Global Dynamics of Three Anticompetitive Systems of Difference Equations in the Plane

Abstract

We investigate the global dynamics of several anticompetitive systems of rational difference equations which are special cases of general linear fractional system of the forms 𝑥_𝑛+1=(𝛼₁+𝛽₁𝑥_𝑛+𝛾₁𝑦_𝑛)/(𝐴₁+𝐵₁𝑥_𝑛+𝐶₁𝑦_𝑛),𝑦_𝑛+1=(𝛼₂+𝛽₂𝑥_𝑛+𝛾₂𝑦_𝑛)/(𝐴₂+𝐵₂𝑥_𝑛+𝐶₂𝑦_𝑛),𝑛=0,1,..., where all parameters and the initial conditions 𝑥₀,𝑦₀ are arbitrary nonnegative numbers, such that both denominators are positive. We find the basins of attraction of all attractors of these systems.

1. Introduction

A first order system of difference equations: 𝑥_𝑛+1=𝑓𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝑔𝑥_𝑛,𝑦_𝑛,𝑛=0,1,…,𝑥₀,𝑦₀∈ℛ, (1) where ℛ⊂ℝ², (𝑓,𝑔)∶ℛ→ℛ,𝑓,𝑔 are continuous functions, is competitive if 𝑓(𝑥,𝑦) is nondecreasing in 𝑥 and nonincreasing in 𝑦, and 𝑔(𝑥,𝑦) is nonincreasing in 𝑥 and nondecreasing in 𝑦.

System (1) where the functions 𝑓 and 𝑔 have monotonic character opposite the monotonic character in competitive system is called anticompetitive; see [1, 2].

In this paper, we consider the following anticompetitive systems of difference equations: 𝑥_𝑛+1=𝛾₁𝑦_𝑛𝐵₁𝑥_𝑛+𝑦_𝑛,𝑦_𝑛+1=𝛽₂𝑥_𝑛𝐵₂𝑥_𝑛+𝑦_𝑛,𝑛=0,1,…, (2) where all parameters are positive numbers and the initial conditions (𝑥₀,𝑦₀) are arbitrary nonnegative numbers such that 𝑥₀+𝑦₀>0. In the classification of all linear fractional systems in [3], System (2) was mentioned as system (18,18). We also consider systems 𝑥_𝑛+1=𝛾₁𝑦_𝑛𝐵₁𝑥_𝑛+𝑦_𝑛,𝑦_𝑛+1=𝛼₂+𝛽₂𝑥_𝑛𝑦_𝑛𝑥,𝑛=0,1,…,_𝑛+1=𝛾₁𝑦_𝑛𝐴₁+𝑥_𝑛,𝑦_𝑛+1=𝛼₂+𝛽₂𝑥_𝑛𝑦_𝑛,𝑛=0,1,…, (3)(4) with 𝑥₀>0, 𝑦₀>0, which were labeled as systems (18,23) and (16,23), respectively, in [3]. Three systems have interesting and different dynamics. While System (2) has all bounded solutions, most of solutions of Systems (3) and (4) are unbounded. Another major difference is the existence of the unique period-two solution for (2) and, in a special case, the abundance of such solutions, while neither (3) nor (4) has period-two solutions. We show that every solution of System (3) converges to the unique equilibrium or is approaching (0,∞) and so System (3) gives an example of a semistable equilibrium point. Finally, we show that all solutions of (4) which start on the stable set converge to the unique equilibrium, while all solutions which start off the stable set are approaching (0,∞) or (∞,0). We also get that for some special values of parameters, Systems (4) and (3) can be decoupled and explicitly solved.

Competitive systems of the form (1) were studied by many authors such as Clark and Kulenović [4], Hess [5], Hirsch and Smith [6], Kulenović and Merino [7], Kulenović and Nurkanović [8], Garić-Demirović et al. [9, 10], and Smith [11, 12]. Precise results about the basins of attraction of the equilibrium points have been obtained in [13].

The study of anticompetitive systems started in [1] and has advanced since then; see [2]. The principal tool of study of anticompetitive systems is the fact that the second iterate of the map associated with anticompetitive system is competitive map and so elaborate theory for such maps developed recently in [6, 14, 15] can be applied.

The major result on a global behavior of System (2) is the following theorem.

Theorem 1. (a) Assume that 𝐵₁<𝐵₂. Then the unique positive equilibrium point 𝐸(𝑥,𝑦) of System (2) is locally asymptotically stable with the basin of attraction ℬ(𝐸)=(0,∞)². The unique period-two solution {𝑃₁,𝑃₂}={(𝛾₁,0),(0,𝛽₂/𝐵₂)} is a saddle point and its basin of attraction is the union of coordinate axes without the origin, that is, ℬ({𝑃₁,𝑃₂})=((𝑥,0)∣𝑥>0)∪(0,𝑦)∣𝑦>0).
(b) Assume that 𝐵₁>𝐵₂. Then every solution of System (2) converges to the period-two solution {𝑃₁,𝑃₂} or to the equilibrium 𝐸. More precisely, there exists a set 𝒞={(𝑥,(𝑦/𝑥)𝑥)∶𝑥>0}⊂ℛ=(0,∞)² which is the basin of attraction of 𝐸. The set 𝒞 has the property that for 𝒲₋∶=𝑥∈ℛ⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑥⪯_se𝑦=𝑥,𝑦∶𝑦>𝑦𝑥,𝒲𝑥≥0₊∶=𝑥∈ℛ⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑦⪯_se𝑥=𝑥,𝑦∶0≤𝑦<𝑦𝑥𝑥, (5) the following holds.
(i) If (𝑥₀,𝑦₀)∈𝒲₊, then lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=𝑃₁,lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=𝑃₂. (6)
(ii) If (𝑥₀,𝑦₀)∈𝒲₋, lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=𝑃₂,lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=𝑃₁. (7)
(c) Assume that 𝐵₁=𝐵₂. Then every solution of System (2) is equal to either the period-two solution {𝐵₁𝑦₀/(𝐵₁𝑥₀+𝑦₀),𝛽₂𝑥₀/(𝐵₁𝑥₀+𝑦₀)}, {𝐵₁𝐷𝑥₀/(𝐵₁𝑦₀+𝐷𝑥₀),𝛽₂𝑦₀/(𝐵₁𝑦₀+𝐷𝑥₀)} or to the equilibrium 𝐸(𝑥,𝑦), where 𝐷=𝛽₂/𝛾₁ and 𝑥=𝛽₂𝛾₁/(𝐵₁+1).

