Global Dynamics of Some Competitive and Cooperative Discrete Dynamical Systems

This dissertation investigates the dynamics of some second-order difference equations and systems of difference equations whose defining functions satisfy certain monotonicity properties. In each study we utilize the theory for specific classes of monotone difference equations to establish local and global dynamics. Manuscript 1 is an introduction that provides fundamental definitions and important results for difference equations that are used throughout the rest of the thesis. Manuscript 2 presents some potential global dynamic scenarios for competitive systems of difference equations in the plane. These results are extended to apply to the class of second-order difference equations whose transition functions are decreasing in the first variable and increasing in the second. In particular, these results are applied to investigate the following equation as a case study: xn+1 = Cxn−1 + Exn−1 axn + dxn + f , n = 0, 1, . . . , (1) where the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C,E, a, d, f ≥ 0, C+E > 0, a+ C > 0, and a+ d > 0. A rich collection of additional dynamical behaviors for Equation (1) are established to provide a nearly complete characterization of its global dynamics with the basins of attraction of equilibria and periodic solutions. Manuscript 3 considers the following second-order generalization of the classical Beverton-Holt equation: xn+1 = af(xn, xn−1) 1 + f(xn, xn−1) , n = 0, 1, . . . . (2) Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that the solution is defined. Local and global dynamics of Equation (2) are presented in the event f is chosen to be a certain type of linear or quadratic polynomial. Particular consideration is given to the existence problem of period-two solutions. Manuscript 4 presents an order-k generalization of Equation (2), xn+1 = af(xn, xn−1, . . . , xn+1−k) 1 + f(xn, xn−1, . . . , xn+1−k) , n = 0, 1, . . . , k ≥ 1, (3) where f remains a function nondecreasing in all of its arguments, a > 0, and x0, x−1, . . . , x1−k ≥ 0. We examine several interesting examples in which f is a transcendental function. This manuscript establishes conditions under which Equation (3) possesses a unique positive equilibrium that is a global attractor of all solutions with positive initial conditions. In particular, results are presented for the special case in which f(x, . . . , x) is chosen to be a concave function.

to the class of second-order difference equations whose transition functions are decreasing in the first variable and increasing in the second. In particular, these results are applied to investigate the following equation as a case study: x n+1 = Cx 2 n−1 + Ex n−1 ax 2 n + dx n + f , n = 0, 1, . . . , where the initial conditions x −1 and x 0 are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C, E, a, d, f ≥ 0, C + E > 0, a + C > 0, and a + d > 0. A rich collection of additional dynamical behaviors for Equation (1) are established to provide a nearly complete characterization of its global dynamics with the basins of attraction of equilibria and periodic solutions.
Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers such that the solution is defined. Local and global dynamics of Equation (2) are presented in the event f is chosen to be a certain type of linear or quadratic polynomial. Particular consideration is given to the existence problem of period-two solutions.
Manuscript 4 presents an order-k generalization of Equation (2), x n+1 = af (x n , x n−1 , . . . , x n+1−k ) 1 + f (x n , x n−1 , . . . , x n+1−k ) , n = 0, 1, . . . , k ≥ 1, where f remains a function nondecreasing in all of its arguments, a > 0, and  Discrete dynamical systems describe the evolution of a quantity or population whose changes are measured over discrete time intervals. Difference equations may be thought of more specifically as recurrence relations that describe a discrete dynamical system by relating the size of the next state (or generation), often denoted x n+1 , to some function of the sizes of several past states x n , x n−1 , . . . . For example, a second-order autonomous difference equation may take the form x n+1 = f (x n , x n−1 ), n = 0, 1, . . . , The paramount goal is to determine the global dynamics of a difference equation by analytically characterizing the end behavior of all solutions as n → ∞.

Local Stability Analysis
To develop the necessary vocabulary we will utilize to study second-order difference equations, we will first reference some fundamental definitions provided in [5]. All definitions will accommodate the second-order Equation (1), but analogous statements will hold for equations of higher order or systems of difference equations. In particular, related preliminary material may be found in [6].
Definition 1 A number x ∈ I satisfying x = f (x, x) is called an equilibrium, or fixed point, of Equation (1).
Definition 2 Let x be an equilibrium of Equation (1).
(ii) x is called locally asymptotically stable if it is locally stable and if there exists γ > 0 such that for all x 0 , x −1 ∈ I with |x 0 − x| + |x −1 − x| < γ, we have lim n→∞ x n = x.
(iii) x is called a global attractor if for every x 0 , x −1 ∈ I we have lim n→∞ x n = x.
(iv) x is called globally asymptotically stable if it is locally stable and a global attractor.
(v) x is called unstable if it is not stable.
(vi) x is called a repeller if there exists r > 0 such that for all x 0 , x −1 ∈ I with 0 < |x 0 − x| + |x −1 − x| < r, there exists N ≥ 1 such that A repeller is an unstable equilibrium.  [6] or Theorem 1.1.1 of [5]) summarizes the potential cases that will be used to classify the local stability of equilibria.
Theorem 1 Consider an equilibrium x of Equation (1).
(i) x is locally asymptotically stable if and only if every solution of Equation  Much of our work will investigate the existence of periodic solutions of prime period two. The general definition of a periodic solution is given below.

Definition 3 A solution {x n } of Equation (1) is said to be periodic with period
p if x n+p = x n for all n ≥ −1.
A solution {x n } is said to be periodic with prime period p, or a minimal period-p solution, if it is periodic with period p and p is the least positive integer for which Equation (4) holds.

Monotone Systems of Difference Equations
One can also consider systems of difference equations of the form x n+1 = g(x n , y n ) y n+1 = h(x n , y n ) , n = 0, 1, . . . , where g and h are given functions and the initial condition (x 0 , y 0 ) comes from some considered set in the intersection of the domains of g and h. A great deal of theory has been established for such systems in the event the defining functions obey certain monotonicity restrictions.
Definition 4 Let R be a subset of R 2 with nonempty interior, and let T : R → R be a continuous map. Set T (x, y) = (g(x, y), h(x, y)). The map T is competitive if g(x, y) is nondecreasing in x and nonincreasing in y while h(x, y) is nonincreasing in x and nondecreasing in y. If both g and h are nondecreasing in x and y, we say that T is cooperative. If T is competitive (resp. cooperative), the associated system of difference equations (5) is said to be competitive (resp. cooperative). The map T and the associated system of difference equations are said to be strongly competitive (resp. strongly cooperative) if the adjectives nondecreasing and nonincreasing are replaced by increasing and decreasing, respectively.
Competitive and cooperative systems have been widely studied, largely due to their applicability to biological modeling. These monotone systems rank among the most important classes of systems that model interspecies relationships. Relevant research in evolutionary biology may be found in [2,4]. The theory developed for such systems provides useful insight into the global dynamics of difference equations such as Equation (1)  system of difference equations. Set x n−1 = u n and x n = v n to obtain the equivalent ). The second iterate T 2 is given by ) .
If f is a function nonincreasing in the first argument and nondecreasing in the second, the second iterate of its corresponding map, T 2 , is competitive; see [8]. General dynamic scenarios for Equation (1) when f exhibits this monotonic character (and, more generally, for competitive systems) will be presented in Manuscript 2.
If f is a function nondecreasing in both arguments, then T 2 is cooperative. Indeed, Manuscripts 3 and 4 will investigate Equation (1) for a class of functions that are always nondecreasing in both arguments.
We now present some general results discussed in [3,8] for order-preserving maps that provide an essential foundation for many consequential results established for competitive and cooperative systems. More specific background material is presented in Manuscript 2 that is tailored to competitive systems.
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 whenever x ≺ y. The next result is stated for order-preserving maps on R n .
Theorem 2 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.
If a T ( a) and T ( b) b, then a, b is invariant and: (i) There exists a fixed point of T in a, b .
(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 .
We say that { x n } n∈Z is an entire orbit of a map T : This orbit is said to join u 1 to u 2 if x n → u 1 as n → −∞ and x n → u 2 as n → ∞. The following result is for strictly orderpreserving maps.
Theorem 3 (Order Interval Trichotomy of Dancer and Hess) Let u 1 u 2 be distinct fixed points of a strictly order-preserving map T : A → A, 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 .
We also have the following powerful corollaries. These results have been utilized in papers such as [1] to determine the basins of attraction of certain fixed points; moreover, they provide a theoretical foundation for the investigation of the dynamics of competitive and cooperative systems. In particular, Kulenović and Merino have proven general results for monotone systems in [7,8,9,10] that establish the existence of certain invariant curves that may separate regions of different dynamical behaviors for special cases of System (5).
In many cases such curves will be classified as the stable or unstable manifolds for saddle-point equilibria. We will utilize such results to establish the global dynamics of several monotone difference equations in Manuscripts 2 and 3.

