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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 295308, 35 pages
http://dx.doi.org/10.1155/2011/295308
Research Article

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

1Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina
2Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
3International University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina

Received 26 April 2011; Accepted 28 August 2011

Academic Editor: YumingΒ Shi

Copyright © 2011 S. Kalabuőić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate global dynamics of the following systems of difference equations π‘₯𝑛+1=𝛽1π‘₯𝑛/(𝐡1π‘₯𝑛+𝑦𝑛), 𝑦𝑛+1=(𝛼2+𝛾2𝑦𝑛)/(𝐴2+π‘₯𝑛), 𝑛=0,1,2,…, where the parameters 𝛽1, 𝐡1, 𝛽2, 𝛼2, 𝛾2, 𝐴2 are positive numbers, and initial conditions π‘₯0 and 𝑦0 are arbitrary nonnegative numbers such that π‘₯0+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.

1. Introduction

In this paper we consider the following rational system of difference equationsπ‘₯𝑛+1=𝛽1π‘₯𝑛𝐡1π‘₯𝑛+𝑦𝑛,𝑦𝑛+1=𝛼2+𝛾2𝑦𝑛𝐴2+π‘₯𝑛,𝑛=0,1,2,…,(1.1) where the parameters 𝛽1, 𝐡1, 𝛽2, 𝛼2, 𝛾2, 𝐴2 are positive numbers, and initial conditions π‘₯0 and 𝑦0 are nonnegative numbers such that π‘₯0+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 π‘₯𝑛 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 π‘₯𝑛 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 𝐡1 and 𝐴2 are related to the density-dependent effects on newborn recruitment. Finally, 𝛼2 affects stocking for species with state variable 𝑦𝑛.

In Section 2, 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 π‘₯𝑛+1ξ€·π‘₯=𝑓𝑛,𝑦𝑛,𝑦𝑛+1ξ€·π‘₯=𝑔𝑛,𝑦𝑛,ξ€·π‘₯𝑛=0,1,2,…,βˆ’1,π‘₯0ξ€Έβˆˆβ„Γ—β„,(2.1) where 𝑓,π‘”βˆΆβ„Γ—β„β†’β„ are continuous functions on an interval β„βŠ‚β„, 𝑓(π‘₯,𝑦) is nondecreasing in π‘₯ and non-increasing in 𝑦, and 𝑔(π‘₯,𝑦) is non-increasing in π‘₯ 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 βͺ― on ℝ2 so that the positive cone is the fourth quadrant, that is, (π‘₯1,𝑦1)βͺ―(π‘₯2,𝑦2) if and only if π‘₯1≀π‘₯2 and 𝑦1β‰₯𝑦2. For 𝐱,π²βˆˆβ„2 the order interval ⟦𝐱,𝐲⟧ is the set of all 𝐳 such that 𝐱βͺ―𝐳βͺ―𝐲. A set π’œ is said to be linearly ordered if βͺ― is a total order on π’œ. If a set π’œβŠ‚β„2 is linearly ordered by βͺ―, then the infimum 𝐒=infπ’œ and supremum 𝐬=supπ’œ of π’œ exist in ℝ2=[βˆ’βˆž,∞]Γ—[βˆ’βˆž,∞]. If both 𝐒 and 𝐬 belong to ℝ2, then the linearly ordered set π’œ is bounded, and conversely. We note that the ordering βͺ― may be extended to the extended plane ℝ2 in a natural way. For example, (0,∞)βͺ―(π‘Ž,𝑏) if π‘Žβ‰₯0 or π‘Ž=∞. If π±βˆˆβ„2, we denote with 𝒬ℓ(𝐱), β„“βˆˆ{1,2,3,4}, the four quadrants in ℝ2 relative to 𝐱, that is, 𝒬1(π‘₯,𝑦)={(𝑒,𝑣)βˆˆβ„2βˆΆπ‘’β‰₯π‘₯,𝑣β‰₯𝑦}, 𝒬2(π‘₯,𝑦)={(𝑒,𝑣)βˆˆβ„2∢π‘₯β‰₯𝑒,𝑣β‰₯𝑦}, and so on.

