# Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane

Authors: S. Kalabušić; M. R. S. Kulenović; E. Pilav
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We investigate global dynamics of the following systems of difference equationsxn+1=β1xn/(B1xn+yn),
yn+1=(α2+γ2yn)/(A2+xn),
n=0,1,2,…, where the parameters β1,
B1,
β2,
α2,
γ2,
A2 are positive numbers, and initial conditions x0 and y0 are arbitrary nonnegative numbers such that x0+y0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.


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## Body

## 1. Introduction

In this paper we consider the following rational system of difference equations(1.1)xn+1=β1xnB1xn+yn,yn+1=α2+γ2ynA2+xn,n=0,1,2,…,
where the parameters β1, B1, β2, α2, γ2, A2 are positive numbers, and initial conditions x0 and y0 are nonnegative numbers such that x0+y0>0. System (1.1) was mentioned in [1] as one of three systems of open problem 3 which asked for the description of global dynamics of some rational systems of difference equations. In notation used to labels systems of linear fractional difference equations used in [1] system (1.1) is known as (3.19) and (4.1). In this paper, we provide the precise description of global dynamics of the system (1.1). We show that the system (1.1) may have between zero and three equilibrium points, which may have different local character. If the system (1.1) has one equilibrium point, then this point is either locally asymptotically stable or saddle point or nonhyperbolic equilibrium point. If the system (1.1) has two equilibrium points, then they are either locally asymptotically stable, and nonhyperbolic, or locally asymptotically stable and saddle point. If the system (1.1) has three equilibrium points then two of the equilibrium points are locally asymptotically stable and the third point, which is between these two points in South-East ordering defined below, is a saddle point. The major problem for global dynamics of the system (1.1) is determining the basins of attraction of different equilibrium points. The difficulty in analyzing the behavior of all solutions of the system (1.1) lies in the fact that there are many regions of parameters where this system possesses different equilibrium points with different local character and that in several cases the equilibrium point is nonhyperbolic. However, all these cases can be handled by using recent results in [2]. The dual of this system is the system where xn and yn replace their role, and it was labeled as system (4.1) and (3.19) in [1]. Dynamics of this system immediately follows from the results proven here, by simply replacing the roles of xn and yn.System (1.1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [2, 3]. System (1.1) has a potential to be used as a mathematical model for competition. In fact, the first equation of (1.1) is of Leslie-Gower type, and the second equation can be considered to be of Leslie-Gower type with stocking (or immigration) represented with the term α2, see [4–7]. Here β1, γ2 are the inherent birth rates while B1 and A2 are related to the density-dependent effects on newborn recruitment. Finally, α2 affects stocking for species with state variable yn.In Section2, we present some general results about competitive systems in the plane. In Section 3 contains some basic facts such as the nonexistence of period-two solution of system (1.1). In Section 4 analyzes local stability which is fairly complicated for this system. Finally, in Section 5 gives global dynamics for all values of parameters. This section finishes with an introduction of a new terminology for different type scenarios for competitive systems that can be used to give a simple classification of all possible global behavior for system (1.1). The interesting feature of this paper is that there are five regions of the parameters in which one of the equilibrium points is nonhyperbolic, and yet we are able to describe the global dynamics in all five cases. To achieve this goal, we use new method of proving stability of nonhyperbolic equilibrium points introduced in [2].


## 2. Preliminaries

Consider a first-order system of difference equations of the form(2.1)xn+1=f(xn,yn),yn+1=g(xn,yn),n=0,1,2,…,(x-1,x0)∈I×I,
where f,g:ℐ×ℐ→ℐ are continuous functions on an interval ℐ⊂ℝ, f(x,y) is nondecreasing in x and non-increasing in y, and g(x,y) is non-increasing in x and nondecreasing in y. Such system is called competitive. One may associate a competitive map T to a competitive system (2.1) by setting T=(f,g) and considering T on ℬ=ℐ×ℐ.We now present some basic notions about competitive maps in plane. Define a partial order⪯ on ℝ2 so that the positive cone is the fourth quadrant, that is, (x1,y1)⪯(x2,y2) if and only if 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. A set 𝒜 is said to be linearly ordered if ⪯ is a total order on 𝒜. If a set 𝒜⊂ℝ2 is linearly ordered by ⪯, then the infimum i=inf𝒜 and supremum s=sup𝒜 of 𝒜 exist in ℝ¯2=[-∞,∞]×[-∞,∞]. If both i and s belong to ℝ2, then the linearly ordered set 𝒜 is bounded, and conversely. We note that the ordering ⪯ may be extended to the extended plane ℝ¯2 in a natural way. For example, (0,∞)⪯(a,b) if a≥0 or a=∞. If x∈ℝ2, we denote with 𝒬ℓ(x), ℓ∈{1,2,3,4}, the four quadrants in ℝ¯2 relative to x, that is, 𝒬1(x,y)={(u,v)∈ℝ¯2:u≥x,v≥y}, 𝒬2(x,y)={(u,v)∈ℝ¯2:x≥u,v≥y}, and so on.AmapT on a set ℬ⊂ℝ2 is a continuous function T:ℬ→ℬ. The map is smooth on ℬ if the interior of ℬ is nonempty and if T is continuously differentiable on the interior of ℬ. A set 𝒜⊂ℬ is invariant for the map T if T(𝒜)⊂𝒜. A point x∈ℬ is a fixed point of T if T(x)=x, and a minimal period-two point if T2(x)=x and T(x)≠x. A period-two point is either a fixed point or a minimal period-two point. The orbit ofx∈ℬ is the sequence {Tℓ(x)}ℓ=0∞. A minimal period two orbit is an orbit {xℓ}ℓ=0∞ for which x0≠x1 and x0=x2. The basin of attraction of a fixed point x is the set of all y such that Tn(y)→x. A fixed point x is a global attractor on a set 𝒜 if 𝒜 is a subset of the basin of attraction of x. A fixed point x is a saddle point if T is differentiable at x, and the eigenvalues of the Jacobian matrix of T at x are such that one of them lies in the interior of the unit circle in ℝ2, while the other eigenvalue lies in the exterior of the unit circle. If T=(T1,T2) is a map on ℛ⊂ℝ2, define the sets ℛT(-,+):={(x,y)∈ℛ:T1(x,y)≤x,T2(x,y)≥y} and ℛT(+,-):={(x,y)∈ℛ:T1(x,y)≥x,T2(x,y)≤y}. For 𝒜⊂ℝ2 and x∈ℝ2, define the distance fromxto𝒜 as dist(x,𝒜):=inf{∥x-y∥:y∈𝒜}.A mapT is competitive if T(x)⪯T(y) whenever x⪯y, and T is strongly competitive if x⪯y implies that T(x)-T(y)∈{(u,v):u>0,v<0}. If T is differentiable, a sufficient condition for T to be strongly competitive is that the Jacobian matrix of T at any x∈ℬ has the sign configuration (2.2)(+--+).
For additional definitions and results (e.g., repeller, hyperbolic fixed points, stability, asymptotic stability, stable and unstable manifolds) see [8, 9] for competitive maps, and [10, 11] for difference equations.If𝒜 is any subset of ℝk, we shall use the notation clos(𝒜) to denote the closure of 𝒜 in ℝk, and 𝒜∘ to denote the interior of 𝒜.The next results are stated for order-preserving maps onℝn and are known but given here for completeness. See [12] for a more general version valid in ordered Banach spaces.Theorem 2.1.
For a nonempty setR⊂ℝn and ⪯ a partial order on ℝn, let T:R→R be an order-preserving map, and let a,b∈R be such that a≺b and ⟦a,b⟧⊂R. If a⪯T(a) and T(b)⪯b, then  ⟦a,b⟧is invariant and(i)
there exists a fixed point ofT in ⟦a,b⟧,(ii)
ifT is strongly order preserving, then there exists a fixed point in⟦a,b⟧  which is stable relative to ⟦a,b⟧,(iii)
if there is only one fixed point in⟦a,b⟧, then it is a global attractor in ⟦a,b⟧ and therefore asymptotically stable relative to ⟦a,b⟧.Corollary 2.2.
If the nonnegative cone of⪯ is a generalized quadrant in ℝn, and if T 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.Define arectangular regionℛ in ℝ2 to be the cartesian product of two intervals in ℝ.Remark 2.3.
It follows from the Perron-Frobenius theorem and a change of variables [9] that, at each point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that if the map is strongly competitive then no eigenvector is aligned with a coordinate axis.Theorem 2.4.
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 matrixJT(x¯) of T 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 axes.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 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.We shall see in Theorem2.7 and in the examples in [2] 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 2.5.
For the curve𝒞 of Theorem 2.4 to have endpoints in  ∂ℛ, it is sufficient that at least one of the following conditions is satisfied.   (i)
The mapT has no fixed points nor periodic points of minimal period two in Δ.(ii)
The mapT has no fixed points in Δ, detJT(x¯)>0, and T(x)=x¯ has no solutions x∈Δ.(iii)
The mapT has no points of minimal period two in Δ, detJT(x¯)<0, and T(x)=x¯ has no solutions x∈Δ.In many cases, one can expect the curve𝒞 to be smooth.Theorem 2.6.
Under the hypotheses of Theorem2.4, suppose that there exists a neighborhood  𝒰 of  x¯ in  ℝ2 such that  T is of class  Ck on  𝒰∪Δ for some  k≥1, and that the Jacobian matrix of  T at each  x∈Δ is invertible. Then, the curve  𝒞 in the conclusion of Theorem 2.4 is of class  Ck.In applications, it is common to have rectangular domainsℛ for competitive maps. If a competitive map has several fixed points, often the domain of the map may be split into rectangular invariant subsets such that Theorem 2.4 could be applied to the restriction of the map to one or more subsets. For maps that are strongly competitive near the fixed point, hypothesis (b) of Theorem 2.4 reduces just to |λ|<1. This follows from a change of variables [9] that allows the Perron-Frobenius theorem to be applied to give that at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrant, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axes.Smith performed a systematic study of competitive and cooperative maps in [9, 13, 14] and in particular introduced invariant manifolds techniques in his analysis [13–15] with some results valid for maps on n-dimensional space. Smith restricted attention mostly to competitive maps T that satisfy additional constraints. In particular, T is required to be a diffeomorphism of a neighborhood of ℝ+n that satisfies certain conditions (this is the case if T is orientation preserving or orientation reversing), and that the coordinate semiaxes are invariant under T. For such class of maps (as well as for cooperative maps satisfying similar hypotheses), Smith obtained results on invariant manifolds passing through hyperbolic fixed points and a fairly complete description of the phase-portrait when n=2, especially for those cases having a unique fixed point on each of the open positive semiaxes. In our results, presented here, we removed all these constraints and added the precise analysis of invariant manifolds of nonhyperbolic equilibrium points. The invariance of coordinate semiaxes seems to be serious restriction in the case of competitive models with constant stocking or harvesting, see [16] for stocking.The next result is useful for determining basins of attraction of fixed points of competitive maps. Compare to Theorem 4.4 in [13], where hyperbolicity of the fixed point is assumed, in addition to other hypotheses.Theorem 2.7.
(A) Assume the hypotheses of Theorem 2.4, and let  𝒞 be the curve whose existence is guaranteed by Theorem 2.4. If the endpoints of  𝒞 belong to  ∂ℛ, then  𝒞 separates  ℛ into two connected components, namely,
(2.3)W-:={x∈R∖C:∃y∈Cwithx⪯sey},W+:={x∈R∖C:∃y∈Cwithy⪯sex},
such that the following statements are true:  (i)
𝒲- is invariant, and dist(Tn(x),𝒬2(x¯))→0 as n→∞ for every x∈𝒲-.(ii)
𝒲+ is invariant, and dist(Tn(x),𝒬4(x¯))→0 as n→∞ for every x∈𝒲+.(B) If, in addition to the hypotheses of part  (A), x¯ is an interior point of ℛ, and T is C2 and strongly competitive in a neighborhood of x¯, then T has no periodic points in the boundary of 𝒬1(x¯)∪𝒬3(x¯) except for x¯, and the following statements are true.  (iii)
For everyx∈𝒲- there exists n0∈ℕ such that Tn(x)∈int𝒬2(x¯) for n≥n0.(iv)
For everyx∈𝒲+ there exists n0∈ℕ such that Tn(x)∈int𝒬4(x¯) for n≥n0.Basins of attraction ofperiod-two solutions or period-two orbits of certain systems or maps can be effectively treated with Theorems 2.4 and 2.7. See [2, 6, 11] for the hyperbolic case; for the nonhyperbolic case, see examples in [2, 17].IfT is a map on a set ℛ and if x¯ is a fixed point of T, the stable set𝒲s(x¯) of x¯ is the set {x∈ℛ:Tn(x)→x¯}, and unstable set𝒲u(x¯) of x¯ is the set (2.4){x∈R:thereexists{xn}n=-∞0⊂Rs.t.T(xn)=xn+1,x0=x,lim⁡n→-∞xn=x¯}.
When T is noninvertible, the set 𝒲s(x¯) may not be connected and made up of infinitely many curves, or 𝒲u(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 𝒲s(x¯) and 𝒲u(x¯) are the stable and unstable manifolds of x¯.Theorem 2.8.
In addition to the hypotheses of part (B) of Theorem2.7, suppose that  μ>1 and that the eigenspace  Eμ associated with  μ is not a coordinate axes. If the curve  𝒞 of Theorem 2.4 has endpoints in  ∂ℛ, then  𝒞 is the stable set  𝒲s(x¯) of  x¯, and the unstable set  𝒲u(x¯) of  x¯ is a curve in  ℛ that is tangential to  Eμ at  x¯ and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of  𝒲u(x¯) in  ℛ are fixed points of  T.The following result gives information onlocal dynamics near a fixed point of a map when there exists a characteristic vector whose coordinates have negative product and such that the associated eigenvalue is hyperbolic. This is a well-known result, valid in much more general setting which we include it here for completeness. A point (x,y) is a subsolution if T(x,y)⪯se(x,y), and (x,y) is a supersolution if (x,y)⪯seT(x,y). An order interval⟦(a,b),(c,d)⟧ is the cartesian product of the two compact intervals [a,c] and [b,d].Theorem 2.9.
LetT be a competitive map on a rectangular set  ℛ⊂ℝ2 with an isolated fixed point  x¯∈ℛ such that  ℛ∩int(𝒬2(x¯)∪𝒬4(x¯))≠∅. Suppose that   T has a  C1 extension to a neighborhood of  x¯. Let  v=(v(1),v(2))∈ℝ2 be an eigenvector of the Jacobian matrix of  T at  x¯, with associated eigenvalue  μ∈ℝ. If  v(1)v(2)<0, then there exists an order interval  ℐ which is also a relative neighborhood of  x¯ such that, for every relative neighborhood  𝒰⊂ℐ of  x¯, the following statements are true.   (i)
Ifμ>1, then 𝒰∩int𝒬2(x¯) contains a subsolution, and 𝒰∩int𝒬4(x¯) contains a supersolution. In this case, for every x∈ℐ∩int(𝒬2(x¯)∪𝒬4(x¯)), there exists N such that Tn(x)∉ℐ for n≥N.(ii)
Ifμ<1, then 𝒰∩int𝒬2(x¯) contains a supersolution and 𝒰∩int𝒬4(x¯) contains a subsolution. In this case, Tn(x)→x¯ for every x∈ℐ.In the nonhyperbolic case, we have the following result.Theorem 2.10.
Assume that the hypotheses of Theorem2.9 hold, that  T is real analytic at  x¯, and that  μ=1. Let  cj,  dj,  j=2,3,… be defined by the Taylor series
(2.5)T(x¯+tv)=x¯+vt+(c2,d2)t2+⋯+(cn,dn)tn+⋯.
Suppose that there exists an index ℓ≥2 such that (cℓ,dℓ)≠(0,0) and (cj,dj)=(0,0) for j<l. If either (a)
cℓdℓ<0, or (b)cℓ≠0, T(x¯+tv)(2)is  affine  in t, or (c)dℓ≠0, T(x¯+tv)(1)is  affine  in t, then there exists an order interval ℐ which is also a relative neighborhood of x¯ such that, for every relative neighborhood 𝒰⊂ℐ of x¯, the following statements are true.
(i)
Ifℓ is odd and (cℓ,dℓ)⪯se(0,0), then 𝒰∩int𝒬4(x¯) contains a supersolution, and 𝒰∩int𝒬2(x¯) contains a subsolution. In this case, for every x∈ℐ∩int(𝒬2(x¯)∪𝒬4(x¯)), there exists N such that Tn(x)∉ℐ for n≥N.(ii)
Ifℓ is odd and (0,0)⪯se(cℓ,dℓ), then 𝒰∩int𝒬4(x¯) contains a subsolution and 𝒰∩int𝒬2(x¯) contains a supersolution. In this case, Tn(x)→x¯ for every x∈ℐ.(iii)
Ifℓ is even and (cℓ,dℓ)⪯se(0,0), then 𝒰∩int𝒬4(x¯) contains a subsolution and 𝒰∩int𝒬2(x¯) contains a subsolution. In this case, Tn(x)→x¯ for every x∈ℐ∩𝒬4(x¯), and for every x∈ℐ∩int(𝒬2(x¯)), there exists N such that Tn(x)∉ℐ for n≥N.(iv)
Ifℓ is even and (0,0)⪯se(cℓ,dℓ), then 𝒰∩int𝒬2(x¯) contains a supersolution and 𝒰∩int𝒬4(x¯) contains a supersolution. In this case, Tn(x)→x¯ for every x∈ℐ∩𝒬2(x¯), and, for every x∈ℐ∩int(𝒬4(x¯)) there exists, N such that Tn(x)∉ℐ for n≥N.