Abstract
In this paper we present some global dynamic scenarios for general competitive maps in the plane. We apply these results to the class of second-order autonomous difference equations whose transition functions are decreasing in the variable x n and increasing in the variable x n−1 . We illustrate our results with the application to the difference equation where the initial conditions x −1 and x 0 are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C, E, a, d, f ≥ 0, C + E > 0, a + C > 0, and a + d > 0. We characterize the global dynamics of this equation with the basins of attraction of its equilibria and periodic solutions.

Introduction
Consider the second-order quadratic-fractional difference equation where the parameters satisfy C, E, a, d, f ≥ 0, C + E > 0, and a + C > 0, and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers such that x −1 x 0 > 0 when f = 0. We also stipulate that a + d > 0 to avoid overlap with the study of quadratic difference equations in [1]. Notice that Equation (1) is a special case of the equation where F = 0. For Equation (1) we will precisely define the basins of attraction of all attractors, which consist of the equilibrium points, period-two solutions, and points at infinity. Our investigation of the global character of Equation (1) will be based on the theory of competitive systems.
The special case of Equation (1) where C = a = 0 is one of the semi-implicit discretizations of the logistic differential equation where r and K are positive constants that represent the growth rate and sustainable population level, respectively. The more general logistic differential equation where r, K, M are positive constants, will have Equation (1) as one of its discretizations. Thus Equation (1) has potential applications in population dynamics. In particular, the special case of Equation (2) with C = a = 0 and d = 1, or was thoroughly studied in [12] and led to the formulation of the global perioddoubling bifurcation result in [18]. We thus exclude the case when both C and a are zero to avoid overlap with previously studied results.
Both Equations (1) and (2) 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 defined. A great deal of special cases of Equation (3) have been studied in [2,11,13,14,21,22] that may engender various different dynamical phenomena. For example, the equation , n = 0, 1, . . . , was studied in [11] and also uses the theory of monotone maps given in [18,19].
However, the global dynamics of this equation is vastly dissimilar to that of Equation (1). Indeed, the authors in [11] reveal the coexistence of a sole locally asymptotically stable equilibrium point and a locally asymptotically stable minimal period-two solution. Equation (1), on the other hand, can have as many as three isolated fixed points with a saddle-point period-two solution. The possible dynamic scenarios for Equation (1) will provide motivation for obtaining corresponding results for general second-order difference equations in Section 2.3.
Many other interesting special cases of Equation (3) have been studied in [13,21,22,23] and exhibit rich dynamical behaviors that include the Allee effect, period-doubling bifurcation, Neimark-Sacker bifurcation, and chaos. More special cases in which the numerator of Equation (3) is quadratic and the denominator is linear are treated in [7,8,14].
The following theorem from [5] applies to Equation (1): Theorem 1 Let I be a set of real numbers and f : I ×I → I be a function which is nonincreasing in the first variable and nondecreasing 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 are eventually monotonic.
The consequence of Theorem 1 is that every bounded solution of Equation (4) converges to either an equilibrium, a period-two solution, or a singular point on the boundary, as in the case of the difference equation

Preliminaries
In this section we provide some basic facts about competitive maps and systems of difference equations in the plane from [18,19,20].
Definition 1 Let R be a subset of R 2 with nonempty interior, and let T : R → R be a continuous map. Set T (x, y) = (f (x, y), g(x, y)). The map T is competitive if f (x, y) is nondecreasing in x and nonincreasing in y while g(x, y) is nonincreasing in x and nondecreasing in y. If both f and g are nondecreasing in x and y, we say that T is cooperative. If T is competitive (resp. cooperative), the associated system of difference equations x n+1 = f (x n , y n ) y n+1 = g(x n , y n ) , n = 0, 1, . . . , is said to be competitive (resp. cooperative). The map T and the associated system of difference equations are said to be strongly competitive (resp. strongly cooperative) if the adjectives nondecreasing and nonincreasing are replaced by increasing and decreasing, respectively.

Definition 3
The southeast partial order on R 2 is defined such that (x 1 , y 1 ) se (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≥ y 2 . A strict inequality between points may be defined such that (x 1 , y 1 ) ≺ se (x 2 , y 2 ) if (x 1 , y 1 ) se (x 2 , y 2 ) and (x 1 , y 1 ) = (x 2 , y 2 ). An even stronger inequality may be defined such that (x 1 , y 1 ) se (x 2 , y 2 ) if x 1 < x 2 and y 1 > y 2 . (Similar orderings may be defined for the northeast partial order defined such that (x 1 , y 1 ) ne (x 2 , y 2 ) if x 1 ≤ x 2 and y 1 ≤ y 2 .) Remark 1 A competitive map T : R → R is monotone with respect to the southeast order; that is, x se y implies that T ( x) se T ( y) for all x and y in R.
A strongly competitive map T satisfies the property that, for all x and y in R, if x ≺ se y, then T ( x) se T ( y).
The following definition comes from [25].
Definition 4 A competitive map T : R → R is said to satisfy condition (O+) if for every x, y ∈ R, T ( x) ne T ( y) implies x ne y.
A result of deMottoni-Schiaffino [9] generalized by Smith [25] yields that all bounded solutions of a competitive map satisfying condition (O+) must converge.
Now we provide some theorems from [18,19,20] that will be of particular importance in our investigation of the global dynamics of Equation (1). The first two results hold for any kind of unstable fixed points of competitive maps; see [20].
, and let T : R → R be a strongly competitive map with a unique fixed pointx ∈ R, and such that T is twice continuously differentiable in a neighborhood ofx. Assume further that at the pointx the map T has associated characteristic values µ and ν satisfying 1 < µ and −µ < ν < µ, with ν = 0, and that no standard basis vector is an eigenvector associated to one of the characteristic values.
Then there exist curves C 1 , C 2 in R and there exist p 1 , p 2 ∈ ∂R with p 1 < < sex < < se p 2 such that (i) For = 1, 2, C is invariant, north-east strongly linearly ordered, such that x ∈ C and C ⊂ Q 3 (x) ∪ Q 1 (x); the endpoints q , r of C , where q ne r , belong to the boundary of R. For , j ∈ {1, 2} with = j, C is a subset of the closure of one of the components of R \ C j . Both C 1 and C 2 are tangential atx to the eigenspace associated with ν.
(ii) For = 1, 2, let B be the component of R \ C whose closure contains p .
Then B is invariant. Also, for x ∈ B 1 , T n (x) accumulates on Q 2 (p 1 ) ∩ ∂R, and for x ∈ B 2 , T n (x) accumulates on Q 4 (p 2 ) ∩ ∂R.
Then D 1 ∪ D 2 is invariant. In the case of a saddle point or nonhyperbolic fixed point of stable type we have more precise results given in [18,19].
Theorem 3 Let T be a competitive map on a rectangular region R ⊂ R 2 . Let 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 throughx that is invariant and a subset of the basin of attraction ofx, such that C is tangential to the eigenspace E λ atx, 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 5 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 4
For the curve C of Theorem 3 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 3 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 5 (A) Assume the hypotheses of Theorem 3, and let C be the curve whose existence is guaranteed by Theorem 3. 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 ofx, then T has no periodic points in the boundary of Q 1 (x) ∪ Q 3 (x) except forx, 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 ifx is a fixed point of T , the stable set W s (x) ofx is the set {x ∈ R : T n (x) →x} and the unstable set W u (x) ofx is the set When T is non-invertible, the set W s (x) may not be connected and be made up 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 ofx.
Theorem 6 In addition to the hypotheses of part (B) of Theorem 5, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis. If the curve C of Theorem 3 has endpoints in ∂R, then C is the stable set W s (x) ofx, and the unstable set W u (x) ofx is a curve in R that is tangential to E µ atx 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 .

