n 0 , n 0 + 1, . . .
From (44) and (47), Combine (45) and (48) to obtain which contradicts the assumptions y n = 0 and y n /x n → 0. Thus (x n , y n ) ∈ C * for all n large enough. To prove (ii), consider p 1 and p 2 as fixed points of the strongly cooperative map T 2 . Set C := C ∩ Q 4 (C), = 1, 2. Assume p 1 is a saddle point.
To prove (iii), note that for T 2 , the points p 1 and p 3 are saddle points and consequently p 2 is locally asymptotically stable. Note that [p 1 , . The argument used in the proof of part (ii) can be used here to conclude that the region between C 1 and C 2 is precisely the basin of p 2 as a fixed point of

Introduction and Preliminaries
In this paper we consider the maps on R n which are coordinate-wise monotonic and we characterize those maps which are monotone to a standard ordering ≤ σ of R n . In two dimensional case the obtained characterization will coincide with two classes of maps known as competiive and cooperative for which there is an extensive theory developped in [9,10,11,17,18,19,20,21,27].
is called 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 and it is called cooperative if both functions f, g are non-decreasing in x and y. This fact can be illustrated by using the signature matrices for cooperative and competitive case respectively. Here 1(resp. −1) means that the corresponding function is non-decraesing (resp. non-increasing) in its argument. As it was shown in sequence of papers [17,18,19,20,21,27] cooperative and competitive maps in the plane have a lot of structure which leads to characterization of the global stable and unstable manifolds of all hyperbolic fixed and periodic points which in turn gives the boundaries of the basins of attraction of such points. In addition the union of global unstable manifolds gives the carrying simplex that majority of solutions follow to the attracting fixed or periodic points.
We were also able to obtain similar results for some non-hyperbolic fixed and periodic points, see [19,20]. In this paper we will try to extend some of the results in [17,18,19,20] to the case of maps in R n , n ≥ 3. The major difficulty in this process is description of the boundary of an invariant surface which plays the role of global stable manifold in two dimensional case. While in two dimensional case the global stable manifolds are continuous curves their boundaries are always either fixed or periodic points or points on the boundary in higher dimensional case the global stable manifolds are surfaces which boundaries could have complicated structure, that makes the problem of their description harder. In this paper we will first characterize the maps monotone with respect to a standard ordering in R n among all coordinate-wise monotonic maps and then we will obtain the description of a certain invariant manifold, which plays the role of global stable manifold. We will illustrate our results with several examples. The next example demonstrate use of techniques from [17,18,19,20] in proving global dynamics of a two-dimensional competitive map: Example 3 Consider the system of difference equations x n+1 = b 1 x n 1 + x n + c 1 y n , y n+1 = b 2 y n 1 + c 2 x n + y n n = 0, 1, . . .
where the parameters b 1 , b 2 , c 1 , and c 2 are positive real numbers and the initial conditions x 0 and y 0 are arbitrary non-negative numbers.
In a modelling setting, system (51) is well-known Leslie-Gower system [3] and is one of discrete version of Lotka-Volterra system of differential equations [16].
The state variables x n and y n denote population sizes during the n-th generation, and the sequence {(x n , y n ) : n = 0, 1, 2, ...} depicts how the populations evolve over time. Competition between the two populations is reflected by the fact that the transition function for each population is a decreasing function of the other population size.
It has been shown in [3,18] that under hypotheses and equation (51) has four equilibrium points: E 0 (0, 0), Theorem 4 in [3] states that E 1 and E 2 are saddle points, E 3 is locally asymptotically stable, and E 0 (0, 0) is a repeller. The same theorem states that E 3 is globally asymptotically stable, but the proof of this statement given in [3] is incomplete and it was completed in [19]. and E 2 satisfy: In view of the uniqueness of the global stable manifold which follows from the Stable Manifold Theorem and the Hartman-Grobman Theorem, see [5], we obtain (54).
It is clear from (51) Next we consider five cases labeled (a)-(e), corresponding to (x 0 , y 0 ) being a member of one of the sets appearing in the right-hand-side of (55).