## 3. Some Basic Facts

In this section, we give some basic facts about the nonexistence of period-two solutions, local injectivity of mapT at the equilibrium point.


### 3.1. Equilibrium Points

The equilibrium points(x̅,y̅) of the system (1.1) satisfy(3.1)x̅=β1x̅B1x̅+y̅,y̅=α2+γ2y̅A2+x̅.
Solutions of System (3.1) are(i)
x̅=0, y̅=α2/A2-γ2 when A2>γ2, that is,
(3.2)E1=(0,α2A2-γ2).(ii)
Ifx̅≠0, then using System (3.1), we obtain
(3.3)y̅=β1-B1x̅,0=x̅2B1-x̅(B1(γ2-A2)+β1)-(β1(A2-γ2)-α2).Solutions of System (3.3) are (3.4)x̅3,2=B1(γ2-A2)+β1±D02B1,y̅2,3=B1(A2-γ2)+β1±D02,
where D0=(B1(γ2-A2)-β1)2-4B1α2, which gives a pair of the equilibrium points E2=(x̅2,y̅2) and E3=(x̅3,y̅3).Geometrically, the equilibrium points are the intersections of two equilibrium curves:C1:x=0∪y=-B1x+β1 and C2:y=α2/(A2-γ2+x). Depending on the values of parameters, C2 may have between 0 and 3 intersection points with two lines which constitutes C1.The algebraic criteria for the existence of the equilibrium points are summarized in  Table1.Table 1E1A2>γ2, α2>(β1+B1(A2-γ2))24B1 orA2>γ2,
α2≤(β1+B1(A2-γ2))24B1,
β1<α2A2-γ2,
B1>β1A2-γ2E1≡E2≡E3A2>γ2, β1=α2A2-γ2, B1=α2(A2-γ2)2E1≡E3A2>γ2, β1=α2A2-γ2, B1>α2(A2-γ2)2E1≡E2, E3A2>γ2,β1=α2A2-γ2,B1<α2(A2-γ2)2E1, E2, E3A2>γ2, B1(γ2-A2)+4B1α2<β1<α2A2-γ2, α2>B1(A2-γ2)2E1, E3A2>γ2, β1>α2A2-γ2E1, E2=E3A2>γ2, (A2-γ2)B1<β1, (B1(γ2-A2)-β1)2-4B1α2=0E2, E3A2<γ2, (B1(γ2-A2)-β1)2-4B1α2>0, β1>B1(γ2-A2)E2=E3A2<γ2, (B1(γ2-A2)-β1)2-4B1α2=0,β1>B1(γ2-A2)e1, e2A2=γ2,β12-4B1α2≥0No equilibriumγ2≥A2 and α2≤(β1+B1(A2-γ2))24B1, β1<B1(γ2-A2) orγ2≥A2 and α2>(β1+B1(A2-γ2))24B1Where(3.5)e1=(β1+β12-4B1α22B1,β1-β12-4B1α22),e2=(β1-β12-4B1α22B1,β1+β12-4B1α22).Remark 3.1.
Observe the following: If the system (1.1) has two or three equilibrium points E1, E2, and E3 then, E1⪯E2⪯E3. Indeed, consider the critical curve C2:y=α2/(A2-γ2+x). Observe that y(0)=α2/(A2-γ2), y(x̅2)=y̅2, and y(x̅3)=y̅3. It is obvious that the following holds 0≤x̅2≤x̅3. Since, the critical curve C2 decreases, we have y(0)≥y(x̅2)≥y(x̅3), that is, E1⪯E2⪯E3.Lemma 3.2.
Assume thatx0=0,  y0∈ℝ+=(0,∞). Then the following statements are true for solutions of the system (1.1).   (i)
IfA2>γ2, then xn=0, for all n∈ℕ, and yn→α2/(A2-γ2), n→∞.(ii)
IfA2<γ2, then xn=0, for all n∈ℕ, and yn→∞, n→∞.(iii)
IfA2=γ2, then yn=y0+(α2/γ2)n, and xn=0, for all  n∈ℕ, yn→∞, n→∞.
Assume thatx0≠0 and (x0,y0)∈ℝ2+. Then, the following statements are true for all n=1,2,…:(iv)
xn≤β1/B1.(v)
yn≤c(γ2/A2)n+(α2/(A2-γ2)) and
(a)
yn≥B1α2/(B1(A2-γ2)+β1)+ϵ1, A2>γ2, where ϵ1 is arbitrarily small positive number.(b)
yn≤α2/(A2-γ2)+ε2, ε>0, A2>γ2, where ϵ2 is arbitrarily small positive number.Proof.
Since  (i)–(iv) are immediate consequences of the system (1.1), we will prove only  (v).
Takex0=0 and y0∈ℝ+. Then, we have xn=0 for all n∈ℕ, and
(3.6)yn+1=α2A2+γ2A2yn.
Solution of (3.6), when A2≠γ2 is
(3.7)yn=c(γ2A2)n+α2A2-γ2
which immediately implies (i) and (ii). Statement (iii) follows from (3.6).
Equation(3.8)xn+1=β1xnB1xn+yn
implies that
(3.9)xn≤β1B1.
Using the last inequality, we have
(3.10)yn+1=α2+γ2ynA2+xn≥α2+γ2ynA2+(β1/B1)=B1α2A2B1+β1+γ2B1B1A2+β1yn,
which by difference inequality theorem [18] implies the following
(3.11)yn≥c(γ2B1A2B1+β1)n+B1α2B1(A2-γ2)+β1≥B1α2B1(A2-γ2)+β1+ϵ1,n⟶∞.
Furthermore, second equation in (1.1) implies that
(3.12)yn+1≤α2A2+γ2A2yn
which, by the difference inequalities argument, see [18], implies that yn≤un, where un satisfies (3.6). In view of (3.7) we obtain our conclusion.


