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

Authors: M. DiPippo; M. R. S. Kulenović
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## 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 formsxn+1=(α1+β1xn+γ1yn)/(A1+B1xn+C1yn),yn+1=(α2+β2xn+γ2yn)/(A2+B2xn+C2yn),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.


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## Body

## 1. Introduction

A first order system of difference equations:(1)xn+1=f(xn,yn),yn+1=g(xn,yn),n=0,1,…,(x0,y0)∈ℛ,
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:(2)xn+1=γ1ynB1xn+yn,yn+1=β2xnB2xn+yn,n=0,1,…,
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
(3)xn+1=γ1ynB1xn+yn,yn+1=α2+β2xnyn,n=0,1,…,(4)xn+1=γ1ynA1+xn,yn+1=α2+β2xnyn,n=0,1,…,
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 thatB1<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 thatB1>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
(5)𝒲-:={x∈ℛ∖𝒞:∃y∈𝒞withx⪯sey}={(x,y):y>y-x-x≥0},𝒲+:={x∈ℛ∖𝒞:∃y∈𝒞withy⪯sex}={(x,y):0≤y<y-x-x},
the following holds.
(i) If(x0,y0)∈𝒲+, then
(6)limn→∞(x2n,y2n)=P1,limn→∞(x2n+1,y2n+1)=P2.
(ii) If(x0,y0)∈𝒲-,
(7)limn→∞(x2n,y2n)=P2,limn→∞(x2n+1,y2n+1)=P1.
(c) Assume thatB1=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 mapT=(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 Tn(x)→E as n→∞ for x∈𝒰; the basin of attraction is the set of all x∈ℛ such that Tn(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 JT(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.
LetT=(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 𝒲locs is the set
(8)𝒲locs((x-,y-))={limn→∞(x,y):Tn(x,y)∈U(∀n≥0)∧limn→∞Tn(x,y)=(x-,y-)}.
The global stable manifold Ws of a saddle point (x-,y-) is the set
(9)𝒲s((x-,y-))={(x,y):limn→∞Tn(x,y)=(x-,y-)}.The mapT 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 pointsv,w∈ℝ+2 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:Intℝ+2→Intℝ+2 is strongly monotone if v≺w implies that f(v)≺≺f(w) for all v,w∈Intℝ+2. 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 ℝ+2 may be used to show that T is monotone, as in [17].Forx=(x1,x2)∈ℝ2, define Ql(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.
Let𝒮 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.
Let𝒮 be a nonempty subset of ℝ2. If T is a competitive map for which (O+) holds, then for all x∈𝒮, {Tn(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, {T2n} 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.
Letℛ⊂ℝ2 be the cartesian product of two intervals in ℝ. Let T:ℛ→ℛ be a C1 competitive map. If T is injective and detJT(x)>0 for all x∈ℛ, then T satisfies (O+). If T is injective and detJT(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.
LetT 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 mapT has a C1 extension to a neighborhood of x¯.(b)
The Jacobian matrix ofT at x¯ has real eigenvalues λ, μ such that 0<|λ|<μ, where |λ|<1, and the eigenspace Eλ associated with λ is not a coordinate axis.
Then there exists a curve𝒞⊂ℛ through x¯ that is invariant and a subset of the basin of attraction of x¯, such that 𝒞 is tangential to the eigenspace Eλ 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.
Letℐ1,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 pointx¯ is the only fixed point of T in (Q1(x¯)∪Q3(x¯))∩int(ℛ).(3)
The mapT 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)
detJT(x¯)>0 and T(x)=x¯ only for x=x¯.
Then the following statements are true.(i)
The stable manifold𝒲s(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 𝒲s(x¯) into two invariant connected regions 𝒲+ (“below the stable set”), and 𝒲- (“above the stable set”), where
(11)𝒲+:={x∈ℛ∖𝒲s(x¯):∃x′∈𝒲s(x¯)withx⪯sex′},𝒲-:={x∈ℛ∖𝒲s(x¯):∃x′∈𝒲s(x¯)withx′⪯sex}.(ii)
The unstable manifold𝒲u(x¯) is connected and it is the graph of a continuous decreasing curve.(iii)
For everyx∈𝒲+, Tn(x) eventually enters the interior of the invariant set Q4(x¯)∩ℛ, and for every x∈𝒲-, Tn(x) eventually enters the interior of the invariant set Q2(x¯)∩ℛ.(iv)
Letm∈Q2(x¯) and M∈Q4(x¯) be the endpoints of 𝒲u(x¯), where m⪯sex¯⪯seM. For every x∈𝒲- and every z∈ℛ such that m⪯sez, there exists m∈ℕ such that Tm(x)⪯sez, and for every x∈𝒲+ and every z∈ℛ such that z⪯seM, there exists m∈ℕ such that M⪯seTm(x).