2. Preliminaries

We now give some basic notions about competitive systems and maps in the plane of the form (1), where 𝑓 and 𝑔 are continuous functions, 𝑓(𝑥,𝑦) is nondecreasing in 𝑥 and nonincreasing in 𝑦, and 𝑔(𝑥,𝑦) is nonincreasing in 𝑥 and nondecreasing in 𝑦 in some domain 𝐴 with nonempty interior.

Consider a map 𝑇=(𝑓,𝑔) on a set ℛ⊂ℝ², and let 𝐸∈ℛ. The point 𝐸∈ℛ is called a fixed point if 𝑇(𝐸)=𝐸. An isolated fixed point is a fixed point that has a neighborhood with no other fixed points in it. A fixed point 𝐸∈ℛ is an attractor if there exists a neighborhood 𝒰 of 𝐸 such that 𝑇^𝑛(𝐱)→𝐸 as 𝑛→∞ for 𝐱∈𝒰; the basin of attraction is the set of all 𝐱∈ℛ such that 𝑇^𝑛(𝐱)→𝐸 as 𝑛→∞. A fixed point 𝐸 is a global attractor on a set 𝒦 if 𝐸 is an attractor and 𝒦 is a subset of the basin of attraction of 𝐸. If 𝑇 is differentiable at a fixed point 𝐸 and if the Jacobian 𝐽_𝑇(𝐸) has one eigenvalue with modulus less than one and a second eigenvalue with modulus greater than one, 𝐸 is said to be a saddle. See [16] for additional definitions.

Definition 2. Let 𝑇=(𝑓,𝑔) be a continuously differentiable vector function and let 𝑈 be a neighborhood of a saddle point (𝑥,𝑦) of (1). The local stable manifold 𝒲^𝑠_loc is the set 𝒲^𝑠_loc𝑥,𝑦=𝑥,𝑦∶𝑇^𝑛𝑥,𝑦∈𝑈(∀𝑛≥0)∧lim_𝑛→∞𝑇^𝑛𝑥,𝑦=𝑥,𝑦. (8) The global stable manifold 𝑊^𝑠 of a saddle point (𝑥,𝑦) is the set 𝒲^𝑠𝑥,𝑦=𝑥,𝑦∶lim_𝑛→∞𝑇^𝑛𝑥,𝑦=𝑥,𝑦. (9)

The map 𝑇 may be viewed as a monotone map if we define a partial order on ℝ² so that the positive cone in this new partial order is the fourth (resp. first) quadrant. Define a south-east (resp. north-east) partial order ⪯_se (resp. ⪯_ne) on ℝ² so that the positive cone is the fourth quadrant (resp. first quadrant), that is, (𝑥¹,𝑦¹)⪯_se(𝑥²,𝑦²) (resp. (𝑥¹,𝑦¹)⪯_ne(𝑥²,𝑦²)) if and only if 𝑥¹≤𝑥² and 𝑦¹≥𝑦² (resp. 𝑥¹≤𝑥² and 𝑦¹≤𝑦²). For 𝐱,𝐲∈ℝ² the order interval ⟦𝐱,𝐲⟧ is the set of all 𝐳 such that 𝐱⪯𝐳⪯𝐲. The map 𝑇 is called competitive (resp. cooperative) on a set 𝑆 if 𝐯⪯_se𝐰 (resp. 𝐯⪯_ne𝐰) implies 𝑇(𝐯)⪯_se𝑇(𝐰) (resp. 𝑇(𝐯)⪯_ne𝑇(𝐰)).

Two points 𝐯,𝐰∈ℝ²₊ are said to be related if 𝐯⪯𝐰 or 𝐰⪯𝐯. Also, a strict inequality between points may be defined as 𝐯≺𝐰 if 𝐯⪯𝐰 and 𝐯≠𝐰. A stronger inequality may be defined as 𝐯≺≺𝐰 if 𝑣₁<𝑤₁ and 𝑤₂<𝑣₂. A map 𝑓∶Intℝ²₊→Intℝ²₊ is strongly monotone if 𝐯≺𝐰 implies that 𝑓(𝐯)≺≺𝑓(𝐰) for all 𝐯,𝐰∈Intℝ²₊. Clearly, being related is invariant under the iteration of a strongly monotone map. Differentiable strongly monotone maps have Jacobian with constant sign configuration +−−+. (10) The mean value theorem and the convexity of ℝ²₊ may be used to show that 𝑇 is monotone, as in [17].

For 𝐱=(𝑥₁,𝑥₂)∈ℝ², define 𝑄_𝑙(𝐱) for 𝑙=1,…,4 to be the usual four quadrants based at 𝐱 and numbered in a counterclockwise direction, for example, 𝑄₁(𝐱)={𝐲=(𝑦₁,𝑦₂)∈ℝ²∶𝑥₁≤𝑦₁,𝑥₂≤𝑦₂}. We now state three results for competitive maps in the plane.

The following definition is from [11].

Definition 3. Let 𝒮 be a nonempty subset of ℝ². A competitive map 𝑇∶𝒮→𝒮 is said to satisfy condition (𝑂+) if for every 𝑥, 𝑦 in 𝒮, 𝑇(𝑥)⪯_ne𝑇(𝑦) implies 𝑥⪯_ne𝑦, and 𝑇 is said to satisfy condition (𝑂−) if for every 𝑥, 𝑦 in 𝒮, 𝑇(𝑥)⪯_ne𝑇(𝑦) implies 𝑦⪯_ne𝑥.

The following theorem was proved by de Mottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [11, 12].

Theorem 4. Let 𝒮 be a nonempty subset of ℝ². If 𝑇 is a competitive map for which (𝑂+) holds, then for all 𝑥∈𝒮, {𝑇^𝑛(𝑥)} is eventually componentwise monotone. If the orbit of 𝑥 has compact closure, then it converges to a fixed point of 𝑇. If instead (𝑂−) holds, then for all 𝑥∈𝒮, {𝑇^2𝑛} is eventually componentwise monotone. 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 following result is from [11], with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (𝑂+) and (𝑂−).

Theorem 5. Let ℛ⊂ℝ² be the cartesian product of two intervals in ℝ. Let 𝑇∶ℛ→ℛ be a 𝐶¹ competitive map. If 𝑇 is injective and det𝐽_𝑇(𝑥)>0 for all 𝑥∈ℛ, then 𝑇 satisfies (𝑂+). If 𝑇 is injective and det𝐽_𝑇(𝑥)<0 for all 𝑥∈ℛ, then 𝑇 satisfies (𝑂−).

The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess, see [5, 15, 18], and is helpful for determining the basins of attraction of the equilibrium points.