Remark 2
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 second partial derivative D 2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Equation (4) 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 (4) is a strongly competitive map on I × I.
Set x n−1 = u n and x n = v n in Equation (4) to obtain the equivalent system ). The second iterate T 2 is given by which is strongly competitive on I × I; see [18,19].

Remark 3 The characteristic equation of Equation (4) at an equilibrium point
has two real roots λ, µ which satisfy µ < 0 < λ and |λ| < µ whenever f is strongly decreasing in the first variable and strongly increasing in the second variable. Thus the applicability of Theorems 3-6 depends on the existence and nonexistence of a minimal period-two solution.

Main Results
In this section we present some global dynamic scenarios for competitive maps which are motivated by some dynamic scenarios for Equation (1). Thus different global dynamic scenarios for Equation (1) will be examples of general global results for competitive maps.
Theorem 7 Consider the competitive map T generated by system (5) on a rectangular region R. Suppose T has no minimal period-two solutions in R, is strongly competitive on int R, and is C 2 in a neighborhood of any fixed point.
(a) Assume T has a saddle fixed point E 2 and either a singular point or another where E 1 is the southwest corner of the region R. If E 1 is a fixed point, assume it is a repeller or nonhyperbolic. Then every nonconstant solution which starts off the stable manifold W s (E 2 ) will approach the boundary of the region R. See Figure 1 for visual illustration.
In Cases (b)-(e), assume T has at least three fixed points are saddle points, and E 2 is locally asymptotically stable and is the southwest corner of the region R. Assume that the Jacobian J T (x) of T evaluated at both E 1 and E 3 has real eigenvalues λ, µ such that 0 < |λ| < 1 < µ and the eigenspace E λ associated with λ is not a coordinate axis. Finally, suppose that the left vertical (resp. bottom horizontal) boundary (b) In addition to the hypotheses listed above, suppose T has two additional fixed points E 4 and E 5 such that E i ne E 4 ne E 5 for i = 1, 2, 3, E 4 is a repeller, and E 5 is a saddle point. Then every solution which starts below (resp. above) the union of the stable manifolds W s (E 3 ) ∪ W s (E 5 ) (resp. W s (E 1 )∪W s (E 5 )) will approach the boundary of the region R. Every solution which starts between the stable manifolds W s (E 1 ) and W s (E 3 ) converges to Figure 2 for visual illustration.
(c) Assume exactly the hypotheses listed above. Then every solution which starts below (resp. above) the manifold W s (E 3 ) (resp. W s (E 1 )) will approach the boundary of the region R. Every solution which starts between the stable manifolds W s (E 1 ) and W s (E 3 ) converges to E 2 . See Figure 3 for visual illustration.
(d) In addition to the hypotheses listed above, suppose T has an additional fixed point E 4 such that E i ne E 4 for i = 1, 2, 3 and E 4 is nonhyperbolic of unstable type. Assume that no standard basis vector is an eigenvector associated to either of the characteristic values of E 4 . Then there exist continuous, nondecreasing, and invariant curves C 1 , C 2 (with C 1 above C 2 ) which emanate from E 4 such that the region between the curves is invariant. The region below (resp. above) the union of invariant curves is invariant, and every solution which starts in either region will approach the boundary of R. If T satisfies condition (O+), for every initial point (x 0 , y 0 ) between C 1 and C 2 the corresponding solution either converges to E 4 or approaches the boundary of R. See Figure 4 for visual illustration.
(e) In addition to the hypotheses listed above, suppose T has an additional fixed point E 4 such that E i ne E 4 for i = 1, 2, 3 and E 4 is a repeller. Assume that no standard basis vector is an eigenvector associated to either of the characteristic values of E 4 . Then there exist continuous, nondecreasing, and invariant curves C 1 , C 2 (with C 1 above C 2 ) which emanate from E 4 such that the region between the curves is invariant. The region below (resp. above) and every solution which starts in either region will approach the boundary of R. If T satisfies condition (O+), for every initial point (x 0 , y 0 ) between C 1 and C 2 the corresponding solution approaches the boundary of R. See Figure   5 for visual illustration. Proof.
(a) The existence of the global stable and unstable manifolds of the saddle point equilibrium is guaranteed by Theorems 3-6. In any case W s (E 2 ) has endpoints on the boundary of R. In view of Theorem 5 every solution which starts in W − eventually enters int Q 2 (E 2 ) and every solution which starts in . Regardless of whether z is above or below W u (E 2 ), one can find u ∈ W u (E 2 ) such that u se z. By monotonicity of the map T , this implies that T n ( u) se T n ( z) for all n ∈ N, and so lim n→∞ In a similar way the case when the initial point x 0 ∈ W − can be handled.
(b) The existence of the global stable manifolds of E 1 , E 3 , E 5 and the global unstable manifold of E 5 is guaranteed by Theorems 3-6; see also [24]. Indeed, by Theorems 3 and 4, both W s (E 1 ) and W s (E 3 ) have endpoints at E 4 , and W s (E 5 ) has endpoints at E 4 and some point on the boundary of R. Since Let a, b be the order interval consisting of all c ∈ R 2 such that a ne c ne b. Consider an arbitrary initial point x 0 = (x 0 , y 0 ) ∈ int E 1 , E 3 . Then there exist some projections onto the unstable manifolds W u (E 1 ) and W u (E 3 ), P y and P x , respectively, such that P y se x 0 se P x , which implies that T ( x 0 ) ∈ int E 1 , E 3 and the result follows.
Now suppose x 0 ∈ B\ E 1 , E 3 , where B denotes the region between the stable manifolds W s (E 1 ) and W s (E 3 ). Then there must exist s l ∈ W s (E 1 ) and s u ∈ W s (E 3 ) such that s l se x 0 se s u . But then and thus T n ( x 0 ) ∈ E 1 , E 3 for n sufficiently large, which implies that and thus the solution approaches the boundary of the region R. The treat- .
. Without loss of generality suppose x 0 is to the right of W s (E 5 ) (otherwise the treatment is analogous) so that there exists some p ∈ W s (E 5 ) such that p se x 0 . We claim that there exists some n such that T n ( x 0 ) ∈ int Q 4 (E 5 ). Certainly for any n it is the case that T n ( p) se T n ( x 0 ). For a contradiction suppose T n ( x 0 ) → E 5 as n → ∞. But then for some n, T n ( x 0 ) ∈ W s loc (E 5 ), the local stable manifold tangential to the , we now have the relation T n ( p) se T n ( x 0 ), but any points on this invariant curve are not comparable with respect to the southeast ordering.
By continuity of T the only finite points to which any solution may converge are fixed points, and therefore it must be the case that eventually the solution .
In any case we can compare x 0 to a point on W s (E 5 ) and show using a similar argument as that used above that the corresponding solution must enter either int Q 4 (E 5 ) or int Q 2 (E 5 ) (in which case we can apply the previous results to establish the long-term behavior of the solution).
Finally suppose x 0 ∈ E 2 , E 4 \B. By comparing x 0 to some point on either as appropriate, we may utilize a similar argument as before to deduce that the corresponding solution cannot converge to E 3 or E 1 .
, and we can apply the results of the previous case to complete the proof.
