Global Dynamics of Some Discrete Dynamical Systems in Mathematical Biology

This thesis will be presented in manuscript format. The first chapter will introduce preliminary definitions and theorems of difference equations that will be utilized in chapters 2, 3, and 4. The second chapter will investigate the global behavior of two difference equations with exponential nonlinearities xn+1 = be −cxn + pxn−1, n = 0, 1, . . . where the parameters b, c are positive real numbers and p ∈ (0, 1) and xn+1 = a+ bxn−1e −xn , n = 0, 1, . . . where the parameters a, b are positive numbers. The initial conditions x−1, x0 are arbitrary nonnegative numbers. The two equations are well known mathematical models in biology, which behavior was studied by other authors and resulted in partial global dynamics behavior. In this manuscript, we complete the results of other authors and give the global dynamics of both equations. In order to obtain our results we will prove several results on global attractivity and boundedness and unboundedness for general second order difference equations xn+1 = f(xn, xn−1), n = 0, 1, . . . which are of interest on their own. The third chapter will investigate the global behavior of the cooperative system xt+1 = min{r11xt + r12yt, K1}, yt+1 = min{r21xt + r22yt, K2}, t = 0, 1, . . . where the initial conditions x0, y0 are arbitrary nonnegative numbers. This system models a population comprised of two subpopulations on different patches of land. The model considers the minimum between the maximum carrying capacity of each patch (K1 or K2 resp.) and the linear combination of the population from patch i from the last time step with those who migrated to patch i for i=1,2. We break the behavior of the system into several cases based on whether the linear combination of the population or maximum carrying capacity is greater. We are able to conclude that either one fixed point will be a global attractor of the interior region of R+ or there will exist a line of fixed points with the stable manifolds as the basins of attractions. We then extend some of these results to the n–dimensional case using similar techniques. We investigate the global behavior of the general cooperative system xt+1 = min{ri1xt + ri2xt + . . .+ riixt + . . .+ rinxt , Ki}, for i = 1, 2, . . ., n, and t=0,1,. . . where the initial conditions of x0 are arbitrary nonnegative numbers for i=1,2, . . ., n. We are able to conclude in some cases that one fixed point will be a global attractor of the interior region of R+. Finally, in the fourth chapter we will prove general results regarding the global stability of monotone systems without minimal period two solutions on a rectangular region R. We will illustrate the general results in two examples of well known systems used in mathematical biology. The first of the systems that will be investigated is a modified Leslie-Gower system of the form xn+1 = αxn+(1−α) cxn a+ cxn + yn and yn+1 = βyn+(1−β) dyn b+ xn + dyn , n = 0, 1, . . . , where the parameters a, b, c, d are positive numbers, α and β are positive values less than 1, and the initial conditions x0, y0 are arbitrary nonnegative numbers. In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when c = d = 1 and a = b, there will exist an infinite number of interior equilibrium points in which case we will find the basin of attraction for each of the equilibrium points. The second system that will be investigated is a version of a Lotka-Volterra model of the form xn+1 = xn(A− yn) K1 + xn and yn+1 = yn(A− xn) K2 + yn , n = 0, 1, 2, ..., where the parameters of A, K1, and K2 are all positive and the initial conditions x0, y0 are arbitrary nonnegative numbers, which is a semi implicit discretization of the continuous version. In most cases, there will be between one and three equilibrium points with solutions converging to one of the points. In one case when A > K1 = K2, however, there will exist an infinite number of equilibrium points. In this case for each equilibrium point, there will be a stable manifold as its basin of attraction.

The model considers the minimum between the maximum carrying capacity of each patch (K 1 or K 2 resp.) and the linear combination of the population from patch i from the last time step with those who migrated to patch i for i=1,2. We break the behavior of the system into several cases based on whether the linear combination of the population or maximum carrying capacity is greater. We are able to conclude that either one fixed point will be a global attractor of the interior region of R 2 + or there will exist a line of fixed points with the stable manifolds as the basins of attractions. We then extend some of these results to the n-dimensional case using similar techniques. We investigate the global behavior of the general cooperative system for i = 1, 2, . . ., n, and t=0,1,. . . where the initial conditions of x i 0 are arbitrary nonnegative numbers for i=1,2, . . ., n. We are able to conclude in some cases that one fixed point will be a global attractor of the interior region of R n + .
Finally, in the fourth chapter we will prove general results regarding the global stability of monotone systems without minimal period two solutions on a rectangular region R. We will illustrate the general results in two examples of well known systems used in mathematical biology. The first of the systems that will be investigated is a modified Leslie-Gower system of the form x n+1 = αx n +(1−α) cx n a + cx n + y n and y n+1 = βy n +(1−β) dy n b + x n + dy n , n = 0, 1, . . . , where the parameters a, b, c, d are positive numbers, α and β are positive values less than 1, and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers.
In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when c = d = 1 and a = b, there will exist an infinite number of interior equilibrium points in which case we will find the basin of attraction for each of the equilibrium points.
The second system that will be investigated is a version of a Lotka-Volterra model of the form x n+1 = x n (A − y n ) K 1 + x n and y n+1 = y n (A − x n ) K 2 + y n , n = 0, 1, 2, . .., where the parameters of A, K 1 , and K 2 are all positive and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers, which is a semi implicit discretization of the continuous version. In most cases, there will be between one and three equilibrium points with solutions converging to one of the points. In one case when A > K 1 = K 2 , however, there will exist an infinite number of equilibrium points. In this case for each equilibrium point, there will be a stable manifold as its basin of attraction.

ACKNOWLEDGMENTS
This thesis is the result of hard work and dedication over the last five years at the University of Rhode Island. I would like to take a moment to thank the incredible people who have helped make this journey possible.
First, I want to express my gratitude for Dr. Mustafa Kulenović. Five years ago, he introduced me to the world of difference equations. His enthusiasm and passion for mathematics inspired me to pursue research. As an advisor, Dr. Kulenović has helped foster my growth in the field and learn the art of solving proofs.
His innovation and intuition motivates me to be the best mathematician I can be.
Meanwhile, his advice and stories continues to teach me invaluable life lessons.
I would also like to thank Dr

Introduction
This thesis will primarily focus on difference equations and mathematical biology. In this chapter, we will outline some of the introductory theory of difference equations.
First we will consider the general system of difference equations of the form x n+1 = f (x n , y n ) y n+1 = g(x n , y n ) , n = 0, 1, . . .
Definition 1 A point (x,ȳ) is said to be an equilibrium point or fixed point if f (x,ȳ) =x and g(x,ȳ) =ȳ.
The following theory is used for local stability analysis. For an equilibrium point in the System (1), we define the stability in the following way.
(b) An equilibrium point (x,ȳ) is unstable if it is not stable.
(d) A periodic point (x p , y p ) of period m is stable if (x p , y p ) is a stable point of F m where F is the map of the equilibrium point.
We define the Jacobian matrix and linearization of the map F of the System (1) to be the following. This will then help us to define hyperbolic and nonhyperbolic.
is called the linearization of the map F at the fixed point (x,ȳ).
(b) An equilibrium point (x,ȳ) of the map F is said to be hyperbolic if the linearization of F is hyperbolic, that is if the Jacobian matrix J F (x,ȳ) at (x,ȳ) has no eigenvalues on the unit circle. If J F (x,ȳ) has at least one eigenvalue on the unit circle, then it is a non-hyperbolic equilibrium point.
Based on the eigenvalues of the Jacobian matrix we can make conclusions of the stability of an equilibrium point.
Theorem 1 Let F = (f, g) be a continuously differentiable function defined on an open set W in R 2 , and let (x,ȳ) in W be a fixed point of F .
(a) If all the eigenvalues of the Jacobian matrix J F (x,ȳ) have modulus less than one, then the equilibrium point (x,ȳ) is asymptotically stable.
(b) If at least one of the eigenvalues of the Jacobian matrix J F (x,ȳ) has modulus greater than one, then the equilibrium point (x,ȳ) is unstable.
The following theorem can be used to check the local stability of an equilibrium point.  Finally we will give the formal definition of a periodic solution.
Definition 5 (a) A solution {x n } is said to be periodic with period p if x n+p = x n for all n ≥ −1.
(b) A solution {x n } is said to be periodic with prime period p, or a p-cycle if it is periodic with period p and p is the least positive integer for which part (a) holds.
All other necessary definitions and theorems for the manuscripts will be self contained within chapters 2, 3, and 4. Abstract. We investigate the global behavior of two difference equations with exponential nonlinearities
where the parameters a, b are positive numbers. The initial conditions x −1 , x 0 are arbitrary nonnegative numbers. The two equations are well known mathematical models in biology, which behavior was studied by other authors and resulted in partial global dynamics behavior. In this paper, we complete the results of other authors and give the global dynamics of both equations. In order to obtain our results we will prove several results on global attractivity and boundedness and unboundedness for general second order difference equations x n+1 = f (x n , x n−1 ), n = 0, 1, . . . which are of interest on their own.

Introduction and Preliminaries
We investigate the global behavior of the system of difference equations x n+1 = be −cxn + py n , y n+1 = x n , n = 0, 1, . . . where the parameters b and c are positive real numbers, p ∈ (0, 1), and the initial conditions x −1 , x 0 are arbitrary nonnegative numbers. This system can be rewritten in the form of the second order difference equation x n+1 = be −cxn + px n−1 , n = 0, 1, . . .
In [5], the authors originally studied this model to describe the synchrony of ovulation cycles of the Glaucous-winged Gulls. The model assumed that there is an infinite breeding season as well as the number of gulls available to breed is infinite.
The value of c is a positive number representing the colony density. The parameter b is the number of birds per day ready to begin ovulating. The parameter p is the probability that a bird will begin to ovulate and 1 − e −cxn is the probability of delaying ovulation. In making the model, the authors assumed that the delay only occurs for birds entering the system, not birds switching between different segments of the cycle. Note the authors state that the bifurcation of two-cycle solutions is the same as ovulation synchrony with the value of c increasing. In [5], they used the local bifurcation theory to come to the conclusion that there exists a unique equilibrium such that for sufficiently small values of c, the equilibrium branch is locally asymptotically stable. Additionally, for large enough values of c, there exists a two-cycle branch that will be locally asymptotically stable. In this paper we will improve these results by making them global. Using the results of Camouzis and Ladas, see [2] and [6], we are able to find the global dynamics of (4), which was not completed in [5]. We will show that Equation (4) exhibits global period doubling bifurcation described by Theorem 5.1 in [11], which shows that global dynamics of Equation (4) changes from global asymptotic stability of the unique equilibrium solution to the global asymptotic stability of the minimal period-two solution within its basin of attraction, as the parameter passes through the critical value.
where the parameters a, b are positive real numbers and the the initial conditions x −1 , x 0 are arbitrary nonnegative numbers. As it was mentioned in [8], Equation (5) could be considered as a mathematical model in biology where a represent the constant immigration and b represent the population growth rate. In this paper, we find a simpler equivalent condition to −a+ √ a 2 +4a a+ √ a 2 +4a e a+ √ a 2 +4a 2 < b in [8] for the existence of a minimal period-two solution. We split the results into the two cases of b ≥ e a and b < e a . While using a similar method as in [9] to establish the existence of a period-two solution when b < e a , we are able to find the global dynamics of Equation (5). By using new results for general second order difference equations we will prove the existence of unbounded solutions for the case when b ≥ e a . Similar as for Equation (4) we will show that Equation (5) exhibits global period doubling bifurcation described by Theorem 5.1 in [11]. In addition, we give the precise description of the basins of attractions of all attractors of both Equations (4) and (5).
The rest of the paper is organized as follows. In the rest of this section we introduce some known results about monotone systems in the plane needed for the proofs of the main results as well as some new results about the existence of unbounded solutions. Section 2 gives the global dynamics of Equation (4) and Section 3 gives the global dynamics of Equation (5).
The next result, which is combination of two theorems from [2] and [6], is important for the global dynamics of general second order difference equation.
Theorem 4 Let I be a set of real numbers and f : I × I → I be a function which is either non-increasing in the first variable and non-decreasing in the second variable or non-decreasing in both variables. Then, for every solution {x n } ∞ n=−1 of the equation the subsequences {x 2n } ∞ n=0 and {x 2n−1 } ∞ n=0 of even and odd terms of the solution are eventually monotonic.
We now give some basic notions about monotone maps in the plane.
A partial ordering on R 2 where x, y ∈ R 2 is said to be related if x y or y x. Also, a strict inequality between points may be defined as x ≺ y if x y and x = y. A stronger inequality may be defined as x = (x 1 , x 2 ) y = (y 1 , y 2 ) if x y with x 1 = y 1 and x 2 = y 2 .
A map T on a nonempty set R ⊂ R 2 is a continuous function T : R → R.
The map T is monotone if x y implies T (x) T (y) for all x, y ∈ R, and it is Throughout this paper we shall use the North-East ordering (NE) for which the positive cone is the first quadrant, i.e. this partial ordering is defined by A map T on a nonempty set R ⊂ R 2 which is monotone with respect to the North-East ordering is called cooperative and a map monotone with respect to the South-East ordering is called competitive.
For x ∈ R 2 , define Q (x) for = 1, . . . , 4 to be the usual four quadrants based at x and numbered in a counterclockwise direction. Basin of attraction of a fixed point (x,ȳ) of a map T , denoted as B((x,ȳ)), is defined as the set of all initial points (x 0 , y 0 ) for which the sequence of iterates T n ((x 0 , y 0 )) converges to (x,ȳ).
Similarly, we define a basin of attraction of a periodic point of period p. The next few results, from [12,11], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L. Smith in [14,13].
Theorem 5 Let T be a competitive map on a rectangular region R ⊂ R 2 . Let x ∈ R be a fixed point of T such that ∆ : x is not the NW or SE vertex of R), and T is strongly competitive on ∆.
Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian J T (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T .
We shall see in Theorem 7 that the situation where the endpoints of C are boundary points of R is of interest. The following result gives a sufficient condition for this case.

