Global Dynamics and Boundedness of Discrete Population Models

Discrete dynamical systems are widely used in biological and entomological applications to model interacting populations. The manuscripts included in this thesis present global dynamic results for three different population models. Manuscript 1 presents basic concepts and definitions for general systems of difference equations in order to lay the theoretical foundation for the remaining sections. Manuscript 2 discusses competitive systems of difference equations with the form xn+1 = b1 xn α1 + xn + c1 yn , yn+1 = b2 yn α2 + c2 xn + yn n = 0, 1, 2, . . . , where the parameters b1, b2 are positive real numbers and α1,α2, c1, c2 and the initial conditions x0, y0 are arbitrary nonnegative numbers. In particular, the special cases when α1 = α2 = 0 and when α1 = 0 and α2 = 0 are investigated. The global behavior of the system in these cases is fully characterized. Global results are also established for general competitive systems of difference equations that have a particular orientation of equilibria and certain local stability characteristics. In Manuscript 3, the system of difference equations xn+1 = α xn 1 + β yn , yn+1 = γ xn yn xn + δ yn , n = 0, 1, 2, . . . , is analyzed, where α, β, γ, δ, x0, y0 are positive real numbers. The system was formulated by P. H. Leslie in 1948 and models a host-parasite type of prey-predator interaction. Manuscript 3 provides the most complete dynamical analysis to date of this classic model. A boundedness and persistence result along with global attractivity results for various parameter regions are established. Numerical evidence of chaotic behavior is also presented for particular solutions of the system. Finally, Manuscript 4 discusses structured models of difference equations with the forms: yn+1 = M ( f1(y (1) n ), . . . , fk(y (k) n ) ) t , n = 0, 1, 2, . . . , y0 ∈ R+, (I) and xn+1 = A xn + k  =1 f(cxn) b, n = 0, 1, 2, . . . , x0 ∈ X+, (II) In (I) and (II), M ∈ Rk×k + , A is a bounded, linear operator on an ordered Banach space X with positive cone X+, and for each  ∈ {1, . . . , k}, b ∈ X+, c is a positive, bounded linear functional on X, and f : [0,∞) → [0,∞) is a continuous function with f(0) = 0. Conditions are established under which there is a oneto-one correspondence between positive equilibrium points (persistence states) of (I) and (II). Under these conditions, and when X = R, the stability type of the zero equilibrium (extinction state) of (I) is shown to be the same as that for (II). Particular attention is given to the case when k = 2. The utility of this analysis is that the dynamics of model (II) on a high dimensionality state space X can be reduced to model (I), where the dimension of the state space is the same as the number k of nonlinearities that appear in (II).

Manuscript 2 discusses competitive systems of difference equations with the form x n+1 = b 1 x n α 1 + x n + c 1 y n , y n+1 = b 2 y n α 2 + c 2 x n + y n n = 0, 1, 2, . . . , where the parameters b 1 , b 2 are positive real numbers and α 1 , α 2 , c 1 , c 2 and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers. In particular, the special cases when α 1 = α 2 = 0 and when α 1 = 0 and α 2 = 0 are investigated. The global behavior of the system in these cases is fully characterized. Global results are also established for general competitive systems of difference equations that have a particular orientation of equilibria and certain local stability characteristics.
In Manuscript 3, the system of difference equations x n+1 = α x n 1 + β y n , y n+1 = γ x n y n x n + δ y n , n = 0, 1, 2, . . . , is analyzed, where α, β, γ, δ, x 0 , y 0 are positive real numbers. The system was for- parents Anne and Tim, as well as my sister Beth, have also always been there to encourage me to continue to pursue my dreams and to enjoy the small victories along the way. I will forever be grateful for the guidance and encouragement that those closest to me have provided during this experience. iv

Introduction
The dynamical behavior of biological populations has been a primary focus of applied mathematicians for hundreds of years. Modeling and predicting population growth can be tracked back to 1202 when Leonardo Fibonacci considered the progression of a hypothetical rabbit population in his book Liber Abaci [1,2].
These models grew more complex in 1798 when Thomas Malthus presented the Malthusian exponential growth model and in 1844 when Pierre François Verhulst introduced the logistic model of population growth [1,2]. The theory in the area of population dynamics has seen steady progress and many complicated biological phenomena have been accurately modeled. These models are referred to as dynamical systems. Dynamical systems mathematically describe the temporal progression of a given quantity and they come in the form of both difference equations and differential equations [3].
Differential equations are continuous dynamical systems that account for the behavior of a system at all times and are useful in modeling phenomena that are continuously evolving [3]. Difference equations, on the other hand, are discrete dynamical systems that are useful when describing populations with non-overlapping generations. These discrete models often exhibit unique dynamical scenarios that cannot otherwise be modeled by differential equations and can more accurately reflect the global character of certain populations. See [4], [5], [6], and [7] for interesting applications.
The focus of this thesis is on systems of difference equations and their applications to population dynamics. In particular, several population models are analyzed and results are presented related to the global dynamics of these sys-tems. The remainder of the introduction is organized as follows: an overview of the types of population models models that are studied in Manuscripts 2, 3, and 4 is presented in Section 1.1 along with a summary of the important results that are attained. Basic theory of difference equations is then presented in Section 1.2, which helps to lay the theoretical foundation for the proceeding manuscripts.

Overview of Population Models
The population models that are studied within this thesis include competitive models, host-parasitoid and host parasite models, and structured population models. Background for each of these models is presented in this section. An overview of the specific results that are attained in Manuscripts 2, 3, and 4 is also provided.
A two-dimensional competitive system of difference equations has the form x n+1 = f (x n , y n ), y n+1 = g(x n , y n ), n = 0, 1, 2, . . . , where the function f is nondecreasing in the first variable and nonincreasing in the second and g is nonincreasing in the first variable and nondecreasing in the second [8]. Systems of this type can be used to model two species living in an environment that are in competition for the same resources. Competitive systems satisfy monotonicty properties (as discussed in Section 1.2.2), which are useful in proving global results. See [9] and [10] for interesting competitive systems of difference equations that have been studied recently. There are many dynamical scenarios that can exist for a competitive system, including competitive exclusion, competitive coexistence, and Allee's effect. Competitive exclusion occurs when one species is driven to extinction, competitive coexistence occurs when two competing species reach an equilibrium state and are able to coexist, and Allee's effect occurs when two species with nonzero populations are both driven to extinction [11].
Many of the original two-species, discrete competition models were based off of the Lotka-Volterra competitive system of differential equations and they exhibited primarily the phenomenon of competitive exclusion [12]. One of the first models that incorporated competitive coexistence was formulated by J. C. Gower and P.
H. Leslie in 1958 [13]. Based on experiments performed by T. Park in 1948 [14], Leslie and Gower presented the competitive model , t = 0, 1, 2 . . . , where λ 1 , λ 2 , α 1 , α 2 , β 1 , β 2 are positive constants. Manuscript 2 focuses on a comparable form of system (1), given by x n+1 = b 1 x n α 1 + x n + c 1 y n , y n+1 = b 2 y n α 2 + c 2 x n + y n , n = 0, 1, 2, . . . , where the parameters b 1 , b 2 are positive real numbers, and α 1 , α 2 , c 1 , c 2 and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers. The case when all parameters are taken to be positive was studied in [12] by J. M. Cushing. The analysis in [12] is not complete, however, as it does not treat the sensitive case when α 1 = 0 or α 2 = 0. In these cases, the extinction equilibrium is replaced by a singular point at the origin, which greatly complicates the problem. Manuscript 2 fully characterizes the global dynamics of this system when α 1 = 0 or α 2 = 0 and the dynamical scenarios are compared to the case when α 1 , α 2 > 0. The proof techniques reflect the sensitivity of the singular point at the origin. Using these techniques, global results are also established for general competitive systems of difference equations that have a particular orientation of equilibria (or singular points) and certain local stability characteristics.
A host-parasitoid model is a type of prey-predator model involving two species, a parasitoid and a host [3]. A parasitoid is a type of parasite that lives freely and lays eggs in the larvae or pupae of the host population. The development of the parasitoid depends on the population of the host and the population of the host depends on how many of its peers survived the infestation [3,15]. The general framework for describing the dynamics of such a model is x n+1 = a x n f (x n , y n ), y n+1 = c x n (1 − f (x n , y n )), n = 0, 1, 2, . . . , where x n and y n represent the size of the host and parasitoid populations in the n th generation, respectively, and f (x n , y n ) represents the fraction of hosts that survive the parasitoid [15,16]. One of the first models of this type was developed by Nicholson and Bailey in 1935 for a host, Trialeurodes vaporariorum, and a parasitoid, Encarsia formosa, and is described by the system x n+1 = a x n e −b yn , y n+1 = x n 1 − e −b yn , n = 0, 1, 2, . . . .
This model assumes that the reproductive rate of the host and the searching efficiency of the parasitoid are constant and that the environment is consistent for all parasitoids [16,17]. A host-parasite model has a similar structure to that of a host-parasitoid model but the parasite may not kill the host [3,18]. One hostparasite model of particular interest, formulated in 1948 by P. H. Leslie, is given by N 1 (t + 1) = λ 1 N 1 (t) 1 + (λ 1 − 1) N 2 (t) , t = 0, 1, 2, . . . , where λ 1 , λ 2 > 1 and K 1 , K 2 are positive constants (see page 239 in [19]). Manuscript 3 takes a closer look at system (2) and presents the most complete analysis to date of this classic model. A boundedness and persistence result along with global attractivity results for various parameter regions are established. Numerical evidence of chaotic behavior is also presented for particular solutions of the system.
The third type of population model that is analyzed in this thesis is referred to as a structured population model. Structured models are useful for populations that include individuals with a variety of physical or physiological characteristics that may interact with the environment differently [4]. These differences play an important role in the dynamics of the entire population and must be taken into account within the model. The basic theory for models of this type is presented by J. M. Cushing in [2]. The general framework involves dividing a population into k classes so that at discrete time n ∈ N the number of individuals (or density) in class is y () n . The vector y n ∈ R k + , defined by y n = (y (1) n , y (2) n , . . . , y (k) n ), tracks the size of each of the k classes. It is typically assumed that the new number of individuals in each class is a function of the previous number of individuals in each of the k classes [2]. This inter-dependence can be condensed in the matrix model: y n+1 = P (y n ) y n , n = 0, 1, 2, . . . , y 0 ∈ R k + , which is explored in [2]. The operator P is referred to as a projection matrix.
Manuscript 4 focuses on a special case of systems of the form (3), given by . . , f k (y (k) n ) ) t , n = 0, 1, 2, . . . , y 0 ∈ R k + , Friedland and Karlin in [20]. Results are established in Manuscript 4 related to the existence, uniqueness, and stability of a positive equilibrium (i.e. persistence state) as well as to the stability of the origin (i.e. extinction state). Particular attention is given to the special case when k = 2. The benefit of analyzing system (I) is that it can be used as a tool to study more complicated structured population models, given by f (c x n ) b , n = 0, 1, 2, . . . , where A is a bounded, linear operator on an ordered Banach space X with positive cone X + , and for each ∈ {1, . . . , k}, b ∈ X + and c is a positive bounded linear functional on X. Model (II) is a generalization of a model studied by Rebarber, Townley, and Tenhumberg in [21] and can be useful to model plant and fishery populations. The matrix A is referred to as the survival operator and it reflects survival and growth of each class. The terms f (·)b are referred to as fecundity operators and they reflect the nonlinear dependence on new offspring and the redistribution of the offspring to each class of the structured model [21].
Model (II) is potentially set in a high dimensionality state space X. The nonlinearities, however, are of a very specific type. Manuscript 4 presents a technique to greatly reduce the complexity of the problem to one where the dimension of the state space is the same as the number k of nonlinearities that appear in (II).
In particular, a k-dimensional model of the form (I) is found to have a significant connection with (II). Conditions are given under which there is a one-to-one correspondence between the positive equilibrium points of (I) and (II). Furthermore, when the state space of (II) is R m , it is established that the stability character of the origin in (I) is the same as that of (II). In this way, all of the results established for system (I) can be used to establish results for (II).
For convenience to the reader, basic theory and results are presented in the coming sections that contribute to establishing global results for the three models introduced above.