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

Figure 1
Global dynamics of System (2).The equilibrium pointE(x-,y-) of System (2) satisfies the following system of equations:
(12)x¯=γ1y¯B1x¯+y-,y¯=β2x¯B2x¯+y-.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:
(13)y=B1x2γ1-x,(14)x=y2β2-B2y.
It is clear that at the point of intersection E curve (13) is steeper than curve (14), that is,
(15)dydx|(13)(E)>dydx|(14)(E),
which gives
(16)B1x-(2γ1-x-)(γ1-x-)2>(β2-B2y-)2y-(2β2-B2y-).
This inequality is equivalent to the following inequality:
(17)(β2+y-2x-)(γ1+B1x-2y-)>B1x-y-,
which is always satisfied.


### 3.1. Linearized Stability Analysis of System (2)

In this section we present the linearized stability analysis of the equilibriumE of System (2).Theorem 9.
(i) IfB1<B2, then E is locally asymptotically stable.
(ii) IfB1=B2, then E is a nonhyperbolic equilibrium point.
(iii) IfB1>B2, then E is a saddle point.Proof.
The mapT associated to System (2) is
(18)T(x,y)=(f(x,y)g(x,y))=(γ1yB1x+yβ2xB2x+y).
The Jacobian matrix of map (18) is
(19)JT(x,y)=(-B1γ1y(B1x+y)2B1γ1x(B1x+y)2β2y(B2x+y)2-β2x(B2x+y)2),
and evaluated at the equilibrium point E=(x-,y-) is
(20)JT(x¯,y¯)=(-B1γ1x-2y-B1γ1x-3y-2y-3β2x-2-y-2β2x-).
The characteristic equation has the following form:
(21)λ2+(B1γ1x-2y-+1β2y-2x-)λ=0,
and the characteristic roots are
(22)λ1=0,λ2=-B1γ1x-2y--1β2y-2x-.
A straightforward calculation shows that the conditions λ2∈(-1,0), λ2=-1, and λ2∈(-∞,-1), respectively, are equivalent to the conditions
(23)(x-y-)2<β2γ1,(x-y-)2=β2γ1,(x-y-)2>β2γ1.
On the other hand, by dividing two equilibrium equations (12) we obtain
(24)(x-y-)2=β2γ1B1x-+y-B2x-+y-,
which implies that the condition (23) is equivalent to the conditions
(25)B1<B2,B1=B2,B1>B2,
which completes the proof of the theorem.