Theorem 6
For the curve C of Theorem 5 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied.
i. The map T has no fixed points nor periodic points of minimal period-two in ∆.
ii. The map T has no fixed points in ∆, det J T (x) > 0, and T (x) = x has no solutions x ∈ ∆.
iii. The map T has no points of minimal period-two in ∆, det J T (x) < 0, and For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 5 reduces just to |λ| < 1. This follows from a change of variables [14] that allows the Perron-Frobenius Theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.
The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 7 (A) Assume the hypotheses of Theorem 5, and let C be the curve whose existence is guaranteed by Theorem 5. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W − := {x ∈ R \ C : ∃y ∈ C with x se y} and (7) such that the following statements are true.
(B) If, in addition to the hypotheses of part (A), x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q 1 (x) ∪ Q 3 (x) except for x, and the following statements are true.
(iii) For every x ∈ W − there exists n 0 ∈ N such that T n (x) ∈ int Q 2 (x) for n ≥ n 0 .
(iv) For every x ∈ W + there exists n 0 ∈ N such that T n (x) ∈ int Q 4 (x) for n ≥ n 0 .
If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set When T is non-invertible, the set W s (x) may not be connected and made up of infinitely many curves, or W u (x) may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map.
If the map is a diffeomorphism on R, the sets W s (x) and W u (x) are the stable and unstable manifolds of x.
Theorem 8 In addition to the hypotheses of part (B) of Theorem 7, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis. If the curve C of Theorem 5 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T .

Remark 1
We say that f (u, v) is strongly decreasing in the first argument and strongly increasing in the second argument if it is differentiable and has first partial derivative D 1 f negative and first partial derivative D 2 f positive in a considered set. The connection between the theory of monotone maps and the asymptotic behavior of Equation (6) follows from the fact that if f is strongly decreasing in the first argument and strongly increasing in the second argument, then the second iterate of a map associated to Equation (6) is a strictly competitive map on I × I, see [11].
Set x n−1 = u n and x n = v n in Equation (6) to obtain the equivalent system The second iterate T 2 is given by and it is strictly competitive on I × I, see [12].

Remark 2
The characteristic equation of Equation (6) at an equilibrium point (x,x): has two real roots λ, µ which satisfy λ < 0 < µ, and |λ| < µ, whenever f is strictly decreasing in first and increasing in second variable. Thus the applicability of and where at least one inequality is strict. If x −1 ≤ L and x 0 ≥ U, then the correspond- If, in addition, f is continuous and Equation and conditions (10) and (11) we obtain By using induction it follows that x 2n−1 ≤ L and x 2n ≥ U for all n = 0, 1, . .  x 2n−1 = −∞.
If I = (−∞, ∞), then similar to the two cases above, at most one subsequence can converge as there is no minimal period-two solution. So either with the option of both occurring. Finally, we will prove that I cannot be I = ] for all n, both subsequences would be convergent. As lim n→∞ x 2n−1 = p < lim n→∞ x 2n = q for some p, q ∈ R, there exists a period-two solution, which is a contradiction. The case when x −1 ≥ U and x 0 ≤ L will follow similarly to the proof used here.
Many examples of the use of Theorem 9 are provided in [4].
Theorem 10 Assume that f : I × I → I is a function which is nondecreasing in both variables, where I is an interval of real numbers that may be infinite. Assume there exists numbers L, U ∈ I such that L < U where and are satisfied, where at least one inequality is strict. If x −1 , x 0 ≤ L, then the corre- If, in addition, f is continuous and Equation (6) has no minimal period-two solution, then either x n converges to an equilibrium point or lim n→∞ x 2n−1 = −∞ and/or lim If x −1 , x 0 ≥ U, then the corresponding solution {x n } ∞ n=−1 satisfies x n ≥ U, n = 0, 1, . . .
If, in addition, f is continuous and Equation (6) has no period-two solution, then either x n converges to an equilibrium point or Proof. Assume that x −1 , x 0 ≤ L. Then by using the monotonicity of f (both variables are nondecreasing) and conditions (12) and (13) we obtain By using induction it follows that x 2n−1 , x 2n ≤ L for all n = 0, 1, . .  a, b ∈ R such that a < L < U < b. As both subsequences are less than L, then x n ∈ [a, L] for every n. As a consequence, both subsequences will be convergent.
Thus, lim n→∞ x 2n−1 = p and lim n→∞ x 2n = q. If p = q, we get a contradiction as the subsequences do not converge to an equilibrium point. Otherwise, p = q, so (p, q) is a period-two solution, which is a contradiction as well.
with the possibility of both options occurring.
Now assume that x −1 , x 0 ≥ U. Then by using the monotonicity of f and conditions (12) and (13) we obtain By using induction it follows that x 2n−1 , x 2n ≥ U for all n = 0, 1, . . a, b ∈ R such that a < L < U < b. As both subsequences are greater than U , then x n ∈ [U, b] for every n. As a consequence, both subsequences will be convergent.
Thus, lim n→∞ x 2n−1 = p and lim n→∞ x 2n = q. If p = q, we get a contradiction as the subsequences do not converge to an equilibrium point. Otherwise, p = q, so (p, q) is a period-two solution, which is a contradiction as well. Thus Example 1 Consider the difference equation increasing in both variables. The equilibrium points are x 0 = 0 and x + = 1/2.
The linearized difference equation is z n+1 = 2xz n + 2xz n−1 and the characteristic equation is λ 2 = 2xλ + 2x. The zero equilibrium x 0 is locally asymptotically stable.
For the equilibrium point x + , λ 2 = λ + 1, so that λ 1,2 = 1± ∈ (−1, 0), then x + is a saddle point. There is no minimal period-two solution as φ = ψ 2 + φ 2 and ψ = φ 2 + ψ 2 implies φ = ψ. Now we want to find a L < U that satisfies the conditions (12) and (13). Condition (12) f (L, L) ≤ L implies 2L 2 ≤ L, which simplifies to L ≤ 1/2. As well, f (U, U ) ≥ U if 2U 2 ≥ U , which simplifies to U ≥ 1/2. We can choose at least one of these inequalities to be strict. From Theorem 10, we can conclude that every solution with x 1 , x 0 ≤ L converges to 0, while every solution with x −1 , x 0 ≥ U is eventually increasing and tends toward ∞. As L < 1/2 < U are arbitrary this conclusion holds for every case where x −1 , x 0 ≤ L or x −1 , x 0 ≥ U . These results do not give conclusions when x −1 ≤ L and x 0 ≥ U or x −1 ≥ U and x 0 ≤ L. In this case one may use theory of monotone maps as in [3].
Example 2 Consider the difference equation There is no equilibrium points as the discriminant of the equilibrium equation 1 − 8a < 0 and no minimal period-two solution exists as φ = ψ 2 + φ 2 + a and ψ = φ 2 + ψ 2 + a implies φ = ψ. We can find U that satisfies the conditions (12) and (13) of Theorem 10. As f (U, U ) ≥ U simplifies to 2U 2 + a ≥ U , which always holds, every solution will be eventually increasing and tends to ∞.
Example 3 Consider the difference equation The equilibrium points are x 0 = 0 and x ± = ±1/ 4 √ 2. The characteristic equation at the equilibrium solutionx is λ 2 = 5x 4 λ + 5x 4 . For the equilibrium point x 0 , λ 2 = 0 so that λ 1,2 = 0 and x 0 is locally asymptotically stable. For the equilibrium point x ± , λ 2 = 5/2λ+5/2, so that λ 1,2 = 5± We can choose at least one of these inequalities to be strict. From Theorem 10, we can conclude that every solution with x 1 , x 0 ≤ L, L > 0 converges to 0, while every solution with x −1 , x 0 ≥ U is eventually increasing and tends toward ∞. As L < 1/ 4 √ 2 < U are arbitrary we conclude that Theorem 10 does not apply when x −1 ≤ L and x 0 ≥ U or x −1 ≥ U and x 0 ≤ L.
In this cases one can use the results from [3].
Example 4 Consider the difference equation there exists x 0 and x, and if a + b > 2, then there exist three equilibrium points For the equilibrium point x 0 , λ 2 = 0 so that λ 1,2 = 0 and thus, x 0 is locally asymptotically stable. The conditions for local stability of the equilibrium pointsx ± are quite involved and can be found in [1]. In particular x − will either be a saddle point, repeller, or non-hyperbolic depending on whether 2a(a + b) + (a − b) (a + b) 2 − 4 is greater than, less than, or equal to 0, and the equilibrium point x + is either locally asymptotically stable or non-hyperbolic when it exists. Now we want to find a L < U that satisfies the conditions (12) and (13) of 1+L 2 ≤ L, which simplifies to 0 ≤ 1 + L 2 − (a + b)L. This will occur when L < L − or L > L + where we can set L − = x − and L + = This occurs when U − < U < U + where we can set U − = x − and U + = x + . For both L and U to exist, we need L < L − to satisfy L < U. From Theorem 10, we can conclude that every solution with x 1 , x 0 ≤ L converges to 0, while every solution with x −1 , x 0 ≥ U converges to x + . Note that in the region where L and U exist, no minimal period-two solutions exists. All the period-two solutions are located in the region which is the union of the second and the fourth quadrant with respect to x − .

Global Dynamics of Equation (4)
In this section we present the global dynamics of Equation (4).

Local stability results
We begin by observing that the function f (u, v) = be −cu + pv is decreasing in Applying local stability test [10] we obtain i) If x < 1 c , then the equilibrium point x is locally asymptotically stable.
ii) If x > 1 c , then the equilibrium point x is a saddle point.
iii) If x = 1 c , then the equilibrium point x is non-hyperbolic of the stable type (with eigenvalues λ 1 = −1 and λ 2 = p).

Proof.
i) Equilibrium point x is locally asymptotically stable if As p ∈ (0, 1) then 1 − p < 2 holds. As (1 − p)cx > 0, then x is stable if Therefore, the equilibrium x is locally asymptotically stable if x < 1 c ii) If |(1 − p)cx| > |1 − p| , then the equilibrium point x is a saddle point. As We see that cx = 1 ⇔ x = 1 c . The characteristic equation at the equilibrium becomes with eigenvalues λ 1 = −1 and λ 2 = p.