### 3.2. Period-Two Solution

In this section, we prove that System (1.1) has no minimal period-two solution which will be essential for application of Theorems 2.5–2.7. The map T associated to System (1.1) is given by (3.13)T(x,y)=(β1xB1x+y,α2+γ2yA2+x).Lemma 3.3.
System (1.1) has no minimal period-two solution.Proof.
We have(3.14)T(T(x,y))=T(β1xB1x+y,α2+γ2yA2+x)=(β12x(A2+x)B1β1x(A2+x)+(α2+γ2y)(B1x+y),(B1x+y)(α2(A2+x)+γ2(α2+γ2y))(A2+x)(A2(B1x+y)+β1x)).
Period-two solution satisfies
(3.15)β12x(A2+x)B1β1x(A2+x)+(α2+γ2y)(B1x+y)-x=0,(3.16)(B1x+y)(α2(A2+x)+γ2(α2+γ2y))(A2+x)(A2(B1x+y)+β1x)-y=0.
We show that this system has no other positive solutions except the equilibrium points.
Equation (3.15) is equivalent to
(3.17)x(yα2+x2B1β1-A2β12+x(-β12+B1(α2+A2β1))+y2γ2+xyB1γ2)=0,(3.18)-xy2A2+x2B1α2+x2y(-A2B1-β1)+y(A2α2+α2γ2)+x(A2B1α2+B1α2γ2)+y2(-A22+γ22)+xy(-A22B1+α2-A2β1+B1γ22)=0.
Ifx=0 then we obtain the fixed point E1. So assume that x≠0. Then, using (3.17), we have
(3.19)yα2+x2B1β1-A2β12+x(-β12+B1(α2+A2β1))+y2γ2+xyB1γ2=0.
Equation (3.19) implies that
(3.20)y2=-yα2-x2B1β1+A2β12-x(-β12+B1(α2+A2β1))-xyB1γ2γ2.
Substituting (3.20) into (3.18), we have
(3.21)x+A2=0
or
(3.22)A2((y+xB1)α2+(x+A2)(xB1-β1)β1)+((y+xB1)α2-xyβ1)γ2+β1(-xB1+β1)γ22=0.
Equation (3.21) implies thatx=-A2, and (3.23) implies that
(3.23)y=A2((x+A2)β12-xB1(α2+(x+A2)β1))-xB1α2γ2+(xB1-β1)β1γ22A2α2+(α2-xβ1)γ2.
Replacing (3.23) into (3.19), we get
(3.24)-x2B1-α2+β1(A2-γ2)+x(-A2B1+β1+B1γ2)=0,(3.25)-(β1(α2+β1(A2-γ2))(A2+γ2)2)+x(α2+β1(A2-γ2))(A2+γ2)(A2B1-β1)+B1γ2)+x2β1(A22B1+A2B1γ2+β1γ2)=0.
Solutions of (3.24) are the equilibrium points.
Consider (3.25). Discriminant of this equation is given by
(3.26)Δ=(4β12(A22B1+(A2B1+β1)γ2)+(α2+β1(A2-γ2))(β1-B1(A2γ2))2)×(α2+β1(A2-γ2)).
Now, Δ≥0 implies that
(3.27)x1=-(A2+γ2)(Δ1+Δ)2β1(A22B1+(A2B1+β1)γ2),x2=-(A2+γ2)(Δ1-Δ)2β1(A22B1+(A2B1+β1)γ2),
where
(3.28)Δ1=(α2+β1(A2-γ2))(-β1+B1(A2+γ2)).
Using (3.23), we obtain
(3.29)y1=-Δ2-Δ2(A22B1+(A2B1+β1)γ2),y2=-Δ2+Δ2(A22B1+(A2B1+β1)γ2),
where
(3.30)Δ2=A22B1β1+α2(β1-B1γ2)+β1γ2(β1+B1γ2)+A2(β12-B1(α2-2β1γ2)).
We prove the following claims.
Claim 3.4.
For all values of parametersy2<0.Proof.
IfΔ2>0, then y2<0. Now, we assume that Δ2≤0. Then,
(3.31)y2<0⟺Δ2+Δ>0⟺Δ-Δ22>0,(3.32)Δ-Δ22=-4β1(A22B1+(A2B1+β1)γ2)(A2B1(-α2+β1γ2)+γ2(β12+B1(-α2+β1γ2))).
Equation (3.32) implies that
(3.33)Δ-Δ22>0⟺α2>A2B1β1γ2+β12γ2+B1β1γ22B1(A2+γ2).
Since Δ2≤0 if and only if
(3.34)B1>β1A2+γ2,α2≥A22B1β1+A2β12+2A2B-1β1γ2+β12γ2+B1β1γ22A2B1-β1+B1γ2,A22B1β1+A2β12+2A2B-1β1γ2+β12γ2+B1β1γ22A2B1-β1+B1γ2-A2B1β1γ2+β12γ2+B1β1γ22B1(A2+γ2)=β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))B1(A2+γ2)(-β1+B1(A2+γ2))>0.
This implies that (3.33) is true. That is y2<0.Claim 3.5.
Assume thatx1≥0. Then that y1<0.Proof.
Assume thatx1>0. This is equivalent to
(3.35)A2<γ2,B1+β1(A2+3γ2)A22-γ22>0,(3.36)α2≤-(β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))).
Now
(3.37)y1<0⟺Δ2≥0,Δ-Δ22<0,
which is equivalent to
(3.38)Δ2≥0⟺B1<β1A2+γ2,α2≤-(β1(A2+γ2)(β1+B1(A2+γ2))β1-B1(A2+γ2))
or
(3.39)B1≥β1A2+γ2.
Also,(3.40)Δ-Δ22<0⟺α2<β1γ2(β1+B1(A2+γ2))B1(A2+γ2).
Since,(3.41)β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))+β1γ2(β1+B1(A2+γ2))B1(A2+γ2)=β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))2B1(A2+γ2)(β1-B1(A2+γ2))2>0.
This inequality and (3.36) imply (3.40).
Since(3.42)β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))2-(β1(A2+γ2)(β1+B1(A2+γ2))β1-B1(A2+γ2))=2β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))(β1-B1(A2+γ2))2>0.
Last inequality, (3.36) and (3.39) imply that Δ2≥0. So we prove, if x1>0, then y1<0.
Assume thatx1=0.
We have(3.43)x1=0⟺Δ1+Δ=0.
Since,(3.44)Δ12-Δ=4β12(α2+β1(A2-γ2))(A2+γ2)2(B1A22+B1γ2A2+β1γ2),
we have that x1=0 if and only if
(3.45)A2<γ2,α2+A2β1=β1γ2,Δ1≤0,
which is true, because
(3.46)Δ1:=(α2+β1(A2-γ2))(-β1+B1(A2+γ2)).
Replacingα2 with β1(γ2-A2) in the formula for y1, we obtain that y1=-β1.
Hence, there does not exist period-two solution.