Basic Notions of Difference Equations
Difference equations are used to describe the progression of a given quantity over discrete time increments. As mentioned above, a common application is tracking the size of a population with discrete generations. If we denote by x n the size of the population in the n th generation and assume that the size of the population in the n + 1 st generation (i.e. x n+1 ) is a function of x n , we arrive at the difference equation If the initial value of the population were known (say x 0 ), then (4) provides all of the information needed to track the population through each generation [22,23]. (4). This type of difference equation is first-order (since x n+1 only depends on one previous generation) and autonomous (since x n+1 does not depend explicitly on n) [22,23].
Of particular interest in this thesis are systems of difference equations modeling two or more populations that depend on one another through each discrete generation. Consider a planar region D ∈ R 2 . A two-dimensional, first-order system of difference equations has the form x n+1 = f (x n , y n ), y n+1 = g(x n , y n ), n = 0, 1, 2, . . .
where f, g : D → R are continuous functions and (x 0 , y 0 ) ∈ D. Systems of this form can be used to model two populations that interact in many different ways, including cooperation, competition, and predator-prey type interactions. Associated If we specify an initial condition (x 0 , y 0 ) ∈ D and repeatedly apply the map T , the resulting sequence of ordered pairs {(x 0 , y 0 ), T (x 0 , y 0 ), T 2 (x 0 , y 0 ), T 3 (x 0 , y 0 ), . . . } is called a solution of (5) (see [22,23]).
If one were to find a solution of a difference equation by specifying initial conditions, the main question to consider is how the resulting sequence behaves as n → ∞. A complete description of the behavior of the solutions for an arbitrary initial condition is referred to as the global behavior of the difference equation. This analysis begins with finding the equilibrium points of the system. An equilibrium point (x,ȳ) of (5) is a point that satisfiesx = f (x,ȳ) andȳ = g(x,ȳ). Equilibrium points of (5) are referred to as fixed points of (6) where (x,ȳ) = T (x,ȳ). For each fixed point, the basin of attraction B(x,ȳ) is the largest set of points in D that are attracted to (x,ȳ) under forward iterations of T (see [23]). That is, There is also the potential for the existence of periodic solutions, which play a role in the global behavior of system (5). A minimal period-two solution is a point (x, y) ∈ D such that T 2 (x, y) = (x, y) and T (x, y) = (x, y) [22,23]. The same principles described above can also be applied to larger dimensional systems.
The allure of difference equations is that the global behavior of a system can be extremely simple or incredibly complex. There can be multiple equilibrium points, periodic points, attracting curves, unbounded behavior, and even chaos.
Determining global behavior is the central focus of much of the research that is done in difference equations. The following subsections provide some of the more specific theory that is used to establish global results.

Local Stability Analysis
To determine the global behavior of a system of difference equations, the first step is understanding the local behavior in a neighborhood of each equilibrium point. This analysis is referred to as local stability analysis. Consider the twodimensional system of difference equations given in (5). Let || · || denote the euclidean norm in R 2 . That is, ||(x, y)|| = x 2 + y 2 . The following definitions can be seen in their original form in introductory textbooks written by M. R. S.
Kulenović and O. Merino [23] or S. Elaydi [22]. The definitions are presented for a two-dimensional system of differences equations, but the same principles apply for higher dimension.
Definition 1 Consider an equilibrium point (x,ȳ) of system (5). Then Consider the map T : D → D given in (6), where f and g are taken to be continuously differentiable on D. In order to determine the local character of a fixed point, one can consider the linearization of the map near each fixed point.
This requires computing the Jacobian matrix of T at (x,ȳ), which is given by The linearization of the map T , denoted by D T , at the fixed point (x,ȳ) is then given by and the characteristic equation associated with the Jacobian matrix (7) is Locally, the map T behaves like the linearization given in (8). Therefore, the eigenvalues λ 1 and λ 2 of J T (x,ȳ) (i.e. roots of the characteristic equation) provide information about the local stability characteristics of (x,ȳ). If both eigenvalues have modulus less than one, then (x,ȳ) is locally asymptotically stable and if at least one of the eigenvalues is greater than one in modulus, then (x,ȳ) is unstable. Furthermore, a fixed point (x,ȳ) of the map T is hyperbolic if J T (x,ȳ) has no eigenvalues on the unit circle, otherwise (x,ȳ) is said to be nonhyperbolic. Hyperbolic fixed points can have three qualitatively distinct classifications, which are described in Definition 2.
Definition 2 Consider an equilibrium point (x,ȳ) of system (5), is locally asymptotically stable then the eigenvalues of J T (x,ȳ) are such that |λ 1 |, |λ 2 | < 1. In this case, there is an open neighborhood U of (x,ȳ) in which all points converge to the equilibrium under forward iterations of the map T . That is, Such an equilibrium point is referred to as a sink. Parts (ii) and (iii) describe the two unstable situations.
(ii) If the eigenvalues of J T (x,ȳ) are such that |λ 1 |, |λ 2 | > 1, then there is an open neighborhood U of (x,ȳ) in which all points converge to the equilibrium point under backward iterations of the map T . That is, Such an equilibrium point is referred to as a source or repeller.
(iii) If the eigenvalues of J T (x,ȳ) are such that |λ 1 | < 1 and |λ 2 | > 1, then in any neighborhood U of (x,ȳ), the forward iterates under T of some points in U converge to (x,ȳ) and the backward iterates under T of some points in U converge to (x,ȳ). Such a point is referred to as a saddle point.
There are shortcuts that can be used to determine the local character of equilibrium points in the two dimensional case, referred to as the Schur-Cohn Criteria.
In the case of a nonhyperbolic equilibrium (i.e. when eigenvalues of J T (x,ȳ) satisfy |λ j | = 1 for j = 1 or 2), more analysis is needed to determine the local stability character. This analysis involves higher order terms in the Taylor expansion, which will not be covered here.
Another theorem that is helpful in analyzing eigenvalues of the Jacobian matrix is the Perron-Frobenius Theorem from [24]. Prior to stating the theorem, a few basic notions of matrices are needed: A matrix A ∈ R m×m is nonnegative (positive), written A ≥ 0 (A > 0) if all of the entries of A are nonnegative (positive).
A nonnegative matrix A is called primitive if there exists an N ∈ N such that [24]. The Perron-Frobenius Theorem specifically treats nonnegative, irreducible matrices.
Theorem 2 Let A ∈ R m×m be nonnegative and irreducible. Then, (i) A has a positive (real) eigenvalue λ max such that all other eigenvalues of A satisfy |λ| ≤ λ max .
(ii) λ max has algebraic and geometric multiplicity one, and has an eigenvector (iii) Any nonnegative eigenvector is a multiple of x.
(iv) Suppose y ∈ R m + , y = 0 and µ ∈ R is such that A y ≤ µ y. Then y > 0 and µ ≥ λ max , with µ = λ max if and only if y is a multiple of x.
(vi) If A is primitive, then all other eigenvalues of A satisfy |λ| < λ max In the following section, basic notions of monotone maps are presented, which can play a critical role in determining the global dynamics of systems of difference equations.

Monotone Systems
A set P ⊂ R m is an order cone if P is closed, convex, and such that λ P ⊂ P for all λ ≥ 0, P ∩ (−P) = {0} and P = {0}. Every order cone P induces a partial order on R m . For points x, y ∈ P, we say x y if and only if y − x ∈ P, x ≺ y if and only if y − x ∈ P \ {0}, and x y if and only if y − x ∈ int(P). The ordered set x, y relative to the partial order is defined by x, y := {u ∈ R m : x u y}.
For D ⊂ R m , a map T : D → D is said to be monotone (with respect to the partial order ) if x y implies that T (x) T (y). The map T is strictly monotone if x ≺ y implies that T (x) ≺ T (y) and strongly monotone if x ≺ y implies that T (x) T (y). See [8] for more detailed information regarding monotone systems. The following result is stated for monotone (i.e. order preserving) maps and appears in [25].
Theorem 3 For a nonempty set D ∈ R m and a partial order on R m , let T : D → D be an order-preserving map and let u, v ∈ D be such that u ≺ v and (ii) If T is strongly order-preserving then there exists a fixed point of T in u, v that is stable relative to u, v.
(iii) If there is only one fixed point in u, v then it is a global attractor in u, v and therefore asymptotically stable relative to u, v.
A direct consequence of Theorem 3 is Corollary 1 proven by Dancer and Hess in [25].
Corollary 1 If the nonnegative cone of a partial ordering is a generalized orthant in R m , and if T has no fixed points in u, v other than u and v, then the interior of u, v is either a subset of the basin of attraction of u or the basin of attraction of v.
If we restrict out attention to R 2 , then there are two standard partial orders, the North-East and South-East partial orders [8].

Definition 3
The North-East partial order ne on R 2 is defined as follows: where the positive cone is taken to be the standard first quadrant. The South-East partial order se on R 2 is defined as follows: where the positive cone is taken to be the standard fourth quadrant.
The partial-orders in Definition 3 are associated with two important monotone with respect to the South-East partial order se . In other words: The map T is strongly competitive if it is strictly increasing with respect to the South-East partial order [8]. A sufficient condition for T to be strongly competitive is that the Jacobian matrix associated with T has the sign configuration Similarly, T is said to be cooperative if it is nondecreasing with respect to the North-East partial order ne and strongly cooperative if it is strictly increasing with respect to the North-East partial order [8]. A sufficient condition for T to be strongly cooperative is that the Jacobian matrix associated with T has the sign The following definition presents another important property that competitive maps can satisfy on R 2 [8].
Sufficient conditions for competitive maps to satisfy the (O+) and (O−) conditions are provided in Theorem 4 [8].
Competitive maps that satisfy either of the conditions from Definition 4 exhibit well behaved global dynamics as evidenced by the following theorem, which was originally proven by deMottoni-Schiaffino and was later generalized for competitive maps by Smith [26]. If the orbit of x has compact closure in D, then its omega limit set is either a period-two orbit or a fixed point.
The above results are utilized throughout Manuscripts 2 and 4, which deal specifically with monotone maps.