Same conclusion holds if the initial point belongs to one of the critical lines.
Same conclusion holds if the initial point belongs to one of the critical lines.
(e) Next, take (x 0 , y 0 ) ∈ Q 1 (E 3 )∩D 2 . Then (x 1 , y 1 ) = T ((x 0 , y 0 )) ≤ (x 0 , y 0 )) that is x 1 ≤ x 0 , y 1 ≤ y 0 because otherwise (x 1 , y 1 ) = T ((x 0 , y 0 )) ≥ (x 0 , y 0 )) which would imply that {x n , y n } is monotonic sequence in D 2 and so is convergent, which is a contradiction. Thus, {x n , y n } is cw-monotonic sequence and so it must eventually enter the region D 1 . Otherwise, this sequence stays in D 2 and so is convergent which is impossible. satisfies so-called (O+) condition [19,27], which implies that every bounded orbit converges to a fixed point. However if condition holds, then the fixed points E 1 , E 2 , E 3 exchange their local stability characters and so this changes global dynamics. In this case the existence and uniqueness of global stable manifold is essential, as the following result demonstrate, [14,18,19]. in South-east ordering, and {x n } converges to E 2 whenever x 0 is above W s (E 3 ) in South-east ordering.
Let A be a subset of R n , and let T : A −→ A be a continuous function. We denote with T the -th coordinate function of T , that is, T = (T 1 , . . . , T n ). We say that T is or cw-monotonic on A, if for 1 ≤ i, j ≤ n, T i (x 1 , . . . , x n ) is monotonic in Let T be coordinate-wise monotonic map on A ⊂ R n that is not constant on any coordinate. Define the signature matrix of T as the n × n matrix M T = {m ij } with entries Thus for cw-monotonic differentiable maps T with nonzero partial derivatives, Definition 7 For each choice of σ = (σ 1 , . . . , σ n ) with σ ∈ {−1, 1}, the standard cone associated to σ is the set and the standard order associated to σ is the relation given by It is clear that there are 2 n distinct standard cones in R n . A map T on a set A ⊂ R n is said to be monotone with respect to the partial ordering ≤ σ if x ≤ y ⇒ T (x) ≤ σ T (y) for x, y ∈ A.

Main Results
Theorem 13 Let A be a subset of R n with nonempty interior, and let T : A → A be a cw-monotonic map. A necessary and sufficient condition for T to be monotone with respect to a standard ordering ≤ σ of R n is that its signature matrix M T has the form M T = σ t σ.
Proof. Suppose T is monotone increasing with respect to ≤ σ ; we wish to prove m ij = σ i σ j for i, j = 1, . . . , n. Now consider x in the interior of A, and let δ ≥ 0 be small enough so that the closed ball centered at x with radius δ is contained in A. For i, j fixed in {1, . . . , n} set y = x + δe j , where e j is coordinate vector which j-th component is 1. Hence y − x = 0 for = j, and y j − x j = δ. We now proceed to analyze the following two cases: σ j = 1 and σ j = −1. If σ j = 1, then x ≤ σ y, which implies that T (x) ≤ σ T (y). In particular, σ i (T i (y)−T i (x)) ≥ 0.
In the second case σ j = −1 we have , that is, T i is nonincreasing in the j-th coordinate, thus giving To prove sufficiency, suppose M T = σ t σ, and let x, y be such that x ≤ σ y.
Then σ (y − x ) ≥ 0, for = 1, . . . , n. That is, x ≤ y whenever σ = 1, and x ≥ y whenever σ = −1. Corollary 6 When n = 2, 3 or 4, a necessary and sufficient condition for the cw-monotonic map T to be monotonically increasing with respect to a standard ordering is that one of the following cases holds: (a) n = 2, and M T is equal to one of the following matrices: (b) n = 3, and M T is equal to one of the following matrices: (c) n = 4, and M T is equal to one of the following matrices: Remark 1 For n = 2 the signature matrices coincide with the competitive and cooperative cases in the sense of M. Hirsch [27]. In the case n = 3 one of the signature matrices coincide with the cooperative case in the sense of M.