(c) The proofs used to show that the region between the stable manifolds W s (E 1 ) and W s (E 3 ) is the basin of attraction of E 2 and that solutions with initial conditions starting outside this region will approach the boundary of the region are similar to those provided in case (b) and will be omitted.
(d) The proof used to show that the region between the stable manifolds W s (E 1 ) and W s (E 3 ) is the basin of attraction of E 2 is the same as in case (b) and will be omitted. In view of the main result in [24] there exists a most unstable manifold W u max (E 4 ), which is the graph of a decreasing function passing through E 4 , which at E 4 is tangent to the eigenspace that corresponds to the largest (in absolute value) eigenvalue. The existence of the invariant curves C 1 , C 2 is guaranteed by Theorem 2 applied to the open rectangular region R = int R, in which T has only the interior fixed point E 4 . The endpoints q 1 and q 2 of the full curves C 1 and C 2 , respectively, should coincide with the fixed points E 1 and E 3 on the boundary. The proofs that nth iterates of points which start in the invariant region below W s (E 3 ) ∪ C 2 (resp. above W s (E 1 ) ∪ C 1 ) are approaching the boundary of the region R are similar to those provided in case (b); also, see Theorem 2 (ii). If an initial point x 0 = (x 0 , y 0 ) ∈ Q 1 (E 4 ) is between C 1 and C 2 , then there exist if T additionally satisfies condition (O+) then the solution approaches the boundary of the region or T n ( x 0 ) → E 4 as n → ∞.
(e) The proof for this case is analogous to that provided in case (d) and will be omitted. Note that if T satisfies condition (O+) then every solution with between the curves C 1 and C 2 must approach the boundary of the region since in this case E 4 is a repeller and has trivial basin of attraction. In the case of Equation (4) we have the following results which are direct applications of Theorem 7. See [10] for similar results.
where b ≤ ∞. Assume that f is decreasing in the first variable and increasing in the second variable on (a, b) 2 such that f is C 2 in a neighborhood of any fixed point.    (a) Assume that Equation (4) has one saddle equilibrium point x > a and that a is either a repelling (or nonhyperbolic) equilibrium point or a singular point of R. If Equation (4) has no minimal period-two solutions, then every nonconstant solution which starts off the stable manifold W s ((x, x)) will approach the boundary of the region R.
In Cases (b)-(e), assume that Equation (4) has a locally asymptotically stable equilibrium point a and the unique minimal period-two solution {a, p, a, p, . . .}, with p > a, such that P 1 = (a, p) and P 2 = (p, a) are saddle points. Assume further that the Jacobian J T 2 (x) of T 2 , where T is the map corresponding to Equation (4), evaluated at both P 1 and P 2 has real eigenvalues λ, µ such that 0 < |λ| < 1 < µ and the eigenspace E λ associated with λ is not a coordinate axis. Finally, suppose that (b) In addition to the hypotheses listed above, assume that Equation (4) has two additional equilibrium points x 2 , x 1 such that x 2 > x 1 > a, x 1 is a repeller, and x 2 is a saddle point. Then every solution which starts between the stable manifolds W s (P 1 ) and W s (P 2 ) converges to (a, a) while every solution which approaching the boundary of the region R.
(c) Assume exactly the hypotheses listed above. Then every solution which starts between the stable manifolds W s (P 1 ) and W s (P 2 ) converges to (a, a) while every solution which starts below W s (P 2 ) (resp. above W s (P 1 )) is approaching the boundary of the region R.
(d) In addition to the hypotheses listed above, assume that Equation (4) has an additional equilibrium point x such that x > a and x is nonhyperbolic of unstable type. Assume that no standard basis vector is an eigenvector associated to either of the eigenvalues of the Jacobian J T 2 (x) evaluated at (x, x). Then there exist two continuous and nondecreasing curves C 1 and C 2 (with C 1 above C 2 ) which start at (x, x) and serve as the boundary of the region containing the basin of attraction of (x, x). Every solution which starts between the stable manifolds W s (P 1 ) and W s (P 2 ) converges to (a, a), while every solution which starts below W s (P 2 ) ∪ C 2 (resp. above W s (P 1 ) ∪ C 1 ) is approaching the boundary of the region R. Every solution which starts between C 1 and C 2 converges to (x, x) or approaches the boundary of the region.
(e) In addition to the hypotheses listed above, assume that Equation (4) has an additional equilibrium point x such that x > a and x is a repeller. Assume that no standard basis vector is an eigenvector associated to either of the eigenvalues of the Jacobian J T 2 (x) evaluated at (x, x). Then there exist two continuous and nondecreasing curves C 1 and C 2 (with C 1 above C 2 ) which start at (x, x). Every solution which starts between the stable manifolds W s (P 1 ) and W s (P 2 ) converges to (a, a), while every solution which starts below W s (P 2 ) ∪ C 2 (resp. above W s (P 1 ) ∪ C 1 ) is approaching the boundary of the region R. Every solution which starts between C 1 and C 2 approaches the boundary of the region. Proof.
In all cases recall that the applicability of Theorem 7 to a map T requires the nonexistence of minimal period-two solutions. Since we seek to apply the results of this theorem to T 2 , where T is the map corresponding to Equation (4), we must rule out the possibility of minimal period-four solutions for Equation (4). However, realize that Theorem 1 specifically precludes the existence of periodic solutions of prime period greater than two.
(a) In view of Remark 2 the second iterate T 2 of the map T associated with Equation (4) is strongly competitive on (a, b) 2 . Applying Theorem 7 part (a) to T 2 we complete the proof.
(b) In view of Remark 2 the second iterate T 2 of the map T associated with Equation (4) is strongly competitive and has five equilibrium points E 1 = for every (x 0 , y 0 ) between the stable manifolds W s (P 1 ) and W s (P 2 ) . Furthermore, we also have that where we utilize continuity of the map T . Consequently lim n→∞ T n ((x 0 , y 0 )) =  (4); see [18,19].
2 Remark 4 As shown in [20], the curves C 1 and C 2 may coincide on one or both sides of the fixed point. Different global dynamic scenarios for competitive or cooperative maps and corresponding difference equations were established in the cases when these maps have a finite or infinite number of period-two solutions in [2,4,20].
Remark 5 Some special cases of Theorems 7 and 8 have appeared in a number of papers. For example, the global dynamics of the system where a, b ∈ (0, 1) and x 0 , y 0 ∈ [0, ∞), as studied in [6], follows from Theorem 7 case (a). Furthermore, several cases of the global dynamics of the system where a, b, c, d ∈ (0, ∞) and x 0 , y 0 ∈ [0, ∞), as studied in [3], follow from Theorem 7 cases (a)-(d).
The global dynamics of the difference equations where B, γ > 0 and B < 4γ + 1, and The global dynamics of the difference equation where b, c, f ≥ 0 and b + c + f > 0, is described by Theorem 8 cases (a)-(d) for several regions of parameters.
It is worth noticing that case (e) in both Theorems 7 and 8 has been identified for the first time in the case of Equation (1). (1) In this section we apply the results of Theorem 8 to the study of the global dynamics of Equation (1). We begin by investigating the existence and local stability of equilibria and periodic solutions. (1) An equilibrium point x of Equation (1) satisfies