Global Manifolds
Many authors have devoted time to developing general theory for determining the global dynamics for systems of difference equations. One important concept involves the existence of global stable and unstable manifolds. The local stable and unstable manifolds for a fixed point of the map (6) are defined in [23] as Kulenović and Merino in [11,27,28] present several theorems that allow the local manifolds to be extended in certain situations.
Theorem 6 Let T be a competitive map on a rectangular region D ⊂ R 2 . Let x is not the NW or SE vertex of D), 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 ofx.
(b) The Jacobian J T (x) of T atx 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 ⊂ D 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 D 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 .
In the particular systems that are studied in this thesis, the endpoints of C are boundary points of the region D. This situation is further discussed in the following theorem from [28].
Theorem 7 For the curve C to have endpoints in ∂D, 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 ∆, detJ T (x) > 0, and T (x) =x has no The map T has no points of minimal period-two in ∆, detJ T (x) > 0, and The existence of the curve C described in the previous two theorems is incredibly helpful in determining basins of attraction of fixed points. The following theorem expounds on this idea [28].
Theorem 8 Assume the hypotheses of Theorem 6, and let C be the curve whose existence is guaranteed by Theorem 6. If the endpoints of C belong to ∂D, then C separates D into two connected components, namely such that the following statements are true (i) W − is invariant and dist(T n (x), Q 2 (x)) → 0 as n → ∞ for every x ∈ W − .
If, in additionx is an interior point of D 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.
(i) For every x ∈ W − there exists n 0 ∈ N such that T n (x) ∈ intQ 2 (x) for n ≥ n 0 .
(ii) For every x ∈ W + there exists n 0 ∈ N such that T n (x) ∈ intQ 4 (x) for n ≥ n 0 .
The manuscripts of this thesis utilize the basic theory presented in the previous sections to establish global results for the population models introduced in Section 1.1.

List of References
Abstract Global dynamic results are obtained for families of competitive systems of difference equations of the form where the parameters b 1 , b 2 are positive real numbers, and α 1 , α 2 , c 1 , and c 2 and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. In particular, we investigate the effect of parameters α 1 , α 2 on the global dynamics of this system.

Introduction
Consider the system of difference equations where the parameters b 1 , b 2 are positive real numbers, and α 1 , α 2 , c 1 , and c 2 and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We consider the effect of terms α 1 , α 2 on the global dynamics of system (1). The global dynamics of (1) was considered in the case where the parameters α 1 , α 2 are positive in [1,2] and the complete description of the dynamics was given in [2], where the following result was obtained: Assuming, without loss of generality, that α 1 = α 2 = 1, it has been shown in [1] that under the condition b 1 > 1 and b 2 > 1, the points are equilibria of equation (1), and that for some values of the parameters there exists an additional equilibrium point E 3 , located in the open positive quadrant, given by Important subsets of parameter space are described in Table 1, together with corresponding behavior of fixed points established in [1]. Table 1: Global behavior of solutions to (1) when b 1 > 1 and b 2 > 1. Equality relations are not represented for the sake of a simpler description.

Condition
An application of Theorem 9 in [2] applies when parameters vary from Case 2 to Case 4 of Table 1. Set and define T α to be the map of system (1) restricted to R = [0, ∞) × (0, ∞), that is, Therefore, Theorem 9 in [2] gives global behavior of solutions to system (1) In particular, a bifurcation occurs when the equilibrium x α changes its local character from a locally stable equilibrium to a saddle point. This happens when the parameters cross the critical surface It is also shown in [1] that the open, positive semiaxis (0, ∞)×{0} is attracted to E 1 , and that the open, positive semiaxis {0} × (0, ∞) is attracted to E 2 . The following two results describe the global dynamics of system (1) in all cases. The first result gives the global dynamics in the hyperbolic case and the second result in the non-hyperbolic case. Theorem 1 Consider system (1).
is the graph of a continuous, increasing function of the first coordinate. Furthermore, a solution {x n } converges to See Figure 1 for graphical interpretation.
The non-hyperbolic case when was not considered in [1]. When (3) holds, a direct calculation gives that the equilibrium points of T are E 0 (0, 0) and all points on the segment E : The eigenvalues of the Jacobian of T at E t are and the corresponding eigenvectors are It is shown in [3] that, for system (1), the hypotheses of Theorem 5 in [4] are satisfied and that all solutions fall inside an invariant rectangular region. Therefore, every solution of (1) converges to an equilibrium point. A direct calculation shows that the origin is a repeller. We conclude that every nonzero solution converges to a point (x, y) ∈ E. Also, with an argument similar to the one used in [5], one has that the equilibrium depends continuously on the initial condition. That is, if T * (x, y) := lim T n (x, y), then T * is continuous. These observations, together with an application of Theorem 1 in [4] lead to the following result.
(ii) For every (x, y) ∈ E with x = 0 and y = 0, the stable set W s (x,y) is an unbounded, increasing curve C with endpoint (0, 0).
(iii) The limiting equilibrium varies continuously with the initial condition.
See Figure 1 for graphical interpretation. Statement (ii) excludes equilibria of the form (0, y) and (x, 0) since the hypotheses of Theorem 1 in [4] are not satisfied at these points.
In this paper we consider two related systems, namely x n x n + c 1 y n , y n+1 = b 2 y n c 2 x n + y n n = 0, 1, . . . Figure 1: Global dynamics of system (1). and x n+1 = b 1 x n α 1 + x n + c 1 y n , y n+1 = b 2 y n c 2 x n + y n n = 0, 1, . . . , where all present coefficients are positive and the initial conditions are nonnegative.
We derive the global dynamics of both systems (4) and (5), which explains the effect of the parameters α 1 , α 2 on the global dynamics.
The paper is organized as follows. The second section presents some basic preliminary results about competitive systems, which is our main tool in proving the results. The third section contains the global dynamics of system (4) and the fourth section gives the global dynamics of system (5). The fifth section presents some results on global dynamic scenarios for general competitive systems. Actually, we show that all global dynamic results that hold for any of the three systems (1), (4), and (5) can be immediately applied to a general competitive system and that the global behavior of all three systems is determined by the linearized dynamics.
We will compare and contrast the global dynamics of (4) and (5) with that of system (1).

Preliminaries
In this section we provide some basic facts about competitive maps and systems of difference equations in the plane.
Definition 5 Let R be a subset of R 2 with nonempty interior, and let T : R → R be a map (i.e. a continuous function). Set T (x, y) = (f (x, y), g(x, y)). The map T is competitive if f (x, y) is nondecreasing in x and nonincreasing in y and 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 (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, 2, . . . , (x 0 , y 0 ) ∈ R is said to be competitive (cooperative). The map T and the associated system of difference equations are said to be strongly competitive (strongly cooperative) if the adjectives nondecreasing and nonincreasing are replaced by increasing and decreasing.
If T is differentiable, a sufficient condition for T to be strongly competitive is that the Jacobian matrix of T at any point (x, y) ∈ R has the sign configuration Competitive systems of the form (6) have been studied by many authors such as Clark, Kulenović, and Selgrade, Hess, Hirsch and Smith, Kulenović, Merino, and Nurkanović, Leonard and May, Smale and others [6,7,8,9,10,11,12,13]. See [14] for interesting applications of this theory to basic models in population dynamics.
Denote with se the South-East partial order in the plane whose nonnegative cone is the standard fourth quadrant {(x, y) : x ≥ 0, y ≤ 0}, that is, (x 1 , y 1 ) (x 2 , y 2 ) if and only if x 1 ≤ x 2 and y 1 ≥ y 2 . The North-East partial order ne is defined analogously with the nonnegative cone given by the standard first quadrant Competitive maps T in the plane preserve the South-East ordering: T (u) se T (v) whenever u se v. Similarly, cooperative maps in the plane preserve the North-East ordering. In fact, the concepts of competitive and cooperative (for maps) may be defined in terms of the order preserving properties of maps. Thus the theory of competitive maps is a special case of the theory of order preserving maps (or monotone operators).
Order preserving maps in R n , and in particular competitive maps in R 2 , may have chaotic dynamics. Smale [12] showed that any continuous time vector field on the standard (n − 1)-simplex in R n can be embedded on a smooth, competitive vector field in R n for which the simplex is an attractor. In the case of a planar system (6), this means that any first order difference equation, including chaotic, can be embedded into a competitive system (6) in the plane. An effective algebraic method to do this is provided by Smith in [13].
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. Also, x y if y − x ∈ int(P). A map T on a subset of R n is order preserving if T (x) T (y) whenever x y, strictly order preserving if T (x) ≺ T (y) 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 R if x is a stable (asymptotically stable) fixed point of the restriction of T to R .

Definition 6
Let R be a nonempty subset of R 2 . A competitive map T : R → R is said to satisfy condition (O+) if for every x, y in R, T (x) ne T (y) implies x ne y, and T is said to satisfy condition (O−) if for every x, y in R, T (x) ne T (y) implies y ne x.
The following theorem was proved by deMottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations.
Smith generalized the proof to competitive and cooperative maps [15].
Theorem 3 Let R be a nonempty subset of R 2 . If T is a competitive map for which (O+) holds, then for all x ∈ R, {T n (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T .
If instead (O−) holds, then for all x ∈ R, {T 2n (x)} is eventually componentwise monotone. If the orbit of x has compact closure in R, then its omega limit set is either a period-two orbit or a fixed point.
The next two results are stated for order-preserving maps on R n . These results are known but are given here for completeness. See [7] for a more general version that is valid in ordered Banach spaces.
Theorem 4 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.

Corollary 2 If the nonnegative cone of is a generalized quadrant in R n , and if
T has no fixed points in u 1 , u 2 other than u 1 and u 2 , then the interior of u 1 , u 2 is either a subset of the basin of attraction of u 1 or a subset of the basin of attraction of u 2 .
Our main tool will be results from [2,4,16] regarding the existence of the global stable and unstable manifolds of competitive maps in the plane.
The non-hyperbolic equilibrium solution of system (6) is said to be of stable (resp. unstable) type if the second eigenvalue of the Jacobian matrix evaluated at the equilibrium solution is by absolute value less than 1 (resp. bigger than 1).

Global Dynamics of System (4)
First we give some basic results about the global behavior of system (4).
Denote by the map associated with system (4). System (4) is homogeneous and was partially investigated in [17].

Lemma 1
The following statements hold: (a) Every solution of system (4) satisfies x n ≤ b 1 , y n ≤ b 2 , n ≥ 1.
(b) det J T (x, y) = 0 for every (x, y), where J T denotes the Jacobian matrix of (d) Every solution of system (4) satisfies the difference equation then the map T has an invariant line Proof. The Jacobian matrix J T of the map T has the form which implies (b). Parts (a) and (c) follow by immediate checking. Part (d) follows by dividing equations of system (4). Part (e) follows from (d) is exactly an equilibrium of the equation in (d). ✷ System (4) always has two equilibrium solutions on the axes, . It can also have either exactly one interior equilibrium solution E or an infinite number of interior equilibrium solutions E t . Since the interior equilibrium solution E is an intersection of two equilibrium curves, smaller than x-intercept of C 2 and y-intercept of C 1 bigger than y-intercept of C 2 ). These two geometrical conditions can be unified as condition (7). Table 2: The equilibrium points for system (4).
Condition (7) implies that c 1 c 2 = 1, in which case the interior equilibrium and every point on the segment x + c 1 y = b 1 , x, y ≥ 0 is an equilibrium solution Table 2 for a summary of the equilibrium points of system (4).
The following result describes the local stability character of all equilibrium solutions. (4). Proof.