### 3.2. Global Results for System (2)

In this section we present the proof of Theorem1 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
(26){P1,P2}={(γ1,0),(0,β2B2)}
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
(27)ℬ={(x,y)∣x>0,y=0orx=0,y>0}
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 mapT is given as(28)T2(x,y)=[xβ2γ1y+xB1xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2yβ2γ1y+xB2β2B1x2+γ1xyB22+β2xy+γ1y2B2].A period-two solution of System (2) satisfies T2(x,y)=(x,y), which immediately leads to the following equations:
(29)(B2-B1)(β2γ1x-B2γ1xy-γ1y2)=0,(30)(B2-B1)(β2γ1y-B2xy-γ1B1x2)=0,
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 thatT(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
(31)JT2(x,y)=[yβ2γ12B1B1B2x2+2B1xy+y2(xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2)2-xβ2γ12B1B1B2x2+2B1xy+y2(xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2)2-yβ22γ1B1B2x2+2B1xy+y2(β2B1x2+γ1xyB22+β2xy+γ1y2B2)2xβ22γ1B1B2x2+2B1xy+y2(β2B1x2+γ1xyB22+β2xy+γ1y2B2)2.]
The Jacobian matrix of T2 evaluated at P1 is
(32)JT2(P1)=[0-1β2γ1B201B1B2]
and the Jacobian matrix of T2 evaluated at P2 is
(33)JT2(P2)=[1B1B20B2-1β2γ1(β22B2+β2γ1B2)0].
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 Theorem1.
First, observe that the rectangleℛ=[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, (xn,yn)∈ℛ for n≥1. The map T2 is a competitive map on ℛ.Case a.
Assume thatB1<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 (xl,yl)∈int⟦P2,E⟧ and (xu,yu)∈int⟦E,P1⟧, such that (xl,yl)⪯se(x,y)⪯se(xu,yu). Consequently, since T2 is competitive T2n((xl,yl))⪯seT2n((x,y))⪯seT2n((xu,yu)) for n≥1 and so limn→∞T2n((x,y))=E, which by continuity of T implies that
(34)limn→∞T2n+1((x,y))=limn→∞T(T2n((x,y)))=T(limn→∞T2n((x,y)))=T(E)=E,
and so limn→∞Tn((x,y))=E.Case b.
Assume thatB1>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 thatB1=B2. Then by dividing two equations of System (2) we obtain that the solution of (2) satisfies
(35)yn+1xn+1=γ2β1xnyn=γ2β11yn/xn.
This means that yn/xn satisfies first order difference equation un+1=D/un, where D=γ2/β1. All nonconstant solutions of un+1=D/un are period-two solutions {u0,D/u0}. Thus yn={u0xn,(D/u0)xn}. In this case, System (2) becomes
(36)xn+1=B1u0B1+u0,yn+1=β2B1+u0,n=1,2,…,xn+1=B1DD+B1u0,yn+1=β2u0D+B1u0,n=1,2,…,
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)=tkf(u,v), g(tu,tv)=tkg(u,v) for all u, v in intersection of domains of f and g and all t≠0. In that case, the ratio zn=yn/xn of every solution of (1) satisfies the first order difference equation:
(37)zn+1=f(1,zn)g(1,zn)=F(zn),n=0,1,…
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:
(38)xn+1=α1+β1xn+γ1ynA1+B1xn+C1yn,yn+1=α2+β2xn+γ2ynA2+B2xn+C2yn,n=0,1,…,
where all parameters and the initial conditions (x0,y0) are arbitrary nonnegative numbers such that Ai+Bix0+Ciy0>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 zn=yn/xn is
(39)zn+1=β2(B1+zn)γ1(B2+zn)zn=F(zn),n=0,1,….
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.


## 3.1. Linearized Stability Analysis of System (2)

In this section we present the linearized stability analysis of the equilibriumE of System (2).Theorem 9.
(i) IfB1<B2, then E is locally asymptotically stable.
(ii) IfB1=B2, then E is a nonhyperbolic equilibrium point.
(iii) IfB1>B2, then E is a saddle point.Proof.
The mapT associated to System (2) is
(18)T(x,y)=(f(x,y)g(x,y))=(γ1yB1x+yβ2xB2x+y).
The Jacobian matrix of map (18) is
(19)JT(x,y)=(-B1γ1y(B1x+y)2B1γ1x(B1x+y)2β2y(B2x+y)2-β2x(B2x+y)2),
and evaluated at the equilibrium point E=(x-,y-) is
(20)JT(x¯,y¯)=(-B1γ1x-2y-B1γ1x-3y-2y-3β2x-2-y-2β2x-).
The characteristic equation has the following form:
(21)λ2+(B1γ1x-2y-+1β2y-2x-)λ=0,
and the characteristic roots are
(22)λ1=0,λ2=-B1γ1x-2y--1β2y-2x-.
A straightforward calculation shows that the conditions λ2∈(-1,0), λ2=-1, and λ2∈(-∞,-1), respectively, are equivalent to the conditions
(23)(x-y-)2<β2γ1,(x-y-)2=β2γ1,(x-y-)2>β2γ1.
On the other hand, by dividing two equilibrium equations (12) we obtain
(24)(x-y-)2=β2γ1B1x-+y-B2x-+y-,
which implies that the condition (23) is equivalent to the conditions
(25)B1<B2,B1=B2,B1>B2,
which completes the proof of the theorem.