Equilibrium Solutions of Equation
In the case when f > 0, it is clear that Equation (1) always has the zero equilibrium. The following cases will investigate the existence of any remaining positive equilibrium points.
Case 1 (af > 0): When f > 0 and a > 0, denote by x + and x − the two possibly remaining positive equilibria: A routine checking will find the parametric conditions under which the above solutions x + and x − are both real and nonnegative. Tables   1 and 2 summarize the values of parameters for which Equation (1) has one, two, or three equilibrium points and possibly period-two solutions (the existence of which we will investigate in Section 2.4.3). Table 1. Existence of equilibria and period-two solutions for a > 0, f > 0, C > 0. Table 2. Existence of equilibria and period-two solutions for a > 0, f > 0, C = 0.
Case 2 (af = 0): When f > 0 but a = 0 notice that Equation (6) reduces to which has the isolated solutions x 0 = 0 and possibly Existence of equilibria is summarized in Table 3.
none Table 3. Existence of equilibria and period-two solutions for a = 0, f > 0.
When a > 0 and f = 0, Equation (6) becomes and since necessarily x = 0 in this case, Descartes' Rule of Signs yields that there may exist at most one positive fixed point x + > 0. See Table 4 for a summary of the parametric conditions under which an equilibrium point exists. Table 4. Existence of equilibria for a > 0, f = 0.
In the case a = f = 0 the solutions of Equation (6) must satisfy Since we must have x > 0, this equation has the isolated solution when d > C and E > 0. All remaining subcases may be summarized in Table 5. Table 5. Existence of equilibria for a = 0, f = 0.

Local Stability Analysis of the Equilibrium Solutions
Define the function g such that . The partial derivatives of g are given by The characteristic equation of the linearization of Equation (1) about x is λ 2 = P λ + Q, where P = g u (x, x) and Q = g v (x, x). Using Equation (6), this becomes Lemma 1 The zero equilibrium x 0 = 0, which exists whenever f > 0, has the following stability: Proof. Notice that, evaluated at (x 0 , x 0 ), P = 0 and Q = E f . Using Theorem 2.13 of [16], the first two results of the claim are immediate by checking the necessary inequalities. If E = f , then the characteristic equation of the linearized equation of Equation (1) (given in Equation (8)) is λ 2 = 1 and hence λ 1 = 1, λ 2 = −1 so that x 0 is nonhyperbolic of resonant type (1, −1). 2 Case 1 (af > 0): , then x ± = C−d 2a is a nonhyperbolic equilibrium point of unstable type.
(b) Suppose one of the following conditions holds: Then the positive equilibrium x + is a saddle point. Proof.
(a) Notice that, for x = 0, we have the following: and From Equation (10) it is clear that, for x ± = C−d 2a , |P | + 1 − Q = 0, and hence this equilibrium is indeed nonhyperbolic. Using the equilibrium equation and the fact that 2ax ± + d = C, notice that the characteristic equation (8) becomes Since λ 1 = 1 and λ 2 = − 2Cx + E Cx + E < −1, this nonhyperbolic equilibrium point is of the unstable type.
(b) Note that in all but the last case x + is the unique positive equilibrium. It is clear from Equation (10) However, by definition in Equation (7) we have that (c) Since by Equation (10) we have that |P | + 1 − Q < 0 and hence |P | < |1 − Q|.
after we use Equation (6). Thus by Theorem 2.13 of [16], x − is indeed a repeller.
1. If C < d and f < E, then x + is a saddle point.
2. If C > d and f > E, then x + is a repeller.
3. If C = d and f = E, then every point x > 0 is a nonhyperbolic equilibrium of unstable type.
(b) If a > 0, f = 0, and either E > 0 or (C > d and E = 0), then x + is a saddle point.
1. If C < d and E > 0, then x + is a saddle point.
2. If C = d and E = 0, then any x > 0 is a nonhyperbolic equilibrium of unstable type. Proof.
(a) By Equation (10), we have that By Equation (9) we have that |P | − 1 + Q > 0. In the case when C > d, we can also check immediately that |Q| > 1. Thus when E > f and C < d, x + is a saddle point, and when E < f and C > d, x + is a repeller, which establishes Cases 1 and 2. In the nonhyperbolic case when E = f and C = d, each point x > 0 is an equilibrium point and the characteristic equation (8) reduces to But then λ 1 = 1 and λ 2 < −1 so that each nonhyperbolic equilibrium is of the unstable type.
Thus we have verified Case 3, and the proof is complete.
(c) 1. When d > C and E > 0, Equation (10) reduces to and coupling this result with Equation (9) shows that x + is a saddle point.
(a) Suppose on the contrary that there exists a strictly positive periodic solution {φ, ψ, φ, ψ, . . .} with φ = ψ. Now φ and ψ satisfy: Since a + d > 0, a + C > 0, and C + E > 0, it is clear that the latter factor of Equation (11)   The following result gives the relation between the equilibria and period-two solutions.
(a) We need the check the following inequality: Notice that the right-hand side of Inequality (12) is positive since a > 0: we may square both sides of Inequality (12) to obtain which is always true by assumption. Thus indeed ψ < x − .
(b) In this case x + = f −E C−d , so ψ < x + by definition since necessarily d > 0. Proof. Using the substitution x n−1 = u n , x n = v n , Equation (1) becomes The corresponding map T is thus given by The second iteration T 2 of the map is given by Notice that the map T 2 is strongly competitive. The Jacobian of T 2 is given by Notice that if Equation (1)  (a) If f > E and C > 0, then ψ = f −E C . Therefore |λ 1 | < 1 ⇐⇒ E < aψ 2 + dψ + f, and  The following result will establish the axes as the unstable manifolds for the isolated period-two points on the axes and will establish the axes as a repelling set when the period-two solution does not exist. (1).
If f > E, then every solution with initial conditions x −1 x 0 = 0 and x −1 + x 0 > 0 will break into two subsequences of odd-and even-indexed terms.
One subsequence will be identically zero, and the other will converge to 0 if x i < f − E C = ψ and will be monotonically increasing (and hence unbounded) If f ≤ E, as above, one subsequence will be identically zero and the other will be unbounded.
Then every solution with initial conditions x −1 x 0 = 0 and x −1 + x 0 > 0 will break into two subsequences of odd-and even-indexed terms. One subsequence will be identically zero, and the other will converge to 0 if E < f and will be monotonically increasing (and hence unbounded) if E > f . Every point on the axes will be a period-two point if E = f . Proof.
(a) Suppose f C > 0. Without loss of generality suppose x −1 = 0 and x 0 > 0. Then Since x 3 = 0, we may show a similar inequality as above for x 4 and x 2 . By induction we may establish the claim.
(b) Now suppose f > 0 and C = 0. Again without loss of generality we may assume x −1 = 0 and x 0 > 0. Now and we again use induction to establish the claim.