Lemma 2 Consider system
(a) In view of (9), we have which implies that the eigenvalues of the Jacobian matrix are The corresponding eigenvectors are as stated.
(b) In view of (9), we have which implies that the eigenvalues of the Jacobian matrix are λ 1 = 0, The corresponding eigenvectors are as stated.
is a saddle point when b 2 < b 1 c 2 and b 1 < b 2 c 1 and is locally asymptotically (d) In this case, the eigenvalues of the Jacobian matrix evaluated at the equilibrium E t are λ 1 = 0, λ 2 = 1. The eigenvector that corresponds to λ 1 = 0 is where y > 0 satisfies x + c 1 y = b 1 and points towards the first quadrant.

✷
Now, global behavior of system (4) is described by the following result: , then the equilibrium solutions E x , E y are locally asymptotically stable and the interior equilibrium E is a saddle point. The separatrix S , which is a graph of a continuous, nondecreasing curve, is the basin of attraction of E and the region below (resp. above) S is the basin of attraction of E x (resp. E y ). y-axis) is attracted by E x (resp. E y ).
, then the equilibrium solution E x (resp. E y ) is locally asymptotically stable and E y (resp. E x ) is a saddle point. The basin of attraction of E x (resp. E y ) is the first quadrant of initial conditions without the positive part of the y-axis (resp. x-axis), which is attracted by E y (resp. E x ).
which is the graph of a continuous, nondecreasing function asymptotic to (0, 0) and is exactly the basin of attraction of E t . The limiting equilibrium varies continuously with the initial condition.
(e) If E x (resp. E y ) is non-hyperbolic and E y (resp. E x ) is locally asymptotically stable, then E y (resp. E x ) attracts the first quadrant of initial conditions except the positive part of x-axis (resp. y-axis) which is attracted by E attracts the first quadrant of initial conditions except the positive part of the y-axis (resp. x-axis), which is attracted by E y (resp. E x ).
See Figure 2 for graphical interpretation. Proof.
(a) First we show that T does not have any period-two solutions. Our condition implies c 1 c 2 > 1. By direct calculation one can show that a period-two solution satisfies the equation which means that both terms of the solution are negative and so there is no period-two solution in the first quadrant.
Taking into account that the Jacobian matrix evaluated at E has all nonzero entries, Theorem 5 of [16] implies the existence and uniqueness of both global stable and unstable manifolds, W s (E) and W u (E), and thus W s (E) = S .
Furthermore, Theorem 5 of [16] implies that every (x 0 , y 0 ) below S will satisfy T n ((x 0 , y 0 )) ∈ E, E x for some n ≥ N . In view of Corollary 2, In a similar way, we can treat the case when (x 0 , y 0 ) is above S .
(b) In view of Lemma 2 part (a), the eigenvectors which correspond to E x and E y point to the interior of the fourth and the second quadrant, which means that the local unstable manifolds W u loc (E x ) and W u loc (E y ) exist and point strictly toward E. Thus, there exist points u, v in the interior of E y , E x , is locally asymptotically stable and E y is a saddle point. In view of Lemma 2 part (b), the eigenvector which corresponds to E y points to the interior of the fourth quadrant, which means that the local unstable manifold W u loc (E y ) exists and points strictly toward E x . Thus, there exists a point u in the interior of E y , E x , arbitrarily close to E y such that u se T (u). Consequently, the map T has a lower solution in every neighborhood of E y , which in view of Theorem 6 in [4] implies that the interior of E y , E x is a subset of the basin of attraction of E x . The result follows.
The proof when b 2 > b 1 c 2 , b 1 < b 2 c 1 is similar and will be omitted.
(d) By Theorem 1 of [4], for each E t there exists the set W s (E t ) passing through E t and asymptotic to (0, 0), which is the graph of a continuous, nondecreasing function and is exactly the basin of attraction of E t . The continuity of the limiting equilibrium solution as a function of initial conditions follows as in [5].
(e) The proof is similar to the proof of part (c) and will be omitted. ✷ Figure 2: Global dynamics of system (4). (5) In this section we present the global behavior of system (5). Denote bỹ

Global Dynamics of System
the map associated with system (5).

Lemma 3
The following statements hold (a) Every solution of system (5) satisfies x n < b 1 , y n ≤ b 2 , n ≥ 1.
This is equivalent to It follows that x 1 = x 2 , y 1 = y 2 and thusT is injective. The Jacobian matrix associated with the mapT has the form Therefore, Note that a direct consequence of the above form of JT (x, y) is thatT is strongly competitive. This will play a pivotal role in the global behavior of the system.
System (5) always has an equilibrium solution on the y-axis, Depending on the values of the parameters α 1 , b 1 , b 2 , c 1 and c 2 , there is also the possibility of either exactly one interior equilibrium solution E or an infinite number of interior equilibrium solutions E t .
The interior equilibrium solution is an intersection of two equilibrium curves . the x-intercept of C 2 is smaller than the x-intercept of C 1 and the y-intercept of C 2 is bigger than the y-intercept of . the x-intercept of C 1 is smaller than the x-intercept of C 2 and the y-intercept of C 1 is bigger than the y-intercept of C 2 ). As in (7), these two geometrical conditions can be unified as that c 1 c 2 = 1 and the interior equilibrium E(x,ȳ) is given as: Note that if ∆ 1 ∆ 2 < 0, ∆ 1 = 0 and ∆ 2 = 0, or ∆ 2 = 0 and ∆ 1 = 0, then there does not exist an interior equilibrium solution. Since we must have α 1 < b 1 in order for an interior equilibrium point to exist.
Furthermore, if ∆ 1 = ∆ 2 = 0, then the two equilibrium curves C 1 and C 2 coincide and every point of the segment c 2 x + y = b 2 , x, y ≥ 0 is an equilibrium The equilibrium points for system (5) are summarized in Table 3.

Condition
Equilibrium Points Table 3: The equilibrium points for system (5).
The local stability character of E x , E y , E and E t are presented in Lemma 4.
The proof requires Proposition 1.

Proposition 1
The eigenvalues λ and µ of JT (E) are positive.
Proof. In view of (11) we have which implies .
(a) The equilibrium solution E x exists if α 1 < b 1 . It is locally asymptotically In each case, the eigenvectors associated with the eigenvalues The equilibrium solution E y always exists and it is locally asymptotically In each case, the eigenvectors associated with the eigenvalues λ 1 = 0 and They are non-hyperbolic of the stable type and the eigenvector Proof.
(a) In view of (11), we have which implies that the eigenvalues of the Jacobian matrix are . The corresponding eigenvectors are as stated.
(b) In view of (11), we have which implies that the eigenvalues of the Jacobian matrix are λ 1 = 0, The corresponding eigenvectors are as stated.

✷
The global behavior of system (5) is described by the following result. Note that the proofs presented for Theorem 6 differ from those of Theorem 5 in order to depict an alternative approach. Theorem 6 Consider system (5).
(a) If α 1 ≥ b 1 , then E y is the unique equilibrium solution of system (5) and it is locally asymptotically stable. Every solution in the first quadrant which starts off of the x-axis converges to E y and every solution which starts on the positive x-axis converges to the singular point (0, 0).
is the first quadrant of initial conditions without the positive part of the y-axis (resp. x-axis), which is equilibrium solutions E x , E y and E. The equilibrium solutions E x and E y are saddle points and E is locally asymptotically stable. Every solution in the first quadrant which starts off of the coordinate axes converges to E and every solution which starts on the positive x-axis (resp. y-axis) converges to , then there is an infinite family of equilibrium solutions E t for which there exists the global stable manifold , which is the graph of a continuous, nondecreasing function asymptotic to (0, 0) and is exactly the basin of attraction of E t . The limiting equilibrium varies continuously with the initial condition.
(f ) If E x (resp. E y ) is non-hyperbolic and E y (resp. E x ) is locally asymptotically stable, then E y (resp. E x ) attracts the first quadrant of initial conditions except the positive part of the x-axis (resp. y-axis), which is attracted by E x (resp. E y ). If E x (resp. E y ) is non-hyperbolic and E y (resp. E x ) is a saddle point, then E x (resp. E y ) attracts the first quadrant of initial conditions except the positive part of the y-axis (resp. x-axis), which is attracted by E y (resp. E x ).
See Figure 5 for graphical interpretation. Proof.
(a) Let α 1 ≥ b 1 . Lemma 3(c) and 3(d) guarantee that for initial conditions on the positive y-axis,T (x 0 , y 0 ) = E y and for initial conditions on the positive x-axis, lim n→∞T n (x 0 , y 0 ) = (0, 0). To treat the dynamics in the interior of By Theorem 2 of [16], R a is invariant. The region R a also attracts the interior of R 2 + . To verify this, suppose that (x 0 , y 0 ) ∈ R a with x 0 , y 0 > 0. In this case c 2 x 0 + y 0 > b 2 and It follows that there exists N > 0 such that for all n ≥ N , (x n , y n ) ∈ R a and thus R a is attracting. To conclude the proof, suppose (x 0 , y 0 ) ∈ R a with x 0 , y 0 > 0. In this case x 1 < x 0 , y 1 ≥ y 0 and as a consequence of the invariance of R a , {x n } is a decreasing sequence while {y n } is a nondecreasing sequence. Therefore, lim n→∞T n (x 0 , y 0 ) = E y . The above arguments prove that the basins of attraction for E y and the singular point guarantee that for all initial conditions on the positive y-axis,T (x 0 , y 0 ) = E y and for all initial conditions on the positive x-axis, lim n→∞T To treat the interior of R 2 Note that R b is an invariant region by Theorem 2 of [16]. Consider (x 0 , y 0 ) ∈ R b with x 0 , y 0 > 0 and notice As a consequence of the invariance of R b , {x n } is a nondecreasing sequence and {y n } is a nonincreasing sequence. Therefore, using basic properties of sequences and the fact thatT is strongly competitive, lim n→∞T Therefore, lim n→∞T n (x 0 , y 0 ) = E x . We have arrived at the desired result that the basins of attraction for E x and E y are The proof for the case when b Note that R c is invariant by Theorem 2 of [16]. Provided that (x 0 , y 0 ) ∈ R c with x 0 , y 0 > 0, monotonicity properties (similar to part (a) and (b)) along with Lemma 3(b) can be used to prove that lim n→∞T n (x 0 , y 0 ) = E.
Suppose (x 0 , y 0 ) ∈ R c with x 0 , y 0 > 0. By Lemma 3(a) we know that Therefore, lim n→∞T n (x 0 , y 0 ) = E for all (x 0 , y 0 ) ∈ R c . We have reached the desired result that the basins of attraction for E, E x and E y are Theorem 1 of [4] guarantees that there exist the global stable and unstable manifolds for E, W s (E) and W u (E), respectively, with the above mentioned properties. An immediate checking shows that E y se E se E x and that the interior of the ordered interval E y , E is a subset of B(E y ), while the interior of the ordered interval E, E x is a subset of B(E x ). Now, take any By Lemma 3(c) and the monotonicity ofT , for n ≥ 1, Since lim n→∞T n ((x W s (E) , y 0 )) = E, (19) implies thatT n ((x 0 , y 0 )) enters the ordered interval E y , E and so converges to E y . In a similar way, one can show that the ordered interval E, E x attracts all points below W s (E), and so all such points converge to E x .
(e) By Theorem 1 of [4], for each E t there exists the set W s (E t ) passing through E t and asymptotic to (0, 0), which is the graph of a continuous, nondecreasing function and is exactly the basin of attraction of E t . The continuity of the limiting equilibrium solution as a function of initial conditions follows as in [5].
(f) The proof is similar to the proof of part (b) and will be omitted here.  (5) is positive and the map satisfies (O+) condition. This condition greatly simplifies the proof for system (5). The qualitative difference between system (1) and systems (4) and (5) is in the case when b 1 ≤ 1, b 2 ≤ 1. In this case, E 0 (0, 0) is a globally asymptotically stable equilibrium for system (1), the basin of attraction of the singular point E 0 (0, 0) is an empty set for system (4), and the basin of attraction of the singular point E 0 (0, 0) is the nonnegative part of the x-axis for system (5). Furthermore, while system (4) always possesses two equilibrium solutions, systems (1) and (5) possess only one equilibrium solution for all parameter regions.