Corollary 6. If the nonnegative cone with respect to the partial order ⪯ is a generalized quadrant in ℝ² and if the competitive map 𝑇∶→ℝ² has no fixed points in ⟦𝑢₁,𝑢₂⟧ other than 𝑢₁ and 𝑢₂, then the interior of ⟦𝑢₁,𝑢₂⟧ is either a subset of the basin of attraction of 𝑢₁ or a subset of the basin of attraction of 𝑢₂.

The next two results are from [15, 18].

Theorem 7. Let 𝑇 be a competitive map on a rectangular region ℛ⊂ℝ². Let 𝑥∈ℛ be a fixed point of 𝑇 such that Δ∶=ℛ∩int(𝑄₁(𝑥)∪𝑄₃(𝑥)) is nonempty (i.e., 𝑥 is not the NW or SE vertex of ℛ), and 𝑇 is strongly competitive on Δ. Suppose that the following statements are true.(a)The map 𝑇 has a 𝐶¹ extension to a neighborhood of 𝑥.(b)The Jacobian matrix 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.

Theorem 8. Let ℐ₁,ℐ₂ be intervals in ℝ with endpoints 𝑎₁,𝑎₂ and 𝑏₁,𝑏₂ with endpoints, respectively, with 𝑎₁<𝑎₂ and 𝑏₁<𝑏₂, where −∞≤𝑎₁<𝑎₂≤∞ and −∞≤𝑏₁<𝑏₂≤∞. Let 𝑇 be a competitive map on a rectangle ℛ=ℐ₁×ℐ₂ and 𝐱∈int(ℛ). Suppose that the following hypotheses are satisfied.(1)𝑇(int(ℛ))⊂int(ℛ) and 𝑇 is strongly competitive on int(ℛ).(2)The point 𝐱 is the only fixed point of 𝑇 in (𝑄₁(𝐱)∪𝑄₃(𝐱))∩int(ℛ).(3)The map 𝑇 is continuously differentiable in a neighborhood of 𝐱, and 𝐱 is the saddle point.(4)At least one of the following statements is true.(a)𝑇 has no minimal period-two orbits in (𝑄₁(𝐱)∪𝑄₃(𝐱))∩int(ℛ).(b)det𝐽_𝑇(𝐱)>0 and 𝑇(𝐱)=𝐱 only for 𝐱=𝐱.
Then the following statements are true.(i)The stable manifold 𝒲^𝑠(𝐱) is connected and it is the graph of a continuous increasing curve with endpoints in 𝜕ℛ. int(ℛ) is divided by the closure of 𝒲^𝑠(𝐱) into two invariant connected regions 𝒲₊ (“below the stable set”), and 𝒲₋ (“above the stable set”), where 𝒲₊∶=𝐱∈ℛ⧵𝒲^𝑠(𝐱)∶∃𝐱^′∈𝒲^𝑠(𝐱)with𝐱⪯_se𝐱^′,𝒲₋∶=𝐱∈ℛ⧵𝒲^𝑠(𝐱)∶∃𝐱^′∈𝒲^𝑠(𝐱)with𝐱^′⪯_se𝐱. (11)(ii)The unstable manifold 𝒲^𝑢(𝐱) is connected and it is the graph of a continuous decreasing curve.(iii)For every 𝐱∈𝒲₊, 𝑇^𝑛(𝐱) eventually enters the interior of the invariant set 𝑄₄(𝐱)∩ℛ, and for every 𝐱∈𝒲₋, 𝑇^𝑛(𝐱) eventually enters the interior of the invariant set 𝑄₂(𝐱)∩ℛ.(iv)Let 𝐦∈𝑄₂(𝐱) and 𝐌∈𝑄₄(𝐱) be the endpoints of 𝒲^𝑢(𝐱), where 𝐦⪯_se𝐱⪯_se𝐌. For every 𝐱∈𝒲₋ and every 𝐳∈ℛ such that 𝐦⪯_se𝑧, there exists 𝑚∈ℕ such that 𝑇^𝑚(𝐱)⪯_se𝑧, and for every 𝐱∈𝒲₊ and every 𝐳∈ℛ such that 𝐳⪯_se𝐌, there exists 𝑚∈ℕ such that 𝐌⪯_se𝑇^𝑚(𝐱).

3. Global Dynamics of System (2) (See Figure 1)

Figure 1 

Global dynamics of System (2).

The equilibrium point 𝐸(𝑥,𝑦) of System (2) satisfies the following system of equations: 𝛾𝑥=¹𝑦𝐵₁𝑥+𝑦,𝛽𝑦=²𝑥𝐵₂𝑥+𝑦. (12)

It is easy to see that System (12) has unique equilibrium point 𝐸 in the first quadrant, for all values of the parameters. Indeed, the positive equilibrium point is an intersection of the following two curves: 𝐵𝑦=¹𝑥²𝛾₁,𝑦−𝑥𝑥=²𝛽₂−𝐵₂𝑦. (13)(14) It is clear that at the point of intersection 𝐸 curve (13) is steeper than curve (14), that is, 𝑑𝑦𝑑𝑥₍₁₃₎(𝐸)>𝑑𝑦𝑑𝑥₍₁₄₎(𝐸), (15) which gives 𝐵₁𝑥2𝛾₁−𝑥𝛾₁−𝑥²>𝛽₂−𝐵₂𝑦²𝑦2𝛽₂−𝐵₂𝑦. (16) This inequality is equivalent to the following inequality: 𝛽₂+𝑦²𝑥𝛾₁+𝐵₁𝑥²𝑦>𝐵₁𝑥𝑦, (17) which is always satisfied.

3.1. Linearized Stability Analysis of System (2)

In this section we present the linearized stability analysis of the equilibrium 𝐸 of System (2).

Theorem 9. (i) If 𝐵₁<𝐵₂, then 𝐸 is locally asymptotically stable.
(ii) If 𝐵₁=𝐵₂, then 𝐸 is a nonhyperbolic equilibrium point.
(iii) If 𝐵₁>𝐵₂, then 𝐸 is a saddle point.