2
If T is the map corresponding to Equation (1), then the strongly competitive map T 2 inherits as equilibria all corresponding fixed points and period-two points of Equation (1). With this in mind, the map T 2 may have as many as five isolated fixed points, listed below: One can verify that no eigenvector associated with either characteristic value of (x + , x + ) (or (x ± , x ± )) is aligned with a coordinate axis. Using Lemmas 1-7 and Theorem 8, we may now deduce the global dynamics of Equation (1). Again, assume C + E > 0, a + C > 0, and a + d > 0.
(a) Suppose one of the following conditions holds: In Cases 1-3, Equation In the following cases, assume E > 0 and d > 0.
Then Equation (1) has three equilibrium points: 0, which is locally asymptotically stable, x − , which is a repeller, and x + , which is a saddle point.
Then Equation (1) possesses only the zero equilibrium, and it is a repeller.
All nonzero solutions are unbounded.
(b) Suppose a = 0 and one of the following conditions holds: In either case Equation (1) possesses every positive number as an equilibrium.
(In the first case, 0 is also an equilibrium.) All non-equilibrium solutions are unbounded and will oscillate between approaching 0 and ∞.   x n = 0. Then subsequences of even-and odd-indexed terms are monotonically decreasing, so there must exist some k ∈ N such that for all n > k, both x n+1 < x n−1 and x n+2 < x n . Since C ≥ d, we may use the first inequality to show that In a similar way, our second assumed inequality implies that dx n +E < dx n+1 .
But then However, this contradicts the fact that Equation (1) has no minimal periodtwo solution. Consequently, every solution has an unbounded subsequence.
(d) Notice that Equation (1) reduces to After taking the logarithm of both sides and seting u n = ln(x n ) and K = ln C d we obtain the linear, second-order, nonhomogeneous equation which, after the substitution v n = u n − u n−1 , reduces to v n+1 + 2v n = K.
Equation (14) is of first order and has the general solution and hence This first-order nonautonomous equation now has solution Finally, Equation (13) has solution Thus we see that, as n → ∞, every solution {x n } will oscillate between approaching 0 and ∞. We should remark that the above solution is valid for Equation (13) for all C, d > 0, even when C = d, the condition treated in part (b). If C = d the solution reduces to Theorem 11 Assume C = 0. (b) Suppose C = 0. In view of E < f , Equation (1) implies: By Inequality (15) it is clear that the susbsequences of even-and odd-indexed terms of Equation (1) are monotonically decreasing, which is consistent with Theorem 1. Since Equation (1) is bounded below, all solutions must converge to x 0 .  Conjecture 2 Suppose f = E = 0, a > 0, and C ≤ d.
Equation (1) has no equilibrium points, and all solutions are unbounded.
Further, we have added the stipulation Ed > 0 in parts (b) through (e) of Theorem 9 to ensure the applicability of Theorem 3, which requires that the eigenspace associated with the eigenvalue λ 1 does not align with a coordinate axis.
We believe the established results for Ed > 0 in which the period-two solution exists will still hold for Ed = 0, and thus we leave the following conjecture.

List of References
where f is a function nondecreasing in both arguments, the parameter a is a positive constant, and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. We will discuss several interesting examples of such equations and present some general theory. In particular, we will investigate the local and global dynamics in the event f is a certain type of linear or quadratic polynomial, and we explore the existence problem of period-two solutions.
Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. Equation (1) is a generalization of the first-order Beverton- Holt equation where a > 0 and x 0 ≥ 0. The global dynamics of Equation (2) may be summarized as follows:

Many variations of Equation (2) have been studied. German biochemist Leonor
Michaelis and Canadian physician Maud Menten used the model in their study of enzyme kinetics in 1913; see [18]. Additionally, Jacques Monod, a French biochemist, happened upon the model empirically in his study of microorganism growth around 1942; see [18]. It was not until 1957 that fisheries scientists Ray Beverton and Sidney Holt used the model in their study of population dynamics.
The so-called Monod differential equation is given by where N (t) is the concentration of bacteria at time t, dN dt is the growth rate of the bacteria, S(t) is the concentration of the nutrient, r is the maximum growth rate of the bacteria, and a is a half-saturation constant (when S = a, the right-hand side of Equation (4) equals r/2). Based on experimental data, the following system of two differential equations for the nutrient S and bacteria N , as presented in [18], is given by where the constant γ is called the growth yield. Both Equation (4) and System (5) contain the function f (x) = rx/(a + x) known as the Monod function, Michaelis-Menten function, Beverton-Holt function, or Holling function of the first kind; see [4,9].
One possible two-generation population model based on Equation (2), where a i > 0 for i = 1, 2 and x −1 , x 0 ≥ 0, was considered in [16]. The global dynamics of Equation (6) may be summarized as follows: This result was extended in [4] to the case of a k-generation population model based on Equation (2) of the form where a i ≥ 0 for i = 0, 1, . . . , k − 1, k−1 i=0 a i > 0, and x 1−k , . . . , x 0 ≥ 0. It was shown that the global dynamics of Equation (7) is given precisely by (3), where a = k−1 i=0 a i and we consider all initial conditions positive.
The simplest model of Beverton-Holt type which exhibits two coexisting attractors and the Allee effect is the sigmoid Beverton-Holt or (second-type Holling) difference equation where a > 0 and x 0 ≥ 0. The dynamics of Equation (8) may be concisely summarized as follows: where x − and x + are the two positive equilibria when a ≥ 2; see [1,4]. One possible two-generation population model based on Equation (8), where a i > 0 for i = 1, 2 and x −1 , x 0 ≥ 0, was considered in [3]. However, the summary of the global dynamics of Equation (10) is not an immediate extension of the global dynamics of Equation (8) as given in (9); see [3]. Equation (10) can have up to three equilibrium solutions and up to three period-two solutions. In the case when Equation (10) has three equilibrium solutions and three period-two solutions, the zero equilibrium, the larger positive equilibrium, and one periodtwo solution are attractors with substantial basins of attraction, which together with the remaining equilibrium and the global stable manifolds of the saddle-point period-two solutions exhaust the first quadrant of initial conditions. This behavior happens when the coefficient a 2 is in some sense dominant to a 1 ; see [3]. Such behavior is typical for other models in population dynamics such as which were also investigated in [3]. In the case of a k-generation population model based on the sigmoid Beverton-Holt difference equation with k > 2, one can expect to have attractive period-k solutions as well as chaos.
The first model of the form given in Equation (1), where f is a linear function in both variables (that is, f (u, v) = cu + dv for c, d, u, v ≥ 0) was considered in [17] and some global dynamics were described in part of the parametric space.
Here we will extend the results from [17] to the whole parametric space. In this paper we will then restrict ourselves to the case when f (u, v) is a quadratic polynomial, which will give similar global dynamics to that presented for Equation (10). The corresponding dynamic scenarios will be essentially the same for any Then Equation (1) becomes x n+1 = F (x n , x n−1 ) for all n = 0, 1, . . . , where F (u, v) is nondecreasing in both of its arguments.
The following theorem from [2] immediately applies to Equation (1).
Theorem 1 Let I be a set of real numbers and F : I ×I → I be a function which is nondecreasing in the first variable and nondecreasing 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 are eventually monotonic.
The consequence of Theorem 1 is that every bounded solution of Equation (12) converges to either an equilibrium, a period-two solution, or to a singular point on the boundary. It should be noticed that Theorem 1 is specific for secondorder difference equations and does not extend to difference equations of order higher than two. Furthermore, the powerful theory of monotone maps in the plane [14,15] can be applied to Equation (1) to determine the boundaries of the basins of attraction of the equilibrium solutions and period-two solutions. Finally, when f (u, v) is a polynomial function, all computation needed to determine the local stability of all equilibrium solutions and period-two solutions is reduced to the theory of counting the number of zeros of polynomials in a given interval, as given in [10]. This theory will give more precise results than the global attractivity and global asymptotic stability results in [6,7]. However, in the case of difference equations of the form where a > 0 and g is nondecreasing in all its arguments, Theorem 1 does not apply for k > 2, but the results from [6,7,11] can give global dynamics in some regions of the parametric space.
The following theorem from [8] is often useful in determining the global attractivity of a unique positive equilibrium.
be some open interval and assume that F ∈ C[I × I, (0, ∞)] satisfies the following conditions: y) is nondecreasing in each of its arguments; (ii) Equation (12) has a unique positive equilibrium point x ∈ I and the function F (x, x) satisfies the negative feedback condition: Then every positive solution of Equation (12) with initial conditions in I converges to x.