Global Dynamics Scenarios for Competitive Systems
In this section we give some general results about the global dynamics of a general competitive system (6). The proofs are analogous to the ones given in Theorems 5 and 6 and will be ommited.
Theorem 7 Consider the competitive map T associated with system (6).
(a) Assume that T has a saddle fixed point E x , locally asymptotically stable point E y , and either another repelling fixed point or a singular point E 0 , which is a South-West corner of the region R and satisfies E y se E 0 se E x . If T has no period-two solutions, then every solution which starts off of the x-axis converges to E y and every solution which starts on the positive part of the x-axis converges to E x .
(b) Assume that T has a saddle fixed point E y , locally asymptotically stable point (c) Assume that T has three fixed points E x , E y , E such that E y se E se E x , where E x , E y are saddle points, and E is locally asymptotically stable.
Assume that T has either another repelling fixed point or a singular point E 0 , which is a South-West corner of the region R and satisfies E y se E 0 se E x .
If T has no period-two solutions, then every solution which starts in the interior of R converges to E. Every solution which starts on the positive part of the x-axis (resp. y-axis) converges to E x (resp. E y ).
(d) Assume that T has three fixed points E where E x , E y are locally asymptotically stable, and E is a saddle point. (e) Assume that T has an infinite number of fixed points which belong to the arc of a continuous decreasing curve C with end points E x and E y on the x and y-axis respectively. Assume that T has either another repelling fixed point or a singular point E 0 , which is a South-West corner of the region R . If T has no period-two solutions, then every solution which starts in the first quadrant belongs to exactly one nondecreasing, continuous curve W s (E C ) that crosses C at the point E C and either has an endpoint at E 0 or is asymptotic to E 0 .
The dynamic behavior described in (a), (b), and (d) of Theorem 7 are called competitive exclusion scenarios and the dynamic behavior described in (c) of Theorem 7 is called a competitive coexistence scenario.

Acknowledgements.
M. R. S. Kulenović is supported in part by Maitland P. Simmons Foundation.

Competing interests
The authors declare that they have no competing interests.

Authors contributions
All the authors contributed equally to this work. They all read and approved the final version of the manuscript.

Introduction
A host-parasitoid model is a type of prey-predator model where the development of the attacking species (parasitoid) depends on the quantity of the food species (host) made available to it and the population of the food species depends on how many of its peers survived the infestation [1,2]. A parasitoid is a term that refers to a parasite that lives in an environment and lays eggs in the larvae or pupae of the host population [1,3]. Once a host is parasitized, it dies off but the eggs of the parasitoid may continue to the next generation [1,3]. Often parasitoids are strategically used as biological pest control agents to kill off unwanted insect populations and applications can be found in [1], [4], [5], and [6]. The general framework for describing the dynamics of such a model is where x n and y n represent the size of the host and parasitoid populations in the n th generation, respectively, and f (x n , y n ) represents the fraction of hosts that survive the parasitoid [2,4]. Many authors have studied host-parasitoid models with the general form given in (1). See [7], [4], [3], [8], [9], [10], and [11] for an analysis of such models.
Host-parasite models have a similar structure to that of host-parasitoid models with the biggest difference being that the parasite may not kill the host [1,12].
These models have attracted the attention of many authors in recent years and several interesting systems are studied in [13], [14], [15], [16] and [17]. One hostparasite model of particular interest, formulated in 1948 by P. H. Leslie, is given by where λ 1 , λ 2 > 1 and K 1 , K 2 are positive constants (see page 239 in [18]). The quantity N 1 represents the population of the host and N 2 represents the population of the parasite. An increase in the parasite population N 2 results in a decrease in the host population N 1 and an increase in the ratio N 2 N 1 results in a decrease in the parasite population as they lack resources to survive. See [18,19] for more information on (2). System (2) can be rewritten as x n+1 = α x n 1 + β y n , y n+1 = γ x n y n x n + δ y n , n = 0, 1, 2, . . . , where α, β, γ, δ, x 0 , y 0 are positive real numbers.
System (3) has been studied by Q. Din and T. Donchev, who claim in Theorem 6 of [12] that when α, γ > 1 the unique positive equilibrium is a global attractor.
However, the proof in [12] is incorrect as we now explain. The analysis of system (3) in [12] relies on Theorem 5 of [12], which is a result that appeared first as Theorems 2.2.9 and 2.2.11 in the PhD thesis of M. Nurkanović [20]. Also see [21].
A generalization of these results is Theorem 3 in [22]. is an invariant subset of the positive quadrant of the plane, then by monotonicity and invariance, From (4) This change of variables allows for the elimination of the parameters β and δ, and after renaming variables, system (3) is transformed to x n+1 = α x n 1 + y n , y n+1 = γ x n y n x n + y n , n = 0, 1, 2, . . . .
An elementary calculation gives that a positive equilibrium for (5) exists if and only if α > 1 and γ > 1. When this equilibrium exists, it is unique and given by It is worth pointing out that whenever the positive equilibrium (6) exists, it is locally asymptotically stable. This can be seen from the characteristic polynomial g(t) of the Jacobian matrix of the map associated with (5) evaluated at the equilibrium (6), which is given by Rather than calculating the roots of g(t) explicitly, we proceed to verify the Schur-Cohn condition for the roots of a quadratic polynomial to be inside of the unit disk, namely |p| < q + 1 < 2 (see [23]). This inequality for the polynomial g(t) becomes the relation (after some simplification) α < 1 + α γ − γ < α γ, which is true for α > 1, γ > 1. Thus (6) is locally asymptotically stable.
The main result of this paper is Theorem 1, which is presented below.
Theorem 1 Assume α, γ are arbitrary positive real numbers. Then system (5) has a positive equilibrium (x + ,ȳ + ) if and only if α > 1 and γ > 1. If it exists, the positive equilibrium is unique and given by (6). For arbitrary positive numbers x 0 and y 0 , let {(x n , y n )} be given by (5). Then the following statements are true: (i) If α < 1, then (x n , y n ) → (0, 0).
The seven dynamical scenarios described in Theorem 1 are depicted in Figure   6, where the various parameter regions are labelled according to the long-term behavior of solutions {(x n , y n )} of system (5).
(∞, 0) (x, 0) It is not known to the authors if chaos is a feature of a substantial portion of the systems associated with the parameter set 1 < γ < α. It is important to note that the arguments used to prove the boundedness and persistence of the solutions of system (5) are not affected by the presence of chaotic orbits. suggest (x n , y n ) → (x,ȳ) = (9000, 9000). In plot (a), the equilibrium (9000, 9000) is shown as a black dot. Plot (b) shows a smaller window, where some lune-shaped regions apparently devoid of points can be seen. One such region seems to contain the point (3500, 1900), which is marked with the symbol +. A total of 300 000 points were generated for these plots.
!" """ #"" """ #!" """ $"" """ #" """ $" """ %" """ &" """ !" """ A unique positive equilibrium also exists when 1 < γ < α, but the sets P µ used in part (vi) fail to have desired properties and another approach is needed to establish boundedness and persistence. The proof of statement (vii) regarding the boundedness and persistence of solutions is presented in Section 3.4. We introduce a useful change of coordinates to treat the problem in the whole plane. The subsequent proof is based on the construction of a family of compact sets K τ . We show that for large enough τ , the sets K τ are invariant under the associated map and the collection forms a cover of the plane. Compared to the construction of P µ , the construction of K τ is significantly more involved, as it requires the introduction of auxiliary maps and certain curves that are invariant for those auxiliary maps.

Global Behavior in the Absence of a Positive Equilibrium
This section presents a proof of statements (i) through (v) of Theorem 1.
Suppose first that α ≤ 1. Choose arbitrary positive real numbers x 0 and y 0 . With (x n , y n ) given by (5) for n > 0, we have x n+1 = α x n 1 + y n < α x n ≤ x n for all n ≥ 0, and thus x n converges to a nonnegative real numberx.
The map R has a real analytic extensionR to a neighborhood N ⊂ R 2 of (x, 0). It is shown in [25] that if N is small enough, then for every point (x, y) ∈ N \{(x, 0)} there exists n > 0 such thatR n (x, y) ∈ N . This contradicts x n →x, sox = 0. We conclude x n → 0 and y n → 0. α = 1, γ > 1: We claimx = 0. Suppose this is not the case, i.e.x > 0. Then 1 + y n = xn x n+1 → 1, so y n → 0. Also, y n+1 y n = γ x n x n + y n → γ > 1, which implies y n → 0. This contradicts the assumption, hence x n → 0. We have, Therefore, y n → 0 and statement (iii) follows. Now, suppose that α > 1 and γ < 1. Using system (5), y n+1 = γ x n y n x n + y n < γ x n y n x n = γ y n for all n ≥ 0, and thus y n → 0 as n → ∞. Furthermore, since α > 1, there exists N > 0 and A > 1 such that α 1+yn > A for all n ≥ N . Then, Consequently, x n → ∞ as n → ∞ and statement (iv) follows.
If α > 1 and γ = 1, we have y n+1 = x n y n x n + y n < y n for all n ≥ 0.
Some properties of the sets P µ are given in Proposition 1 below. Figure 9: (a) The sets P 2 and P 4 . (b) The boundary of a set P µ (solid) and its image S(∂P µ ) (dashed).
Since (1, 1) is the only accumulation point of the bounded sequence {S n (x, y)}, it follows that S n (x, y) → (1, 1). ✷ Now, for the proof of (iv) and (v) of Proposition 1, let µ > 1 be fixed but arbitrary, and let P 1 , . . . P 6 be the extreme points or vertices of P µ given in (15).
To prove (iv), it is sufficient to prove S(∂P µ ) ⊂ P µ . We have Hence S([P 1 , P 2 ]) is a line segment with endpoints in the set P µ , which is convex.

Boundedness and Persistence of Solutions
A proof of boundedness and persistence of solutions of system (5) for 1 < γ < α is presented in this section, which corresponds to statement (vii) of Theorem 1.

Structure of the Proof of Statement (vii) of Theorem 1
Throughout the section we shall assume the inequality Under this assumption, there exists a unique positive equilibrium (6) for system (5). The change of variables conjugates system (5) to x n+1 = a x n + (1 − a) x n y n , y n+1 = y n (1 − b) x n y n + b , n = 0, 1, 2, . . . , (19) where a and b are as in (12). The map corresponding to (19) is given by in which case the map T has a unique positive fixed point, namely (1, 1). We shall prove (vii) of Theorem 1 by proving a similar statement for (19) under assumption (21).
SetT := L • T • E. That is, ThusT is a conjugate of T for which the origin is the (unique) fixed point. An immediate consequence of the definition ofT is Proposition 2 presented below. It can also be shown that bounded subsets of R 2 are contained inT -invariant compact sets, as described in Proposition 3.
Proposition 3 Suppose 0 < a < b < 1. Then for any bounded set B ⊂ R 2 there exists aT -invariant compact set K such that B ⊂ K.
Propositions 2 and 3 have the following corollary, which is precisely statement (vii) of Theorem 1.
The remainder of this section is devoted to proving Proposition 3. The proof involves constructing a family of compact sets K τ that satisfy the properties set forth in Proposition 3 for τ taken to be sufficiently large. First, we present basic results about T andT as well as results related to two auxiliary maps that are useful in the construction of K τ and for the arguments that follow.