## 3.2. Global Results for System (2)

In this section we present the proof of Theorem1 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
(26){P1,P2}={(γ1,0),(0,β2B2)}
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
(27)ℬ={(x,y)∣x>0,y=0orx=0,y>0}
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 mapT is given as(28)T2(x,y)=[xβ2γ1y+xB1xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2yβ2γ1y+xB2β2B1x2+γ1xyB22+β2xy+γ1y2B2].A period-two solution of System (2) satisfies T2(x,y)=(x,y), which immediately leads to the following equations:
(29)(B2-B1)(β2γ1x-B2γ1xy-γ1y2)=0,(30)(B2-B1)(β2γ1y-B2xy-γ1B1x2)=0,
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 thatT(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
(31)JT2(x,y)=[yβ2γ12B1B1B2x2+2B1xy+y2(xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2)2-xβ2γ12B1B1B2x2+2B1xy+y2(xyβ2+x2β2B1+y2γ1B1+xyγ1B1B2)2-yβ22γ1B1B2x2+2B1xy+y2(β2B1x2+γ1xyB22+β2xy+γ1y2B2)2xβ22γ1B1B2x2+2B1xy+y2(β2B1x2+γ1xyB22+β2xy+γ1y2B2)2.]
The Jacobian matrix of T2 evaluated at P1 is
(32)JT2(P1)=[0-1β2γ1B201B1B2]
and the Jacobian matrix of T2 evaluated at P2 is
(33)JT2(P2)=[1B1B20B2-1β2γ1(β22B2+β2γ1B2)0].
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 Theorem1.
First, observe that the rectangleℛ=[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, (xn,yn)∈ℛ for n≥1. The map T2 is a competitive map on ℛ.Case a.
Assume thatB1<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 (xl,yl)∈int⟦P2,E⟧ and (xu,yu)∈int⟦E,P1⟧, such that (xl,yl)⪯se(x,y)⪯se(xu,yu). Consequently, since T2 is competitive T2n((xl,yl))⪯seT2n((x,y))⪯seT2n((xu,yu)) for n≥1 and so limn→∞T2n((x,y))=E, which by continuity of T implies that
(34)limn→∞T2n+1((x,y))=limn→∞T(T2n((x,y)))=T(limn→∞T2n((x,y)))=T(E)=E,
and so limn→∞Tn((x,y))=E.Case b.
Assume thatB1>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 thatB1=B2. Then by dividing two equations of System (2) we obtain that the solution of (2) satisfies
(35)yn+1xn+1=γ2β1xnyn=γ2β11yn/xn.
This means that yn/xn satisfies first order difference equation un+1=D/un, where D=γ2/β1. All nonconstant solutions of un+1=D/un are period-two solutions {u0,D/u0}. Thus yn={u0xn,(D/u0)xn}. In this case, System (2) becomes
(36)xn+1=B1u0B1+u0,yn+1=β2B1+u0,n=1,2,…,xn+1=B1DD+B1u0,yn+1=β2u0D+B1u0,n=1,2,…,
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)=tkf(u,v), g(tu,tv)=tkg(u,v) for all u, v in intersection of domains of f and g and all t≠0. In that case, the ratio zn=yn/xn of every solution of (1) satisfies the first order difference equation:
(37)zn+1=f(1,zn)g(1,zn)=F(zn),n=0,1,…
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:
(38)xn+1=α1+β1xn+γ1ynA1+B1xn+C1yn,yn+1=α2+β2xn+γ2ynA2+B2xn+C2yn,n=0,1,…,
where all parameters and the initial conditions (x0,y0) are arbitrary nonnegative numbers such that Ai+Bix0+Ciy0>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 zn=yn/xn is
(39)zn+1=β2(B1+zn)γ1(B2+zn)zn=F(zn),n=0,1,….
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 pointE(x-,y-) of System (3) satisfies the following system of equations:
(40)x¯=γ1y¯B1x¯+y-,y¯=α2+β2x¯y-.
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:
(41)y=B1x2γ1-x,(42)x=y2-α2β2.
It is clear that at the point of intersection E curve (41) is steeper than curve (42), that is,
(43)dydx|(41)(E)>dydx|(42)(E),
which gives
(44)B1x-(2γ1-x-)(γ1-x-)2>β22y-.
This inequality is equivalent to the following inequality:
(45)2β1x-y-(2γ1-x-)>β2(γ1-x-)2,
which in turn is equivalent to
(46)2+2B1γ1x-2y->β2B1γ1x-3y-3.