## 3.1. Equilibrium Points

The equilibrium points(x̅,y̅) of the system (1.1) satisfy(3.1)x̅=β1x̅B1x̅+y̅,y̅=α2+γ2y̅A2+x̅.
Solutions of System (3.1) are(i)
x̅=0, y̅=α2/A2-γ2 when A2>γ2, that is,
(3.2)E1=(0,α2A2-γ2).(ii)
Ifx̅≠0, then using System (3.1), we obtain
(3.3)y̅=β1-B1x̅,0=x̅2B1-x̅(B1(γ2-A2)+β1)-(β1(A2-γ2)-α2).Solutions of System (3.3) are (3.4)x̅3,2=B1(γ2-A2)+β1±D02B1,y̅2,3=B1(A2-γ2)+β1±D02,
where D0=(B1(γ2-A2)-β1)2-4B1α2, which gives a pair of the equilibrium points E2=(x̅2,y̅2) and E3=(x̅3,y̅3).Geometrically, the equilibrium points are the intersections of two equilibrium curves:C1:x=0∪y=-B1x+β1 and C2:y=α2/(A2-γ2+x). Depending on the values of parameters, C2 may have between 0 and 3 intersection points with two lines which constitutes C1.The algebraic criteria for the existence of the equilibrium points are summarized in  Table1.Table 1E1A2>γ2, α2>(β1+B1(A2-γ2))24B1 orA2>γ2,
α2≤(β1+B1(A2-γ2))24B1,
β1<α2A2-γ2,
B1>β1A2-γ2E1≡E2≡E3A2>γ2, β1=α2A2-γ2, B1=α2(A2-γ2)2E1≡E3A2>γ2, β1=α2A2-γ2, B1>α2(A2-γ2)2E1≡E2, E3A2>γ2,β1=α2A2-γ2,B1<α2(A2-γ2)2E1, E2, E3A2>γ2, B1(γ2-A2)+4B1α2<β1<α2A2-γ2, α2>B1(A2-γ2)2E1, E3A2>γ2, β1>α2A2-γ2E1, E2=E3A2>γ2, (A2-γ2)B1<β1, (B1(γ2-A2)-β1)2-4B1α2=0E2, E3A2<γ2, (B1(γ2-A2)-β1)2-4B1α2>0, β1>B1(γ2-A2)E2=E3A2<γ2, (B1(γ2-A2)-β1)2-4B1α2=0,β1>B1(γ2-A2)e1, e2A2=γ2,β12-4B1α2≥0No equilibriumγ2≥A2 and α2≤(β1+B1(A2-γ2))24B1, β1<B1(γ2-A2) orγ2≥A2 and α2>(β1+B1(A2-γ2))24B1Where(3.5)e1=(β1+β12-4B1α22B1,β1-β12-4B1α22),e2=(β1-β12-4B1α22B1,β1+β12-4B1α22).Remark 3.1.
Observe the following: If the system (1.1) has two or three equilibrium points E1, E2, and E3 then, E1⪯E2⪯E3. Indeed, consider the critical curve C2:y=α2/(A2-γ2+x). Observe that y(0)=α2/(A2-γ2), y(x̅2)=y̅2, and y(x̅3)=y̅3. It is obvious that the following holds 0≤x̅2≤x̅3. Since, the critical curve C2 decreases, we have y(0)≥y(x̅2)≥y(x̅3), that is, E1⪯E2⪯E3.Lemma 3.2.
Assume thatx0=0,  y0∈ℝ+=(0,∞). Then the following statements are true for solutions of the system (1.1).   (i)
IfA2>γ2, then xn=0, for all n∈ℕ, and yn→α2/(A2-γ2), n→∞.(ii)
IfA2<γ2, then xn=0, for all n∈ℕ, and yn→∞, n→∞.(iii)
IfA2=γ2, then yn=y0+(α2/γ2)n, and xn=0, for all  n∈ℕ, yn→∞, n→∞.
Assume thatx0≠0 and (x0,y0)∈ℝ2+. Then, the following statements are true for all n=1,2,…:(iv)
xn≤β1/B1.(v)
yn≤c(γ2/A2)n+(α2/(A2-γ2)) and
(a)
yn≥B1α2/(B1(A2-γ2)+β1)+ϵ1, A2>γ2, where ϵ1 is arbitrarily small positive number.(b)
yn≤α2/(A2-γ2)+ε2, ε>0, A2>γ2, where ϵ2 is arbitrarily small positive number.Proof.
Since  (i)–(iv) are immediate consequences of the system (1.1), we will prove only  (v).
Takex0=0 and y0∈ℝ+. Then, we have xn=0 for all n∈ℕ, and
(3.6)yn+1=α2A2+γ2A2yn.
Solution of (3.6), when A2≠γ2 is
(3.7)yn=c(γ2A2)n+α2A2-γ2
which immediately implies (i) and (ii). Statement (iii) follows from (3.6).
Equation(3.8)xn+1=β1xnB1xn+yn
implies that
(3.9)xn≤β1B1.
Using the last inequality, we have
(3.10)yn+1=α2+γ2ynA2+xn≥α2+γ2ynA2+(β1/B1)=B1α2A2B1+β1+γ2B1B1A2+β1yn,
which by difference inequality theorem [18] implies the following
(3.11)yn≥c(γ2B1A2B1+β1)n+B1α2B1(A2-γ2)+β1≥B1α2B1(A2-γ2)+β1+ϵ1,n⟶∞.
Furthermore, second equation in (1.1) implies that
(3.12)yn+1≤α2A2+γ2A2yn
which, by the difference inequalities argument, see [18], implies that yn≤un, where un satisfies (3.6). In view of (3.7) we obtain our conclusion.


## 3.2. Period-Two Solution

In this section, we prove that System (1.1) has no minimal period-two solution which will be essential for application of Theorems 2.5–2.7. The map T associated to System (1.1) is given by (3.13)T(x,y)=(β1xB1x+y,α2+γ2yA2+x).Lemma 3.3.
System (1.1) has no minimal period-two solution.Proof.
We have(3.14)T(T(x,y))=T(β1xB1x+y,α2+γ2yA2+x)=(β12x(A2+x)B1β1x(A2+x)+(α2+γ2y)(B1x+y),(B1x+y)(α2(A2+x)+γ2(α2+γ2y))(A2+x)(A2(B1x+y)+β1x)).
Period-two solution satisfies
(3.15)β12x(A2+x)B1β1x(A2+x)+(α2+γ2y)(B1x+y)-x=0,(3.16)(B1x+y)(α2(A2+x)+γ2(α2+γ2y))(A2+x)(A2(B1x+y)+β1x)-y=0.
We show that this system has no other positive solutions except the equilibrium points.
Equation (3.15) is equivalent to
(3.17)x(yα2+x2B1β1-A2β12+x(-β12+B1(α2+A2β1))+y2γ2+xyB1γ2)=0,(3.18)-xy2A2+x2B1α2+x2y(-A2B1-β1)+y(A2α2+α2γ2)+x(A2B1α2+B1α2γ2)+y2(-A22+γ22)+xy(-A22B1+α2-A2β1+B1γ22)=0.
Ifx=0 then we obtain the fixed point E1. So assume that x≠0. Then, using (3.17), we have
(3.19)yα2+x2B1β1-A2β12+x(-β12+B1(α2+A2β1))+y2γ2+xyB1γ2=0.
Equation (3.19) implies that
(3.20)y2=-yα2-x2B1β1+A2β12-x(-β12+B1(α2+A2β1))-xyB1γ2γ2.
Substituting (3.20) into (3.18), we have
(3.21)x+A2=0
or
(3.22)A2((y+xB1)α2+(x+A2)(xB1-β1)β1)+((y+xB1)α2-xyβ1)γ2+β1(-xB1+β1)γ22=0.
Equation (3.21) implies thatx=-A2, and (3.23) implies that
(3.23)y=A2((x+A2)β12-xB1(α2+(x+A2)β1))-xB1α2γ2+(xB1-β1)β1γ22A2α2+(α2-xβ1)γ2.
Replacing (3.23) into (3.19), we get
(3.24)-x2B1-α2+β1(A2-γ2)+x(-A2B1+β1+B1γ2)=0,(3.25)-(β1(α2+β1(A2-γ2))(A2+γ2)2)+x(α2+β1(A2-γ2))(A2+γ2)(A2B1-β1)+B1γ2)+x2β1(A22B1+A2B1γ2+β1γ2)=0.
Solutions of (3.24) are the equilibrium points.
Consider (3.25). Discriminant of this equation is given by
(3.26)Δ=(4β12(A22B1+(A2B1+β1)γ2)+(α2+β1(A2-γ2))(β1-B1(A2γ2))2)×(α2+β1(A2-γ2)).
Now, Δ≥0 implies that
(3.27)x1=-(A2+γ2)(Δ1+Δ)2β1(A22B1+(A2B1+β1)γ2),x2=-(A2+γ2)(Δ1-Δ)2β1(A22B1+(A2B1+β1)γ2),
where
(3.28)Δ1=(α2+β1(A2-γ2))(-β1+B1(A2+γ2)).
Using (3.23), we obtain
(3.29)y1=-Δ2-Δ2(A22B1+(A2B1+β1)γ2),y2=-Δ2+Δ2(A22B1+(A2B1+β1)γ2),
where
(3.30)Δ2=A22B1β1+α2(β1-B1γ2)+β1γ2(β1+B1γ2)+A2(β12-B1(α2-2β1γ2)).
We prove the following claims.
Claim 3.4.
For all values of parametersy2<0.Proof.
IfΔ2>0, then y2<0. Now, we assume that Δ2≤0. Then,
(3.31)y2<0⟺Δ2+Δ>0⟺Δ-Δ22>0,(3.32)Δ-Δ22=-4β1(A22B1+(A2B1+β1)γ2)(A2B1(-α2+β1γ2)+γ2(β12+B1(-α2+β1γ2))).
Equation (3.32) implies that
(3.33)Δ-Δ22>0⟺α2>A2B1β1γ2+β12γ2+B1β1γ22B1(A2+γ2).
Since Δ2≤0 if and only if
(3.34)B1>β1A2+γ2,α2≥A22B1β1+A2β12+2A2B-1β1γ2+β12γ2+B1β1γ22A2B1-β1+B1γ2,A22B1β1+A2β12+2A2B-1β1γ2+β12γ2+B1β1γ22A2B1-β1+B1γ2-A2B1β1γ2+β12γ2+B1β1γ22B1(A2+γ2)=β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))B1(A2+γ2)(-β1+B1(A2+γ2))>0.
This implies that (3.33) is true. That is y2<0.Claim 3.5.
Assume thatx1≥0. Then that y1<0.Proof.
Assume thatx1>0. This is equivalent to
(3.35)A2<γ2,B1+β1(A2+3γ2)A22-γ22>0,(3.36)α2≤-(β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))).
Now
(3.37)y1<0⟺Δ2≥0,Δ-Δ22<0,
which is equivalent to
(3.38)Δ2≥0⟺B1<β1A2+γ2,α2≤-(β1(A2+γ2)(β1+B1(A2+γ2))β1-B1(A2+γ2))
or
(3.39)B1≥β1A2+γ2.
Also,(3.40)Δ-Δ22<0⟺α2<β1γ2(β1+B1(A2+γ2))B1(A2+γ2).
Since,(3.41)β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))+β1γ2(β1+B1(A2+γ2))B1(A2+γ2)=β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))2B1(A2+γ2)(β1-B1(A2+γ2))2>0.
This inequality and (3.36) imply (3.40).
Since(3.42)β1(A2(A2B1+β1)+3β1γ2-B1γ22)(β1+B1(A2+γ2))(β1-B1(A2+γ2))2-(β1(A2+γ2)(β1+B1(A2+γ2))β1-B1(A2+γ2))=2β1(A22B1+(A2B1+β1)γ2)(β1+B1(A2+γ2))(β1-B1(A2+γ2))2>0.
Last inequality, (3.36) and (3.39) imply that Δ2≥0. So we prove, if x1>0, then y1<0.
Assume thatx1=0.
We have(3.43)x1=0⟺Δ1+Δ=0.
Since,(3.44)Δ12-Δ=4β12(α2+β1(A2-γ2))(A2+γ2)2(B1A22+B1γ2A2+β1γ2),
we have that x1=0 if and only if
(3.45)A2<γ2,α2+A2β1=β1γ2,Δ1≤0,
which is true, because
(3.46)Δ1:=(α2+β1(A2-γ2))(-β1+B1(A2+γ2)).
Replacingα2 with β1(γ2-A2) in the formula for y1, we obtain that y1=-β1.
Hence, there does not exist period-two solution.