Hirsch. Second and fourth signature matrix could be described as the matrices that describe the competition between two groups: one group consisting of two species and one group consisting of third species. In the case n = 4 one of the signature matrices coincide with the cooperative case in the sense of M. Hirsch.
Second, fourth, and eighth signature matrix could be described as the matrices that describe the competition between two groups: one group consisting of three species and one group consisting of fourth species (second and eighth signature matrix) and two groups consisting of two species each (fourth signature matrix).
We now turn our attention to the properties of certain invariant surfaces of codimension 1 in R n . LetR = R ∪ {−∞, +∞} be the extended real numbers.
Then a partial order σ on R n with positive cone given by a generalized octant extends in a natural way to a partial order onR n , which we shall denote with the same symbol σ .
iv. If T is differentiable on R and such that the n-th column of T (z) has positive entries for z ∈ R, then φ is Lipschitz on (graph(φ) ∩ R) π .
Proof. For x, y ∈R n such that x = y, the line segment L(x, y) : Claim 9 The closure of graph(φ) ∩ R does not contain any vertical line segments.
Proof. Suppose [s, t] is a vertical line segment in the closure of graph(φ) ∩ R.
Every small neighborhood of either one of the points s or t contains points in B(p) and in its complement R \ B(p). Thus a similar statement is true for the points T (s) and T (t). Strong monotonicity of T and s σ t imply T (s) < σ T (t) and furthermore, positive real numbers δ 1 and δ 2 exist such that T (B(s; δ 1 )) < σ T (B(t; δ 2 )). This relation together with the fact that T (B(t); δ 2 ) contains points in B(p), imply T (B(s; δ 1 )) ⊂ B(p), and consequently, B(s; δ 1 ) ⊂ B(p). But this contradicts B(s; δ 1 ) having elements not in B(p). 2 Proof. There exist points x and y in R such that p n ≤ φ(x π ) < φ(y π ) ≤ q n , by the hypothesis on B(p). If p n < φ(x π ) then (x π , φ(x π )) ∈ graph(φ) ∩ R and if φ(y π ) < q n , then (y π , φ(y π )) ∈ graph(φ) ∩ R, and in either case there is nothing else to prove. In the case when p n = φ(x π ) and φ(y π ) = q n , set f (t) : Then there exist t * ∈ [0, 1] and sequences {t } and {s } in [0, 1] such that t → t * , f (t ) → p n and s → t * , f (s ) → q n . It follows that the vertical line segment with endpoints (((1 − t * )x + t * y) π , p n ) and (((1 − t * )x + t * y) π , q n ) is a subset of the closure of graph(φ) ∩ R, which contradicts Claim 9. 2

Claim 11
The restriction of φ to (graph(φ) ∩ R) π is decreasing in all variables.
Proof. Let x ∈ graph(φ) ∩ R and e be an element of the standard basis of R n different from (0, . . . , 0, 1). Note that e π is an element of the standard basis of R n−1 , and for h ∈ R, (x+h e) π = x π +h e π . Choose h > 0 small enough so that x+h e ∈ R and consequently x π + h e π ∈ R π In this case, x σ x + h e, and strict monotonicity of T implies T (x) < σ T (x + h e). The latter relation and the definition of φ give the relation φ((x + h e) π ) ≤ φ(x π ), that is, φ(x π + h e π ) ≤ φ(x π ). Now strong monotonicity of T and the argument used in Claim 9 imply φ(x π + h e π ) < φ(x π ).
Putting together relations (63), (64) and (65) we obtain Since T x * (e) is precisely the n-th column of T x * , equation (46) implies that for m large enough, T (x * ) < σ T (x m ). This contradicts iii.

Examples
Example 4 Consider the following difference equation where a 1 , d 1 > 0, φ 1 is such that The equilibrium equation of (67) is which gives E 0 = 0 and E x = d 1 a 1 as the two fixed points to (67). (67), E 0 is locally asymptotically stable and E x is repeller for all parameter values.