Ancillary Properties and Maps
If F = (f 1 , f 2 ) is a map on a planar region R, the equilibrium curves of F are the sets {(x, y) ∈ R : f 1 (x, y) = x} and {(x, y) ∈ R : f 2 (x, y) = y}. The equilibrium curves of the maps T andT given in (20) and (23) play a prominent role in our proof. Before we go any further, we adopt the following convention in order to simplify notation use: The equilibrium curves of the maps T are as follows: That is, Similarly, the equilibrium curves of the mapT arê The curvesĈ 1 andĈ 2 have (0, 0) as their only common point, and the complement in the plane of their union consists of four disjoint connected componentŝ That is, The sets R andR , 1 ≤ ≤ 4, are depicted in Figure 10. Now, denote with se the South-East partial order on R 2 whose nonnegative cone is the standard Similarly, denote with ne the North-East partial order on R 2 whose nonnegative cone is the standard first quadrant {(u, v) : u, v ≥ 0}.

Proposition 4
The following statements are true: We shall need the maps We shall need the constant r given by Under assumption (21), r satisfies 0 < r < 1.
Lemma 1, given below, details an invariant curve corresponding to the map M along with properties of its image underT .
Lemma 1 Let τ be a fixed but otherwise arbitrary positive real number. Letf 1 : (−∞, τ] → R be the function given bŷ and letD 1 andD 1 be the setŝ Thenf 1 (·) is a convex smooth function, Figure 11 shows the curveD 1 described in Lemma 1 along with its image under the mapT . An extension ofD 1 and its corresponding image in the third quadrant are also included to illustrate the relationT ( , which is needed in the arguments used in Section 3.4.4.

Proof.
A straightforward calculation giveŝ  (21) and (25), imply thatf 1 (u) is defined for u ≤ τ , and consequently v is a convex function of u for u ≤ τ .
With the change of coordinates x = e u , y = e v , together with x 0 := e τ and the inclusion M (D 1 ) ⊂ D 1 is equivalent toM (D 1 ) ⊂D 1 . We prove the former.
Suppose (x, y) ∈ D 1 , and set (x , y ) := M (x, y) = a x, Then (x , y ) ∈ D 1 if and only if x ≤ e τ and Through algebraic manipulation, equation (28) may be rewritten as The equality a r = b and further simplification in (29) give the equation Since by assumption (x, y) ∈ D 1 , we have (30) is true. It is also the case that x = a x ≤ a e τ < e τ . This proves (x , y ) ∈ D 1 .
To prove the second inclusion in (27), consider (u, v) ∈D 1 , and set (u , v ) = From the definition ofM and T we have Consider the function ψ(t) with t ≤ τ , given by This fact, (31), and the increasing character of ψ give complete the proof of the second inclusion in (27). See Figure 11. ✷ Lemma 2, given below, details a property of the image underT of certain line segments.
Lemma 2 Let p and q be arbitrary negative numbers such that q p < r, where r is defined in (24). LetD 2 be the line in the plane through (p, 0) and (0, q), and letD  We claim It can be easily shown that for fixed v ≤ 0, φ(u, v) is increasing in u. Therefore, it is sufficient to verify (33) for φ(0, v). Equivalently, with y := e v , we will verify that f (y) ≤ −r for all y ∈ (0, 1), where Notice, t + ln(1 − t) < 0 for t ∈ (0, 1).

The final lemma in this subsection details an invariant curve corresponding
to the mapN along with properties of its image underT . Prior to stating the lemma, we verify that Consider (u, v) ∈ {(s, t) : s + t ≥ 0, s ≤ 0, t > 0} such that u + v = 0 and noticê T (u, v) = (ln(a e u + (1 − a)), v).
Since v > 0,T (u, v) is in the first quadrant of the plane and thus belongs toR 1 .
By continuity ofT , (38) follows. This relation will be helpful in proving Lemma 3 below.
Lemma 3 Let c 0 be a fixed but otherwise arbitrary negative real number, and set and c 2 := − 1 2 ln(b) .
by the first part of this proof and the relationT (u, v) ∈ { (s, t) : s > c 2 t 2 + c 1 t + c 0 , t > 0} follows. The curveD 3 along with its image underT can be seen in Figure 13. ✷

Construction of a Family of Compact Sets
We begin by establishing some useful inequalities. We shall need the following values, which can be obtained from equation (26): Lemma 4, presented below, is easily established from relations (21), (25), (46) and (47).
Lemma 4 There exists τ 1 > 0 such that The sets K τ are introduced next.
Definition 7 Let τ ∈ R + be such that τ ≥ τ 1 with τ 1 as in Lemma 4, and set , and   Remark 1 In Definition 7, q 1 < 0 and p 2 < 0 by Lemma 4. Therefore, K τ is a compact and convex neighborhood of the origin such that ∂K τ = ∪ 4
Remark 2 In order to simplify notation, dependence on τ has been suppressed in the terms q 1 , p 1 , q 2 , andD , 0 ≤ ≤ 4. Figure 14: A set K τ whose boundary consists of the setsD for 0 ≤ ≤ 4 The proof of Proposition 3 involves an asymptotic argument on the parameter τ as it relates to the compact set K τ . It is useful for us to first describe the asymptotic behavior of q 1 , q 2 , and p 2 when τ → +∞.

Proof of Proposition 3
To prove Proposition 3, we establish first that any given bounded set B ⊂ R 2 is contained in K τ for τ large enough.
Claim 2 Let B ⊂ R 2 be bounded. Then for all τ large enough, B ⊂ K τ .
Proof. Since K τ is convex, the quadrilateral S whose endpoints are (τ, 0), (0, q 1 ), (p 2 , 0) and (0, q 2 ) is such that S ⊂ K τ (see Figure 15). Therefore, Claim 1 implies that for all large enough τ , K τ contains B. ✷ Next we prove that for all τ large enough,T (D ) ⊂ K τ for 0 ≤ ≤ 4. Once this has been established, it follows that K τ isT -invariant for large τ and the proof Τ Τ Figure 15: The quadrilateral S τ with S τ ⊂ K τ of Proposition 3 will be complete. The boundary of K τ along with its image under the mapT can be seen in Figure 16. We assume in Claims 3 through 7 that τ ≥ τ 1 .
We now show thatT (D 0 ) is a curve linearly ordered in the ne partial order. We may writeD Thus bothũ(t) andṽ(t) are decreasing functions of t in [0, 1], soT (D 0 ) is a curve linearly ordered in the ne partial order. It follows thatT (D 0 ) is a subset of the closed rectangular region R determined by the verticesT (τ, 0) andT (τ, −τ ). Since the second coordinate ofT (τ, −τ ) is equal to −τ andD 1 is the graph of a convex function, it follows from (27) that R ⊂ K τ , and consequently, Figure 16: Boundary of the set K τ (solid) and its image underT (dashed).
Proof. For (u, v) ∈D 1 arbitrary but fixed, let (ũ,ṽ) be given by By the second relation in (27) of Lemma 1, and convexity of K τ andD 1 , it is sufficient to verify thatṽ < 0. Notice (u, v) ∈D 1 implies u > 0, v < 0, and Proof. The line segmentD 2 has slope − q 1 p 2 =f 1 (0). Noŵ Thus the hypothesis − q 1 p 2 > −r of Lemma 2 is satisfied. Now, let L be the line through (0, q 1 ) and (p 2 , 0) and let L 0 be the connected component of R 2 \ L that contains the origin. Lemma 2 guaranteesT (D 2 ) ⊂ L 0 . Also, noteT (D 2 ) is linearly ordered in the se partial order, which we verify next. We may writeD Using statements (i) and (ii) of Claim 1 and (52) we conclude that for τ large enough,ũ(t) is a decreasing function of t ∈ [0, 1] andṽ(t) is an increasing function of t ∈ [0, 1]. Consequently,T (D 2 ) is a curve linearly ordered in the se partial order and is thus a subset of the rectangular region R determined by the initial and final points. HenceT Note thatT (0, q 1 ) ∈ K τ by Claim 4 andT (p 2 , 0) ∈ K τ by Lemma 3. It follows from (53) and the convexity of K τ thatT (D 2 ) ⊂ K τ . ✷ Claim 6 For all τ large enough,T (D 3 ) ⊂ K τ .
We also haveT (τ, 0) = ln (ae τ + (1 − a)e τ ) , ln 1 In light of (64) and (65), to prove the claim, it is sufficient to verify thatT (D 4 ) is in a suitable component of the complement of the line through (0, q 2 ) and (τ, 0), for τ large enough. More precisely, we wish to verify For fixed τ , define We have, Finally, ψ τ (0) < 0 follows from Claim 6 and thus (68) implies (67). ✷ This completes the proof of Proposition 3 and thus, by Corollary 4, of statement (vii) from Theorem 1. for ∈ {1, . . . , k}, A is a bounded, linear operator on an ordered Banach space X with positive cone X + such that AX + ⊂ X + , and for each ∈ {1, . . . , k}, b ∈ X + and c is a positive bounded linear functional on X.

List of References
Consider the following systems of difference equations: . . , f k (y (k) n ) ) t , n = 0, 1, 2, . . . , y 0 ∈ R k + (I) and x Conditions are established under which there is a correspondence between equilibrium points of (I) and (II). Under these conditions, and when X = R m , the stability type of the zero equilibrium of (I) is shown to be the same as that for (II). When k = 2 and the functions f have certain monotonicity and convexity characteristics, sufficient conditions are given for the existence of a unique positive equilibrium for system (I). In this case, the stability of the equilibrium at the origin is also established. Examples are included.

Introduction
Discrete dynamical systems are used to model populations in biological, epidemiological, and entomological applications. Many of these populations are nonhomogeneous in the sense that individuals vary in physiological characteristics and may interact with the environment differently. These differences play an important role in the dynamics of the entire population and necessitate a special type of mathematical model, referred to as a structured population model. Structured population models divide a population into specific classes or categories based on, among other things, chronological age, body size, or genetic differences [1].
The model then tracks the size/density through each generation for the variety of classes by utilizing information related to the growth within each class as well as rates of transfer from one class to another [1,2]. Models of this type have also been referred to as compartmental models and are discussed in detail in [1] and [2].
For interesting applications of structured models that have been studied recently, see [3], [4], [5], and [6]. The focus of this manuscript will be on two specific types of structured population models, introduced next.
Suppose that a population is divided into k classes so that at (discrete) time where f : [0, ∞) → [0, ∞) is a continuous function with f (0) = 0 for ∈ {1, 2, . . . , k}. The terms m ,j are nonnegative constants that reflect birth and death rates of the class and take into account transfer rates from the j to class.
If we denote by M = (m ,j ) the k × k coefficient matrix, then we may write (1) as the system . . , f k (y (k) n ) ) t , n = 0, 1, 2, . . . , y 0 ∈ R k + . (I) Systems of the form (I) were introduced for a specific genetics example by Friedland and Karlin in [7] and can also be thought of as a special case of the more general equation y n+1 = P (y n ) y n , n = 0, 1, 2, . . . , y 0 ∈ R k + , which is studied by J. M. Cushing in [1]. Model (I) is a primary focus of this manuscript.
Model (I) is useful in analyzing other types of structured population models.
In particular, (I) is utilized in this paper to study systems of the form where A is a bounded, linear operator on an ordered Banach space X with positive cone X + such that A X + ⊂ X + and for each ∈ {1, . . . , k}, b ∈ X + , c is a positive bounded linear functional on X, and f is a real valued function, as defined above.