Periodic solutions
In this section we present results about existence and uniqueness of the minimal period-two solution of Equation (4).
where φ = ψ. This implies Note that F (0) = be −cb 1−p > 0 since b > 0. Additionally, as φ approaches ∞, then F (φ) approaches −∞. Notice graphically, the the function F begins above the x-axis and ends approaching −∞. As the function F crosses the x-axis at least once at x, then F must cross the x-axis at least three times when F (x) > 0. This will result in the existence of a minimal period-two solution. We want to prove the values of parameters that F (x) > 0 holds true. Observe that the derivative of F is so that when x is substituted F (x) = xbc 2 e −cx + (p − 1). Then F (x) > 0 when x > 1 c as Thus when x > 1 c , there will be a minimal period-two solution. Next we want to prove that the period-two solution is unique. Rewritting (14) we obtain Let g(x) = xe −cx . As g (x) = e −cx (1 − cx), then the global maximum of g is attatined at x = 1 c . For each y value there will be two corresponding x values when g(x) < g( 1 c ) = 1 ce . This will happen when Let G(x) = e cx − ecx and notice that G(0) = 1. The derivative of G will be Thus when the period-two solution exists, it is unique.

Global stability results
In view of Theorem 4 every bounded solution of Equation (4) converges to either an equilibrium solution or a minimal period-two solution.
Lemma 2 The solutions of Equation (4) are bounded.

Consider the difference equation of
The solution of Equation (15) is As n → ∞, then In view of the difference inequality result, see [7] x n ≤ u n ≤ b 1−p + = U for n = 0, 1, ... and some > 0 when x 0 ≤ u 0 .
Theorem 12 (i) Ifx > 1 c , then the equilibrium solutionx is a saddle point and the minimal period-two solution {φ, ψ}, φ < ψ is globally asymptotically stable within the basin of attraction B(φ, ψ) is the global stable manifold of (x,x).
(ii) Ifx ≤ 1 c , then the equilibrium solutionx is globally asymptotically stable. and global unstable W u (x, x) manifolds, where W s (x, x) is the graph of a nondecreasing function and W u (x, x) is the graph of a non-increasing function, which has endpoints at (φ, ψ) and (ψ, φ). Every initial point (x −1 , x 0 ) which starts south east of W s (x, x) is attracted to (ψ, φ), while every initial point (x −1 , x 0 ) which starts north west of W s (x, x) is attracted to (φ, ψ), see Theorems 5-7. In this case in view of Theorem 4 global attractivity of period-two solution implies its local stability since the convergence is monotonic.
When x ≤ 1 c , the equilibrium solution is locally and so globally asymptotically stable by Lemma 1 part (i) and part (iii) .

Global Dynamics of Equation (5)
In this section we present global dynamics of Equation (5).

Local stability results
First, notice that the function f (u, v) = a + bve −u is decreasing in the first variable and increasing in the second variable. By Theorem 4, for all solutions The characteristic equation of Equation (5) is , then the equilibrium solution x is locally asymptotically , then the equilibrium solution x is a saddle point.
, then the equilibrium solution x is non-hyperbolic of stable type (with eigenvalues λ 1 = −1 and λ 2 = be −x ).

Proof.
i) The equilibrium point x is locally asymptotically stable if As be −x > 0, then 1 − be −x < 2 holds true. So rearranging the other inequality we Therefore, the equilibrium x is locally asymptotically stable if x < e x b − 1. As Then we can equivalently write the condition to be locally asymptotically stable as then the equilibrium solution x is a saddle point. Note that be −x < 1 since always holds as a > 0. The condition for x to be a saddle point yields By using (16), the inequality can then equivalently be written as iii) The equilibrium point x is non-hyperbolic point if We see that In view of (16) this can be rewritten as The characteristic equation at the equilibrium point will become with eigenvalues λ 1 = −1 and λ 2 = be −x ∈ (0, 1).

Periodic solutions
In this section we present results about existence and uniqueness of minimal period-two solutions of Equation (5).
, then Equation (5) has minimal period-two solution: Proof. We want to find for which values of x there exists a minimal periodtwo solution (φ, ψ) where φ and ψ are distinct positive real numbers. A period-two where φ and ψ are distinct real numbers. Rewritting ψ and then substituting into Let will be a zero of F as is positive as a and b are positive constants. As φ approaches ∞, then F approaches −∞ assuming that b < e a . When F (x) > 0 then F will cross the x−axis at least three times resulting in a minimal period-two solution. Thus, we want to prove when F (x) > 0 holds. Taking the derivative of F we have , there will be a minimal period-two solution.
Next we want to prove that the minimal period-two solution is unique. By rewriting (17) we find that There exists a unique value of m where 1 m+1 = be −m for which this holds. Using the first-derivative theorem we can check that m is a local minima. Note it suffices to check the numerator of g (m − 1) as the denominator is always positive. Using the fact that 1 m+1 = be −m This proves that g (m − 1) < 0. Next using the same method taking the numerator of g (m + 1) we see that This proves that g (m + 1) > 0. As the derivative changes from negative to positive around the critical point, it will be a local minima. Note that g(a) > 0 and as x approaches ∞, g(x) approaches ∞. Since m is the only critical point, each y value will have two x values with the exception at m. This results in the fact that there can only be one period-two solution.
there are no minimal period-two solutions.
Proof. Assume that {φ, ψ} is a period-two solution. Then {φ, ψ} satisfies (17) and so it satisfies (18) as well. Let will be a zero of F as We see that which is a positive value as a and b are positive constants. As φ approaches ∞, then F approaches ∞ as b ≥ e a . As the function begins above the x-axis at a and approaches ∞, F will cross the x-axis an even number of times. Since F (x) = 0 is one of the points that lie on the x-axis and the only equilibrium point, there cannot be a minimal period-two solution.
The result of proposition 1 has been verified through Mathematica simulations as well.

Global stability results
By Theorem 4 every bounded solution of Equation (5) converges to either an equilibrium solution or a minimal period-two solution.

Lemma 4 The solutions of Equation
Proof. By Equation (5),

Consider the difference equation of
Suppose that b < e a . The solution of Equation (19) Theorem 14 Consider Equation (5).
, then there exists a period-two solution that is locally asymptotically stable and the equilibrium point, x, that is is a saddle point. The unique period-two solution attracts all solutions which start off the global stable manifold of W s (E(x, x)).
(ii) If b < e a and x < a+ √ a 2 +4a 2 , then the equilibrium solution, x, is globally asymptotically stable.
(iii) If b < e a and x = a+ √ a 2 +4a 2 , then the equilibrium solution, x, is non-hyperbolic of the stable type and is global attractor.

Proof.
(i) Using Theorem 4 every bounded solution of Equation (5) converges to an equilibrium solution or period-two solution. By Lemma 4, when b < e a every solution of Equation (5) is bounded such that all solutions will converge to either an equilibrium solution or period-two solution. If b < e a and x > a+ √ a 2 +4a 2 , then x will be a saddle point by Lemma 3 part (ii), and there will be a minimal period-two solution by Theorem 13. In view of Theorems 5-7 there exist the global stable manifold W s (x, x) and global unstable is the graph of a non-decreasing function and W u (x, x) is the graph of a non-increasing function, which has endpoints (ii) When b < e a and x < a+ Proof. We will use Theorem 9 to prove this theorem. The conditions of (10) and (11) of Theorem 9 become These inequalities can be reduced to Any value of L and U such that There is a vertical asymptote at 1 − be −x = 0 that is at x = ln(b). In the interval (ln(b), ∞) we can find L and U that satisfies these inequalities.
As b ≥ e a , then ln(b) ≥ a so that (ln(b), ∞) is part of the domain of difference equation (5). An example of where this holds is when L = a + . Using the fact that b ≥ e a and is small, then b ≥ e a+ . By condition (11) the inequality holds true as We will use condition (10) and b ≥ e a to find the criteria for U based on our L. Thus, Let U be such that U > a + ln a+ . It holds that U ≥ L.
Overall, as f is continuous and there is no minimal period-two solution by Proposition 1, using Theorem (9) some solutions will approach ∞.

Remark 4 For instance, case i) of Theorem 14 holds when
holds when a = 4, b = 2 and case iii) holds when a = 2, b = In conclusion, Equations (4) and (5) exhibit the global period doubling bifurcation described by Theorem 5.1 in [11]. Checking the conditions of Theorem 5.1 in [11] is exactly the content of Lemmas 1-3 and Theorems 10-12.
Abstract We will investigate the global behavior of the cooperative system where the initial conditions x 0 , y 0 are arbitrary nonnegative numbers. This system models a population comprised of two subpopulations on different patches of land. The model, introduced in [4], considers the minimum between the maximum carrying capacity of each patch (K 1 or K 2 resp.) and the linear combination of the population from patch i from the last time step with those who migrated to patch i for i=1,2. We break the behavior of the system into several cases based on whether the linear combination of the population or maximum carrying capacity is greater. We are able to conclude that either one fixed point will be a global attractor of the interior region of R 2 + or there will exist a line of fixed points with the stable manifolds as the basins of attractions. We then extend some of these results to the n-dimensional case, first introduced in [2], using similar techniques.
We investigate the global behavior of general cooperative system for i = 1, 2, . . ., n, and t=0,1,. . . where the initial conditions of x i 0 are arbitrary nonnegative numbers for i=1,2, . . ., n. We are able to conclude in some cases that one fixed point will be a global attractor of the interior region of R n + .

Introduction
We investigate the global behavior of the cooperative system where the initial conditions where b 1 and b 2 are the probabilities that a female will give birth, f 1 and f 2 are the probabilities that the baby is female, µ 1 and µ 2 are the probabilities of death, α is the probability of a successful migration, and m 1 and m 2 are the probabilities of migration for each patch respectively. The linear models in the system are where x t+1 and y t+1 represent the current populations at time step t+1. Here r 11 x t is the population of patch 1 that remained from time step t, r 12 y t is the population that migrated to patch 1, r 22 y t is the population of patch 2 that remained from time step t, and r 21 x t is the population that migrated to patch 2. Thus, system (20) will compute the minimum of the maximum carrying capacity and the linear combination of the females remaining in the population from the last time step with those who migrated to the patch. It is assumed that each subpopulation will increase until it reaches the maximum carrying capacity, also known as the ceiling density dependence.
In this paper we will present the global dynamics of the model for all cases of the parameters. In particular, we obtain the basins of attraction in the two cases of an infinite number of fixed points, which was not covered in [4]. For the four cases of a finite number of fixed points, we will use different, simpler techniques of monotone discrete dynamical systems to find the basins of attraction of all fixed points and so to give an alternative proof of the results in [4].
The cooperative system (20) can be generalized to study a metapopulation consisting of n subpopulations who live on n different patches of land. This system was originally studied in [2]. Such a system is as follows that each population will grow linearly until reaching the carrying capacity. We will present the global dynamics of some of the cases using similar techniques to the 2-dimensional system (20). We are able to give global results for three cases, in which one fixed point will be a global attractor of the interior region of R n + using simpler, different techniques than originally used in [2]. The remaining case will be left as conjecture.
The next section of this paper contains some basic results on the basins of attraction of the equilibrium solutions of monotone systems and order preserving maps. Section 3 contains the main results of the global dynamics of system (20) in all 6 cases, and section 4 contains the main results of the generalized n-dimensional system (21). It is important to mention that no local or global dynamics of system (20) was obtained in [4] in cases 5 and 6. In fact the global dynamics in these two cases is very interesting and has been observed in number of monotone systems, see [1,9,10].