Proof. The map 𝑇 associated to System (2) is 𝑇=𝑓𝑔=𝛾𝑥,𝑦𝑥,𝑦𝑥,𝑦¹𝑦𝐵₁𝛽𝑥+𝑦₂𝑥𝐵₂𝑥+𝑦. (18) The Jacobian matrix of map (18) is 𝐽_𝑇=−𝐵𝑥,𝑦¹𝛾₁𝑦𝐵₁𝑥+𝑦²𝐵₁𝛾₁𝑥𝐵₁𝑥+𝑦²𝛽₂𝑦𝐵₂𝑥+𝑦²−𝛽₂𝑥𝐵₂𝑥+𝑦², (19) and evaluated at the equilibrium point 𝐸=(𝑥,𝑦) is 𝐽_𝑇𝑥,𝑦=−𝐵₁𝛾₁𝑥²𝑦𝐵₁𝛾₁𝑥³𝑦²𝑦³𝛽₂𝑥²−𝑦²𝛽₂𝑥. (20) The characteristic equation has the following form: 𝜆²+𝐵₁𝛾₁𝑥²𝑦+1𝛽₂𝑦²𝑥𝜆=0, (21) and the characteristic roots are 𝜆₁=0,𝜆₂𝐵=−¹𝛾₁𝑥²𝑦−1𝛽₂𝑦²𝑥. (22) A straightforward calculation shows that the conditions 𝜆₂∈(−1,0), 𝜆₂=−1, and 𝜆₂∈(−∞,−1), respectively, are equivalent to the conditions 𝑥𝑦²<𝛽₂𝛾₁,𝑥𝑦²=𝛽₂𝛾₁,𝑥𝑦²>𝛽₂𝛾₁. (23) On the other hand, by dividing two equilibrium equations (12) we obtain 𝑥𝑦²=𝛽₂𝛾₁𝐵₁𝑥+𝑦𝐵₂𝑥+𝑦, (24) which implies that the condition (23) is equivalent to the conditions 𝐵₁<𝐵₂,𝐵₁=𝐵₂,𝐵₁>𝐵₂, (25) which completes the proof of the theorem.

3.2. Global Results for System (2)

In this section we present the proof of Theorem 1 on the global dynamics of System (2). First, we prove the following result on existence and local behavior of the period-two solutions.

Lemma 10. System (2) has the minimal period-two solution 𝑃₁,𝑃₂=𝛾₁,𝛽,00,²𝐵₂ (26) for all values of parameters. If 𝐵₁≠𝐵₂, then the solution (26) is the unique period-two solution and when 𝐵₁=𝐵₂ there are infinitely many period-two solutions. The set ℬ=𝑥,𝑦∣𝑥>0,𝑦=0or𝑥=0,𝑦>0 (27) is a subset of the basin of attraction of the solution (26). The period-two solution (26) is locally stable if 𝐵₁>𝐵₂ and a saddle point if 𝐵₁<𝐵₂. Finally the period-two solution (26) is nonhyperbolic if 𝐵₁=𝐵₂.

Proof. The second iterate of the map 𝑇 is given as𝑇²𝑥,𝑦=𝑥𝛽₂𝛾₁𝑦+𝑥𝐵₁𝑥𝑦𝛽₂+𝑥²𝛽₂𝐵₁+𝑦²𝛾₁𝐵₁+𝑥𝑦𝛾₁𝐵₁𝐵₂𝑦𝛽₂𝛾₁𝑦+𝑥𝐵₂𝛽₂𝐵₁𝑥²+𝛾₁𝑥𝑦𝐵²₂+𝛽₂𝑥𝑦+𝛾₁𝑦²𝐵₂. (28)A period-two solution of System (2) satisfies 𝑇²(𝑥,𝑦)=(𝑥,𝑦), which immediately leads to the following equations: 𝐵₂−𝐵₁𝛽₂𝛾₁𝑥−𝐵₂𝛾₁𝑥𝑦−𝛾₁𝑦²𝐵=0,₂−𝐵₁𝛽₂𝛾₁𝑦−𝐵₂𝑥𝑦−𝛾₁𝐵₁𝑥²=0, (29)(30) which have either unique solution if 𝐵₁≠𝐵₂ or it has infinitely many solutions if 𝐵₁=𝐵₂. In the first case, (29) gives immediately (13) and (30) gives (14), which means that in this case the only minimal period-two solution is (26).
A straightforward calculation shows that 𝑇(𝑃₁)=𝑃₂,𝑇(𝑃₂)=𝑃₁, which shows that {𝑃₁,𝑃₂} is a minimal period-two solution. Moreover, 𝑇(𝑎,0)=𝑃₂, 𝑇(0,𝑏)=𝑃₁ for every 𝑎>0, 𝑏>0, which shows that the set ℬ is a subset of the basin of attraction of {𝑃₁,𝑃₂}. The Jacobian matrix of 𝑇² is 𝐽_𝑇²=𝑥,𝑦𝑦𝛽₂𝛾²₁𝐵₁𝐵₁𝐵₂𝑥²+2𝐵₁𝑥𝑦+𝑦²𝑥𝑦𝛽₂+𝑥²𝛽₂𝐵₁+𝑦²𝛾₁𝐵₁+𝑥𝑦𝛾₁𝐵₁𝐵₂²−𝑥𝛽₂𝛾²₁𝐵₁𝐵₁𝐵₂𝑥²+2𝐵₁𝑥𝑦+𝑦²𝑥𝑦𝛽₂+𝑥²𝛽₂𝐵₁+𝑦²𝛾₁𝐵₁+𝑥𝑦𝛾₁𝐵₁𝐵₂²−𝑦𝛽²₂𝛾₁𝐵₁𝐵₂𝑥²+2𝐵₁𝑥𝑦+𝑦²𝛽₂𝐵₁𝑥²+𝛾₁𝑥𝑦𝐵²₂+𝛽₂𝑥𝑦+𝛾₁𝑦²𝐵₂²𝑥𝛽²₂𝛾₁𝐵₁𝐵₂𝑥²+2𝐵₁𝑥𝑦+𝑦²𝛽₂𝐵₁𝑥²+𝛾₁𝑥𝑦𝐵²₂+𝛽₂𝑥𝑦+𝛾₁𝑦²𝐵₂². (31) The Jacobian matrix of 𝑇² evaluated at 𝑃₁ is 𝐽_𝑇²𝑃₁=10−𝛽₂𝛾₁𝐵₂01𝐵₁𝐵₂ (32) and the Jacobian matrix of 𝑇² evaluated at 𝑃₂ is 𝐽_𝑇²𝑃₂=1𝐵₁𝐵₂0𝐵₂−1𝛽₂𝛾₁𝛽²₂𝐵₂+𝛽₂𝛾₁𝐵₂0. (33) In both cases, the eigenvalues of the Jacobian matrix of 𝑇² are 𝜆₁=0, 𝜆₂=𝐵₂/𝐵₁, which implies the result on local stability of the minimal period-two solution {𝑃₁,𝑃₂}.

Proof of Theorem 1. First, observe that the rectangle ℛ=[0,𝛾₁]×[0,𝛽₂/𝐵₂]⧵{0,0}=⟦𝑃₂,𝑃₁⟧⧵{0,0} is an invariant and attracting set for the map 𝑇 and so is for the map 𝑇². More precisely, (𝑥_𝑛,𝑦_𝑛)∈ℛ for 𝑛≥1. The map 𝑇² is a competitive map on ℛ.