A map 𝑇 on a set β„¬βŠ‚β„2 is a continuous function π‘‡βˆΆβ„¬β†’β„¬. 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 π’œβŠ‚β„¬ is invariant for the map 𝑇 if 𝑇(π’œ)βŠ‚π’œ. A point π±βˆˆβ„¬ 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 of π±βˆˆβ„¬ is the sequence {𝑇ℓ(𝐱)}βˆžβ„“=0. A minimal period two orbit is an orbit {𝐱ℓ}βˆžβ„“=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 π’œ if π’œ 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 β„›βŠ‚β„2, define the sets ℛ𝑇(βˆ’,+)∢={(π‘₯,𝑦)βˆˆβ„›βˆΆπ‘‡1(π‘₯,𝑦)≀π‘₯,𝑇2(π‘₯,𝑦)β‰₯𝑦} and ℛ𝑇(+,βˆ’)∢={(π‘₯,𝑦)βˆˆβ„›βˆΆπ‘‡1(π‘₯,𝑦)β‰₯π‘₯,𝑇2(π‘₯,𝑦)≀𝑦}. For π’œβŠ‚β„2 and π‘₯βˆˆβ„2, define the distance from π‘₯ to π’œ as dist(π‘₯,π’œ)∢=inf{β€–π‘₯βˆ’π‘¦β€–βˆΆπ‘¦βˆˆπ’œ}.

A map 𝑇 is competitive if 𝑇(𝐱)βͺ―𝑇(𝐲) whenever 𝐱βͺ―𝐲, and 𝑇 is strongly competitive if 𝐱βͺ―𝐲 implies that 𝑇(𝐱)βˆ’π‘‡(𝐲)∈{(𝑒,𝑣)βˆΆπ‘’>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 π±βˆˆβ„¬ has the sign configuration +βˆ’βˆ’+.(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.

If π’œ is any subset of β„π‘˜, we shall use the notation clos(π’œ) to denote the closure of π’œ in β„π‘˜, and π’œβˆ˜ to denote the interior of π’œ.

The next results are stated for order-preserving maps on ℝ𝑛 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 set π‘…βŠ‚β„π‘› and βͺ― a partial order on ℝ𝑛, let π‘‡βˆΆπ‘…β†’π‘… be an order-preserving map, and let π‘Ž,π‘βˆˆπ‘… be such that π‘Žβ‰Ίπ‘ and βŸ¦π‘Ž,π‘βŸ§βŠ‚π‘…. If π‘Žβͺ―𝑇(π‘Ž) and 𝑇(𝑏)βͺ―𝑏, then βŸ¦π‘Ž,π‘βŸ§is invariant and(i)there exists a fixed point of 𝑇 in βŸ¦π‘Ž,π‘βŸ§, (ii)if 𝑇 is strongly order preserving, then there exists a fixed point inβŸ¦π‘Ž,π‘βŸ§ which is stable relative to βŸ¦π‘Ž,π‘βŸ§, (iii)if there is only one fixed point in βŸ¦π‘Ž,π‘βŸ§, then it is a global attractor in βŸ¦π‘Ž,π‘βŸ§ and therefore asymptotically stable relative to βŸ¦π‘Ž,π‘βŸ§.

Corollary 2.2. If the nonnegative cone of βͺ― is a generalized quadrant in ℝ𝑛, and if 𝑇 has no fixed points inβŸ¦π‘’1,𝑒2⟧ other than 𝑒1 and 𝑒2, then the interior of βŸ¦π‘’1,𝑒2⟧is either a subset of the basin of attraction of 𝑒1 or a subset of the basin of attraction of 𝑒2.

Define a rectangular region β„› 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. Let 𝑇 be a competitive map on a rectangular region β„›βŠ‚β„2. Let π‘₯βˆˆβ„› be a fixed point of 𝑇 such that Ξ”βˆΆ=β„›βˆ©int(𝒬1(π‘₯)βˆͺ𝒬3(π‘₯)) is nonempty (i.e., π‘₯ is not the NW or SE vertex of β„›), and 𝑇 is strongly competitive on Ξ”. Suppose that the following statements are true. (a)The map 𝑇 has a 𝐢1 extension to a neighborhood of x. (b)The Jacobian matrix 𝐽𝑇(x) of 𝑇 at x has real eigenvalues πœ†, πœ‡ such that 0<|πœ†|<πœ‡, where |πœ†|<1, and the eigenspace πΈπœ† associated with πœ† is not a coordinate axes. Then there exists a curve π’žβŠ‚β„› through x that is invariant and a subset of the basin of attraction of x, such that π’ž is tangential to the eigenspace πΈπœ† at x, and π’ž is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of π’ž in the interior of β„› are either fixed points or minimal period-two points. In the latter case, the set of endpoints of π’ž is a minimal period-two orbit of 𝑇.

We shall see in Theorem 2.7 and in the examples in [2] that the situation where the endpoints of π’ž are boundary points of β„› is of interest. The following result gives a sufficient condition for this case.

Theorem 2.5. For the curve π’ž of Theorem 2.4 to have endpoints in πœ•β„›, it is sufficient that at least one of the following conditions is satisfied. (i)The map 𝑇 has no fixed points nor periodic points of minimal period two in Ξ”. (ii)The map 𝑇 has no fixed points in Ξ”, det𝐽𝑇(x)>0, and 𝑇(x)=x has no solutions xβˆˆΞ”. (iii)The map 𝑇 has no points of minimal period two in Ξ”, det𝐽𝑇(x)<0, and 𝑇(x)=x has no solutions xβˆˆΞ”.