Local Stability
In this section we provide general conditions to determine the local stability of equilibrium solutions and period-two solutions.
It is clear that x n ≤ a for all n ≥ 1. An equilibrium x of Equation (1) satisfies then in view of Corollary 2 of [11] we may conclude that Further, Theorem 2.13 of [13] implies that if x is unstable, then Let (φ, ψ) be a period-two solution of Equation (1). The Jacobian matrix of the corresponding map T = G 2 , where G(u, v) = (v, F (v, u)) and F is given by Equation (11), is given in Theorem 12 of [5]. The linearized equation evaluated at (φ, ψ) is and DetJ T (φ, ψ) = D 2 F (F (ψ, φ), ψ) · D 2 F (ψ, φ).

Examples 3.3.1 Linear-Linear: f (u, v) = cu + dv
We consider the difference equation where c ≥ 0 and d > 0. If d = 0, then Equation (16) becomes Equation (2)  Since λ(x 0 ) = a(c + d), we have that Further, notice that for all values of parameters for which x + exists. Therefore Note that there is an exchange in stability from x 0 to x + as the parametric value a(c + d) passes through 1.
Since ψ + φ > 0, it must be the case that c = 0, and then 2d 2 ψφ = 0 so that one and therefore by Theorem 2 we have that all solutions with initial conditions in I converge to x + . Now suppose one initial condition is zero, so without loss of generality assume x −1 = 0 and x 0 > 0. Then x 1 = 0 and Further, one can show (c) When c > 0 and a(c + d) > 1, x + is locally asymptotically stable while x 0 is unstable. As in the proof of (b) we can employ Theorem 2 to show that all solutions with positive initial conditions must converge to x + . Since c > 0 and > 0 (and also x 2 > 0), so the solution eventually has consecutive positive terms and must converge to x + . 2

Translated Linear-Linear
We briefly consider the difference equation x n+1 = a(cx n + dx n−1 + k) 1 + cx n + dx n−1 + k , n = 0, 1, . . . , where c ≥ 0, d ≥ 0, c + d > 0, and k > 0. (It is clear that Equation (18)  Proof. The result follows from an application of Theorem 1.4.8 of [12]. We consider the difference equation Remark 1 For the analysis that follows, we will consider Equation (19) with c > 0 and d > 0. Notice that when c = 0 Equation (19) is a special case of Equation (16), and the global dynamics for this case is discussed in Theorem 3. When d = 0, Equation (19) is essentially Equation (8), the dynamics of which may be seen in (9).
An equilibrium of (19) satisfies so that all nonzero equilibria satisfy whence we easily deduce the possible solutions which are real if and only if R = (d − ac) 2 + 4c(ad − 1) satisfies R ≥ 0.
Notice that Here we have that (19) always has the zero equilibrium x 0 = 0, and
Then Equation (19) has no positive equilibria.
Then Equation (19) has the positive equilibrium x + , and it is locally asymptotically stable.
(3) Suppose d < ac, 1 > ad, and R = 0. Then Equation (19) has the positive equilibrium x ± , and it is nonhyperbolic of stable type.
(4) Suppose d < ac, 1 > ad, and R > 0. Then Equation (19) has the two positive equilibria x + , which is locally asymptotically stable, and x − , which is a saddle point.
Proof. The positivity of solutions of Equation (20) follows from Descartes' Rule of Signs. Using Equation (14), notice that Further, for the parametric values for which x + exists, which is immediately true by Inequality (21). Thus if R > 0, x + is locally asymptotically stable, and if R = 0, x ± is nonhyperbolic. In the latter case the characteristic equation of the linearization of Equation (19) about x ± , , reduces to acy 2 − (ac − d)y − d = 0, which has characteristic values y 1 = 1 and y 2 = − d ac , where −1 < y 2 < 0 since ac > d. Thus in this case x ± is nonhyperbolic of stable type.
which is of course true since c > 0. To show more specifically that x − is a saddle point when R > 0, we must show that δ(x − ) < 1, where δ is defined by Equation (15). Notice and so we have that The right-hand side of the latter inequality is positive since a(ac + d) 2 − 4d > 4ac − 4d = 4(ac − d) > 0 by assumption. But then which is automatically true since the latter factor is strictly positive and ad < 1.

But then
Thus we have that necessarily a clear contradiction since c > 0.
(3) Suppose d < ac, 1 > ad, and R = 0. Then the system corresponding to Equation (19) has the equilibria E 0 = (0, 0), which is locally asymptotically stable, and E = (x ± , x ± ), which is nonhyperbolic of stable type. Then there exists a continuous curve C passing through E such that C is the graph of a decreasing function. The set of initial conditions (4) Suppose d < ac, 1 > ad, and R > 0. Then the system corresponding to Equation (19) has the equilibria E 0 = (0, 0), which is locally asymptotically stable, which is a saddle point, and E 2 = (x + , x + ), which is locally asymptotically stable. Then there exist two continuous curves W s (E 1 ) and W u (E 1 ), both passing through E 1 , such that W s (E 1 ) is the graph of a decreasing function and W u (E 1 ) is the graph of an increasing function. The set of initial conditions Proof.
(1) The proof in this case follows from Theorem 1 as well as Lemmas 1, 2, and 3 since x 0 = 0 is the sole equilibrium of Equation (19). (3) The proof follows from an application of Theorems 1-4 of [15] applied to the cooperative second iterate of the map corresponding to Equation (19). The proof is completely analogous to the proof of Theorem 5 in [3], so we omit the details.
(4) The proof follows from an immediate application of Theorem 5 in [3]. We consider the difference equation Remark 2 For the analysis that follows, we will consider Equation (23)  An equilibrium of (23) satisfies so that all nonzero equilibria satisfy whence we easily deduce the possible solutions which are real if and only if R = (c − ad) 2 + 4d(ac − 1) ≥ 0.
Notice that Here we have that Lemma 4 Equation (23) always has the zero equilibrium x 0 = 0, and Proof. The proof follows from the fact that λ(x 0 ) = ac. 2 Lemma 5 Assume c > 0 and d > 0.
Then Equation (23) has no positive equilibria.
Then Equation (23) has the positive equilibrium x + , and it is locally asymptotically stable.
(3) Suppose c < ad, 1 > ac, and R = 0. Then Equation (23) has the positive equilibrium x ± , and it is nonhyperbolic of stable type.
(4) Suppose c < ad, 1 > ac, and R > 0. Then Equation (19) has the two positive equilibria x + , which is locally asymptotically stable, and x − , which is unstable.
(i) If K < 0, then x − is a saddle point.
(ii) If K > 0, then x − is a repeller.
(iii) If K = 0, then x − is nonhyperbolic of unstable type.