## 4. Linearized Stability Analysis

The Jacobian matrix of the mapT, given by (3.13), has the form(4.1)JT=(β1y(B1x+y)2-β1x(B1x+y)2-α2+γ2y(A2+x)2γ2A2+x).
The determinant of (4.1) is given by (4.2)det⁡⁡JT(x,y)=β1(yA2γ2-xα2)(x+A2)2(y+xB1)2.
The value of the Jacobian matrix of T at the equilibrium point E=(x̅,y̅), x̅≠0 is(4.3)JT(x̅,y̅)=(y̅B1x̅+y̅-x̅B1x̅+y̅-y̅A2+x̅γ2A2+x̅).
The determinant of (4.3) is given by (4.4)det⁡⁡JT(x̅,y̅)=y̅(γ2-x̅)(A2+x̅)(B1x̅+y̅),
and the trace of (4.3) is(4.5)TrJT(x̅,y̅)=y̅B1x̅+y̅+γ2A2+x̅.
The characteristic equation has the form (4.6)λ2-λ(y̅B1x̅+y̅+γ2A2+x̅)+y̅(γ2-x̅)(A2+x̅)(B1x̅+y̅)=0.Theorem 4.1.
Assume thatA2>γ2. Then there exists the equilibrium point  E1 and  (i)
E1(0,α2/(A2-γ2)) is locally asymptotically stable if β1<α2/(A2-γ2),(ii)
E1 is a saddle point if β1>α2/(A2-γ2). The corresponding eigenvalues are(4.7)λ1=γ2A2∈(0,1),λ2=β1(A2-γ2)α2∈(1,+∞),(iii)
E1 is nonhyperbolic if β1=α2/(A2-γ2). The corresponding eigenvalues are
(4.8)λ1=γ2A2∈(0,1),λ2=1,
and the corresponding eigenvectors are (0,1) and (-1/α2,1), respectively.Proof.
The Jacobian matrix (4.1) at the equilibrium point E1(0,α2/(A2-γ2)),
(4.9)JT(E1)=(β1y̅0-α2+γy̅A22γ2A2).
Note that the Jacobian matrix (4.9) implies that the map T is not strongly competitive at the equilibrium point E1.
The determinant of (4.9) is given by
(4.10)det⁡JT(x̅,y̅)=β1y̅γ2A2=β1γ2(A2-γ2)α2A2.
Note that, under the hypothesis of Theorem, the determinant is greater than zero.
The trace of (4.9) is
(4.11)TrJT(x̅,y̅)=β1y̅+γ2A2=β1(A2-γ2)α2+γ2A2.
An equilibrium point is locally asymptotically stable if the following conditions are satisfied(4.12)|TrJT(x̅,y̅)|<1+det⁡JT(x̅,y̅)<2.
Now, these two conditions become(4.13)β1(A2-γ2)α2+γ2A2<1+β1γ2(A2-γ2)α2A2<2.
Condition(4.14)β1(A2-γ2)α2+γ2A2<1+β1γ2(A2-γ2)α2A2
implies that
(4.15)(A2-γ2)β1<α2.
Ifβ1<α2/(A2-γ2), then this condition is satisfied.
Condition1+(β1γ2(A2-γ2))/α2A2<2 is equivalent to
(4.16)β1γ2(A2-γ2)α2A2<1.
It is easy to see that the condition is satisfied if β1<α2/(A2-γ2).
Next, we prove (ii).
An equilibrium point is a saddle if and only if the following conditions are satisfied(4.17)|TrJT(x̅,y̅)|>|1+det⁡⁡JT(x̅,y̅)|,Tr2JT(x̅,y̅)-4det⁡⁡JT(x̅,y̅)>0.
The first condition is equivalent to(4.18)(A2-γ2)β1>α2,
which is satisfied if β1>α2/(A2-γ2). The second condition is equivalent to
(4.19)(β1(A2-γ2)α2-γ2A2)2>0.
Finally, we prove (iii).
An equilibrium point is nonhyperbolic if the following conditions are satisfied(4.20)|TrJT(x̅,y̅)|=|1+det⁡JT(x̅,y̅)|,(det⁡JT(x̅,y̅)=1or|TrJT(x̅,y̅)|≤2).
The first condition is equivalent to(4.21)(A2-γ2)β1=α2,
which is satisfied if β1=α2/A2-γ2.
The second condition becomes(4.22)β1(A2-γ2)α2+γ2A2=1+γ2A2<2,
establishing part (iii).We now perform a similar analysis for the other cases in Table1.Theorem 4.2.
Assume that(4.23)A2>γ2,B1(γ2-A2)+4B1α2<β1<α2A2-γ2,α2>B1(A2-γ2)2.
Then E1, E2, and E3 exist and(i)
the equilibrium pointE1 is locally asymptotically stable,(ii)
the equilibrium pointE2 is a saddle point. Furthermore, if B1(γ2-A2)+4B2α2<β1<α2/(A2+B1γ2) and α2>B1A22, then the smaller eigenvalue belongs to the interval (-1,0), and the larger eigenvalue belongs to (1,+∞). In all other cases, the smaller eigenvalues is in (0,1).That is |λ1|<1 is given by
(4.24)λ1=β1γ2+(A2+x̅2)y̅2-(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅22β1(A2+x̅2),
and the corresponding eigenvector is
(4.25)v1=((β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅2β1γ2-(A2+x̅2)y̅2+(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅2,2β1y̅2).
Eigenvalue λ2, where |λ2|>1, is given by
(4.26)λ2=β1γ2+(A2+x̅2)y̅2+(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅22β1(A2+x̅2).(iii)
The equilibrium pointE3 is locally asymptotically stable.Proof.
By Theorem4.1, (i) holds.
Evaluating the Jacobian matrix (4.3) at the equilibrium point E2, we obtain
(4.27)JT=(y̅β1-x̅β1-y̅A2+x̅γ2A2+x̅).
Note that the Jacobian matrix (4.27) implies that the map T is strongly competitive.
The determinant of (4.27) is given by
(4.28)det⁡=y̅γ2β1(A2+x̅)-x̅y̅β1(A2+x̅),
and the trace of (4.27) is given by
(4.29)TrJT(x̅,y̅)=y̅β1+γ2A2+x̅.
The equilibrium point E2 is a saddle if and only if (4.17) is satisfied. The first condition is equivalent to
(4.30)y̅β1+γ2A2+x̅>|1+y̅γβ1(A2+x̅)-x̅y̅β1(A2+x̅)|
which is equivalent to
(4.31)y̅(A2+x̅)+β1γ2>β1(A2+x̅)+y̅(γ2-x̅)⟹y̅(A2+x̅)+β1(x̅-(A2-γ2))-y̅(γ2-x̅)>0⟹y̅(A2+x̅-γ2)-β1(A2-γ2+x̅)>-x̅y̅⟹(A2-γ2+x̅)(y̅-β1)>-x̅y̅⟹(β1-y̅)(A2-γ2+x̅)<x̅y̅.
In light of (3.3) β1-y̅2=B1x̅2, and by using (3.4), A2-γ2+x̅2=y̅3/B1. Now, we have
(4.32)B1x̅2y̅3B1<x̅2y̅2.
This implies that  y̅3<y̅2 which is true.
Condition(4.33)Tr2JT(x̅,y̅)-4det⁡JT(x̅,y̅)>0
is equivalent to
(4.34)(y̅2β1-γ2β1(A2+x̅2))2+x̅2y̅2β2(A2+x̅2)>0,
which is true.
To prove the second part of the statement (ii), we use the characteristic equation (4.6) of System (1.1) at the equilibrium point. Now, we have
(4.35)λ1+λ2=y̅β1+y̅γ2(A2+x̅)β1,λ1λ2=y̅(γ2-x̅)(A2+x̅)β1.
Since the mapT is strongly competitive, the Jacobian matrix (4.27) has two real and distinct eigenvalues, the larger one in absolute value being positive.
The first equation implies that either both eigenvalues are positive, or the smaller one is negative. First, we show that, under hypothesis (ii) of theorem, the product of these two eigenvalues is less than zero. In order to prove that, it is enough to prove thatγ2-x̅2<0.
We have(4.36)γ2-x̅2=γ2-B1(γ2-A2)+β1-(B1(γ2-A2)-β1)2-4B1α22B1=B1(A2+γ2)-β1+(B1(γ2-A2)-β1)2-4B1α22B1.
Now γ2-x̅2<0 if and only if
(4.37)B1(A2+γ2)-β1+(B1(γ2-A2)-β1)2-4B1α2<0,
which holds if
(4.38)β1>B1(A2+γ2),(B1(A2+γ2)-β1)2-(B1(γ2-A2)-β1)2+4B1α2>0,B1(A2+γ2)<β1<α2A2+B1γ2,α2A2+B1γ2-(A2B1+γ2B1)=α2-A22B1A2>0⟺B1<α2A22.
Also, we have(4.39)α2A2+B1γ2-(B1(γ2-A2)+2B1α2)≥0.
In all other casesγ2-x̅2>0.
This proves that the smaller eigenvalue is negative. Since the equilibrium point is a saddle point, it has to belong to(-1,0). The larger one belongs to (1,+∞). The proof of second statement is similar.
Now, we prove thatE3 is locally asymptotically stable.
Notice that(4.40)|TrJT(x̅,y̅)|<1+det⁡JT(x̅,y̅)<2
implies that y̅2>y̅3 which is true.Theorem 4.3.
Assume that(4.41)A2>γ2,β1>(A2-γ2)B1,(B1(A2-γ2)+β1)2-4α2B1=0.
Then E1,E2=E3 exist and:(i)
The equilibrium pointE1 is locally asymptotically stable.(ii)
The equilibrium pointE2=E3=(B1(γ2-A2)+β1/2B1,B1(A2-γ2)+β1/2) is nonhyperbolic. The eigenvalues of the Jacobian matrix evaluated at E2 are(4.42)λ1=1,λ2=A22B12-β12+2B1β1γ2-B12γ222β1(A2B1+β1+B1γ2),
and the corresponding eigenvectors, respectively, are
(4.43)(-1B1,1),((-A2B1+β1+B1γ2)(A2B1+β1+B1γ2)2B1β1(A2B1+β1-B1γ2),1).
Furthermore, if β1>B1(A2+γ2), then λ2∈(-1,0). If B1(A2-γ2)<β1<B1(A2+γ2), then λ2∈(0,1).Proof.
By Theorem4.1, E1 is a locally asymptotically stable.
Now, we prove thatE2 is nonhyperbolic.
The Jacobian matrix (4.3) at the equilibrium point E2=E3=(B1(γ2-A2)+β1/2B1,B1(A2-γ2)+β1/2) is
(4.44)JT(E2)=(y̅β1-x̅β1-y̅A2+x̅γ2A2+x̅).
The eigenvalues of (4.44) satisfy