Model (II) is a generalization of the population model formulated by Rebarber,
Tenhumberg, and Townley in [8]. They introduce the nonlinear, density dependent model given by where A, x n , c, b and f are as stated above when k = 1 (see [8]). System (2) successfully takes into account two important biological processes, survival/growth and fecundity, and is useful in modeling stage structured plant and fishery populations [8,9]. The operator A is referred to as the survival operator and f (·) b is referred to as the fecundity operator . The term c x n describes the number (or density) of offspring, f (c x n ) reflects the nonlinear density dependence on c x n , and b describes how the offspring are distributed amongst the classes. The density dependent fecundity structure f (c x n ) b is common for single-species, structured populations. Specific examples can be seen in [9], [10], and [11]. Some common density dependencies that are used in applications include f (t) = β t α , α ∈ (0, 1) and β > 0, where (3) is a power-law nonlinearity, (4) is of Beverton-Holt type, and (5) is a Ricker nonlinearity [9]. Smith and Thieme in [12] also study (2)  In this paper, we give an answer to the question by finding a k-dimensional model of the form (I) that can be used to study (II). In particular, in Section 4.5, we give conditions under which there is a one-to-one correspondence between the positive equilibrium points of (I) and (II). Positive equilibrium points represent persistence of the population and are an important feature of the models. Furthermore, when the state space of (II) is R m , we establish that the stability character of the origin (which represents the extinction state) in (I) is the same as that of (II). With these similarities between (I) and (II), it is useful to explore how stability of the zero equilibrium is related to the existence, uniqueness, and stability of a positive

Background
For convenience, basic notions and definitions are provided here that are utilized within the main sections of the paper.
Let X be a Banach space over R. A set X + ⊂ X is an order cone if X + is closed, convex, and such that An order cone is solid if int(X + ) = ∅ (see [13]). If X * is the dual space of X, the dual cone of X + is the set If X + is solid, then the set X * + is a cone in X * . A functional c ∈ X * + is said to be positive. The functional is strictly positive if c(x) > 0 for x ∈ X + \ {0}. If x ∈ int(X + ), then c(x) > 0 for every c ∈ X * + \ {0} ( [13], Proposition 19.3). Every order cone X + induces a partial order on X as follows: x y if and only if y − x ∈ X + . In this case, X is an ordered Banach space with order cone X + . For points x, y ∈ X + , we say x y if and only if y − x ∈ X + , x ≺ y if and only if y − x ∈ X + \ {0}, and x y if and only if y − x ∈ int(X + ) (see [14]).
The partial order is compatible with addition, multiplication by a nonnegative scalar, and convergence. The order interval x, y, relative to the partial order , is defined by x, y := {u ∈ X : x u y}.
Let T : X → X be an operator. We say that T is monotone (with respect to the partial order ) if x y =⇒ T (x) T (y), strictly monotone if x ≺ y =⇒ T (x) ≺ T (y) and strongly monotone if x ≺ y =⇒ T (x) T (y). If T is linear, monotonicity is equivalent to T X + ⊂ X + and strong monotonicity is equivalent to T (X + \ {0}) ⊂ int(X + ). If an operator T is monotone with respect to a partial order, it is referred to as order-preserving [14].
Theorem 1 For a nonempty set U ∈ R m and a partial order on R m , Let T : U → U be an order-preserving map and let u, v ∈ U be such that u ≺ v and u, v ⊂ U . If u T (u) and T (v) v, then u, v is an invariant set and (i) There exists a fixed point of T in u, v.
(ii) If T is strongly order-preserving, then there exists a fixed point of T in u, v that is stable relative to u, v.
(iii) If there is only one fixed point in u, v, then it is a global attractor in u, v and therefore asymptotically stable relative to u, v.
Properties of matrices with real entries also play a role in the coming sections.
A matrix A ∈ R m×m is nonnegative (positive), written A ≥ 0 (A > 0) if all of the entries of A are nonnegative (positive). If A > 0, then A is strongly monotone as a linear operator on R m with order cone R m

General Results for System (I)
In this section we investigate systems of the form (I) on R k + with regards to stability of the zero equilibrium and existence, uniqueness, and stability of a positive equilibrium. Let be a partial order on R k , where the order cone is taken to be the standard positive orthant R k + . In other words, for u, v ∈ R k , u v if and only if u () ≤ v () for all ∈ {1, 2, . . . , k}. Let T : R k + → R k + be the map associated with system (I). That is, for y ∈ R k + , For notational simplicity, let e := (1, 1, . . . , 1) t ∈ R k + .
Lemmas 1 and 2, presented next, help to establish results related to the existence of a positive fixed point for the map T as well as to the stability of the origin.
Lemma 1 provides sufficient conditions for the existence of a strict supersolution (i.e. y ∈ int(R k + ) such that T (y) ≺ y) or a strict subsolution (i.e. y ∈ int(R k + ) such that y ≺ T (y)) where ||y|| > r for r arbitrarily large. These solutions provide valuable information about the behavior of the map T at infinity. Lemma 2 presents a similar result with y as close to the origin as we wish.
Lemma 1 Let r > 0. The following properties hold: , then there exists t * > r such that for all t > t * , y = t e satisfies T (y) ≺ y.
(ii) If f (∞) h − (M ) > 1, then there exists t * > r such that for all t > t * , y = t e satisfies y ≺ T (y).
Choose t * such that t * > r and max for all t > t * .
Therefore, for all t > t * , Also, by the definition of h + (M ), M h + (M ) e ≺ e. Consequently, defining y = t e and using the form of T given in (7) for all t > t * .
Therefore, for all t > t * , Also, by the definition of h − (M ), M h − (M ) e e. Consequently, defining y = t e and using the form of T given in (7), we have The result follows. ✷ Lemma 2 Let δ > 0. The following properties hold: , then there exists t * < δ such that for all t < t * , y = t e satisfies T (y) ≺ y.
Proof. The proofs of (i) and (ii) in Lemma 2 are similar to the proofs of (i) and
This α can also be chosen in such a way that ||α v|| < . Let y = α v and this proves the claim. Suppose that the positive fixed point x of the map T is unique. By statement (iii) of Theorem 1, x is a global attractor in the order interval y , y. By Lemma 1, y can be chosen such that ||y|| > r for any r > 0 and by the above claim, y can be chosen such that ||y || < for any > 0. Letting → 0 and r → ∞, it follows that x is a global attractor on int(R k + ). ✷ We now explore the special case when k = 2 for system (I).

The Two-Dimensional Case for System (I)
In the two-dimensional case, more information can be determined about the fixed points of the map T in (6). Denote byT the map in (6) when k = 2. That In the proceeding arguments, when it is assumed that f 1 ,

Conditions for Uniqueness of the Positive Equilibrium
In this section we give conditions for which the mapT in (12) Fixed points of the mapT occur at the intersection points of the equilibrium curves C 1 and C 2 . We focus on two special cases: Case 1: f 1 , f 2 are twice differentiable, strictly increasing, and convex.
Two theorems are now presented, which provide necessary and sufficient conditions for a unique positive fixed point ofT to exist in Cases 1 and 2 from (14).
We begin with Theorem 4, which involves Case 1.
Theorem 4 Let M > 0 and suppose that f 1 , f 2 are twice differentiable, strictly increasing, and convex. Consider the following conditions: .
(B3) The limits L 1 and L 2 exist and L 2 < L 1 , where The where x 2 (t) = m 12 f 2 (t), and The curves C 1 and C 2 intersect if and only if φ 1 and φ 2 intersect. We need the following derivatives: Under the hypotheses imposed on f 1 , f 2 , it is clear from (18) that Let Q 1 be the standard first quadrant. We claim that if m 11 f 1 (0) ≥ 1, then If Since (B1) and (B2) are not satisfied, then dy 1 dx 1 , dy 1 dx 1 > 0 for all t ≥ 0. From (19), (21) and the fact that (B3) is not satisfied, it follows that φ 1 and φ 2 do not have a positive intersection, a contradiction. We conclude that ifT has a unique positive fixed point (t 1 , t 2 ) ∈ (0, ∞) × (0, ∞), then (A) is satisfied along with at least one of (B1), (B2) and (B3).
Proof. The proof of Theorem 5 is similar to that of Theorem 4 and is omitted here.  (14). In these special cases, more can be said about the stability of the origin as a fixed point of the mapT . The Jacobian matrix associated withT is .
When f 1 (t), f 2 (t) ≥ 0 for all t ≥ 0, all entries of JT are nonnegative on R 2 + and thus the mapT is monotone (i.e. cooperative). Evaluating the Jacobian matrix at the fixed point (0, 0), we have , from which it follows that the eigenvalues associated with JT (0, 0) are the roots of the characteristic equation In Case 1 from (14), ifT has a unique positive fixed pointx, then it is shown that x is a repeller or a saddle point and that the fixed point at the origin is locally asymptotically stable. In Case 2, ifT has a unique positive fixed pointx, then under one additional assumption (to rule out nonhyperbolic cases) it is verified thatx is locally asymptotically stable and that the fixed point at the origin is unstable. These statements are formally presented and proven in Theorems 6 and 7.
Theorem 6 Suppose that f 1 , f 2 are twice differentiable, strictly increasing, and convex. If the mapT has a unique positive fixed pointx, thenx is a repeller or a saddle point and the fixed point at the origin is locally asymptotically stable.
In this case, condition (24) is equivalent to .
The irreducibility of M , along with (13), implies that JT (x) is irreducible. By the Perron-Frobenius theorem, ρ * = ρ(JT (x)) is a positive eigenvalue of JT (x) associated with a positive eigenvector v. That is, For α < 0, take y =x + αv. Consequently, Since the interior of the order interval 0,x is a subset of the basin of attraction ofx, for α < 0 chosen sufficient close to zero, the left hand side of (32) is positive, which implies that ρ * < 1. Hencex is locally asymptotically stable. ✷ The above results provide a strong connection between the existence of a positive fixed point and the stability of the origin when f 1 , f 2 satisfy the conditions set forth in (14). A natural extension is to investigate the stability of the origin in these same cases when it is assumed that a positive fixed point does not exist.
This will be investigated in the future.