Proof. The positivity of solutions of Equation (24) follows from Descartes'
Rule of Signs. Much of the local stability analysis is symmetric to the considerations in the proof of Lemma 2. Notice that For the parametric values for which x + exists, which is immediately true by Inequality (25). Thus if R > 0, x + is locally asymptotically stable, and if R = 0, x ± is nonhyperbolic. In the latter case the characteristic equation of the linearization of Equation (19) about x ± , , reduces to ady 2 − cy + c − ad = 0, which has characteristic values y 1 = 1 and y 2 = c−ad ad , where −1 < y 2 < 0 since ad > c. Thus in this case x ± is nonhyperbolic of stable type.
which is of course true since d > 0. To more specifically classify x − , we must calculate δ(x − ). Notice and so we have that Notice that R > 0 automatically implies a(ad + c) > 2, as since c < ad. Therefore we may square both sides to obtain The investigation of the existence of periodic solutions of Equation (23) is an interesting one that involves a thorough analysis of potential parametric cases.
This analysis will reveal the potential for the existence of several nonzero periodic solutions. The juxtaposition of Equation (19) with Equation (23) illustrates an interesting phenomenon in which, loosely speaking, the dominance of the delay term x 2 n−1 contributes to the possibility of periodic solutions arising.
(2) If ∆ j ≤ 0 for some 2 ≤ j ≤ 5 and ∆ i > 0 for i = j, then Equation (29) has two distinct real roots and two pairs of conjugate imaginary roots. Consequently, Equation (23) has one minimal period-two solution.
Proof. The proof of the first statement follows from Descartes' Rule of Signs.
Then the values of ∆ k are listed above, and the veracity of the statements above may now be verified by employing Theorem 1 of [10]. Notice that ∆ 1 > 0 for all

Remark 3
The parametric conditions discussed above do not exhaust all of the parametric space but cover a substantial region of parameters for which Equation (23) possesses hyperbolic dynamics.
We will use the sufficient conditions provided in Lemmas 4, 5, and 6 to realize some global dynamic scenarios provided in [3]. We will not investigate the dynamics of Equation (23) when it has one or no positive fixed point since in such cases the dynamics should be similar to the dynamics of Equation (19) discussed in Theorem 5. The following theorem relies on results from [3] and summarizes potential hyperbolic dynamic scenarios for Equation (23) in the event it possesses three fixed points and zero, one, or three pairs of hyperbolic period-two points.
See also the statement and proof of Theorem 11 in [3].
(iii) If ∆ i ≤ 0 and ∆ i+1 ≥ 0 (such that at least one of these is strict) for some 2 ≤ i ≤ 4, and if ∆ 6 < 0, then Eq. Moreover, we give sufficient conditions to guarantee this equation has a unique positive and globally attracting fixed point.

Introduction and Preliminaries
Consider the following order-k difference equation: x n+1 = af (x n , x n−1 , . . . , x n+1−k ) 1 + f (x n , x n−1 , . . . , x n+1−k ) , n = 0, 1, . . . , k ≥ 1, where f is a continuous function nondecreasing in all arguments, the parameter a is a positive real number, and the initial conditions x 0 , x −1 , . . . , x 1−k are arbitrary nonnegative numbers such that the solution is defined. We assume f is never identically equal to the zero function.
Equation (1) is a generalization of the first-order Beverton-Holt equation where a > 0 and x −1 , x 0 ≥ 0. Global dynamics are known and may be summarized as follows: lim n→∞ x n = 0 if a ≤ 1 a − 1 if a > 1 and x 0 > 0. (2) have been studied. The form of the model actually predates its use by Beverton and Holt; see [12]. For instance, the so-called Monod system of differential equations is given by

Many variations of Equation
where N (t) is the concentration of bacteria at time t, dN dt is the growth rate of the bacteria, S(t) is the concentration of the nutrient, r is the maximum growth rate of the bacteria, k is a half-saturation constant, and the constant γ is called the growth yield; see [12]. Both Equation (2) and System (3) contain the function f (x) = rx/(a + x) known as the Monod function, Michaelis-Menten function, Beverton-Holt function, or Holling function of the first kind; see [3,9]. Some global dynamic scenarios of several two-generation models using this function were investigated in [2].
The Beverton-Holt function is an increasing and concave function and we will prove some global attractivity results for general difference equations with a transition function that is increasing and concave along the diagonal. More precisely, we will prove some global attractivity results for Equation (1), where f (x, . . . , x) is an increasing and concave function.
Theorem 1 Let I be a set of real numbers and F : I × I → I be a function which is nondecreasing in both variables. 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 are eventually monotonic.
The consequence of Theorem 1 is that every bounded solution of Equation (4) converges to either an equilibrium, a period-two solution, or to a singular point on the boundary. Notice that Theorem 1 does not apply if k > 2, but the results from [5,7,10] can give global dynamics in some regions of the parametric space.
In the case k > 2, Equation (1) may have periodic solutions of different periods and even chaos; see [6].
The following theorem from [8] applies to the kth-order Equation (1) and will be instrumental in establishing our main result.
It will be useful to examine the multivariable functions f and F along the diagonal. For convenience, make the following definitions: G(x) = F (x, . . . , x).
Despite the utility of the above results, there certainly exist scenarios in which neither the function g(x) nor G(x) is concave on the interval (0, a). In such situations it is useful to have the following theorem, which provides a sufficient condition to guarantee the existence (or nonexistence) of a unique positive fixed point that is a global attractor of positive solutions..
Set u(x) = x g(x) + x. If u(x) is an injective (or monotone) function, then it intersects the line y = a at most once. Setting u (x) > 0 and rearranging will establish the main claim.
Now lim x→0 + u(x) < a implies there exists exactly one positive fixed point of Equation (1), so Equation (13) establishes the hypothesis of (1). As in the proof of Theorem 4, the global attractivity of the unique fixed point will again follow from Theorem 2.

If lim
x→0 + u(x) ≥ a, then Equation (1) has only the zero equilibrium since u is increasing, and Equation (13) establishes the hypothesis of (2). Again we may employ the order-k generalization of Theorems 1.4.8 and A.0.1 of [11] to obtain the global attractivity of the zero equilibrium, and the proof is complete. 2

Remark 4
In some cases the veracity of Inequality (12) of Theorem 6 may imply the concavity condition required by Theorems 4 or 5 or Corollary 1, but the hypotheses of the latter results may be easier to verify.

Examples
In most of the provided examples we will focus on equations of second order for concision. However, all results can be generalized to corresponding equations of any order.
The following results give the global dynamics of Equation (14).
Theorem 7 (1) If a(p + q) > 1, then there exists a unique positive equilibrium x + , and it is a global attractor of all nonzero solutions.
Remark 5 There are a wealth of other functions f such that g(x) is concave and Corollary 1 applies to Equation (1). Second-order examples include the logarithmic function f 1 (u, v) = log ((1 + u) p (1 + v) q ) as well as the shifted sigmoid function
The applicability of Corollary 1 is limited by the fact that g(x) = (p + q)(sin(x) + x) is strictly concave only when sin(x) > 0, and therefore global results can only be obtained for a ≤ π. Using the full strength of Theorems 4 and 5 will also have limitations for any choice of a > 0; the interval [0, a] is always invariant for Equation (17), but a larger value of a would prescribe the need for a larger interval over which G(x) should be concave. Instead we may consider applying Theorem 6.
(1) If 2a(p + q) > 1, then Equation (17) has precisely one positive fixed point, and it is a global attractor of all solutions with positive initial conditions.
(2) If 2a(p + q) ≤ 1, then Equation (17) has only the zero equilibrium, and it is a global attractor of all solutions.

Remark 6
Verifying Inequality (18) in general appears to be difficult, although for specific values of p and q this hypothesis should be able to be easily checked. For example, if p = q = 1, this condition is immediately satisfied and leads to a global exchange of stability result as a passes through the critical value 1 4 . In general, Mathematica verifies this inequality should hold for all x > 0 when approximately p + q > 0.2015. For p and q smaller than this threshold, multiple equilibria or even interior periodic solutions may exist.

Order-k Linear
Consider the equation where c i ≥ 0 for i = 1, . . . , k − 1. An equilibrium x of Equation (19) satisfies the following: Using Equation (10) we see that x 0 = 0 has λ(x 0 ) k = a k−1 i=0 c i , so If a k−1 i=0 c i > 1, then Equation (19) has the unique positive equilibrium we have that x + is locally asymptotically stable whenever it exists. Proof. (1) The proof is the same as that of Theorem 5.