(4.45)(B1(A2-γ2)+β12β1-λ)(2B1γ22B1A2+B1(γ2-A2)+β1-λ)-B1(γ2-A2)+β12β1B1B1(B1(A2-γ2)+β1)2B1A2+B1(γ2-A2)+β1=0,
and are given as
(4.46)λ1=1,λ2=A22B12-β12+2B1β1γ2-B12γ222β1(A2B1+β1+B1γ2).
Hence, E2 is nonhyperbolic.
Notice that|λ2|<1. Now, we show that λ2 can be in (-1,0) or (0,1).
We have(4.47)λ2=A22B12-β12+2B1β1γ2-B12γ222β1(A2B1+β1+B1γ2)=(B1(A2+γ2)-β1)(B1(A2-γ2)+β1)2β1(A2B1+β1+B1γ2),
which is negative if β1>B1(A2+γ2), and positive if β1<B1(A2+γ2).Theorem 4.4.
Assume that(4.48)A2>γ2,β1>α2A2-γ2.
Then, E1 and E3 exist and  (i)
the equilibrium pointE1 is a saddle point,(ii)
the equilibrium pointE3 is locally asymptotically stable.Proof.
By Theorem4.1, (i) holds. Observe that the assumption of Theorem 4.4 implies that the x coordinate of the equilibrium point E2 is less than zero.
The proof that the equilibrium pointE3 is locally asymptotically stable is similar to the corresponding proof of Theorem 4.2.Theorem 4.5.
The following statements are true.(a)
Assume(4.49)A2<γ2,(B1(γ2-A2)-β1)2-4B1α2>0,β1>B1(γ2-A2).
Then E2 and E3 exist and (i)
the equilibrium pointE2 is a saddle. If β1>B1(γ2-A2), then the larger eigenvalue is in (1,+∞), the smaller eigenvalue is in (-1,0). If B1(γ2-A2)<β1<B1(γ2+A2), then the smaller eigenvalue is in (0,1). That is, |λ1|<1 is given by
(4.50)λ1=β1γ2+(A2+x̅2)y̅2-(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅22β1(A2+x̅2),
and the corresponding eigenvector is
(4.51)v1=((β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅2β1γ2-(A2+x̅2)y̅2+(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅2,2β1y̅2).
Eigenvalue λ2, where |λ2|>1, is given by
(4.52)λ2=β1γ2+(A2+x̅2)y̅2+(β1γ2+(A2+x̅2)y̅2)2+4β1(A2+x̅2)(x̅2-γ2)y̅22β1(A2+x̅2).(ii)
The equilibrium pointE3 is locally asymptotically stable.(b)
Assume thatA2<γ2,(B1(γ2-A2)-β1)2-4B1α2=0, β1>B1(γ2-A2).Then there exists a unique positive equilibrium E2=E3=(B1(γ2-A2)+β1/2B1,B1(A2-γ2)+β1/2)which is nonhyperbolic. The eigenvalues are λ1=1 and λ2=A22B12-β12+2B1β1γ2-B12γ22/2β1(A2B1+β1+B1γ2), and the corresponding eigenvectors, respectively, are(4.53)(-1B1,1),((-A2B1+β1+B1γ2)(A2B1+β1+B1γ2)2B1β1(A2B1+β1-B1γ2),1).
If β1>B1(A2+γ2), then λ2∈(-1,0). If B1(γ2-A2)<β1<B1(γ2+A2) then λ2∈(0,1).Proof.
The proof of statements (a) is similar to the proof of the statements (ii) and (iii) of the Theorem4.2.
Now, we prove statement (b).
The characteristic equation of the system (1.1) at the equilibrium point E2=E3 has the form
(4.54)(B1(A2-γ2)+β12β1-λ)(2B1γ22B1A2+B1(γ2-A2)+β1-λ)-B1(γ2-A2)+β12β1B1B1(B1(A2-γ2)+β1)2B1A2+B1(γ2-A2)+β1=0in which solutions are eigenvalues of JT(E2)(4.55)λ1=1,λ2=A22B12-β12+2B1β1γ2-B12γ222β1(A2B1+β1+B1γ2).
Hence, E2=E3 is nonhyperbolic. Notice that |λ2|<1.
Now, we determine that the sign ofλ2(4.56)λ2=A22B12-β12+2B1β1γ2-B12γ222β1(A2B1+β1+B1γ2)=(B1(A2+γ2)-β1)(B1(A2-γ2)+β1)2β1(A2B1+β1+B1γ2)
is negative if β1>B1(A2+γ2), and positive if B1(γ2-A2)<β1<B1(γ2+A2).Now, we consider the special case of the system (1.1) when A2=γ2.In this case, the system (1.1) becomes
(4.57)xn+1=β1xnB1xn+yn,yn+1=α2+A2ynA2+xn,n=0,1,2,….If the following condition holds(4.58)β12-4B1α2≥0,
then the system (4.57) has two positive equilibrium points(4.59)e1=(β1+β12-4B1α22B1,β1-β12-4B1α22),e2=(β1-β12-4B1α22B1,β1+β12-4B1α22).We prove the following.Theorem 4.6.
Assume that(4.60)A2=γ2.
Then the following statements hold.  (i)
Ifβ12-4B1α2=0, then the system (4.57) has the unique equilibrium point e=(β1/2B1,β1/2) which is nonhyperbolic. The following holds:
(a)
Ifα2<B1A22, then λ1=1, and λ2=2A2B1-β1/2(2A2B1+β1)∈(0,1), and the corresponding eigenvectors, respectively, are
(4.61)(-1B1,1),((-2A2B1-β1)2B1β1,1).(b)
Ifα2>B1A22, then λ1=1 and λ2=2A2B1-β1/2(2A2B1+β1)∈(-1,0), and the corresponding eigenvectors, respectively, are
(4.62)(-1B1,1),((-2A2B1-β1)2B1β1,1).(c)
Ifα2=B1A22, then λ1=1 and λ2=0, and the corresponding eigenvectors, respectively, are(4.63)(-1B1,1),(1B1,1).(ii)
Ifβ12-4B1α2>0, then the system (4.57) has two positive equilibrium points: e1 is locally asymptotically stable and e2 is a saddle point. The following holds.
(d)
If2A2B1>β1 or 2A2B1≤β1 and B1A22+α2<A2β1, then λ2∈(1,+∞) and λ1∈(0,1).(e)
If2A2B1<β1 and B1A22+α2>A2β1, then λ2∈(1,+∞) and λ1∈(-1,0), where
(4.64)λ1=2α2B1+3A2β1B1+A2B1β12-4B1α2-F2β1(2A2B1+β1-β12-4B1α2),
where
(4.65)F=(2α2B1+3A2β1B1+A2β12-4B1α2B1)2-8B1β1(B1β1A22-α2β1+(B1A22+α2)β12-4B1α2),
and |λ1|<1.
The corresponding eigenvector for both cases (c) and (d) isv1=(v1(1),v2(1)), where
(4.66)v1(1)=(2A2B1+β1-β12-4B1α2)(-2α2B1+A2β1B1-A2B1β12-4B1α2+D),v2(1)=4A2β12B12+8α2β1B12+4A2β1B12β12-4B1α2,
where
(4.67)D=2B1(β12-4B1α2-2A22α2B12+2α22B1+A22β12B1+6A2α2β1B1+4α2β12+(-B1β1A22+2B1α2A2-4α2β1)β12-4B1α2).Proof.
Assume thatβ12-4B1α2=0. Then we have that
(4.68)e1=e2=e=(β12B1,β12).
The characteristic equation associated to the system (4.57) at the equilibrium point e is given by
(4.69)λ2-λ(12-2B1A22B1A2+β1)+B1A22B1A2+β1-β12(2B1A2+β1)=0.
Solutions of (4.69) are
(4.70)λ1=1,λ2=2A2B1-β12(2A2B1+β1).
It is easy to see that|λ2|<1.
Now, assume thatα2<B1A22. Since β1>0, from β12-4B1α2=0, we have β1=2B1α2. The numerator of λ2 is 2A2B1-2B1α2>0. Assume the opposite, that is, 2A2B1-2B1α2<0. Then, we have
(4.71)A22B12<B1α2⟹α2>B1A22
which is a contradiction. So, we confirmed (a).
Assume thatα2>B1A22. Then the numerator of λ2<0. If 2A2B1-2B1α2>0, then we have
(4.72)A22B12>B1α2⟹α2<B1A22
which is a contradiction. So, (b) holds.
Assume thatβ12-4B1α2>0. Then there are two positive equilibrium points
(4.73)e1=(β1+β12-4B1α22B1,β1-β12-4B1α22),e2=(β1-β12-4B1α22B1,β1+β12-4B1α22).
Now, we prove thate1 is a locally asymptotically stable equilibrium point.
We check the conditions for locally asymptotically stable equilibrium point. We have(4.74)y̅β1+A2A2+x<1+y̅A2β1(A2+x)-x̅y̅β1(A2+x̅).
This implies that(4.75)y(A2+x̅)+β1A2<β1(A2+x̅)+y̅A2-x̅y̅,
which is equivalent to x̅(2y̅-β1)<0 and
(4.76)2y̅-β1=β1-β12-4B1α2-β1=-β12-4B1α2<0,
which is true.
Now, we check condition1+detJT(x̅,y̅)<2. We have that
(4.77)y̅A2β1(A2+x̅)-x̅y̅β1(A2+x̅)<1.
This implies thaty̅A2-x̅y̅<β1A2+β1x̅ which is true, since
(4.78)-A2B12-4B1α22-α2<β1A22+β12+β1β12-4B1α22B1.
Hence, e1 is a locally asymptotically stable equilibrium point.
Now, we prove thate2 is a saddle. We check the condition (4.17).
Condition|TrJT(x̅,y̅)|>|1+detJT(x̅,y̅)| is equivalent to x̅(2y̅-β1)>0. This is true, since
(4.79)2y̅-β1=β1+β12-4B1α2-β1=β12-4B1α2>0.
Condition(4.80)Tr2JT(x̅,y̅)-4det⁡JT(x̅,y̅)>0
is equivalent to
(4.81)(y̅β1-A2A2+x̅)2+x̅y̅β1(A2+x̅)>0.
Hence, e2 is a saddle.
Now, we prove the statements (c) and (d).
The characteristic equation associated to the system (4.57) at the equilibrium point has the following form
(4.82)λ2-λ(y̅B1x̅+y̅+A2A2+x̅)+y̅(A2-x̅)(A2+x̅)(B1x̅+y̅)=0.
Now, we have that(4.83)λ1+λ2=y̅B1x̅+y̅+A2A2+x̅>0,λ1λ2=y̅(A2-x̅)(A2+x̅)(B1x̅+y̅).
Consider A2-x̅. We have that
(4.84)A2-x̅=2B1A2-β1+β12-4B1α22B1,β12-4B1α2-(β1-2B1A2)2=-4B1(B1A22-β1A2+α2)
which implies statements (c) and (d).