Case a. Assume that 𝐵₁<𝐵₂. Then in view of Theorem 9 and Lemma 10, the map 𝑇² has three equilibrium points 𝑃₁, 𝑃₂, and 𝐸, where 𝑃₂⪯_se𝐸⪯_se𝑃₁. The equilibrium points 𝑃₁ and 𝑃₂ are saddle points and 𝐸 is a local attractor. The ordered intervals ⟦𝑃₂,𝐸⟧ and ⟦𝐸,𝑃₁⟧ are both invariant sets of 𝑇² and in view of Corollary 6, their interiors are attracted to 𝐸. If we take the point (𝑥,𝑦)∈ℛ⧵⟦𝑃₂,𝐸⟧∪⟦𝐸,𝑃₁⟧, we can find the points (𝑥_𝑙,𝑦_𝑙)∈int⟦𝑃₂,𝐸⟧ and (𝑥_𝑢,𝑦_𝑢)∈int⟦𝐸,𝑃₁⟧, such that (𝑥_𝑙,𝑦_𝑙)⪯_se(𝑥,𝑦)⪯_se(𝑥_𝑢,𝑦_𝑢). Consequently, since 𝑇² is competitive 𝑇^2𝑛((𝑥_𝑙,𝑦_𝑙))⪯_se𝑇^2𝑛((𝑥,𝑦))⪯_se𝑇^2𝑛((𝑥_𝑢,𝑦_𝑢)) for 𝑛≥1 and so lim_𝑛→∞𝑇^2𝑛((𝑥,𝑦))=𝐸, which by continuity of 𝑇 implies that lim_𝑛→∞𝑇^2𝑛+1𝑥,𝑦=lim_𝑛→∞𝑇𝑇^2𝑛𝑥,𝑦=𝑇lim_𝑛→∞𝑇^2𝑛𝑥,𝑦=𝑇(𝐸)=𝐸, (34) and so lim_𝑛→∞𝑇^𝑛((𝑥,𝑦))=𝐸.

Case b. Assume that 𝐵₁>𝐵₂. Then in view of Theorem 9 and Lemma 10, the map 𝑇² has three equilibrium points 𝑃₁ and 𝑃₂ which are local attractors and 𝐸 which is a saddle point. The ordered intervals ⟦𝑃₂,𝐸⟧ and ⟦𝐸,𝑃₁⟧ are both invariant sets for 𝑇² and in view of Corollary 6, their interiors are attracted to 𝑃₂ and 𝑃₁, respectively. In view of Theorems 7 and 8, there is the set 𝒞 with described properties. Direct calculation shows that the half-line 𝑦=(𝑦/𝑥)𝑥, 𝑥>0 is an invariant set which in view of a uniqueness of stable manifold implies that this half-line is exactly stable manifold mentioned in Theorems 7 and 8. It should be observed that because of the fact that one of the characteristic values at the equilibrium point 𝐸 is 0, this equilibrium is superattractive, that is, 𝑇(𝑥₀,𝑦₀)=(𝑥,𝑦), for every (𝑥₀,𝑦₀)∈𝒞.

Case c. Assume that 𝐵₁=𝐵₂. Then by dividing two equations of System (2) we obtain that the solution of (2) satisfies 𝑦_𝑛+1𝑥_𝑛+1=𝛾₂𝛽₁𝑥_𝑛𝑦_𝑛=𝛾₂𝛽₁1𝑦_𝑛/𝑥_𝑛. (35) This means that 𝑦_𝑛/𝑥_𝑛 satisfies first order difference equation 𝑢_𝑛+1=𝐷/𝑢_𝑛, where 𝐷=𝛾₂/𝛽₁. All nonconstant solutions of 𝑢_𝑛+1=𝐷/𝑢_𝑛 are period-two solutions {𝑢₀,𝐷/𝑢₀}. Thus 𝑦_𝑛={𝑢₀𝑥_𝑛,(𝐷/𝑢₀)𝑥_𝑛}. In this case, System (2) becomes 𝑥_𝑛+1=𝐵₁𝑢₀𝐵₁+𝑢₀,𝑦_𝑛+1=𝛽₂𝐵₁+𝑢₀𝑥,𝑛=1,2,…,_𝑛+1=𝐵₁𝐷𝐷+𝐵₁𝑢₀,𝑦_𝑛+1=𝛽₂𝑢₀𝐷+𝐵₁𝑢₀,𝑛=1,2,…, (36) which completes the proof of Case  (c).

Remark 11. System (2) is an example of the homogeneous system which is a special case of a general System (1), where both functions 𝑓 and 𝑔 are homogeneous functions of the same degree 𝑘, that is, 𝑓(𝑡𝑢,𝑡𝑣)=𝑡^𝑘𝑓(𝑢,𝑣), 𝑔(𝑡𝑢,𝑡𝑣)=𝑡^𝑘𝑔(𝑢,𝑣) for all 𝑢, 𝑣 in intersection of domains of 𝑓 and 𝑔 and all 𝑡≠0. In that case, the ratio 𝑧_𝑛=𝑦_𝑛/𝑥_𝑛 of every solution of (1) satisfies the first order difference equation: 𝑧_𝑛+1=𝑓1,𝑧_𝑛𝑔1,𝑧_𝑛=𝐹𝑧_𝑛,𝑛=0,1,… (37) whose analysis gives valuable information about the dynamics of System (1), but does not provide the global dynamics. In particular, this approach cannot determine precisely the basins of attraction of different types of attractors such as equilibrium points and periodic solutions. This approach in the case of the system of linear fractional equations: 𝑥_𝑛+1=𝛼₁+𝛽₁𝑥_𝑛+𝛾₁𝑦_𝑛𝐴₁+𝐵₁𝑥_𝑛+𝐶₁𝑦_𝑛,𝑦_𝑛+1=𝛼₂+𝛽₂𝑥_𝑛+𝛾₂𝑦_𝑛𝐴₂+𝐵₂𝑥_𝑛+𝐶₂𝑦_𝑛,𝑛=0,1,…, (38) where all parameters and the initial conditions (𝑥₀,𝑦₀) are arbitrary nonnegative numbers such that 𝐴_𝑖+𝐵_𝑖𝑥₀+𝐶_𝑖𝑦₀>0, 𝑖=1,2, was first used in [19] and was systematically developed in the recent paper [20]. In [20], the authors studied all possible homogeneous systems of the form (38) and they proved that every bounded solution converges to either an equilibrium solution or to period-two solution. They were able to find a part of the basin of attraction of the period-two solution but not the complete basin of attraction. In the case of system (2) the auxiliary equation for 𝑧_𝑛=𝑦_𝑛/𝑥_𝑛 is 𝑧_𝑛+1=𝛽₂𝐵₁+𝑧_𝑛𝛾₁𝐵₂+𝑧_𝑛𝑧_𝑛=𝐹𝑧_𝑛,𝑛=0,1,…. (39) Since 𝐹 is decreasing, every solution of the auxiliary equation is approaching not necessarily minimal period-two solution. Further analysis can be continued either by checking negative feedback condition for 𝐹² or by using Theorem  3.2 from [20]. In neither case the complete description of the basins of attraction of the equilibrium and the period-two solution is possible. We prefer our approach because it is more precise and also apply equally well to anticompetitive systems which are not homogeneous. The approach which is making use of homogeneous properties of functions is applicable also to the systems which are neither competitive nor anticompetitive.