Some Basic Results for Order Preserving Maps
In this section we give some basic attractivity results for order preserving maps and systems from [9,10], which will be used in the rest of the paper. See also [5,6,12].
We will begin with some definitions and vocabulary. A first order system of difference equations where S ⊂ R 2 has nonempty interior, (f, g) : S → S, f , g are continuous func- Let be a partial order on R n with a nonnegative cone P . For x, y ∈ R n , the ordered interval x, y is the set of all z such that x z y. We say that x ≺ y if x y and x = y, and x y if y − x ∈ int P , where int P denotes the interior of a set 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 Furthermore, we define the south-east partial order as se on R 2 where (x, y) se (s, t) if and only if x ≤ s and y ≥ t. Similarly, we define the north-east partial order as ne on R 2 where (x, y) ne (s, t) if and only if x ≤ s and y ≤ t.
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 .
The next result is stated for order-preserving maps on R n and is given here for completeness. See [5] for a more general version valid in ordered Banach spaces.
Theorem 16 For a nonempty set R ⊂ R n and a partial order on R n , let T : ii. If T is strongly order preserving, then there exists a fixed point in a, b which is stable relative to a, b .
iii. If there is only one fixed point in a, b , then it is a global attractor in a, b and therefore asymptotically stable relative to a, b .
We say that {x n } n∈Z is an entire orbit of a map T : for all n ∈ Z. This orbit is said to join u 1 to u 2 if x n → u 1 as n → −∞ and x n → u 2 as n → ∞. The following result of the order interval trichotomy of Dancer and Hess is for strictly order preserving maps [3,5]. The result is stated for a partial order in R n , but it also holds in Banach spaces.
Theorem 17 Let u 1 u 2 be distinct fixed points of a strictly order preserving map T : A → A, where A ⊂ R n , and let I = u 1 , u 2 ⊂ A. Then at least one of the following holds.
(a) T has a fixed point in I distinct from u 1 and u 2 .
(b) There exists an entire orbit {x n } n∈Z of T in I joining u 1 to u 2 and satisfying x n x n+1 .
(c) There exists an entire orbit {x n } n∈Z of T in I joining u 2 to u 1 and satisfying x n+1 x n . The following result is a direct consequence of Theorem 17, see [9,10]. Theorem 18 Let T be a competitive (resp. cooperative) map on a rectangular x is not the NW or SE vertex of R"), and T is strongly competitive (resp. cooperative) on ∆. Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian J T (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing (resp. decreasing) continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period two points. In the latter case, the set of endpoints of C is a minimal period two orbit of T .

The 2-Patch System
We will prove the global dynamics of system (20) for two patches of land. Let J = r 11 r 12 r 21 r 22 be the matrix consisting of the constants of system (20), and let T be the cooperative map associated with system (20),

Theorem 19
The following results for system (20) hold: 1. Suppose that ρ(J) < 1 and where at least one of the inequalities is strict. The fixed point (0, 0) is a global attractor. 2. Suppose that ρ(J) > 1, 1 / ∈ σ(J), and where at least one of the inequalities is strict. The fixed point (0, 0) is unstable while the fixed point (K 1 , K 2 ) is a global attractor of the interior region of Then the fixed point (0, 0) is unstable while the fixed point ( is a global attractor of the interior region of R 2 + where the basin of attraction is B(K 1 , K 1 r 21 /(1 − r 22 )). 4. Suppose that ρ(J) > 1, 1 / ∈ σ(J), r 11 < 1, and Then the fixed point of (0, 0) is unstable while the fixed point of (K 2 r 12 /(1 − System (20) has an infinite number of fixed points, which has the stable manifolds W s (E x ) as its basins of attraction. 6. Suppose that ρ(J) ≥ 1, 1 ∈ σ(J), r 11 < 1 and System (20) has an infinite number of fixed points, E y = {(yr 12 /(1 − r 11 ), y)|0 ≤ y ≤ K 2 }, which has the stable manifolds W s (E y ) as its basins of attraction.
Proof. 1. Clearly (0, 0) is always a fixed point. Using the knowledge of inequalities (23), the cooperative map T can be rewritten as Note that no other fixed points will exist as would have to hold true for some fixed point (x, y). This implies that x = r 11 x + r 12 y and y = r 21 x + r 22 y ⇔ This cannot be the case as ρ(J) < 1. The map T has an invariant interval (0, 0), (K 1 , K 2 ) . This can be seen as the map is defined on R 2 + and the system (20) has a maximum carrying capacity (K 1 , K 2 ). As (0, 0) is the unique fixed point of the invariant interval and ρ(J) < 1, then This indeed holds by Theorem 16 so (0, 0) is a global attractor. (24) can be rewritten as

The inequalities in
Therefore as K 1 K 2 is the minimum, we can conclude that This results in the fact that (K 1 , K 2 ) is a fixed point in addition to (0, 0).
Another fixed point will only exist if either x = K 1 and y = K 2 or x = K 1 and y = K 2 . Without loss of generality suppose that x = K 1 and y = K 2 .
The map T gives us that However, this creates a contradiction as the inequality of (24) can be rewritten as Therefore, there are no other fixed points. As T x y ≤ K 1 K 2 for all (x, y) ∈ R 2 + , T has an invariant interval (0, 0), (K 1 , K 2 ) . The fixed point, (0, 0) is unstable as ρ(J) > 1 (see [7]). By Corollary 2, (K 1 , K 2 ) is a global attractor for the interior of (0, 0), (K 1 , K 2 ) , that is 3. Through rearranging one of the inequalities of (25) we have We will have the fixed point of (K 1 , K 1 r 21 /(1 − r 22 )) when By the inequalities of (29) and the fact that the second inequality of (30) indeed holds. The first inequality of (30) can be reduced to Therefore this inequality holds as ρ(J) > 1 and furthermore, ( and E + = (K 1 , K 1 r 21 /(1 − r 22 )). As ρ(J) > 1, then E 0 is unstable. By Corollary 2, E + is an attractor for the interior of E 0 , E + , that is Since (K 1 , K 2 ) is the maximum carrying capacity for the population and E + is a fixed point, using the knowledge that a, b is an invariant set for monotone map T when a is fixed point and b is an end point, then is an invariant set. By Theorem 16 as E + is the only fixed point in the region, then E + is an attractor for the interior of E + , (K 1 , K 2 ) , For all (x, y) / Using the fact that T is monotone, In view of (31) and (32), lim n→∞ T n (x L , y L ) = E + and lim n→∞ T n (x U , y U ) = E + . Using (33) we can conclude that lim n→∞ T n (x, y) = E + . Thus, E + is a global attractor of the interior of E 0 , (K 1 , K 2 ) . 4. Through rearranging the inequalities of (26) we have We will have the fixed point of ( holds true. This will happen if The first inequality of (35) is valid using (34) and the knowledge that The second inequality of (35) can be reduced to Therefore this inequality holds as ρ(J) > 1, and furthermore, (K 2 r 12 /(1 − r 11 ), K 2 ) is a fixed point. The remainder of the proof is analogous to case 3 where we use the fixed point of E = ((K 2 r 12 /(1 − r 11 ), K 2 ) instead of E + = (K 1 , K 1 r 21 /(1 − r 22 )). 5. If we have that As a reminder, in case 3 we proved that (K 1 , K 1 r 21 /(1 − r 22 )) is a fixed point. When x = K 1 , there will exist infinite fixed points whenever .
The remainder of the proof is analogous to case 5, and thus concludes the proof.

The n-Patch System
We will prove the global dynamics in some cases of system (21) for n patches of land. Let M be the matrix comprised of constants of (21): Define T to be the cooperative map associated with system (21) Additionally, letx be an equilibrium point of the system (21). By Lemmas 3.2,
We state the results of Theorem 3.5 in [2].  We can now formulate a theorem about the global stability in some cases and provide new proofs based on the theory of monotone maps.

Theorem 20
The following results for system (21) holds: for each i = 1, 2, . . . , n with at least one strict inequality. Then ρ(M ) < 1 and the fixed point (0, . . . , 0) is a global attractor. is a global attractor of the interior region of R n + . Proof.
1. The point (0, . . . , 0) will always be a fixed point of (21). We can rewrite the inequalities of (38) as We can rewrite the cooperative map as The map T will have an invariant interval (0, . . . , 0), (K 1 , . . . , K n ) as the map is defined on the region of R n + and has a maximum carrying capacity (K 1 , . . . , K n ). As (0, . . . , 0) is the unique fixed point of the invariant interval and ρ(M ) < 1, then Indeed by Theorem 16, (0, . . . , 0) is a global attractor. 2. As a result of the inequalities of (39), Thus in addition to (0, . . . , 0), (K 1 , . . . , K n ) is a fixed point of (21) under the conditions of (39) as we can conclude The map T has an invariant interval (0, . . . , 0), (K 1 , . . . , K n ) since the map is defined on R n + and for all (x 1 , . . . , Again as T is a monotonic map and E + is a fixed point while (K 1 , . . . , K n ) is the maximum carrying capacity, E + , (K 1 , . . . , K n ) is an invariant interval.
By Theorem 16 since E + is the only fixed point in the interval, then E + is an attractor for the interior of E + , (K 1 , . . . , K n ) , that is It remains to prove that E + is the attractor for the interior of the invariant interval E 0 , (K 1 , . . . , K n ) . From above we know that for all (x 1 , . . . , As T is monotone, Using (40) and (41), we can conclude that lim k→∞ T k ((x L 1 , . . . , x L n )) = E + and lim k→∞ T k ((x U 1 , . . . , x U n )) = E + . From the inequalities of (42) we conclude that lim k→∞ T k (x 1 , . . . , x n ) = E + . Therefore E + is a global attractor of the interior of E 0 , (K 1 , . . . , K n ) and thus concludes the proof.
As at this point in time the stable manifold theory does not extend to R n + , for n > 2. We will leave cases 5 and 6 from the 2-dimensional case as conjectures.
Conjecture 1 If ρ(M ) = 1, 1 ∈ σ(M ), and In this paper we will prove general results regarding the global stability of monotone systems without minimal period two solutions on a rectangular region R. We will illustrate the general results in two examples of well known systems used in mathematical biology. The first of the systems that will be investigated is a modified Leslie-Gower system of the form where the parameters a, b, c, d are positive numbers, α and β are positive values less than 1, and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers [32].
In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when c = d = 1 and a = b, there will exist an infinite number of interior equilibrium points in which case we will find the basin of attraction for each of the equilibrium points.
The second system that will be investigated is a version of a Lotka-Volterra model of the form where the parameters of A, K 1 , and K 2 are all positive and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers, which is a semi implicit discretization of the continuous version [1]. In most cases, there will be between one and three equilibrium points with solutions converging to one of the points. In one case when A > K 1 = K 2 , however, there will exist an infinite number of equilibrium points.
In this case for each equilibrium point, there will be a stable manifold as its basin of attraction.

Introduction and Preliminaries
In this paper we will give global dynamic results for monotone systems with no minimal period two solutions on a rectangular region R. These results will be found using the theory of global invariant manifolds developed by Kulenović and Merino in [28,29,30,31].
We will illustrate the general results with two examples of systems that have rational functions as transition functions. The first monotone system that will be considered is where the parameters a, b, c and d are positive numbers and both α and β are positive numbers less than 1. The initial conditions x 0 , y 0 are arbitrary nonnegative numbers. This system was originally outlined in [32] by Pakes and Maller as a way to model the application of plant growth. In particular, the system came as a result of the study of the subterranean clover and its various strains found in the southwest of Western Australia. The motivation around this study of the clover and competing strains can be found in [37] by Rossiter. Rossiter et al. [34] and Pakes and Maller [32] formally derived the model from the experimental data. Their aim was to explore a binary mixture of two strains and observe the competition between desirable and undesirable strains of the clover to see which would endure.
One trait that was considered is the hardness of the seeds, where seeds that soften begin to grow while hard seeds become part of the seed pool for the following year.
The hardness has been studied by Taylor, Rossiter, and Palmer in [36] as well as by others. It was found that in some strains the seeds soften at a faster rate as the years pass while for other strains the rate remains steady. The other quality observed is whether the seed has burrs or became a free seed, which will effect the rate of softening. It is assumed that the seed becomes free within a year once it softens.
where c, d > 0 and the initial conditions x 0 and y 0 are non negative. This system has been studied by many authors such as Kulenović and Clark in [5] and H. L.
Smith [40]. The system (44) has an explicit solution of the form The following theorem summarizes the well known results regarding system (44).