### 4.1. Linearized Stability Analysis of System (3)

In this section we prove the following result.Theorem 12.
The unique equilibriumE of System (3) is a saddle point.Proof.
The mapS associated to System (3) is
(47)S(x,y)=(f(x,y)g(x,y))=(γ1yB1x+yα2+β2xy).
The Jacobian matrix of map (47) is
(48)JS(x,y)=(-B1γ1y(B1x+y)2B1γ1x(B1x+y)2β2y-α2+β2xy2),
and evaluated at the equilibrium point E=(x-,y-) is
(49)JS(x¯,y¯)=(-B1γ1x-2y-B1γ1x-3y-2β2y--1).
The characteristic equation has the following form:(50)λ2+(1+B1γ1x-2y-)λ+B1γ1x-2y--β2B1γ1x-3y-2=0.
Set
(51)P=1+B1γ1x-2y-,Q=B1γ1x-2y--β2B1γ1x-3y-2.
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
(52)2+2B1γ1x-2y->β2B1γ1x-3y-3,
which is condition (46).
The conditionP2-4Q>0 becomes
(53)(1+B1γ1x-2y-)2-4(B1γ1x-2y--β2B1γ1x-3y-3)>0,
which is equivalent to
(54)(1-B1γ1x-2y-)2+4β2B1γ1x-3y-3>0,
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 mapS is given as
(55)S2(x,y)=(y(y+xB1)(α2+xβ2)γ1(y+xB1)α2+xβ2β2B1x2+β2xy+α2B1x+γ1B1y2+α2y,y(y+xB1)(α2+xβ2)(yα2+yβ2γ1+xα2B1)).
Period-two solution satisfies S2(x,y)=(x,y) which reduces to the following two equations:
(56)γ1(y+xB1)(α2+xβ2)-x(β2B1x2+β2xy+α2B1x+γ1B1y2+α2y)=0,(57)yα2+yβ2γ1+xα2B1-(y+xB1)(α2+xβ2)=0.
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 mapsS and S2 associated with System (3) have the following properties. (i)
The mapsS and S2 are injective.(ii)
detJS2(x,y)>0 for all (x,y), y>0.Consequently, S2 satisfies (O+) condition and so {S2n(x0,y0)} is eventually componentwise monotone.Proof.
(i) We will prove thatS is injective and the injectivity of S2 will follow immediately. The condition
(58)S(x1,y1)=S(x2,y2)
is reduced to the following two conditions:
(59)x2y1=x1y2,α2(y2-y1)=β2(x2y1-x1y2),
which immediately implies y1=y2 and so x1=x2.
(ii) A direct calculation shows that(60)detJS2=B12α22γ12y2(α2+xβ2)(yα2+xyβ2+xα2B1+x2β2B1+y2γ1B1)2,
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,
(61)𝒲-:={x∈𝒟∖𝒞:∃y∈𝒞withx⪯sey},𝒲+:={x∈𝒟∖𝒞:∃y∈𝒞withy⪯sex},
which satisfy the following.
(i) If(x0,y0)∈𝒲+, then
(62)limn→∞(x2n,y2n)=E,limn→∞(x2n+1,y2n+1)=(0,∞).
(ii) If(x0,y0)∈𝒲-,
(63)limn→∞(x2n,y2n)=(0,∞),limn→∞(x2n+1,y2n+1)=E.Proof.
Clearly, the rectangle[0,γ1]×(0,∞) is an invariant and attracting set for the map S. In particular, xn≤γ for n≥1.
Then by Lemma14 and Theorem 4, every solution {(xn,yn)} of System (3) has eventually monotone components {(x2n,y2n)} and {(x2n+1,y2n+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 Wu(E).