4. Global Dynamics of System (3) (See Figure 2)

Figure 2 

Global dynamics of System (3).

The equilibrium point 𝐸(𝑥,𝑦) of System (3) satisfies the following system of equations: 𝛾𝑥=¹𝑦𝐵₁𝑥+𝑦,𝛼𝑦=²+𝛽₂𝑥𝑦. (40) It is easy to see that System (40) has the unique equilibrium point 𝐸 in the first quadrant, for all values of the parameters. Indeed, the positive equilibrium point is an intersection of the following two curves: 𝐵𝑦=¹𝑥²𝛾₁,𝑦−𝑥𝑥=²−𝛼₂𝛽₂. (41)(42) It is clear that at the point of intersection 𝐸 curve (41) is steeper than curve (42), that is, 𝑑𝑦𝑑𝑥₍₄₁₎(𝐸)>𝑑𝑦𝑑𝑥₍₄₂₎(𝐸), (43) which gives 𝐵₁𝑥2𝛾₁−𝑥𝛾₁−𝑥²>𝛽₂2𝑦. (44) This inequality is equivalent to the following inequality: 2𝛽₁𝑥𝑦2𝛾₁−𝑥>𝛽₂𝛾₁−𝑥², (45) which in turn is equivalent to 𝐵2+2¹𝛾₁𝑥²𝑦>𝛽₂𝐵₁𝛾₁𝑥³𝑦³. (46)

4.1. Linearized Stability Analysis of System (3)

In this section we prove the following result.

Theorem 12. The unique equilibrium 𝐸 of System (3) is a saddle point.

Proof. The map 𝑆 associated to System (3) is 𝑆=𝑓𝑔=𝛾𝑥,𝑦𝑥,𝑦𝑥,𝑦¹𝑦𝐵₁𝛼𝑥+𝑦₂+𝛽₂𝑥𝑦. (47) The Jacobian matrix of map (47) is 𝐽_𝑆=−𝐵𝑥,𝑦¹𝛾₁𝑦𝐵₁𝑥+𝑦²𝐵₁𝛾₁𝑥𝐵₁𝑥+𝑦²𝛽₂𝑦−𝛼₂+𝛽₂𝑥𝑦², (48) and evaluated at the equilibrium point 𝐸=(𝑥,𝑦) is 𝐽_𝑆𝑥,𝑦=−𝐵₁𝛾₁𝑥²𝑦𝐵₁𝛾₁𝑥³𝑦²𝛽₂𝑦−1. (49)
The characteristic equation has the following form: 𝜆²+𝐵1+¹𝛾₁𝑥²𝑦𝐵𝜆+¹𝛾₁𝑥²𝑦−𝛽₂𝐵₁𝛾₁𝑥³𝑦²=0. (50) Set 𝐵𝑃=1+¹𝛾₁𝑥²𝑦𝐵,𝑄=¹𝛾₁𝑥²𝑦−𝛽₂𝐵₁𝛾₁𝑥³𝑦². (51) Then the necessary and sufficient condition for (50) to have one root inside the unit circle and one root outside the unit circle is |𝑃|>|1+𝑄|, 𝑃²−4𝑄>0; see [7, 21]. The condition |𝑃|>|1+𝑄| leads to 𝑃>−1−𝑄 which is equivalent to 𝐵2+2¹𝛾₁𝑥²𝑦>𝛽₂𝐵₁𝛾₁𝑥³𝑦³, (52) which is condition (46).
The condition 𝑃²−4𝑄>0 becomes 𝐵1+¹𝛾₁𝑥²𝑦²𝐵−4¹𝛾₁𝑥²𝑦−𝛽₂𝐵₁𝛾₁𝑥³𝑦³>0, (53) which is equivalent to 𝐵1−¹𝛾₁𝑥²𝑦²+4𝛽₂𝐵₁𝛾₁𝑥³𝑦³>0, (54) which is clearly satisfied.

4.2. Global Results for System (3)

Lemma 13. System (3) has no minimal period-two solution.

Proof. The second iterate of the map 𝑆 is given as 𝑆²=𝛾𝑥,𝑦₁𝑦+𝑥𝐵₁𝛼₂+𝑥𝛽₂𝛽₂𝐵₁𝑥²+𝛽₂𝑥𝑦+𝛼₂𝐵₁𝑥+𝛾₁𝐵₁𝑦²+𝛼₂𝑦,𝑦𝑦+𝑥𝐵₁𝛼₂+𝑥𝛽₂𝑦𝛼₂+𝑦𝛽₂𝛾₁+𝑥𝛼₂𝐵₁. (55) Period-two solution satisfies 𝑆²(𝑥,𝑦)=(𝑥,𝑦) which reduces to the following two equations: 𝛾₁𝑦+𝑥𝐵₁𝛼₂+𝑥𝛽₂−𝑥𝛽₂𝐵₁𝑥²+𝛽₂𝑥𝑦+𝛼₂𝐵₁𝑥+𝛾₁𝐵₁𝑦²+𝛼₂𝑦=0,𝑦𝛼₂+𝑦𝛽₂𝛾₁+𝑥𝛼₂𝐵₁−𝑦+𝑥𝐵₁𝛼₂+𝑥𝛽₂=0. (56)(57) Equation (57) leads immediately to 𝑦𝛾₁−𝑦𝑥−𝑥²𝐵₁=0 which is exactly the equilibrium equation (41). Using (41) in (56), we obtain after some elementary simplification that period-two solution satisfies (42). This shows that System (3) has no minimal period-two solution.

Lemma 14. The maps 𝑆 and 𝑆² associated with System (3) have the following properties. (i)The maps 𝑆 and 𝑆² are injective.(ii)det𝐽_𝑆²(𝑥,𝑦)>0 for all (𝑥,𝑦), 𝑦>0.Consequently, 𝑆² satisfies (𝑂+) condition and so {𝑆^2𝑛(𝑥₀,𝑦₀)} is eventually componentwise monotone.