Theorem 21
The following statements are true for system (44).
(1) All solutions (x n , y n ) are component-wise monotonic (x n and y n are increasing or decreasing sequences). Both axes are invariant sets.
The Beverton-Holt equations, system (44), can be modified to create a coupled system known as the Leslie-Gower model. This model is the two-species competition model of the form x n+1 = cx n 1 + a 11 x n + a 12 y n , y n+1 = dy n 1 + a 21 x n + a 22 y n , n = 0, 1, . . . , where c, d, a ij ≥ 0 and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers, such that solution is defined for every n. As it was shown in [27] system (45) is semi implicit discretization of the classical Lotka-Volterra system of differential equations. The system (45) is a well known system that has been studied by numerous authors [6,7,28,29]. Note that the terms a 12 and a 21 are the constants added to couple the system as they represent the interspecific competition. System (43) is a modified version of this Leslie-Gower model where the linear factors αx n and βy n represent the stockings for two species in competition, see [11,12,13].
In this paper for system (43), we begin by finding the local stability results as well as proving both the (O + ) condition and boundedness of solutions. Then, we use the global dynamic results to prove that solutions will converge to one of the equilibrium points in most cases. In one case, however, when c = d = 1 and a = b there will exist an infinite number of interior equilibrium solutions. We can conclude that there is a stable manifold which is the basin of attraction for each of the infinite equilibrium points.
The second system that will be considered is where the parameters of A, K 1 , and K 2 are positive and the initial conditions x 0 , y 0 are arbitrary nonnegative numbers. As we will show in Section 4, system (46) is another semi implicit discretization of the Lotka-Volterra differential equation model. We will give a local stability analysis of system (46) in which we are able to find all the eigenvalues of the Jacobian matrix for each equilibrium point.
Additionally, we prove boundedness in order to use the global dynamics results.
System (46) has between one and three equilibrium solutions, where we will prove that solutions will converge to one of the equilibrium points. In addition and of particular interest to us is one case when K 1 = K 2 < A. Here there will exist an infinite number of equilibrium points. In this case, there is a stable manifold for each equilibrium point as its basin of attraction.
In this paper, we will first give some basic definitions and results of monotone systems needed throughout the paper. In the second section, we will prove some general global dynamic results regarding monotone systems without minimal period two solutions. In the third section, we will prove the global dynamics of all cases of system (43), and in the fourth section, we will prove the global dynamic results of all cases of system (46).
The theory of monotone maps will be used to help prove global dynamic results for system (43) and (46). We will begin by giving some basic definitions and information regarding monotone maps in the plane.

A first order system of difference equations
x n+1 = f (x n , y n ) y n+1 = g(x n , y n ) , n = 0, 1, . . . Given a partial ordering on R 2 two points x, y are said to be related if x y or y x, and is said to be a strictly related if x ≺ y if x y and x = y. A stronger inequality is defined as x = (x 1 , x 2 ) y = (y 1 , y 2 ) if x y with x 1 = y 1 and x 2 = y 2 .
We define a map T on a nonempty set R ⊂ R 2 to be a continuous function and furthermore is strongly monotone on for all x, y ∈ R. This implies that being related is invariant under iteration for a strongly monotone map.
A North-East ordering (NE) is an ordering for which the positive cone is the first quadrant, i.e. this partial ordering is defined by (x 1 , y 1 ) ne (x 2 , y 2 ) if 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 [40]. If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point.
The next results, from [30,29], are useful for determining basins of attraction of fixed points of competitive maps. Related results have been obtained by H. L.
Theorem 23 Let T be a competitive map on a rectangular region R ⊂ R 2 . Let x ∈ R be a fixed point of T such that ∆ : x is not the NW or SE vertex of R), and T is strongly competitive on ∆.
Suppose that the following statements are true.
a. The map T has a C 1 extension to a neighborhood of x.
b. The Jacobian J T (x) of T at x has real eigenvalues λ, µ such that 0 < |λ| < µ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of C in the interior of R are either fixed points or minimal period-two points. In the latter case, the set of endpoints of C is a minimal period-two orbit of T .
In the theorem below criteria is given for the curve C to have endpoints on the boundary of the region R, that is ∂R.

Theorem 24
For the curve C of Theorem 23 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied.
i. The map T has no fixed points nor periodic points of minimal period two in ∆.
ii. The map T has no fixed points in ∆, det J T (x) > 0, and T (x) = x has no solutions x ∈ ∆.
iii. The map T has no points of minimal period-two in ∆, det J T (x) < 0, and For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 23 reduces just to |λ| < 1. This follows from a change of variables [42] that allows the Perron-Frobenius Theorem to be applied. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis. The next result is useful for determining basins of attraction of fixed points of competitive maps.
Theorem 25 (A) Assume the hypotheses of Theorem 23, and let C be the curve whose existence is guaranteed by Theorem 23. If the endpoints of C belong to ∂R, then C separates R into two connected components, namely W − := {x ∈ R\C : ∃y ∈ C with x se y}and W + := {x ∈ R\C : ∃y ∈ C with y se x} , such that the following statements are true.

(B) If, in addition to the hypotheses of part (A), x is an interior point of R
and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q 1 (x) ∪ Q 3 (x) except for x, and the following statements are true.
(iii) For every x ∈ W − there exists n 0 ∈ N such that T n (x) ∈ int Q 2 (x) for n ≥ n 0 .
(iv) For every x ∈ W + there exists n 0 ∈ N such that T n (x) ∈ int Q 4 (x) for n ≥ n 0 .
If T is a map on a set R and if x is a fixed point of T , the stable set W s (x) of x is the set {x ∈ R : T n (x) → x} and unstable set W u (x) of x is the set The following result gives a description of the stable and unstable sets of a saddle point of a competitive map.
Theorem 26 In addition to the hypotheses of part (B) of Theorem 25, suppose that µ > 1 and that the eigenspace E µ associated with µ is not a coordinate axis.
If the curve C of Theorem 23 has endpoints in ∂R, then C is the stable set W s (x) of x, and the unstable set W u (x) of x is a curve in R that is tangential to E µ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of W u (x) in R are fixed points of T .

Theorem 27
Let T be a monotone map on a closed and bounded rectangular region R ⊂ R 2 . Suppose that T has a unique fixed pointx in R. Thenx is a global attractor of T on R.
The next result is stated for order-preserving maps on R n . See [16] for a more general version that is valid in ordered Banach spaces.

Corollary 3 If the non-negative 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 .
The following result gives conditions for the existence of the boundary curves of the basin of attraction. The complete results proved by Kulenović and Merino can be found in [31].
Theorem 28 Let p, q ∈ R 2 be such that p se q, and R ⊂ R 2 such that int( p, q se ) ⊂ R ⊂ p, q se . Let T be a competitive map defined on R that is strongly competitive on int(R). If there exist r ∈ {p, q}, and x, y ∈ int (R) such that T n (x) → r and T n (y) → r, then there exists a curve C in R which is strongly north-east linearly ordered and whose endpoints are in ∂R such that the connected components A and B of int(R)\C chosen so that x ∈ A, satisfy T n (z) → r for z ∈ A, and T n (w) → r for w ∈ B ∪ C. If the point r is in R, then r is a fixed point of T .
, and let T : R → R be a strongly competitive map with a unique fixed pointx ∈ R, and such that T is twice continuously differentiable in a neighbourhood ofx. Assume further that at the pointx the map T has associated characteristic values µ and ν satisfying 1 < µ and −µ < ν < µ, with ν = 0, and that no standard basis vector is an eigenvector associated to one of the characteristic values.
Then there exists curves C 1 , C 2 in R and there exist p 1 , p 2 ∈ ∂R with p 1 sē x se p 2 such that i. For l = 1, 2, C l is invariant, north-east strongly linearly ordered, such that x ∈ C l and C l ⊂ Q 3 (x) ∪ Q 1 (x); the endpoints q l , r l of C l , where q l ne r l , belong to the boundary of R. For l, j ∈ 1, 2 with l = j, C l is a subset of the closure of one of the components of R\C j . Both C 1 and C 2 are tangential at x to the eigenspace associated with ν.
ii. For l = 1, 2, let B l be the component of R\C l whose closure contains p l . Then B l is invariant. Also, for x ∈ B 1 , T n (x) accumulates on Q 2 (p 1 ) ∩ ∂R, and for x ∈ B 2 , T n (x) accumulates on Q 4 (p 2 ) ∩ ∂R.
We will use the results from [29,30,31] to prove the general global dynamics results of competitive maps in the plane.