Lemma 5 Given
For the global dynamics of (67) consider the sets of points U = {x : x < E x } and V = {x : x > E x }. If x ∈ U , then lim n→∞ T n (x) = 0 since 0 is a locally asymptotically stable fixed point. Now consider x ∈ V . For a contradiction assume for x ∈ V , the solution is bounded. Then lim n→∞ x n = x * . So, lim n→∞ T (x n ) = T (x * ), which implies, lim n→∞ x n+1 = T (x * ) = x * . Thus x * is a fixed point of (67), a contradiction. Thus for x ∈ V , lim n→∞ T n (x) = ∞.
Example 5 Consider the following system of equations where a i , b i , d i > 0 for i = 1, 2, x 0 , y 0 ≥ 0 and φ 1 , φ 2 are as described in (68). Also, let S : R 2 + → R 2 + be the map associated to (69). The Jacobian of S is Since all of the entries of J(S) are positive, we see that (69) is a cooperative system.
The equilibrium equations of (69) are given by, The solutions of (70) give the following four equilibrium points, E 0 = (0, 0), Notice E 0 , E x and E y will always exist, however for E xy to exist, Notice for these inequalities to hold, the numerator and denominator of these fractions must be either be both positive or both negative.  Proof. The roots of the characteristic polynomial given by J(S(E 0 )) are λ 1 = . Since d 1 , d 2 > 0, it follows that |λ 1 |, |λ 2 | < 1, so E 0 is locally asymptotically stable.
Next for the stability character of the equilibrium points on the axes. Starting with E x , the roots of the characteristic polynomial of J(S(E x )) are λ 1 = 1+d 1 , λ 2 = φ 2 ( a 2 d 1 a 1 − d 2 ). Since d 1 > 0, |λ 1 | > 1 always. So E x is always unstable. Now if a 2 d 1 > a 1 d 2 , then E x is a source and if a 2 d 1 < a 1 d 2 , then E x is a saddle. By a similar argument for E y , we get E y is also always unstable. It is a source when Lastly we will investigate the stability character of E xy . Using the computer algebra system Mathematia, we can find the solutions to the characteristic polynomial of J(S(E xy )), we will call them λ 1 and λ 2 . Now λ 1 , λ 2 have different values depending on whether or not E x , E y are sources or saddles. If E x , E y are sources, then λ 1 < 1 while λ 2 > 1, thus E xy is a saddle. If E x , E y are saddles, then λ 1 , λ 2 > 1, thus E xy is a source. Thus we have the condition that E xy is a source if Notice that it is not possible for E y to be a saddle and E x to be a source, or the other way around, else the condition for the existence of E xy is violated. Thus we have shown that E x , E y and E xy are always unstable, completing the proof.
The equilibrium equations of (71) are given by, The solutions of (72) gives the following equilibrium points, E 0 = (0, 0, 0), Similar to system (69) E 0 , E x , E y and E z will always exist. While for E xy , E xz , E yz and E xyz to exist, each of the coordinates must be non-negative.
Thus, the numerators and denominators of these fractions must be either be both positive or both negative.
First notice that if we consider only the points on the x-axis, y-axis, or z-axis the results of Lemma 1 apply because if we restrict (71) to one variable, we have (67). Similarly if we consider points only in the xy-plane, xz-plane or yz-plane the results of lemma 2 apply because if we restrict (71) to two variables we have (69).
So we only need consider points in the interior of the positive octant.
The global dynamics of (71) are described in the following theorem which follows from Theorem 15.
Next the case when A ≤ 0. By (73) there is only one equilibrium point therefore all roots of G(λ) must be in the interval (0, 1). Since all roots of the characteristic polynomial are less than 1, E 0 is a locally asymptotically stable in this case. Now A = 0 if and only if G(1) = 0, so in this case E 0 is non-hyperbolic, but we will show that E 0 is still an attractor in this instance. Consider A ≤ 0, and the previous claim. Since E 0 is the only equilibrium point and the inequalities from the claim still hold, E 0 is an attractor. Thus lim n→∞ (x n , y n , z n ) = (0, 0, 0) for A ≤ 0, completing the proof.
To discuss the dynamics of (78), we will first consider the case when x = 0.
This restriction gives the system below in the yz plane.