Proof. (i) We will prove that 𝑆 is injective and the injectivity of 𝑆² will follow immediately. The condition 𝑆𝑥₁,𝑦₁=𝑆𝑥₂,𝑦₂ (58) is reduced to the following two conditions: 𝑥₂𝑦₁=𝑥₁𝑦₂,𝛼₂𝑦₂−𝑦₁=𝛽₂𝑥₂𝑦₁−𝑥₁𝑦₂, (59) which immediately implies 𝑦₁=𝑦₂ and so 𝑥₁=𝑥₂.
(ii) A direct calculation shows that det𝐽_𝑆²=𝐵²₁𝛼²₂𝛾²₁𝑦²𝛼₂+𝑥𝛽₂𝑦𝛼₂+𝑥𝑦𝛽₂+𝑥𝛼₂𝐵₁+𝑥²𝛽₂𝐵₁+𝑦²𝛾₁𝐵₁², (60) which implies our statement.
The statement on (𝑂+) condition follows from Theorem 5.

Theorem 15. Consider System (3). Then there exists a set 𝒞⊂𝒟=[0,∞)×(0,∞) which is invariant subset of the basin of attraction of 𝐸. The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates 𝒟 into two connected and invariant components, namely, 𝒲₋∶=𝑥∈𝒟⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑥⪯_se𝑦,𝒲₊∶=𝑥∈𝒟⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑦⪯_se𝑥, (61) which satisfy the following.
(i) If (𝑥₀,𝑦₀)∈𝒲₊, then lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=𝐸,lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=(0,∞). (62)
(ii) If (𝑥₀,𝑦₀)∈𝒲₋, lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=(0,∞),lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=𝐸. (63)

Proof. Clearly, the rectangle [0,𝛾₁]×(0,∞) is an invariant and attracting set for the map 𝑆. In particular, 𝑥_𝑛≤𝛾 for 𝑛≥1.
Then by Lemma 14 and Theorem 4, every solution {(𝑥_𝑛,𝑦_𝑛)} of System (3) has eventually monotone components {(𝑥_2𝑛,𝑦_2𝑛)} and {(𝑥_2𝑛+1,𝑦_2𝑛+1)}, which shows that every bounded solution converges to period-two solution. In view of Lemma 13, there are no minimal period-two solutions and so every bounded solution of System (3) converges to the equilibrium 𝐸. In view of Theorems 7 and 8, there is a set 𝒞 with described properties. Consequently, every solution with an initial point (𝑥₀,𝑦₀)∈𝒲₊ converges to 𝐸, while every solution which starts in 𝒲₋ approaches (0,∞) and is asymptotic to the global unstable manifold 𝑊^𝑢(𝐸).

5. Global Dynamics of System (4) (See Figure 3)

Figure 3 

Global dynamics of System (4).

The equilibrium point 𝐸(𝑥,𝑦) of System (4) satisfies the following system of equations: 𝛾𝑥=¹𝑦𝐴₁+𝑥,𝛼𝑦=²+𝛽₂𝑥𝑦. (64) It is easy to see that System (64) has the unique equilibrium point 𝐸 in the first quadrant, which is an intersection of two parabolas: 𝑥𝐵𝑦=¹+𝑥𝛾₁,𝑦𝑥=²−𝛼₂𝛽₂. (65)(66) It is clear that at the point of intersection 𝐸, curve (65) is steeper than curve (66), that is, 𝑑𝑦𝑑𝑥₍₆₅₎(𝐸)>𝑑𝑦𝑑𝑥₍₆₆₎(𝐸), (67) which gives 2𝑦𝐴₁+2𝑥>𝛽₂𝛾₁. (68)

5.1. Linearized Stability Analysis of System (4)

In this section we prove the following result.

Theorem 16. The unique equilibrium 𝐸 of System (4) is a saddle point.

Proof. The map 𝑆 associated to System (4) is 𝑈=𝑓𝑔=𝛾𝑥,𝑦𝑥,𝑦𝑥,𝑦¹𝑦𝐴₁𝛼+𝑥₂+𝛽₂𝑥𝑦. (69) The Jacobian matrix of map (69) has the form: 𝐽_𝑈=−𝛾𝑥,𝑦¹𝑦𝐴₁+𝑥²𝛾₁𝐴₁𝛽+𝑥₂𝑦−𝛼₂+𝛽₂𝑥𝑦², (70) which evaluated at the equilibrium point 𝐸=(𝑥,𝑦) is 𝐽_𝑈𝑥,𝑦=−𝑥𝐴₁+𝑥𝛾₁𝐴₁+𝑥𝛽₂𝑦−1. (71) The characteristic equation of System (4) has the following form: 𝜆²+1+𝑥𝐴₁+𝑥𝜆+𝑥𝐴₁+𝑥−𝛽₂𝛾₁𝑦𝐴₁+𝑥=0. (72) Set 𝑃=1+𝑥𝐴₁+𝑥,𝑄=𝑥𝐴₁+𝑥−𝛽₂𝛾₁𝑦𝐴₁+𝑥. (73) The necessary and sufficient condition for (72) to have one root inside the unit circle and one root outside the unit circle is |𝑃|>|1+𝑄|, 𝑃²−4𝑄>0; see [7, 21]. In view of the fact that 𝑃>1+𝑄, the condition |𝑃|>|1+𝑄| leads to 𝑃>−1−𝑄 which is equivalent to 2+2𝑥𝐴₁+𝑥>𝛽₂𝛾₁𝑦𝐴₁+𝑥, (74) which is equivalent to the condition (68).
The condition 𝑃²−4𝑄>0 becomes 1+𝑥𝐴₁+𝑥²−4𝑥𝐴₁+𝑥−𝛽₂𝛾₁𝑦𝐴₁+𝑥>0, (75) which is equivalent to 1−𝑥𝐴₁+𝑥²𝛽+4²𝛾₁𝑦𝐴₁+𝑥>0, (76) which is clearly satisfied.

5.2. Global Results for System (4)

Lemma 17. System (4) has no minimal period-two solution.

Proof. The second iterate of the map 𝑈 is given as 𝑈²=1𝑥,𝑦𝑦𝛾₁𝑥+𝐴₁𝛼₂+𝑥𝛽₂𝐴²₁+𝑥𝐴₁+𝑦𝛾₁,𝑦𝑥+𝐴₁𝛼₂+𝑥𝛽₂𝑥𝛼₂+𝛼₂𝐴₁+𝑦𝛽₂𝛾₁. (77) Period-two solution satisfies 𝑈²(𝑥,𝑦)=(𝑥,𝑦) which reduces to the following two equations: 𝑥𝑦𝐴₁𝑥+𝑦=𝛾₁𝛼₂+𝛽₂𝑥,𝛾₁𝑦=𝑥𝐴₁+𝑥. (78)(79) Equation (79) is exactly the equilibrium equation (65). Using (65) in (78) we obtain after some elementary simplification that period-two solution satisfies (66). This shows that System (4) has no minimal period-two solution.