Global Dynamic Results
We will prove some global dynamic results of the general monotone system (47).
Theorem 30 Consider the map T generated by system (47) on a rectangular region R where the fixed point E 0 = (0, 0) is on the bottom left corner of R. Suppose that T is a strongly competitive map with no minimal period two solutions on R.
Furthermore, we will assume that conditions a and b of Theorem 23 holds for any saddle fixed point.    the unstable manifold, W u (E x ) the endpoint will be E y and for the stable manifold, W s (E x ) the endpoint will be the x-axis. Any point on the stable manifold, which in this case is the x-axis will converge to E x . As E y is locally asymptotically stable, points on the y-axis will converge to E y . We will consider the global dynamics in two cases based on the location of the For the first case, suppose the initial point B = (x 0 , y 0 ) is inside the rectan- There will exists two projections of B onto the unstable manifold, W u (E x ), that is P x = (x, y 0 ) and P y = (x 0 , y) such that P y se B se P x . By monotonicity, T n (P y ) se T n (B) se T n (P x ).
Taking the limit we obtain which implies that This yields that lim n→∞ T n (B) = E y . Here we use that the unstable manifold W u (E x ) has an endpoint at E y , lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E y .
Next suppose the initial point B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and above the unstable manifold W U (E x ) of E x . There will exists two projections of B onto the unstable manifold, W u (E x ), that is P x = (x, y 0 ) and P y = (x 0 , y) such that P x se B se P y . By monotonicity, Taking the limit we obtain which implies that This yields that lim n→∞ T n (B) = E y . Here we use that the unstable manifold W u (E x ) has an endpoint at E y , lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E y .
Finally suppose that B = (x 0 , y 0 ) ∈ intR\R 0 . There exists a projection P y = (0, y 0 ) of B onto the y-axis and a projection P x = (x 0 , 0) of B onto the x-axis such that P y se B se P x . By monotonicity this implies that T n (P y ) se T n (B) se T n (P x ).
As the x-axis is the stable manifold of E x , then lim n→∞ T n (P x ) = E x . Furthermore, as E y is locally asymptotically stable, lim n→∞ T n (P y ) = E y . So when the limit of the inequalities is taken implies that We can conclude that as n → ∞ then T n (B) → R 0 . Once T n (B) enters the rectangular region R 0 , the global behavior will follow from the previous cases.
(b) This proof is analogous to the proof of case (a). The difference is that now we consider the stable and unstable manifolds of E y instead of E x as E y is a saddle point and E x is locally asymptotically stable.
(c) By Theorems 23, 24, 25, and 26 as E x and E y are saddle points, there exist the global stable manifolds, W s (E x ) and W s (E y ), and global unstable manifolds, W u (E x ) and W u (E y ). As E + is the interior fixed point, it will be the endpoint of W u (E x ) and W u (E y ). The y-axis will be the stable manifold W s (E y ) of E y . Thus for any initial point that begins on the y-axis will converge to E y . The x-axis will be the stable manifold W s (E x ) of the E x . So we can conclude that any point that begins on the x-axis will converge to E x .
We will consider the global dynamics in a few cases based on the location of the initial point B = (x 0 , y 0 ) ∈ int R.
First suppose that B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and below both the unstable manifolds W u (E x ) and W u (E y ) of E x and E y respectively. There will exists two projections P x = (x, y 0 ) and P y = (x 0 , y) of B, which will either be on the unstable manifold of E x (W u (E x )) or E y (W u (E y )) depending on the initial location of the point B. As the proof holds regardless of whether the projections are onto W u (E x ), W u (E y ), or both, without loss of generality we can suppose that P x is on W u (E x ) and P y is on W u (E y ) such that P y se B se P x . By monotonicity, In view of lim n→∞ T n (P x ) = E + and lim n→∞ T n (P y ) = E + we obtain Next suppose that B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and above both the unstable manifolds W u (E x ) and W u (E y ) of E x and E y respectively. There will exists two projections P x = (x, y 0 ) and P y = (x 0 , y) of B, which will either be on the unstable manifold of E x (W u (E x )) or E y (W u (E y )) depending on the initial location of the point B. Without loss of generality suppose that P x is on W u (E x ) and P y is on W u (E y ) such that P x se B se P y . By monotonicity, T n (P x ) se T n (B) se T n (P y ).
In view of lim n→∞ T n (P x ) = E + and lim n→∞ T n (P y ) = E + we obtain and so lim n→∞ T n (B) = E + .
Finally suppose that B = (x 0 , y 0 ) ∈ int R\R 0 . There exists a projection P y = (0, y 0 ) of B onto the y-axis and a projection P x = (x 0 , 0) of B onto the x-axis such that P y se B se P x . By monotonicity this implies that T n (P y ) se T n (B) se T n (P x ).
As the x-axis is the stable manifold of E x , then lim n→∞ T n (P x ) = E x . Furthermore, as the y-axis is the stable manifold of E y , lim n→∞ T n (P y ) = E y .
So when the limit of the inequalities is taken implies that We can conclude that as n → ∞ then T n (B) → R 0 . Once T n (B) enters the rectangular region R 0 , the global behavior will follow from the previous cases. will be E x and E y on the boundary of the region. The endpoint of W s (E + ) will be E 0 . As E y is locally asymptotically stable, solutions on the y-axis will converge to E y and as E x is locally asymptotically stable, solutions on the x-axis will converge to E x . Any point that begins on the stable manifold W s (E + ) of E + will converge to E + . We will describe the global dynamics in a few cases based on the location of the initial point B = (x 0 , y 0 ) ∈ int R.
First suppose that B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and is both to the left of the stable manifold W s (E + ) and below the unstable manifold W u (E + ) of E + . There will exist a projection P y = (x 0 , y) of B onto the unstable manifold W u (E + ) of E + as well as another projection P x = (x, y 0 ) of B such that P y se B se P x . The projection P x will either be on the unstable manifold W u (E + ) or on the stable manifold W s (E + ) depending on the initial point B. We will first suppose that the projection P x is on the unstable manifold W u (E + ). By monotonicity, Once the limit of the inequalities is taken which implies that E y se lim n→∞ T n (B) se E y as both P x and P y are on the unstable manifold so that lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E y . We can conclude that lim n→∞ T n (B) = E y . Next suppose that the projection P x is on the stable manifold W s (E + ). By monotonicity, T n (P y ) se T n (B) se T n (P x ) still holds. Once the limit of the inequalities is taken we have lim n→∞ T n (P y ) se lim n→∞ T n (B) se lim n→∞ T n (P x ).
As P y is on the unstable manifold, lim n→∞ T n (P y ) = E y , and as P x is on the stable manifold, then lim n→∞ T n (P x ) = E + . This implies that We can conclude that lim n→∞ T n (B) = E y as B does not begin on the stable manifold of E + .
For the next case, suppose that B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and is both to the left of the stable manifold W s (E + ) and above the unstable manifold W u (E + ) of E + . There will exist a projection P x = (x, y 0 ) of B onto the unstable manifold W u (E + ) of E + as well as another projection P y = (x 0 , y) of B such that P x se B se P y . The projection P y will either be on the unstable manifold W u (E + ) or on the stable manifold W s (E + ) depending on the initial point B. We will first suppose that the projection P y is on the unstable manifold W u (E + ). By monotonicity, T n (P x ) se T n (B) se T n (P y ).
Once the limit of the inequalities is taken which implies that E y se lim n→∞ T n (B) se E y as both P x and P y are on the unstable manifold so that lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E y . We can conclude that lim n→∞ T n (B) = E y . Next suppose that the projection P y is on the stable manifold W s (E + ). By monotonicity, T n (P x ) se T n (B) se T n (P y ) still holds. Once the limit of the inequalities is taken we have lim n→∞ T n (P x ) se lim n→∞ T n (B) se lim n→∞ T n (P y ).
As P x is on the unstable manifold, lim n→∞ T n (P x ) = E y , and as P y is on the stable manifold, then lim n→∞ T n (P y ) = E + . This implies that We can conclude that lim n→∞ T n (B) = E y as B does not begin on the stable and is both to the right of the stable manifold W s (E + ) and above the unstable manifold W u (E + ) of E + . There will exist a projection P y = (x 0 , y) of B onto the unstable manifold W u (E + ) of E + as well as another projection P x = (x, y 0 ) of B such that P x se B se P y . The projection P x will either be on the unstable manifold W u (E + ) or on the stable manifold W s (E + ) depending on the initial point B. We will first suppose that the projection P x is on the unstable manifold W u (E + ). By monotonicity, T n (P x ) se T n (B) se T n (P y ).
Once the limit of the inequalities is taken which implies that as both P x and P y are on the unstable manifold so that lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E x . We can conclude that lim n→∞ T n (B) = E x . Next suppose that the projection P x is on the stable manifold W s (E + ). By monotonicity, T n (P x ) se T n (B) se T n (P y ) still holds. Once the limit of the inequalities is taken we have As P y is on the unstable manifold, lim n→∞ T n (P y ) = E x , and as P x is on the stable manifold, then lim n→∞ T n (P x ) = E + . This implies that We can conclude that lim n→∞ T n (B) = E x as B does not begin on the stable manifold of E + .
For the next case, suppose that B = (x 0 , y 0 ) is inside the rectangular region R 0 = E y , E x and is both to the right of the stable manifold W s (E + ) and below the unstable manifold W u (E + ) of E + . There will exist a projection P x = (x, y 0 ) of B onto the unstable manifold W u (E + ) of E + as well as another projection P y = (x 0 , y) of B such that P y se B se P x . The projection P y will either be on the unstable manifold W u (E + ) or on the stable manifold W s (E + ) depending on the initial point B. We will first suppose that the projection P y is on the unstable manifold W u (E + ). By monotonicity, Once the limit of the inequalities is taken which implies that E x se lim n→∞ T n (B) se E x as both P x and P y are on the unstable manifold so that lim n→∞ T n (P x ) = lim n→∞ T n (P y ) = E x . We can conclude that lim n→∞ T n (B) = E x . Next suppose that the projection P y is on the stable manifold W s (E + ). By monotonicity, T n (P y ) se T n (B) se T n (P x ) still holds. Once the limit of the inequalities is taken we have As P x is on the unstable manifold, lim n→∞ T n (P x ) = E x , and as P y is on the stable manifold, then lim n→∞ T n (P y ) = E + . This implies that We can conclude that lim n→∞ T n (B) = E x as B does not begin on the stable manifold of E + .
Finally, suppose that B = (x 0 , y 0 ) ∈ intR\R 0 . There exists a projection P y = (0, y 0 ) of B onto the y-axis and a projection P x = (x 0 , 0) of B onto the x-axis such that P y se B se P x . By monotonicity this implies that T n (P y ) se T n (B) se T n (P x ).
As E x is locally asymptotically stable, then lim n→∞ T n (P x ) = E x . Furthermore, as E y is locally asymptotically stable, lim n→∞ T n (P y ) = E y . So when the limit of the inequalities is taken implies that We can conclude that as n → ∞, then T n (B) → R 0 . Once T n (B) enters the rectangular region, R 0 , the global behavior will follow from one of the previous cases.
(e) As E 0 is a saddle point, there exists a global stable manifold W s (E 0 ), and global unstable manifold W u (E 0 ), by Theorems 23, 24, 25, and 26. As there are no interior fixed points or minimal period-two solutions, the endpoints of both the stable and unstable manifolds will be on the boundary of R. The unstable manifold W u (E 0 ) will be the y-axis with the endpoint of E y . So if a point begins on the y-axis, it will converge to E y . The stable manifold W s (E 0 ) will be the x-axis. Thus, if a point begins on the x-axis, it will converge to E 0 .
Suppose that B = (x 0 , y 0 ) ∈ int R. There will exists two projections P x = (x 0 , 0) and P y = (0, y 0 ) of B onto the x and y axis such that P y se B se P x .

By monotonicity,
T n (P y ) se T n (B) se T n (P x ).
As E y is locally asymptotically stable, lim n→∞ T n (P y ) = E y . Additionally, as P x is on the stable manifold of E 0 , lim n→∞ T n (P x ) = E 0 . Taking the limit of the inequalities which implies As B does not begin on the stable manifold and by the monotone system theory [30] as the stable manifold is unique, we can conclude that lim n→∞ T n (B) = E y .
(f) This proof is analogous to the proof of case (e). The difference is that we consider the existence of E x instead of E y , where E x will be locally asymptotically stable.
(g) Let the map T contain one fixed point E 0 = (0, 0) that is locally asymptotically stable. As Furthermore, we will assume that conditions a and b of Theorem 23 holds for any saddle fixed point.
(a) Assume that the map T has the fixed points E y = (0, y) which is nonhyperbolic of the stable type, E x = (x, 0) which is locally asymptotically stable, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the y-axis converges to E x and every solution which begins on the y-axis without E 0 converges to E y .
(b) Assume that the map T has the fixed points E y = (0, y) which is locally asymptotically stable, E x = (x, 0) which is non-hyperbolic of the stable type, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the x-axis converges to E y and every solution which begins on the x-axis without E 0 converges to E x .
(c) Assume that the map T has the fixed points E y = (0, y) which is nonhyperbolic of the stable type, E x = (x, 0) which is a saddle point, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the x-axis converges to E y and every solution which begins on the x-axis without E 0 converges to E x .
(d) Assume that the map T has the fixed points E y = (0, y) which is a saddle point, E x = (x, 0) which is non-hyperbolic of the stable type, and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution which begins off the y-axis converges to E x and every solution which begins on the y-axis without E 0 converges to E y .
(e) Assume that the map T has the fixed points E y = (0, y) and E x (x, 0) which are both non-hyperbolic of the stable type and E 0 = (0, 0) which is a repeller where E y se E 0 se E x . Every solution on the x-axis without E 0 will converge to E x and every solution on the y-axis without E 0 will converge to E y . Every solution which begins off the x and y axis will converge to exactly one of E x or E y .
(f ) Assume that the map T has the fixed points E y = (0, y) which is locally asymptotically stable and E 0 = (0, 0) which is non-hyperbolic of the unstable type where E y se E 0 . Then there will exists two curves, C 1 and C 2 , C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C 1 and C 2 coincide with each other or C 2 / ∈ R, every solution which begins off the x-axis will converge to E y . Every solution which begins on the x-axis will converge to E 0 . If there exists both C 1 , C 2 ∈ R then every solution to the left of C 2 will converge to E y and every solution to the right of C 2 will converge to E 0 .
(g) Assume that the map T has the fixed points E x = (x, 0) which is locally asymptotically stable and E 0 = (0, 0) which is non-hyperbolic of the unstable type where E 0 se E x .Then there will exists two curves, C 1 and C 2 , C 2 se C 1 , that are continuous and non-decreasing with an endpoint at E 0 . If the curves C 1 and C 2 coincide with each other or C 2 / ∈ R, every solution which begins off the y-axis will converge to E x . Every solution which begins on the y-axis will converge to E 0 . If there exists both C 1 , C 2 ∈ R then every solution to the left of C 1 will converge to E 0 and every solution to the right of C 1 will converge to E x .
(h) Assume that the map T has one fixed point E 0 = (0, 0) which is nonhyperbolic. Then every solution converges to E 0 . there is a stable manifold as its basin of attraction. Each stable manifold will have an end point at E 0 and they are graphs of continuous and non-decreasing functions. The points will depend continuously on the initial point (x 0 , y 0 ).  (a) As E y is non-hyperbolic of the stable type, there exists a stable manifold W s (E y ) by Theorems 23 and 24, which in this case will be the y-axis. Any point on the y-axis will converge to E y . As E x is locally asymptotically stable, an initial point on the x-axis will converge to E x . Suppose there exist an initial point B = (x 0 , y 0 ) ∈ int R. There will exist two projections of B, P x = (x 0 , 0) onto the x-axis and P y = (0, y 0 ) onto the y-axis such that P y se B se P x . By monotonicity this implies Furthermore, taking the limits we have which implies that This step was obtained using the fact that E x is locally asymptotically stable so lim n→∞ T n (P x ) = E x and the y-axis is the stable manifold W s (E y ) so that lim n→∞ T n (P y ) = E y . We can conclude that lim n→∞ T n (B) = E x as the stable manifold of E y is unique and B does not begin on it.
(b) This case is analogous to case (a) where in this case we use the stable manifold of E x , and E y is now locally asymptotically stable.
(c) This proof is analogous to case (a) of Theorem 30 where instead of claiming E y is locally asymptotically stable, E y is now non-hyperbolic of the stable type. We can instead use the fact that the y-axis is the stable manifold W s (E y ) of E y to proceed with proof using the same technique.