In many cases, one can expect the curve π’ž to be smooth.

Theorem 2.6. Under the hypotheses of Theorem 2.4, suppose that there exists a neighborhood 𝒰 of π‘₯ in ℝ2 such that 𝑇 is of class πΆπ‘˜ on 𝒰βˆͺΞ” for some π‘˜β‰₯1, and that the Jacobian matrix of 𝑇 at each π‘₯βˆˆΞ” is invertible. Then, the curve π’ž in the conclusion of Theorem 2.4 is of class πΆπ‘˜.

In applications, it is common to have rectangular domains β„› 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 |πœ†|<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 ℝ𝑛+ 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 π’ž be the curve whose existence is guaranteed by Theorem 2.4. If the endpoints of π’ž belong to πœ•β„›, then π’ž separates β„› into two connected components, namely, π’²βˆ’ξ€½βˆΆ=π‘₯βˆˆβ„›β§΅π’žβˆΆβˆƒπ‘¦βˆˆπ’žwithπ‘₯βͺ―se𝑦,𝒲+ξ€½βˆΆ=π‘₯βˆˆβ„›β§΅π’žβˆΆβˆƒπ‘¦βˆˆπ’žwith𝑦βͺ―seπ‘₯ξ€Ύ,(2.3) such that the following statements are true: (i)π’²βˆ’ is invariant, and dist(𝑇𝑛(π‘₯),𝒬2(π‘₯))β†’0 as π‘›β†’βˆž for every π‘₯βˆˆπ’²βˆ’. (ii)𝒲+ is invariant, and dist(𝑇𝑛(x),𝒬4(x))β†’0 as π‘›β†’βˆž for every xβˆˆπ’²+. (B) If, in addition to the hypotheses of part (A), x is an interior point of β„›, and 𝑇 is 𝐢2 and strongly competitive in a neighborhood of x, then 𝑇 has no periodic points in the boundary of 𝒬1(x)βˆͺ𝒬3(x) except for x, and the following statements are true. (iii) For every xβˆˆπ’²βˆ’ there exists 𝑛0βˆˆβ„• such that 𝑇𝑛(x)∈int𝒬2(x) for 𝑛β‰₯𝑛0. (iv)For every xβˆˆπ’²+ there exists 𝑛0βˆˆβ„• such that 𝑇𝑛(x)∈int𝒬4(x) for 𝑛β‰₯𝑛0.

Basins of attraction of period-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].

If 𝑇 is a map on a set β„› and if π‘₯ is a fixed point of 𝑇, the stable set 𝒲𝑠(π‘₯) of π‘₯ is the set {π‘₯βˆˆβ„›βˆΆπ‘‡π‘›(π‘₯)β†’π‘₯}, and unstable set 𝒲𝑒(π‘₯) of π‘₯ is the set ξ‚»ξ€½π‘₯π‘₯βˆˆβ„›βˆΆthereexists𝑛0𝑛=βˆ’βˆžξ€·π‘₯βŠ‚β„›s.t.𝑇𝑛=π‘₯𝑛+1,π‘₯0=π‘₯,limπ‘›β†’βˆ’βˆžπ‘₯𝑛=π‘₯ξ‚Ό.(2.4) When 𝑇 is noninvertible, the set 𝒲𝑠(π‘₯) may not be connected and made up of infinitely many curves, or 𝒲𝑒(π‘₯) 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 β„›, the sets 𝒲𝑠(π‘₯) and 𝒲𝑒(π‘₯) are the stable and unstable manifolds of π‘₯.

Theorem 2.8. In addition to the hypotheses of part (B) of Theorem 2.7, suppose that πœ‡>1 and that the eigenspace πΈπœ‡ associated with πœ‡ is not a coordinate axes. If the curve π’ž of Theorem 2.4 has endpoints in πœ•β„›, then π’ž is the stable set 𝒲𝑠(π‘₯) of π‘₯, and the unstable set 𝒲𝑒(π‘₯) of π‘₯ is a curve in β„› that is tangential to πΈπœ‡ at π‘₯ and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of 𝒲𝑒(π‘₯) in β„› are fixed points of 𝑇.

The following result gives information on local 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 (π‘₯,𝑦) is a subsolution if 𝑇(π‘₯,𝑦)βͺ―se(π‘₯,𝑦), and (π‘₯,𝑦) is a supersolution if (π‘₯,𝑦)βͺ―se𝑇(π‘₯,𝑦). An order interval ⟦(π‘Ž,𝑏),(𝑐,𝑑)⟧ is the cartesian product of the two compact intervals [π‘Ž,𝑐] and [𝑏,𝑑].