## 4.1. Linearized Stability Analysis of System (3)

In this section we prove the following result.Theorem 12.
The unique equilibriumE of System (3) is a saddle point.Proof.
The mapS associated to System (3) is
(47)S(x,y)=(f(x,y)g(x,y))=(γ1yB1x+yα2+β2xy).
The Jacobian matrix of map (47) is
(48)JS(x,y)=(-B1γ1y(B1x+y)2B1γ1x(B1x+y)2β2y-α2+β2xy2),
and evaluated at the equilibrium point E=(x-,y-) is
(49)JS(x¯,y¯)=(-B1γ1x-2y-B1γ1x-3y-2β2y--1).
The characteristic equation has the following form:(50)λ2+(1+B1γ1x-2y-)λ+B1γ1x-2y--β2B1γ1x-3y-2=0.
Set
(51)P=1+B1γ1x-2y-,Q=B1γ1x-2y--β2B1γ1x-3y-2.
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
(52)2+2B1γ1x-2y->β2B1γ1x-3y-3,
which is condition (46).
The conditionP2-4Q>0 becomes
(53)(1+B1γ1x-2y-)2-4(B1γ1x-2y--β2B1γ1x-3y-3)>0,
which is equivalent to
(54)(1-B1γ1x-2y-)2+4β2B1γ1x-3y-3>0,
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 mapS is given as
(55)S2(x,y)=(y(y+xB1)(α2+xβ2)γ1(y+xB1)α2+xβ2β2B1x2+β2xy+α2B1x+γ1B1y2+α2y,y(y+xB1)(α2+xβ2)(yα2+yβ2γ1+xα2B1)).
Period-two solution satisfies S2(x,y)=(x,y) which reduces to the following two equations:
(56)γ1(y+xB1)(α2+xβ2)-x(β2B1x2+β2xy+α2B1x+γ1B1y2+α2y)=0,(57)yα2+yβ2γ1+xα2B1-(y+xB1)(α2+xβ2)=0.
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 mapsS and S2 associated with System (3) have the following properties. (i)
The mapsS and S2 are injective.(ii)
detJS2(x,y)>0 for all (x,y), y>0.Consequently, S2 satisfies (O+) condition and so {S2n(x0,y0)} is eventually componentwise monotone.Proof.
(i) We will prove thatS is injective and the injectivity of S2 will follow immediately. The condition
(58)S(x1,y1)=S(x2,y2)
is reduced to the following two conditions:
(59)x2y1=x1y2,α2(y2-y1)=β2(x2y1-x1y2),
which immediately implies y1=y2 and so x1=x2.
(ii) A direct calculation shows that(60)detJS2=B12α22γ12y2(α2+xβ2)(yα2+xyβ2+xα2B1+x2β2B1+y2γ1B1)2,
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,
(61)𝒲-:={x∈𝒟∖𝒞:∃y∈𝒞withx⪯sey},𝒲+:={x∈𝒟∖𝒞:∃y∈𝒞withy⪯sex},
which satisfy the following.
(i) If(x0,y0)∈𝒲+, then
(62)limn→∞(x2n,y2n)=E,limn→∞(x2n+1,y2n+1)=(0,∞).
(ii) If(x0,y0)∈𝒲-,
(63)limn→∞(x2n,y2n)=(0,∞),limn→∞(x2n+1,y2n+1)=E.Proof.
Clearly, the rectangle[0,γ1]×(0,∞) is an invariant and attracting set for the map S. In particular, xn≤γ for n≥1.
Then by Lemma14 and Theorem 4, every solution {(xn,yn)} of System (3) has eventually monotone components {(x2n,y2n)} and {(x2n+1,y2n+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 Wu(E).


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

Figure 3
Global dynamics of System (4).The equilibrium pointE(x-,y-) of System (4) satisfies the following system of equations:
(64)x¯=γ1y¯A1+x¯,y¯=α2+β2x¯y-.
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:
(65)y=x(B1+x)γ1,(66)x=y2-α2β2.
It is clear that at the point of intersection E, curve (65) is steeper than curve (66), that is,
(67)dydx|(65)(E)>dydx|(66)(E),
which gives
(68)2y-(A1+2x-)>β2γ1.