(d) This proof is analogous to the proof of case (b) of Theorem 30 where instead
of claiming E x is locally asymptotically stable, E x is now non-hyperbolic of the stable type. We can instead use the fact that the x-axis is the stable manifold W s (E x ) of E x to proceed with proof using the same technique.
(e) As E x and E y are non-hyperbolic of the stable type, there exist stable manifold W s (E x ), which is the x-axis, and W s (E y ), which is the y axis respectively in this case by Theorems 23 and 24. Any point on the y-axis without E 0 will converge to E y and any point on the x-axis without E 0 will converge to the E x . For both E x and E y there will exist a center manifold. This manifold can be used to show all solutions in the interior of R will either converge to E x or E y .
(f) By Theorem 29, there exist two curves C 1 and C 2 , where C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 . The curve C 1 is the boundary of the basin of attraction of a point at infinity, and the curve C 2 is the boundary of the basin of attraction of E y . This proof is analogous to case (e) of Theorem 30 when the two curves C 1 and C 2 coincide or C 2 / ∈ R where instead of claiming E 0 is a saddle point, E 0 is now non-hyperbolic of the unstable type. We can instead use the curve C 1 , that is a stable manifold W s (E 0 ) of E 0 to proceed with proof using the same technique.
If both C 1 , C 2 ∈ R, then, any points on C 1 and C 2 converge to E 0 . Suppose there exists a point B 0 = (x 0 , y 0 ) ∈ R and to the right of C 2 . There there will exist two projections of B, P 2 onto the curve C 2 and P 1 onto the curve C 1 such that P 2 se B 0 se P 1 . By monotonicity this implies T n (P 2 ) se T n (B 0 ) se T n (P 1 ), which once the limits are taken becomes and furthermore The last inequalities were obtained using the fact that P 1 is on C 1 so that lim n→∞ T n (P 1 ) = E 0 and P 2 is on C 2 so that lim n→∞ T n (P 2 ) = E 0 . Therefore, lim n→∞ T n (B 0 ) = E 0 .
Next suppose that there exists a point B 0 = (x 0 , y 0 ) ∈ R and to the left of C 2 . Then B 0 is in the region of the basin of attraction to E y . As B 0 does not begin on C 1 or C 2 , then the point will converge to E y .
(g) This proof is analogous to case (f ) of Theorem 30 when the two curves C 1 and C 2 coincide or if C 2 / ∈ R where instead of claiming E 0 is a saddle point, E 0 is now non-hyperbolic of the unstable type. We can instead use the curve C 1 , that is the stable manifold W s (E 0 ) of E 0 to proceed with proof using the same technique. When the two curves C 1 and C 2 do not coincide, the proof will be analogous to case (f ) given above.

Example 1
We will investigate the global dynamics of system (43): where the parameters a, b, c and d are positive numbers and 0 < α, β < 1.

Local Stability Results
To begin let us find the local stability results of system (43). Additionally, we will prove that the (O + ) condition is satisfied as well as the fact that system (43) is bounded, which will help in proving the global results.
is an equilibrium point. To find the local stability of each equilibrium point we find the Jacobian matrix. The map corresponding to system (43)    Proof. The results follow from the eigenvalues. When both eigenvalues lie within the unit circle, the equilibrium point will be locally asymptotically stable. Note that |λ 1 | < 1 and |λ 2 | < 1 when This will hold true when a > c and b > d. When both eigenvalues lie outside the unit circle, the equilibrium point will be a repeller. So |λ 1 | > 1 and |λ 2 | > 1 when This will hold true when a < c and b < d. Note that in this case λ 1 > 1 and λ 2 > 1, When one eigenvalue lies outside the unit circle and the other within the unit circle the equilibrium point will be a saddle point. Based on the previous calculations this will occur when a < c and b > d or a > c and b < d. Finally, when at least one eigenvalue lies on the unit circle, the equilibrium point is non-hyperbolic. So This will hold true when a = c or b = d. Note that as a, b, c, d > 0 then λ 1 = 1 or λ 2 = 1. It cannot happen that either is equal to -1 as −(a + c) = α(a − c) and Next we evaluate the Jacobian matrix at the point of E y = (0, d−b d ): The (c) non-hyperbolic of the stable type if b = d(a + 1 − c).

Proof.
The results follow from the eigenvalues. As d > b, then we can conclude |λ 2 | < 1 will always hold true as When both eigenvalues lie within the unit circle, the equilibrium point will be locally asymptotically stable. Note that |λ 1 | < 1 when This will hold true when d(a+1−c) > b. When one eigenvalue lies outside the unit circle and the other within the unit circle the equilibrium point will be a saddle point. Note that |λ 1 | > 1 when This will hold true when d(a + 1 − c) < b. Note again that as α < 1, then cannot hold resulting in the fact that λ 2 > 1. Finally, when at least one eigenvalue (in this case |λ 2 | = 1) lies on the unit circle, the equilibrium point is non-hyperbolic. This will happen when This holds true when d(a + 1 − c) = b. As a, b, c, d > 0, then −(−b + d + ad + cd) = α(−b + d + ad − cd) does not hold true resulting in the fact that λ 2 cannot be -1.
Next we evaluate Jacobian matrix at the equilibrium point E x = ( c−a c , 0). (c) non-hyperbolic of the stable type if a = c(b + 1 − d).
Proof. The results follow from the eigenvalues. As a reminder, c > a so that |λ 1 | < 1 will always be true. Indeed this result holds as When both eigenvalues lie within the unit circle, the equilibrium point will be locally asymptotically stable. Note that |λ 2 | < 1 when This holds when c(b + 1 − d) < a. Note that as β < 1, then −(bc + c − a + cd) > β(bc + c − a − cd) cannot hold. So λ 2 > 1 in this case. Finally, when at least one eigenvalue (in this case |λ 2 | = 1) lies on the unit circle, the equilibrium point is non-hyperbolic. This will happen when This holds when c(b+1−d) = a. Note that as a, b, c, d > 0, then −(bc+c−a+cd) = β(bc + c − a − cd) cannot hold so λ 2 = 1 in this case.
Assume the initial conditions of x 0 ≤ u 0 and y 0 ≤ v 0 hold. When iterated u n and v n become Therefore, we have that x n ≤ u n ≤ u 0 and y n ≤ v n ≤ v 0 . Next, we want to see when x n+1 ≤ x n and when y n+1 ≤ y n holds. By reexamining (49) we see that Thus x n+1 ≤ x n when (α + (1 − α) c a ) ≤ 1. This will happen when c a ≤ 1 that is c ≤ a. Similarly, by rewriting (50) we see that Then y n+1 ≤ y n when (β − (1 − β) d b ) ≤ 1. This will hold true when d ≤ b. Next, we will use an alternative method to prove the local stability in two cases for the equilibrium point Furthermore, suppose that E x and E y are locally asymptotically stable while E 0 is a repeller. Then E y se E + se E x and E + will either be non-hyperbolic of the unstable type or a saddle point.

Proof. First note that
which implies that E y se E + se E x . By Corollary 3, int E y , E + is a subset of the basin of attraction of either E y or E + . Since the equilibrium point E y is locally asymptotically stable, the interior of E y , E + is a subset of the basin of attraction of E y . This means that E + cannot be locally asymptotically stable. By Theorem 29, there exists an invariant, north-east strongly linearly ordered curve C x that is the boundary of the basin of attraction for E x with an endpoint of E 0 that passes through E + , and an invariant, north-east strongly linearly ordered curve C y that is the boundary of the basin of attraction for E y with an endpoint of E 0 that passes through E + where C x may coincide with C y . Any points on the curves C x and C y will be attracted to E + as they are on the boundary, E 0 is a repeller, and by the theorem cannot cross over to the other boundary. If C y and C x do not coincide, we can suppose that there exists a point B 0 = (x, y) that is in the region between C x and C y . Additionally, there exists the points of B 1 ∈ C y and B 2 ∈ C x such However, as lim n→∞ T n (B 1 ) = lim n→∞ T n (B 2 ) = E + , we conclude that lim n→∞ T n (B 0 ) = E + . As E + attracts some points, E + cannot be a repeller. Thus, E + is either a saddle point or a non-hyperbolic point. Based on Mathematica calculations the eigenvalues of E + cannot be 1, however, one eigenvalue can potentially be −1. The other eigenvalue will always be greater than 1. If E + exists, it will be non-hyperbolic of the unstable type or a saddle point.
The following lemma will give results regarding the spectral radius based on the slopes of the tangent lines at E + . Let ρ(J) be the spectral radius of J(E + ).  (iii) If m 1 − m 2 < 0, then ρ(J) < 1.
Proof. For the proof let the equilibrium E + be represented as (x,ȳ). Note the Jacobian matrix of J(E + ) can be rewritten as The Taylor expansion of the map T (x, y) is We will consider the linear part to find the slope of the tangent lines. Rewritten this becomes x Let ∆x = x −x and ∆y = y −ȳ. Substituting this in we have ∆x = f x (x,ȳ)∆x + f y (x,ȳ)∆y ∆y = g x (x,ȳ)∆x + g y (x,ȳ)∆y.
From the two equations we have that As the map is competitive, then f x , g y > 0, f y , g x < 0, as well as m 1 , m 2 < 0. This implies that f x , g y < 1. The term m 1 − m 2 is equivalent to Note the characteristic polynomial is p(λ) = λ 2 − (f x + g y )λ + (f x g y − f y g x ), and p(1) = 1−(f x +g y )+f x g y −g x f y . As f y (1−g y ) < 0, then m 1 −m 2 will either be less than, greater than, or equal to zero based on p(1). The characteristic polynomial at 1 is equivalent to p(1) = 1 − tr(J) + det(J). Then we have that p(1) > 0 when ρ(J) < 1, p(1) < 0 when ρ(J) > 1, and p(1) = 0 when ρ(J) = 1.
For system (43), x = f (x, y) means that and y = g(x, y) means that The Furthermore, suppose that E x and E y are both saddle points while E 0 is a repeller.
Then E y se E + se E x and E + will be locally asymptotically stable.  (13) ρ(J) < 1 and therefore, E + will not be non-hyperbolic and furthermore, must be locally asymptotically stable.

Proof. First note that
Next we will prove that the (O + ) condition holds. The (O + ) condition tells us that there will be no minimal period two solutions.

Global Stability Results
In this section we will compile the local stability results and use both Theorems 30 and 31 to give conclusions regarding the global dynamics of system (43).

Proof.
(a) Rewriting the two inequalities give us This implies that cd(c − a) < c − a. As c > a this inequality can be reduced to cd < 1.
(b) Rewriting the two inequalities give us This implies that cd(c − a) > c − a. As c > a this inequality can be reduced to cd > 1.