## 5. Global Behavior

In this section,  we present the results on global behavior of the system (1.1).Theorem 5.1.
Table2 describes the global behavior of the system  (1.1)Table 2
Global behavior of system (1.1). GAS stands for globally asymptotically stable.
RegionGlobal behaviorℛ1A2>γ2, α2>(β1-B1(A2-γ2))24B1,orThere exists a unique equilibriumE1(0,α2/(A2-γ2)), and it is G.A.S.. The basin of attraction of E1 is [0,+∞)2∖{(0,0)}ℛ2A2>γ2,α2≤(β1-B1(A2-γ2))24B1,β1<α2A2-γ2, B1>β1A2-γ2ℛ3A2>γ2, β1=α2(A2-γ2), B1<α2(A2-γ2)2There exist two equilibrium pointsE=E1=E2=(0,α2/(A2-γ2))=(0,β1) which is nonhyperbolic, and E3, which is locally asymptotically stable. Furthermore, the positive part of the y-axes is the basin of attraction ℬ(E1) of E1. The equilibrium point E3 is globally asymptotically stable with the basin of attraction ℬ(E3)=(0,∞)×[0,∞)ℛ4A2>γ2, β1=α2A2-γ2, B1<α2(A2-γ2)2There exist two equilibrium pointsE1 which are a saddle, and E3, which is a locally asymptotically stable equilibrium point. Furthermore, the positive part of y-axes is the global stable manifold of 𝒲s(E1). The equilibrium point E3 is G.A.S. with the basin of attraction ℬ(E3)=(0,∞)×[0,∞)ℛ5γ2<A2, B1(γ2-A2)+4B1α2<β1<α2A2-γ2,  B1<α2(A2-γ2)2There exist three equilibrium pointsE1 and E3 which are locally asymptotically stable and E2 which is a saddle. The global stable manifold 𝒲s(E2) separates the positive quadrant so that all orbits which start below this manifold are attracted to the point E3, and all orbits which start above this manifold are attracted to the equilibrium point E1. All orbits that start on 𝒲s(E2) are attracted to E2. The global unstable manifold 𝒲u(E2) is the graph of a continuous strictly decreasing function, and 𝒲u(E2) has endpoints Ea and E3ℛ6γ2<A2, (A2-γ2)B1<β1,  (B1(A2-γ2)+β1)2-4α2B1=0There exist two equilibrium pointsE1, which is locally asymptotically stable, and E2≡E3, which is nonhyperbolic. Furthermore, there exists an unbounded increasing invariant curve 𝒲E2 which is a part of the basin of attraction of E2. Every solution that starts above this curve is attracted to the equilibrium point E1; every solution that starts below this curve converges to E2ℛ7γ2<A2, β1=α2A2-γ2, B1≥α2(A2-γ2)2There exists a unique equilibrium pointE1=E2=E3 which is nonhyperbolic. All orbits are attracted to the equilibrium point E1ℛ8A2<γ2, (B1(γ2-A2)-β1)2-4B1α2>0,β1>B1(γ2-A2)There exist two equilibrium pointsE2, which is a saddle, and E3, which is a locally asymptotically stable equilibrium point. Furthermore, there exists the global stable manifold 𝒲2(E2) that separates the positive quadrant so that all orbits below this manifold are attracted to the equilibrium point E3, and all orbits above this manifold are asymptotic to (0,∞). All orbits that start on 𝒲s(E2) are attracted to E2. The global unstable manifold 𝒲u(E2) is the graph of a continuous strictly decreasing function, and 𝒲u(E2) has endpoints E2 and E3ℛ9A2<γ2, (B1(γ2-A2)-β1)2-4B1α2>0,β1>B1(γ2-A2)There exists a unique equilibrium pointE=E2=E3 which is nonhyperbolic. Furthermore, there exists an unbounded increasing invariant curve 𝒲E which is a part of the basin of attraction of E. Every solution that stars below this curve is attracted to E; every solution that starts above this curve is asymptotic to (0,∞)ℛ10A2=γ2, β12-4B1α2=0,There exists a unique equilibriume=(β1/2B1,β1/2) which is nonhyperbolic. Furthermore, there exists an unbounded increasing invariant curve 𝒲E, which is a part of the basin of attraction of E. Every solution that starts below this curve is attracted to the equilibrium point: every solution that starts above this curve is asymptotic to (0,∞)ℛ11A2=γ2, β12-4B1α2>0There exist two equilibrium pointse1, which is locally asymptotically stable, and e2, which is a saddle equilibrium point. Furthermore, there exists the global stable manifold Ws(e2) that separates the positive quadrant so that all orbits below this manifold are attracted to the equilibrium point e1, and allorbits above this manifold are asymptotic to (0,∞). All orbits that starts on Ws(e2)are attracted to e2. The global unstable manifold Wu(e2) is the graph of a continuous strictly decreasing function, and Wu(e2) has an endpoint at e1ℛ12γ2≥A2 and either  α2≤(β1+B1(A2-γ2))24B1,β1<B1(γ2-A2) orThe system (1.1) does not possess an equilibrium point. Its global behavior is described as followsℛ13α2>(β1+B1(A2-γ2))24B1xn→0yn→∞, n→∞Proof.
Throughout the proof of theorem⪯ will denote ⪯se.(ℛ1,ℛ2)
In view of Lemma3.2, the map T which corresponds to the system (1.1) has an attractive and invariant box ℬ=[0,β1/B1]×[Ly,Uy], where Ly=B1α2/B1(A2-γ2)+β1, Uy=α2/(A2-γ2), which contains a unique fixed point E1. By Theorem 2.1, every solution of the system (1.1) converges to E1. Clearly, the basin of attraction of the equilibrium point E1 is given by [0,+∞)2∖{(0,0)}.(ℛ3)
By Lemma3.2x0=0 implies that xn=0, for all n∈ℕ, and yn→α2/(A2-γ2), n→∞, which shows that y-axes is a subset of the basin of attraction ℬ(E1). Furthermore, every solution of (1.1) enters and stays in the box ℬ, and so we can restrict our attention to solutions that starts in ℬ. Clearly the set Q4(E3)∩ℬ is an invariant set with a single equilibrium point E3, and so every solution that starts there is attracted to E3. In view of Corollary 2.2, the interior of rectangle ⟦E1,E3⟧ is attracted to either E1 or E3, and because E3 is the local attractor, it is attracted to E3. If (x,y)∈ℬ∖(⟦E1,E3⟧∪(Q4(E3)∩ℬ)∪{(0,y):y≥0}), then there exist the points (xl,yl)∈⟦E1,E3⟧ and (xu,yu)∈Q4(E3)∩ℬ such that (xl,yl)⪯se(x,y)⪯se(xu,yu). Consequently, Tn((xl,yl))⪯seTn((x,y))⪯seTn((xu,yu)) for all n=1,2,… and so Tn((x,y))→E3 as n→∞, which completes the proof.(ℛ4)
The first part of this Theorem is proven in Theorem4.4.Now, we prove a global result(5.1)JT(E1)=(y̅β10-y̅A2γ2A2).
The eigenvalues of JT(E1) are given by λ1=β1/y and λ2=γ2/A2 and so
(5.2)β1>α2A2-γ2⟹λ1>1,A2>γ2⟹λ2<1.
The eigenvector of T at E1 that corresponds to the eigenvalue λ2<1 is (0,y).The rest of the proof is similar to the proof of part (ℛ3) and uses some continuity arguments.(ℛ5)
The first part of this Theorem is proven in Theorem4.2. Lemma 3.3 states that the system (1.1) has no minimal period-two solution. Take that  ℛ=ℝ+2∖{(0,0)}.T is strongly monotone in ℛ and differentiable in intℛ=ℛ∘ (interior of ℛ). E2 is a saddle point and E2∈ℛ∘. Then, all hypothesis (a)–(d) of Theorem 2.6 are satisfied. In light of Theorems 2.6 and 2.7, there exist the global stable manifold Ws(E2) and the global unstable manifold Wu(E2) which are the graphs of a continuous strictly monotonic functions. The global stable manifold Ws(E2) separates the first quadrant into two invariant regions W- (below the stable) manifold and W+ (above the stable manifold) which are connected. Each orbit starting above Ws(E2) remains above and is asymptotic to E1. Each orbit starting below Ws(E2) remains below and is asymptotic to E3. This implies that E1 and E3 are global attractors. Theorem 4.2 implies that they are globally asymptotically stable.(ℛ6)
Notice that in this case the eigenvector which corresponds to nonhyperbolic eigenvalueλ1=1 at E2 is v2=(-1/B1,1), see Theorem 4.3. Thus, the hypotheses of Theorems 2.4 and 2.7 are satisfied at the equilibrium point E2, and the conclusions of Theorems 2.4, 2.5, and 2.7 follow. Let 𝒞, 𝒲- and 𝒲+ be the sets given in the conclusion of Theorems 2.4 and 2.7. Let S:={(x,y):0≤x≤β1/B1,0≤y}. Since β1x/B1x+y≤β1/B1 for x≥0, y≥0, x+y>0, the map T satisfies T([0,∞)2∖(0,0))⊂S. Thus, T(𝒞∪𝒲+)⊂(𝒞∪𝒲+)∩S, which implies that T(𝒞∪𝒲+) is bounded. In view of Theorem 2.7, every solution which starts in 𝒲+ eventually enters 𝒬4(E2), and so is in rectangle S∩𝒬4(E2), which by Theorem 2.1, implies that all such solutions converge to the equilibrium point E2.If(x,y) is in 𝒲-, by Theorem 2.7, the orbit of (x,y) eventually enters 𝒬2(E2). Assume (without loss of generality) that (x,y)∈int𝒬2(E2).In view of Corollary2.2 and the fact that E1 is a local attractor ⟦E1,E2⟧ is a subset of the basin of attraction of E1. Let (x0,y0) be any point in 𝒲+. Then there exists ỹ0≥max{y0,α2/A2-γ2} such that (0,ỹ0)⪯se(x0,y0). Now (0,ỹ0)⪯se(0,ỹ0)=(0,α2+γ2ỹ0/A2), which implies that {Tn(0,ỹ0)}={(0,ỹn)} is an increasing sequence, and so {ỹn} is a decreasing sequence and thus is convergent to α2/A2-γ2. In view of (0,ỹ0)⪯se(x0,y0), we conclude that Tn((0,ỹ0))⪯seTn((x0,y0)) and so Tn((x0,y0)) eventually enters ⟦E1,E2⟧, and so it converges to E1.(ℛ7)
Let(x0,y0) be any point in [0,∞)2∖{(0,0)}. Then there exist points (x̃0,0) and (0,ỹ0),x̃0≥max{x0,β1/B1}, ỹ0≥max{y0,α2/(A2-γ2)} such that (0,ỹ0)⪯se(x0,y0)⪯(x̃0,0). This gives Tn((0,ỹ0))⪯seTn((x0,y0))⪯Tn((x̃0,0)) for all n≥1. Clearly, T(x̃0,0)=(β1/B1,α2/(A2+x̃0))⪯(x̃0,0), which implies that {Tn(x̃0,0)}={(x̃n,ỹn)} is a decreasing sequence bounded below by E1 and so is convergent to E1. Proof that {Tn(0,ỹ0)} is convergent to E1 is carried in a same way as in the proof of (ℛ6). In view of Tn((0,ỹ0))⪯seTn((x0,y0))⪯Tn((x̃0,0)) we conclude that {Tn((x0,y0))} converges to E1.(ℛ8)