### 5.1. Linearized Stability Analysis of System (4)

In this section we prove the following result.Theorem 16.
The unique equilibriumE of System (4) is a saddle point.Proof.
The mapS associated to System (4) is
(69)U(x,y)=(f(x,y)g(x,y))=(γ1yA1+xα2+β2xy).
The Jacobian matrix of map (69) has the form:
(70)JU(x,y)=(-γ1y(A1+x)2γ1A1+xβ2y-α2+β2xy2),
which evaluated at the equilibrium point E=(x-,y-) is
(71)JU(x¯,y¯)=(-x-A1+x-γ1A1+x-β2y--1).
The characteristic equation of System (4) has the following form:
(72)λ2+(1+x-A1+x-)λ+x-A1+x--β2γ1y-(A1+x-)=0.
Set
(73)P=1+x-A1+x-,Q=x-A1+x--β2γ1y-(A1+x-).
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
(74)2+2x-A1+x->β2γ1y-(A1+x-),
which is equivalent to the condition (68).
The conditionP2-4Q>0 becomes
(75)(1+x-A1+x-)2-4(x-A1+x--β2γ1y-(A1+x-))>0,
which is equivalent to
(76)(1-x-A1+x-)2+4β2γ1y-(A1+x-)>0,
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 mapU is given as
(77)U2(x,y)=(1yγ1(x+A1)α2+xβ2A12+xA1+yγ1,y(x+A1)(α2+xβ2)(xα2+α2A1+yβ2γ1)).
Period-two solution satisfies U2(x,y)=(x,y) which reduces to the following two equations:
(78)xy(A1x+y)=γ1(α2+β2x),(79)γ1y=x(A1+x).
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 mapsU and U2 associated with System (4) have the following properties.(i)
IfA1β2≠α2, then the maps U and U2 are injective.(ii)
IfA1β2≠α2, then detJU2(x,y)>0 for all (x,y), y>0.Consequently, U2 satisfies (O+) condition, when A1β2≠α2.Proof.
(i) We will prove thatU is injective which will imply the injectivity of U2. The condition
(80)U(x1,y1)=U(x2,y2)
is reduced to the following two conditions:
(81)A1(y2-y1)=x2y1-x1y2,α2(y2-y1)=β2(x2y1-x1y2),
which implies that y2-y1=(1/A1)(x2y1-x1y2)=(β2/α2)(x2y1-x1y2)(82)(1A1-β2α2)(x2y1-x1y2)=0,
and x2y1-x1y2=0, when A1β2≠α2. This implies y1=y2 and so x1=x2.
(ii) A direct calculation shows that(83)detJU2=(α2-A1β2)2γ1(α2+β2x)(A12+A1x+γ1y)2,
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,
(84)𝒲-:={x∈𝒟∖𝒞:∃y∈𝒞withx⪯sey},𝒲+:={x∈𝒟∖𝒞:∃y∈𝒞withy⪯sex},
which satisfy the following.
(i) If(x0,y0)∈𝒲+, then
(85)limn→∞(x2n,y2n)=(∞,0),limn→∞(x2n+1,y2n+1)=(0,∞).
(ii) If(x0,y0)∈𝒲-,
(86)limn→∞(x2n,y2n)=(0,∞),limn→∞(x2n+1,y2n+1)=(∞,0).Proof.
Assume thatA1β2≠α2. Then by Lemma 18 and Theorem 4, every solution {(xn,yn)} of System (4) has eventually monotone components {(x2n,y2n)} and {(x2n+1,y2n+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 𝒲u(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
(87)xn+1=β2γ12xn(A1+xn),yn+1=β1β2γ1+A1ynyn2,n=0,1,….
Every solution {(xn,yn)} of System (87) has eventually monotone subsequences {(x2n,y2n)} and {(x2n+1,y2n+1)}. Every bounded solution converges to the unique positive equilibrium. Every unbounded solution {(xn,yn)} approaches either (∞,0) or (0,∞).Proof.
Using the conditionA1β2=α2 in the second equation of System (4) gives
(88)yn+1=β2A1+xnyn,
and so xn+1yn+1=β2γ1 which shows that System (4) has an invariant of the form:
(89)xnyn=β2γ1,n=1,2,….
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
(90)x3+A1x2-β2γ12=0,
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).