Lemma 17
The equilibrium point E + will not exist when , and a > c(1 + b − d).
(a) Clearly E + cannot exist by Lemma 7 as either d(1−c+a) < b and c(1+b−d) < a or d(1 − c + a) > b and c(1 + b − d) > a must hold.
(b) For the same reasons as case (a) clearly E + cannot exist.
(c) As a = c(1 + b − d), then E + can be reduced to However (e) First suppose that cd > 1. Then This can be rewritten as c(b − d) < a − c, which is not true in this case as c > a and b > d. Next suppose that cd < 1. Then, Note that the inequality can be rewritten as d(a − c) > b − d, which is false and therefore, E + does not exist.
(f) Suppose that cd > 1. Then This inequality does not hold true as a > c and d > b since the inequality can be rewritten as d(a − c) < b − d. Next suppose that cd < 1. Then, This inequality can be rewritten as c(b − d) > a − c, and so will not hold. Therefore, E + does not exist.
(g) Let cd > 1. This implies that This will provide a contradiction as this inequalities imply cd(a − c) < c(b − d) < a − c . However, this can be reduced to cd < 1 since a > c. Next suppose that cd < 1. This gives us that Again combining the two inequalities, cd(a − c) > c(b − d) > a − c, which in turn can be reduced to cd > 1. This again provides a contradiction, and therefore E + cannot exist.
(h) As a = c then E + can be reduced to Note that one coordinate will be negative or if d = b, then E + = E 0 . Therefore, E + cannot exist.
(i) Similar to case (h) we can reduce E + to This point will therefore not exist as one coordinate will be negative or if c = a, then E + = E 0 .

Theorem 32
Consider system (43), and let a, b, c, d > 0, 0 < α, β < 1, and , and E + is locally asymptotically stable, then E 0 is a repeller, and both E x and E y are saddle points such that E y se E + se E x . Every solution which begins off the x and y axes converges to E + . Every solution which begins on the x-axis without E 0 converges to E x , and every solution which begins on the y-axis − c), and a < c(1 + b − d), then E y is a saddle point, E x is locally asymptotically stable, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to E y . − c), and a > c (1 + b − d), then E y is locally asymptotically stable, E x is a saddle point, E 0 is a repeller, and E + does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis without E 0 , converges to E x .
(e) Suppose that c > a > 0 and b > d > 0. The equilibrium point E 0 will be a saddle point, while E x will be locally asymptotically stable,and both E y and E + will not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis converges to E 0 .
(f ) Suppose that a > c > 0 and d > b > 0. The equilibrium point E 0 will be a saddle point, while E y will be locally asymptotically stable, and both E x and E + will not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis converges to E 0 .
(g) Suppose that 0 < c < a and 0 < d < b. The equilibrium points of E x , E y , and E + do not exist and E 0 is globally asymptotically stable.
Proof. For each case the existence of the equilibrium points is given by  (43), and let a, b, c, d > 0, 0 < α, β < 1, and , then E y is non-hyperbolic of the stable type, E x is locally asymptotically stable, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to E y .
, then E y is a saddle point, E x is non-hyperbolic point of the stable type, E 0 is a repeller, and E + does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis without E 0 converges to , then E y is non-hyperbolic of the stable type, E x is a saddle point, E 0 is a repeller, and E + does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis without E 0 , converges to and E x are non-hyperbolic of the stable type, E 0 is a repeller, and E + does not exist. Every solution on the x-axis without E 0 will converge to E x and every solution on the y-axis without E 0 will converge to E y . Every solution which begins off the x and y axis will converge to exactly one of E x or E y .
(f ) Suppose that c = d = 1 and a = b. Then c > a > 0 and d > b > 0, E 0 will be a repeller, E x and E y will be non-hyperbolic of the stable type, and there will exist infinite number of solutions of the form For each of the equilibrium points of the form E K , there is a stable manifold W s (E K ) as its basins of attraction. All W s (E K ) have an end point at E 0 and they are graphs of continuous and non-decreasing functions. The equilibrium points E K depends continuously on the initial point (x 0 , y 0 ).
(g) Suppose that c = a and d > b > 0. The equilibrium point E 0 will be nonhyperbolic of the unstable type, while E y will be locally asymptotically stable, and both E x and E + will not exist. Then there will exists two curves, C 1 and C 2 , C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C 1 and C 2 coincide with each other or C 2 is not in the region, every solution which begins off the x-axis will converge to E y . Every solution which begins on the x-axis will converge to E 0 . If there exists both C 1 , C 2 in the region then every solution to the left of C 2 will converge to E y and every solution to the right of C 2 will converge to E 0 .
(h) Suppose that c = a and b > d > 0 or b = d and a > c > 0. The equilibrium point E 0 will be non-hyperbolic of the stable type, while E y , E x , and E + will not exist. Every solution will converge to E 0 .
(i) Suppose that b = d and c > a > 0. The equilibrium point E 0 will be nonhyperbolic of the unstable type, while E x is locally asymptotically stable, and both E y and E + do not exist. Then there will exists two curves, C 1 and C 2 , C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 .
If the curves C 1 and C 2 coincide with each other or C 2 is not in the region, every solution which begins off the y-axis will converge to E x . Every solution which begins on the y-axis will converge to E 0 . If there exists both C 1 , C 2 in the region then every solution to the left of C 1 will converge to E 0 and every solution to the right of C 1 will converge to E x .
(j) Suppose that c = a and b = d. Then E 0 is non-hyperbolic of resonance type (1,1), and E x , E y , and E + will not exist. Every solution will converge to E 0 .
Proof. For each case the existence of the equilibrium points is given by For case (f ), we need to check the eigenvalues of the Jacobian matrix for each of the infinite equilibrium points of the form E K . As c = d = 1 and a = b substituting in E K into the Jacobian matrix will yield The eigenvalues of this matrix will be λ 1 = 1 and λ 2 = K + αx + (1 − K − x)β.

Example 2 4.4.1 Derivation of the System
We will begin by deriving the discrete system for example 2. The original continuous Lotka-Volterra system is see [1]. Let x(t) = N 1 (t) and y(t) = N 2 (t) Using the method of semi implicit discretization this system can be converted to the discrete form x n+1 − x n h = r 1 x n − x n x n+1 + a 1 y n k 1 , y n+1 − y n h = r 2 y n − y n y n+1 + a 2 x n k 2 .

Local Stability Results
We will begin by proving local stability results for system (46). Additionally, we will prove that this system is bounded provided 0 ≤ x 0 , y 0 < A.
Suppose thatx = 0 andȳ = 0. Then the system can be reduced tō Thus, when A > K 2 , there will be the equilibrium point of E y = (0, A − K 2 ). Next suppose thatx = 0 andȳ = 0. Using this information the system can be reduced Thus, when A > K 1 , there will be the equilibrium point of E x = (A − K 1 , 0).

This can be rewritten as
When A > K 1 , K 2 and furthermore, K 1 = K 2 then this will exist and result in infinite equilibrium points. Let K = K 1 = K 2 , the infinite equilibrium points will be of the form To find the local stability of E 0 , E x , and E y we find the Jacobian matrix. For system (46) let f and g be defined as f (u, v) = u(A−v) K 1 +u and g(u, v) = v(A−u) K 2 +v . Then the Jacobian matrix of When evaluated at an equilibrium point, the eigenvalues of λ 1 and λ 2 can be found from the matrix. If |λ 1 | < 1 and |λ 2 | < 1, then the equilibrium point is locally asymptotically stable. If |λ 1 | < 1 and |λ 2 | > 1 or |λ 1 | > 1 and |λ 2 | < 1, then the equilibrium point is a saddle point. If |λ 1 | > 1 and |λ 2 | > 1, then the equilibrium point is a repeller. Finally, if either |λ 1 | = 1 or |λ 2 | = 1, then the equilibrium point is non-hyperbolic.
When the Jacobian matrix is evaluated at the equilibrium point of E 0 = (0, 0), Clearly we have the eigenvalues of λ 1 = A K 1 and λ 2 = A K 2 .
(d) non-hyperbolic if K 1 = A or K 2 = A.
Proof. The results follow from the eigenvalues.
Next we will investigate the Jacobian matrix at the equilibrium point of E y = (0, A − K 2 ). This will give the matrix where the eigenvalues will be λ 1 = K 2 K 1 and λ 2 = K 2 A .
Lemma 20 When A > K 2 , the equilibrium point of E y = (0, A − K 2 ) is (a) locally asymptotically stable if K 1 > K 2 .
Proof. The results follow from the eigenvalues.
Finally, we evaluate Jacobian matrix at the equilibrium point of E x = (A − K 1 , 0), which gives The eigenvalues of the matrix are λ 1 = K 1 A and λ 2 = K 1 K 2 .
Lemma 21 When A > K 1 the equilibrium point of E x = (A − K 1 , 0) is (a) locally asymptotically stable if K 1 < K 2 .
Proof. The results follow from the eigenvalues.
Lemma 18 gives the necessary conditions for the equilibrium points to exist.
Using Lemmas 19, 20, and 21 we can make the following conclusions regarding the local stability analysis of system (46). NHST stands for non-hyperbolic of stable type, and IEP stands for infinite equilibrium points.

Stability of Equilibrium Points
The following lemma will prove the criteria of boundedness needed for global analysis given that 0 ≤ x 0 , y 0 < A.
Proof. Note that if x 0 , y 0 ≤ A, then x 1 , y 1 < A. Indeed as using inequalities we can conclude that So it holds that x 1 < A. Additionally, as A > x 0 , y 0 ≥ 0, then x 1 ≥ 0. Similarly as we can conclude using inequalities that resulting in the fact that y 1 < A. Additionally, as A > x 0 , y 0 ≥ 0, then y 1 ≥ 0.
Continuing with this technique for n = 2, 3, . . . we can conclude that 0 ≤ x n < A and 0 ≤ y n < A.

Global Stability Results
In this section we will compile the local stability results and use Theorems 30 and 31 to give conclusions regarding the global dynamics of system (46). We will assume that 0 ≤ x 0 , y 0 < A so that the solutions are bounded. Otherwise, unbounded solutions could go to infinity or negative infinity. System (46) has no minimal period two solutions as was proved by Mathematica.
(a) If A < K 1 , K 2 , then E x and E y will not exist, and every solution converges to E 0 .
(b) If K 1 < A < K 2 , then E 0 is a saddle point, E x is locally asymptotically stable, and E y does not exist. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis converges to E 0 .
(c) If K 2 < A < K 1 , then E 0 is a saddle point, E y is locally asymptotically stable, and E x does not exist. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis converges to E 0 .
(d) If K 2 < K 1 < A, then E 0 is a repeller, E x is a saddle point, and E y is locally asymptotically stable. Every solution which begins off the x-axis converges to E y , and every solution which begins on the x-axis converges to E x .
(e) If K 1 < K 2 < A, then E 0 is a repeller, E x is locally asymptotically stable, and E y is a saddle point. Every solution which begins off the y-axis converges to E x , and every solution which begins on the y-axis converges to E y .
(a) If A = K 1 = K 2 , then E 0 is non-hyperbolic of resonance type (1, 1), and both E x and E y do not exist. Every solution converges to E 0 .
(b) If A = K 1 < K 2 , or A = K 2 < K 1 , then E 0 is non-hyperbolic of stable type and both E x and E y do not exist. Every solution converges to E 0 .
(c) If K 1 < A = K 2 then E 0 is non hyperbolic of the unstable type, E x is locally asymptotically stable, and E y does not exist. Then there will exists two curves, C 1 and C 2 , C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C 1 and C 2 coincide with each other or C 2 is not in the region, every solution which begins off the y-axis will converge to E x . Every solution which begins on the y-axis will converge to E 0 . If there exists both C 1 , C 2 in the region, then every solution to the left of C 1 will converge to E 0 and every solution to the right of C 1 will converge to E x .
(d) If K 2 < A = K 1 then E 0 is non hyperbolic of unstable type, E y is locally asymptotically stable, and E x does not exist. Then there will exists two curves, C 1 and C 2 , C 2 se C 1 that are continuous and non-decreasing with an endpoint at E 0 . If the curves C 1 and C 2 coincide with each other or C 2 is not in the region, every solution which begins off the x-axis will converge to E y . Every solution which begins on the x-axis will converge to E 0 . If there exists both C 1 , C 2 in the region, then every solution to the left of C 2 will converge to E y and every solution to the right of C 2 will converge to E 0 . For case (e), we need to check the eigenvalues of the Jacobian matrix for the infinite equilibrium points of the form E K where The eigenvalues will be λ 1 = 1 and λ 2 = x 2 +(K−A)x+K 2 (A−x)(x+K) . Note that |λ 2 | < 1 as A > K and A − K > x. Therefore, 0 < |λ 2 | < λ 1 .
As the solutions are bounded by Lemma 22 and there are no minimal period two solutions, Theorem 31 will give the global dynamics in cases (a) − (e).