PutT1(x,y)=β1x/(B1x+y), T2(x,y)=(α2+γ2y)/(A2+x). Take x=(x0,y0)∈W+(E2)∩ℛ(-,+), where ℛ(-,+)={(x,y)∈ℛ:T1(x,y)<x,T2(x,y)>y}. It is known that ℛ(-,+) is an invariant set, see [11].Then we have(5.3)T1(x0,y0)=β1x0B1x0+y0<x0,T2(x0,y0)=α2+γ2y0A2+x0>y0.This implies(5.4)(T1(x0,y0),T2(x0,y0))⪯se(x0,y0)⟺T(x0,y0)⪯se(x0,y0).By using monotonicityT2(x0,y0)⪯seT(x0,y0). By using induction Tn+1(x0,y0)⪯seTn(x0,y0). This implies that sequence {xn} is non-increasing and {yn} is nondecreasing. By Lemma 3.2, {xn} is bounded, hence it must converges. By using equation for xn+1 we see that the limit is zero. Since, {yn} is unbounded and nondecreasing then yn→∞, n→∞.By Theorems2.4 and 2.7, all orbits below this manifold are attracted to the equilibrium point E3.(ℛ9)
Since the hypotheses of Theorems2.4, and 2.7 are satisfied at the equilibrium point E2, the conclusions of Theorems 2.4, 2.5, and 2.7 follow. Let 𝒞, 𝒲- and 𝒲+ be the sets given in the conclusion of Theorems 2.4 and 2.7. Let S:={(x,y):0≤x≤β1/B1,0≤y}. Since β1x/B1x+y≤β1/B1 for x≥0, y≥0, x+y>0, the map T satisfies T([0,∞)2∖(0,0))⊂S. Thus T(𝒞∪𝒲+)⊂(𝒞∪𝒲+)∩S, which implies that T(𝒞∪𝒲+) is bounded. In view of Theorem 2.7 every solution which starts in 𝒲+ eventually enters 𝒬4(E2) and so is in rectangle S∩𝒬4(E2), which by Theorem 2.1, implies that all such solutions converge to the equilibrium point E2.If(x,y) is in 𝒲-, by Theorem 2.7 the orbit of (x,y) eventually enters 𝒬2(E2). Assume (without loss of generality) that (x,y)∈int𝒬2(E2).A calculation gives(5.5)T(E2+tv2)=(β1+B1(γ2-A2)-2t2B1,B1(2α2+γ2(β1+B1(A2-γ2)+2t)β1+B1(A2+γ2)-2t)
for all t and
(5.6)ddtT(E2+tv2)=(-1B1,2B1(β1+2α2+B1(A2+γ2)+β1γ2+B1γ2(A2-γ2))(β1+B1(A2+γ2)-2t)2),12d2dt2T(E2+tv2)|t=0=(0,4B1(β1+2α2+B1(A2+γ2)+β1γ2+B1γ2(A2-γ2))(β1+B1(A2+γ2)-2t)3).Since in expansion (2.5) we have (c2,d2) with d2>0 and T(E2+tv2)(1) is affine in t, by Theorem 2.10 in any relative neighborhood of E2 there exists a subsolution (w,z)∈Q2(E2), that is, T(w,z)⪯se(w,z). Choose one such (w,z) so that (x,y)⪯se(w,z). Since T is competitive, Tn+1(w,z)⪯Tn(w,z) for n=0,1,2,…. The monotonically decreasing sequence {Tn(w,z)} in 𝒬2(E) is unbounded below, since if it were not it would converge to the unique fixed point in 𝒬2(E), namely E, which is not possible. Let (wn,zn):=Tn(w,z), n=0,1,…. Then (wn,zn)∈S for n=1,2,…, hence {wn} is bounded. It follows that {zn} is monotone and unbounded. From (1.1) it follows that wn→0. Since Tn(x,y)⪯se(wn,zn), it follows that Tn(x,y)→(0,∞).(ℛ10)
The eigenvalues of the mapT are λ1=1 and λ2=2A2B1-β1/2(2A2B1+β1). The corresponding eigenvectors are
(5.7)(-1B1,1),(2A2B1+β12B1β1,1).
The existence of the continuous curve 𝒲E follows as in the proof of (ℛ9) as well as convergence of all points which start in 𝒲+.If(x,y) is in 𝒲-, by Theorem 2.7 the orbit of (x,y) eventually enters 𝒬2(E2). Assume (without loss of generality) that (x,y)∈int𝒬2(E2).A straightforward calculation gives(5.8)T(E2+tv2)=(β1-2t2B1,B1(2α2+A2(β1+2t))β1+2B1A2-2t)
for all t and
(5.9)ddtT(E2+tv2)=(-1B1,(β1+2B1A2)2(β1+2B1A2-2t)2),12d2dt2T(E2+tv2)|t=0=(0,2β1+2B1A2).Since in expansion (2.5) we have (c2,d2) with d2>0 and T(E2+tv2)(1) is affine in t,  by Theorem 2.10 in any relative neighborhood of E2 there exists a subsolution (w,z)∈Q2(E2), that is, T(w,z)⪯se(w,z). The rest of the proof is same as the proof of the case (ℛ10).(ℛ11)
The proof, which is similar to the proof of (ℛ8), follows as an immediate application of Lemmas 3.2 and 3.3, and Theorems 2.4, 2.7, and 4.6.(ℛ12)
For every point(x0,y0)∈[0,∞)2∖{(0,0)}, there exists x̃0≥β1/B1 such that (x0,y0)⪯(x̃0,0). Clearly, T(x̃0,0)=(β1/B1,α2/(A2+x̃0))⪯(x̃0,0), which implies that {Tn(x̃0,0)}={(x̃n,ỹn)} is a decreasing sequence, and so ỹn is an increasing sequence. If ỹn would be convergent, then {(x̃n,ỹn)} would converge to an equilibrium of system (1.1) which is impossible. Thus, limn→∞ỹn=∞, which implies that limn→∞ỹn=0. In view of (x0,y0)⪯(x̃0,0), we obtain that Tn((x0,y0))⪯Tn((x̃0,0)) and so
(5.10)xn⟶0,yn⟶∞,n⟶∞
follows.Remark 5.2.
We can see from Theorem5.1 that system (1.1) exhibits variety of behaviors in different ranges of parameters. These behaviors can be classified in a few categories that verbally describe situation. Here, we use some terminology introduced in [19].
Acoexistence attractor is one in which both species are present. An exclusion attractor is one in which one species is absent and the other species is present. By multiple mixed-type attractors, we mean a scenario that includes at least one coexistence attractor and at least one exclusion attractor. Park in [20, 21] observed the coexistence case in an experimental treatment that also included cases of competitive exclusion, that is, he observed a case termed to be multiple mixed-type attractors. Competition theory is primarily an equilibrium theory that is exemplified, by its limited number of asymptotic outcomes: a globally attracting coexistence equilibrium; a globally attracting exclusion equilibrium; at least two attracting exclusion equilibria; at least two attracting coexistence equilibria; a continuum of nonhyperbolic equilibria. (In this context, globally attracting means within the positive cone of state space.) Four of these five asymptotic alternatives are illustrated by the LeslieGower model (the discrete analog of the famous LotkaVolterra differential equation model), see [2, 5, 17]
(5.11)xn+1=b1xn1+xn+c1yn,yn+1=b2yn1+c2xn+yn,n=0,1,…
and fifth alternative, precisely two attracting coexistence equilibria are illustrated by the LeslieGower model with stockings, see [16]
(5.12)xn+1=b1xn1+xn+c1yn+h1,yn+1=b2yn1+c2xn+yn+h2,n=0,1,…,
where the parameters b1, b2, c1, c2, h1, and h2 are positive numbers, and the initial conditions x0, y0 are arbitrary nonnegative numbers. Here, b1 and b2 are the inherent birth rates, c1 and c2 the density-dependent effects on newborn recruitment, and h1andh2 are constant stockings. With this in mind, we introduce the following terminology. Let E1, E2 and E3 be three equilibrium points of general competitive system (2.1) in south-east ordering E1⪯seE2⪯seE3 and assume that E1 and E3 are attractors, and E2 is a saddle point or nonhyperbolic equilibrium having one characteristic values in (-1,1). If E1 and E3 belong to y-axes and x-axes and E2 is a saddle point (resp., nonhyperbolic equilibrium having one characteristic values in (-1,1)) then we say that system (2.1) exhibits saddle competitive exclusion (resp., nonhyperbolic competitive exclusion). If E1 and E3 belong to the interior of first quadrant and E2 is a saddle point (respectively, nonhyperbolic equilibrium having one characteristic values in (-1,1)) then we say that system (2.1) exhibits saddle competitive coexistence (resp., nonhyperbolic competitive coexistence). If one of the equilibrium points E1 and E3 is on the axes and the other is in the interior of first quadrant and E2 is a saddle point (resp., nonhyperbolic equilibrium having one characteristic values in (-1,1)), then we say that system (2.1) exhibits saddle competitive exclusion to coexistence (resp., nonhyperbolic competitive exclusion-to-coexistence). If there exists a single attractor in the interior of first quadrant which attracts all points where it is defined except eventually the points on the axes, we say that system (2.1) exhibits global competitive coexistence or global competitive exclusion depending on whether the attractor is on the axes or in the interior of first quadrant. If any of the attractors one the axes is the point (0,∞) or (∞,0), such a situation will be named singular.
Using this terminology, we can describe the global behavior of system (1.1) in a concise way as follows. System (1.1) exhibits global competitive exclusion if the parameters belong to the regions (ℛ1), (ℛ2), or (ℛ7) and global competitive coexistence if the parameters belong to the region (ℛ4).
System (1.1) exhibits saddle competitive exclusion to coexistence if the parameters belong to the region (ℛ5). System (1.1) exhibits nonhyperbolic competitive exclusion to coexistence if the parameters belong to the region (ℛ6). System (1.1) exhibits singular saddle competitive exclusion to coexistence if the parameters belong to the regions (ℛ8) or (ℛ9). System (1.1) exhibits singular nonhyperbolic competitive exclusion to coexistence if the parameters belong to the region (ℛ10) or (ℛ11). Finally, System (1.1) exhibits singular competitive exclusion when the parameters belong to the region (ℛ12). With this terminology the possible scenarios for this system are limited, and transition from one scenario to another could be possible explained by using the bifurcation theory. In particular, transition from global competitive exclusion to global competitive coexistence was explained in [3], and some related results can be found in [19], where an attempt has been made to explain the transitions from one scenario to another by using evolutionary game theory.



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