## 5.1. Linearized Stability Analysis of System (4)

In this section we prove the following result.Theorem 16.
The unique equilibriumE of System (4) is a saddle point.Proof.
The mapS associated to System (4) is
(69)U(x,y)=(f(x,y)g(x,y))=(γ1yA1+xα2+β2xy).
The Jacobian matrix of map (69) has the form:
(70)JU(x,y)=(-γ1y(A1+x)2γ1A1+xβ2y-α2+β2xy2),
which evaluated at the equilibrium point E=(x-,y-) is
(71)JU(x¯,y¯)=(-x-A1+x-γ1A1+x-β2y--1).
The characteristic equation of System (4) has the following form:
(72)λ2+(1+x-A1+x-)λ+x-A1+x--β2γ1y-(A1+x-)=0.
Set
(73)P=1+x-A1+x-,Q=x-A1+x--β2γ1y-(A1+x-).
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
(74)2+2x-A1+x->β2γ1y-(A1+x-),
which is equivalent to the condition (68).
The conditionP2-4Q>0 becomes
(75)(1+x-A1+x-)2-4(x-A1+x--β2γ1y-(A1+x-))>0,
which is equivalent to
(76)(1-x-A1+x-)2+4β2γ1y-(A1+x-)>0,
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 mapU is given as
(77)U2(x,y)=(1yγ1(x+A1)α2+xβ2A12+xA1+yγ1,y(x+A1)(α2+xβ2)(xα2+α2A1+yβ2γ1)).
Period-two solution satisfies U2(x,y)=(x,y) which reduces to the following two equations:
(78)xy(A1x+y)=γ1(α2+β2x),(79)γ1y=x(A1+x).
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 mapsU and U2 associated with System (4) have the following properties.(i)
IfA1β2≠α2, then the maps U and U2 are injective.(ii)
IfA1β2≠α2, then detJU2(x,y)>0 for all (x,y), y>0.Consequently, U2 satisfies (O+) condition, when A1β2≠α2.Proof.
(i) We will prove thatU is injective which will imply the injectivity of U2. The condition
(80)U(x1,y1)=U(x2,y2)
is reduced to the following two conditions:
(81)A1(y2-y1)=x2y1-x1y2,α2(y2-y1)=β2(x2y1-x1y2),
which implies that y2-y1=(1/A1)(x2y1-x1y2)=(β2/α2)(x2y1-x1y2)(82)(1A1-β2α2)(x2y1-x1y2)=0,
and x2y1-x1y2=0, when A1β2≠α2. This implies y1=y2 and so x1=x2.
(ii) A direct calculation shows that(83)detJU2=(α2-A1β2)2γ1(α2+β2x)(A12+A1x+γ1y)2,
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,
(84)𝒲-:={x∈𝒟∖𝒞:∃y∈𝒞withx⪯sey},𝒲+:={x∈𝒟∖𝒞:∃y∈𝒞withy⪯sex},
which satisfy the following.
(i) If(x0,y0)∈𝒲+, then
(85)limn→∞(x2n,y2n)=(∞,0),limn→∞(x2n+1,y2n+1)=(0,∞).
(ii) If(x0,y0)∈𝒲-,
(86)limn→∞(x2n,y2n)=(0,∞),limn→∞(x2n+1,y2n+1)=(∞,0).Proof.
Assume thatA1β2≠α2. Then by Lemma 18 and Theorem 4, every solution {(xn,yn)} of System (4) has eventually monotone components {(x2n,y2n)} and {(x2n+1,y2n+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 𝒲u(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
(87)xn+1=β2γ12xn(A1+xn),yn+1=β1β2γ1+A1ynyn2,n=0,1,….
Every solution {(xn,yn)} of System (87) has eventually monotone subsequences {(x2n,y2n)} and {(x2n+1,y2n+1)}. Every bounded solution converges to the unique positive equilibrium. Every unbounded solution {(xn,yn)} approaches either (∞,0) or (0,∞).Proof.
Using the conditionA1β2=α2 in the second equation of System (4) gives
(88)yn+1=β2A1+xnyn,
and so xn+1yn+1=β2γ1 which shows that System (4) has an invariant of the form:
(89)xnyn=β2γ1,n=1,2,….
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
(90)x3+A1x2-β2γ12=0,
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).



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