Lemma 18. The maps 𝑈 and 𝑈² associated with System (4) have the following properties.(i)If 𝐴₁𝛽₂≠𝛼₂, then the maps 𝑈 and 𝑈² are injective.(ii)If 𝐴₁𝛽₂≠𝛼₂, then det𝐽_𝑈²(𝑥,𝑦)>0 for all (𝑥,𝑦), 𝑦>0.Consequently, 𝑈² satisfies (𝑂+) condition, when 𝐴₁𝛽₂≠𝛼₂.

Proof. (i) We will prove that 𝑈 is injective which will imply the injectivity of 𝑈². The condition 𝑈𝑥₁,𝑦₁=𝑈𝑥₂,𝑦₂ (80) is reduced to the following two conditions: 𝐴₁𝑦₂−𝑦₁=𝑥₂𝑦₁−𝑥₁𝑦₂,𝛼₂𝑦₂−𝑦₁=𝛽₂𝑥₂𝑦₁−𝑥₁𝑦₂, (81) which implies that 𝑦₂−𝑦₁=(1/𝐴₁)(𝑥₂𝑦₁−𝑥₁𝑦₂)=(𝛽₂/𝛼₂)(𝑥₂𝑦₁−𝑥₁𝑦₂)1𝐴₁−𝛽₂𝛼₂𝑥₂𝑦₁−𝑥₁𝑦₂=0, (82) and 𝑥₂𝑦₁−𝑥₁𝑦₂=0, when 𝐴₁𝛽₂≠𝛼₂. This implies 𝑦₁=𝑦₂ and so 𝑥₁=𝑥₂.
(ii) A direct calculation shows that det𝐽_𝑈²=𝛼₂−𝐴₁𝛽₂²𝛾₁𝛼₂+𝛽₂𝑥𝐴²₁+𝐴₁𝑥+𝛾₁𝑦², (83) which proves our statement.
The statement on (𝑂+) condition follows from Theorem 5.

Theorem 19. Consider System (4). Assume that 𝐴₁𝛽₂≠𝛼₂. Then there exists a set 𝒞⊂𝒟=[0,∞)×(0,∞) which is invariant subset of the basin of attraction of 𝐸. The set 𝒞 is a graph of a strictly increasing continuous function of the first variable on an interval and separates 𝒟 into two connected and invariant components, namely, 𝒲₋∶=𝑥∈𝒟⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑥⪯_se𝑦,𝒲₊∶=𝑥∈𝒟⧵𝒞∶∃𝑦∈𝒞𝑤𝑖𝑡ℎ𝑦⪯_se𝑥, (84) which satisfy the following.
(i) If (𝑥₀,𝑦₀)∈𝒲₊, then lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=(∞,0),lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=(0,∞). (85)
(ii) If (𝑥₀,𝑦₀)∈𝒲₋, lim_𝑛→∞𝑥_2𝑛,𝑦_2𝑛=(0,∞),lim_𝑛→∞𝑥_2𝑛+1,𝑦_2𝑛+1=(∞,0). (86)

Proof. Assume that 𝐴₁𝛽₂≠𝛼₂. Then by Lemma 18 and Theorem 4, every solution {(𝑥_𝑛,𝑦_𝑛)} of System (4) has eventually monotone components {(𝑥_2𝑛,𝑦_2𝑛)} and {(𝑥_2𝑛+1,𝑦_2𝑛+1)}, which shows that every bounded solution converges to a period-two solution. In view of Lemma 17, there are no minimal period-two solutions and so every bounded solution of System (4) converges to the equilibrium 𝐸. In view of Theorems 7 and 8, there is a set 𝒞 with described properties. Consequently, the regions 𝒲₋ and 𝒲₊ are invariant for 𝑈² and every solution in 𝒲₋(𝒲₊) is asymptotic to the unstable manifold 𝒲^𝑢(𝐸) and so the statement of the theorem follows.

Theorem 20. Consider System (4). Assume that 𝐴₁𝛽₂=𝛼₂. Then System (4) can be decoupled and written as 𝑥_𝑛+1=𝛽₂𝛾²₁𝑥_𝑛𝐴₁+𝑥_𝑛,𝑦_𝑛+1=𝛽₁𝛽₂𝛾₁+𝐴₁𝑦_𝑛𝑦²_𝑛,𝑛=0,1,…. (87) Every solution {(𝑥_𝑛,𝑦_𝑛)} of System (87) has eventually monotone subsequences {(𝑥_2𝑛,𝑦_2𝑛)} and {(𝑥_2𝑛+1,𝑦_2𝑛+1)}. Every bounded solution converges to the unique positive equilibrium. Every unbounded solution {(𝑥_𝑛,𝑦_𝑛)} approaches either (∞,0) or (0,∞).

Proof. Using the condition 𝐴₁𝛽₂=𝛼₂ in the second equation of System (4) gives 𝑦_𝑛+1=𝛽₂𝐴₁+𝑥_𝑛𝑦_𝑛, (88) and so 𝑥_𝑛+1𝑦_𝑛+1=𝛽₂𝛾₁ which shows that System (4) has an invariant of the form: 𝑥_𝑛𝑦_𝑛=𝛽₂𝛾₁,𝑛=1,2,…. (89) Using (89), System (4) is reduced to System (87). Both equations of System (87) are first order difference equations with decreasing functions and so by Theorem  1.19 [7], the subsequences of even and odd indexes are eventually monotonic and so every bounded solution converges to a period-two solution. An immediate checking shows that neither one of the two equations of System (87) has period-two solutions. For example, the unique equilibrium 𝑥 of the first equation of System (87) satisfies 𝑥³+𝐴₁𝑥²−𝛽₂𝛾²₁=0, (90) while period-two solution satisfies 𝑓²(𝑥)=𝑥 which becomes 𝑥²((𝐴+𝑥)²/(𝐴²𝑥+𝐴𝑥²+𝐶))=𝑥 and so is reduced to (90). Thus every bounded solution converges to the unique equilibrium.
The result for unbounded solutions follows immediately from (89).

Acknowledgment

The authors are grateful to one of the anonymous referees for pointing out the equation of the stable manifold for (2) and for observing the alternative method of the proof of Theorem 1 mentioned in Remark 11.