Examples
To illustrate Theorems 4, 5, 6, and 7 presented in where m 11 , m 12 , m 21 , m 22 , δ 1 , δ 2 > 0 and the initial condition (x 0 , y 0 ) ∈ (0, ∞) × (0, ∞). The map T 1 associated with system (33) is and has the form given in (12), where f (t) = t δ +t for = 1, 2. A simple calculation shows that f 1 , f 2 are strictly increasing and concave. Therefore, Theorem 5 can be applied to determine necessary and sufficient conditions for which a unique positive fixed point exists for the map T 1 . When it is known that T 1 has a unique positive  When it exists,x is unique and a global attractor on int(R 2 + ). When there does not exist a positive fixed point, the fixed point at the origin is a global attractor.
Proof. In this case, which implies that (A) of Theorem 5 is satisfied. Now, assume (H1) is satisfied.
Furthermore, a simple calculation shows that there exists t * = Consequently,x is a global attractor on int(R 2 + ) by Theorem 3. Now, ifx does not exist then (H1), (H2), and (H3) are not satisfied. This immediately implies that m 11 ≤ δ 1 , m 22 ≤ δ 2 and (δ 2 − m 22 )(δ 1 − m 11 ) > m 21 m 12 . We will verify that the origin is locally asymptotically stable by applying Shur-Cohn criterion to the Jacobian of the map evaluated at 0. That is, we will verify conditions (25), given in this example as Since which verifies the second inequality from (37). The first inequality from (37) then follows directly from (δ 2 − m 22 )(δ 1 − m 11 ) > m 21 m 12 . Consequently, the origin is locally asymptotically stable.
Using (36), Lemma 1 guarantees that there exists a point y ∈ (0, ∞) × (0, ∞), with ||y|| as large as we like, such that T 1 (y) ≺ y. Therefore, from Theorem 1, the order interval 0, y is invariant under T 1 . With no other interior fixed points, a result from [18] ensures that the interior of 0, y is a subset of the basin of attraction of the origin. Since y can be chosen as large as we like, the origin is a global attractor. ✷ It is worth mentioning that the global dynamics of (33) can be completely characterized using general theory of two-dimensional cooperative systems of difference equations. The example is presented only to illustrate the main theorems.
The next example illustrates both Theorems 4 and 5.
It follows from Theorem 3 thatx is a global attractor on int(R 2 + ). ✷ It is worth mentioning that the global dynamics of (38) can be completely characterized using general theory of cooperative systems of difference equations.
The example is presented only for illustrative purposes.

Analysis of System (II)
The map T : R k + → R k + , given in (6), can be used to study more general structured population models. We specifically show that T can be used to analyze systems of the form (II).
Let X be an ordered Banach space over R, with solid order cone X + . An element x ∈ X is said to be positive if x ∈ int(X + ). Let A be a nontrivial monotone bounded linear operator (so A X + ⊂ X + ) such that the spectral radius of A is less than one. Let b 1 , . . . , b k be k ≥ 1 linearly independent elements of X + , let c 1 , . . . , c k be positive bounded linear functionals on X, and let f 1 , . . . , f k be real valued functions as defined above. Consider the difference equation Denote by F the map corresponding to (II), i.e., Thus 0 ∈ X is a fixed point of F . Let C : X → R k and B : R k → X + be the linear Define M : R k → R k from (6) as and refer back to the map T in (6), where now we have T (y) = M (f 1 (y (1) ), . . . , f k (y (k) )) t = C(I − A) −1 B(f 1 (y (1) ), . . . , f k (y (k) )) t (46) We explore the relationship between fixed points of the map T in (46) and the map F in (44).

Correspondence of Fixed Points
The first result in this section establishes a correspondence between the fixed points of T and the fixed points of F .  By assumption, for some 0 we haves 0 =t 0 , and since f 0 is one-to-one, f 0 (s 0 ) = f 0 (t 0 ). By this relation and linear independence of the terms b , k =1 (f (s ) − f (t )) b = 0. Thus the left-hand side of equation (50) is not zero. That is, x = y.

✷
There is a corollary to Theorem 8 that is useful when it is known that the map T has either no positive fixed points or a unique positive fixed point.

Corollary 5
If T has no fixed points in int(R k + ), then F has no fixed points in int(X + ). If T has a unique fixed pointt ∈ int(R k + ), then F has a unique fixed point x ∈ int(X + ), namelyx

A Linear Algebra Result
We will see that in the case when X = R m in (II), there is a strong connection between the stability of the origin in X as a fixed point of F and the stability of origin as a fixed point of T . To establish this connection, we shall prove a linear algebra result given in this subsection.
We will need the following lemma. We denote with I m the m × m identity matrix, adj(A) the adjugate of a square matrix A (i.e., the transpose of the matrix of cofactors of A), ρ(A) the spectral radius of A, and σ(A) the spectrum of A. Proof. For t > λ 1 the matrix adj(t I m − A) is invertible, and, Note det(t I m − A) = (t − λ 1 ) · · · (t − λ m ) > 0. Also, the Neumann series expansion (I m − 1 t A) −1 = I + 1 t A + 1 t 2 A 2 + · · · and A irreducible imply that (t I m − A) −1 has positive entries.. Since adj(λ 1 I m − A) = lim For t ≥ 0, let ρ 1 (t) denote the spectral radius of A+t B C, and let ρ 2 (t) denote the spectral radius of t C E. We claim that ρ (t) is a monotonically increasing function of t for t ≥ 0. Since for each t, ρ (t) is a simple root of the characteristic polynomial, by the Implicit Function Theorem ρ (·) is a C (1) function of t. Define We now verify that ρ 1 (t) takes values smaller than one and also larger than one. Note that ρ 1 (0) = ρ(A) < 1. Choose t > ρ(B C) −1 . By Corollary 8.1.19 in [19], ρ(A + t B C) ≥ ρ(t B C) = t ρ(B C) > 1, that is, ρ 1 (t ) ≥ 1.
Since ρ 1 (t) takes values less than, greater than, and (by continuity) equal to one, by statement (i) and monotonicity of ρ , = 1, 2, it follows that ρ 1 (t) > 1 if and only if ρ 2 (t) > 1, ρ 1 (t) < 1 if and only if ρ 2 (t) < 1, and ρ 1 (t) = 1 if and only if ρ 2 (t) = 1. Statement (ii) of the proposition now follows from the latter relations and the equalitiesρ 1 = ρ 1 (1) andρ 2 = ρ 2 (1). ✷ Proposition 4 can now be utilized to discuss the relationship between the stability character of the origin as a fixed point of the map T and as a fixed point of the map F .

Stability of the Origin
We now take X = R m with the standard order cone R m + = {(x 1 , . . . , x m ) : x ≥ 0, = 1, . . . m }. Let A ∈ R m×m + , B ∈ R m×k + and C ∈ R k×m + . For = 1, . . . , k, denote with c the -th row of C, and define c x to be the inner product of c and x, that is c x = c · x. Suppose ρ(A) < 1, and that B has full column rank.
The next result states that for system (I), the local stability character of 0 ∈ R m as a fixed point of F is the same as the local stability character of 0 ∈ R k as a fixed point of T in (46). By J F (x) and J T (t) we denote the Jacobian matrices of F and T at x and t respectively.

Examples
To illustrate the connection between all of the results of the preceeding sections, we consider an example of a system with the form (44) for k = 2, where the functions are taken to be of Beverton-Holt type (4). This example requires results established in Example 1.
Example 3: Consider the system of difference equations where x n , x 0 , b 1 , b 2 ∈ R m + , A ∈ R m + × R m + , δ 1 , δ 2 > 0, and c 1 , c 2 are positive row vectors in R m + . This system has the form given in (44) where f (t) = t t+δ is of Beverton-Holt type. Denote by F 3 : R m + → R m + the map corresponding to (59). That is, F 3 (y) = A y + c 1 y c 1 y + δ 1 b 1 + c 2 y c 2 y + δ 2 b 2 (60) Define v = (I − A) −1 b for = 1, 2 and let M ∈ R 2 + × R 2 + be as in (45). That is, Consider the map T 3 : R 2 + → R 2 + defined in (46), given by Compare (61) to the map studied in Example 1, where m i,j = c i v j . There exists a correspondence between the fixed points of T 3 and the fixed points of F 3 as discussed in the previous section. This leads to the following claim: The map F 3 has a positive fixed pointȳ if and only if at least one of the following conditions is satisfied: (i) c 1 v 1 ≤ δ 1 and c 2 v 2 > δ 2 or c 1 v 1 > δ 1 and c 2 v 2 ≤ δ 2 .
Furthermore, if it exists,ȳ is unique and locally asymptotically stable.
Proof. Under any of the conditions (i) through (iii), it follows from Proposition 1 that the map T 3 from (61) has a unique positive fixed pointx. The existence and uniqueness of a positive fixed pointȳ for the map F 3 then follows from Corollary 5. Now, the local stability character of 0 ∈ R 2 + as a fixed point of T 3 is the same as the local stability character of 0 ∈ R m + as a fixed point of F 3 by Theorem 9. By Theorem 7, 0 ∈ R 2 is an unstable, hyperbolic fixed point of T 3 and thus 0 ∈ R m is an unstable, hyperbolic fixed point of  Figure 19: The curves φ 1 and φ 2 corresponding to system (62) can be seen in (a). A solution (x n , y n ) to (62) with (x 0 , y 0 ) = (2, 6) can be seen in (b). The solution quickly approaches the equilibrium pointx given in (63). m 21 = 1, m 22 = 2, δ 1 = 2, and δ 2 = 1. That is, consider x n+1 = 4 x n 2 + x n + 3 y n 1 + y n , y n+1 = x n 2 + x n + 2 y n 1 + y n , n = 0, 1, 2, . . . , where (x 0 , y 0 ) ∈ (0, ∞) × (0, ∞). It can be easily verified that the conditions set forth in Proposition 1 are satisfied. It follows that system (62) has a unique positive equilibrium point. This equilibrium is given bȳ x ≈ (4.849, 2.053), where the coordinates have been rounded to three decimal places. The equilibrium x is a global attractor on int(R 2 + ) by Proposition 1. The equilibrium point, along with the parametric curves φ 1 and φ 2 for this case (defined in (15)), can be seen in (a) of Figure 19. A solution to (62) for (x 0 , y 0 ) = (2, 6) can be seen in (b) of Figure 19.
It can be verified that system (64)  Theorem 7 guarantees that the fixed point at the origin is locally asymptotically stable and Theorem 6 allows us to conclude thatx is unstable. In fact, the eigenvalues of the Jacobian evaluated atx are λ 1 ≈ 1.272 and λ 2 ≈ 0.126, and thusx is a saddle point. It can also be verified that there do not exist any minimal period two solutions in R 2 + . Therefore, applying results from [20,21,22], there exists the global stable manifold W s (x) and the global unstable manifold W u (x) passing throughx and extending to the boundary of the first quadrant (depicted in (b) of Figure 20). The region in the first quadrant below (resp. above) the curve W s (x) is the basin of attraction of the origin (resp. the point (+∞, +∞)) and the curve W s (x) is precisely the basin of attraction of E.
The equilibrium point, along with the parametric curves φ 1 and φ 2 for this case (defined in (15)), can be seen in (a) of Figure 20. A solution of (64) with an initial condition chosen from below the global stable manifold can be seen in (a) of Figure 21 and a solution of (64) with an initial condition chosen from above the global stable manifold be seen in (b) of Figure 21.   Figure 21: A solution (x n , y n ) to (64) with (x 0 , y 0 ) = ( 3 4 , 1 4 ) can be seen in (a). The solution quickly converges to the origin. A solution (x n , y n ) to (64) with (x 0 , y 0 ) = (1, 1) can be seen in (b). The solution approaches (+∞, +∞).