Theorem 2.9. Let 𝑇 be a competitive map on a rectangular set β„›βŠ‚β„2 with an isolated fixed point π‘₯βˆˆβ„› such that β„›βˆ©int(𝒬2(π‘₯)βˆͺ𝒬4(π‘₯))β‰ βˆ…. Suppose that 𝑇 has a 𝐢1 extension to a neighborhood of π‘₯. Let 𝑣=(𝑣(1),𝑣(2))βˆˆβ„2 be an eigenvector of the Jacobian matrix of 𝑇 at π‘₯, with associated eigenvalue πœ‡βˆˆβ„. If 𝑣(1)𝑣(2)<0, then there exists an order interval ℐ which is also a relative neighborhood of π‘₯ such that, for every relative neighborhood π’°βŠ‚β„ of π‘₯, the following statements are true. (i)If πœ‡>1, then π’°βˆ©int𝒬2(x) contains a subsolution, and π’°βˆ©int𝒬4(x) contains a supersolution. In this case, for every xβˆˆβ„βˆ©int(𝒬2(x)βˆͺ𝒬4(x)), there exists 𝑁 such that 𝑇𝑛(x)βˆ‰β„ for 𝑛β‰₯𝑁. (ii)If πœ‡<1, then π’°βˆ©int𝒬2(x) contains a supersolution and π’°βˆ©int𝒬4(x) contains a subsolution. In this case, 𝑇𝑛(x)β†’x for every xβˆˆβ„.

In the nonhyperbolic case, we have the following result.

Theorem 2.10. Assume that the hypotheses of Theorem 2.9 hold, that 𝑇 is real analytic at π‘₯, and that πœ‡=1. Let 𝑐𝑗, 𝑑𝑗, 𝑗=2,3,… be defined by the Taylor series 𝑇=x+𝑑v𝑐x+v𝑑+2,𝑑2𝑑2𝑐+β‹―+𝑛,𝑑𝑛𝑑𝑛+β‹―.(2.5) Suppose that there exists an index β„“β‰₯2 such that (𝑐ℓ,𝑑ℓ)β‰ (0,0) and (𝑐𝑗,𝑑𝑗)=(0,0) for 𝑗<𝑙. If either (a)𝑐ℓ𝑑ℓ<0, or (b)𝑐ℓ≠0, 𝑇(x+𝑑v)(2)is affine in t, or (c)𝑑ℓ≠0, 𝑇(x+𝑑𝑣)(1)is affine in 𝑑, then there exists an order interval ℐ which is also a relative neighborhood of x such that, for every relative neighborhood π’°βŠ‚β„ of x, the following statements are true. (i)If β„“ is odd and (𝑐ℓ,𝑑ℓ)βͺ―se(0,0), then π’°βˆ©int𝒬4(x) contains a supersolution, and π’°βˆ©int𝒬2(x) contains a subsolution. In this case, for every xβˆˆβ„βˆ©int(𝒬2(x)βˆͺ𝒬4(x)), there exists 𝑁 such that 𝑇𝑛(x)βˆ‰β„ for 𝑛β‰₯𝑁. (ii)If β„“ is odd and (0,0)βͺ―𝑠𝑒(𝑐ℓ,𝑑ℓ), then π’°βˆ©int𝒬4(x) contains a subsolution and π’°βˆ©int𝒬2(x) contains a supersolution. In this case, 𝑇𝑛(x)β†’x for every xβˆˆβ„. (iii)If β„“ is even and (𝑐ℓ,𝑑ℓ)βͺ―𝑠𝑒(0,0), then π’°βˆ©int𝒬4(x) contains a subsolution and π’°βˆ©int𝒬2(x) contains a subsolution. In this case, 𝑇𝑛(x)β†’x for every xβˆˆβ„βˆ©π’¬4(x), and for every xβˆˆβ„βˆ©int(𝒬2(x)), there exists 𝑁 such that 𝑇𝑛(x)βˆ‰β„ for 𝑛β‰₯𝑁.(iv)If β„“ is even and (0,0)βͺ―𝑠𝑒(𝑐ℓ,𝑑ℓ), then π’°βˆ©int𝒬2(x) contains a supersolution and π’°βˆ©int𝒬4(x) contains a supersolution. In this case, 𝑇𝑛(x)β†’x for every xβˆˆβ„βˆ©π’¬2(x), and, for every xβˆˆβ„βˆ©int(𝒬4(x)) there exists, 𝑁 such that 𝑇𝑛(x)βˆ‰β„ for 𝑛β‰₯𝑁.

3. Some Basic Facts

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

3.1. Equilibrium Points

The equilibrium points (π‘₯,𝑦) of the system (1.1) satisfy𝛽π‘₯=1π‘₯𝐡1π‘₯+𝑦,𝛼𝑦=2+𝛾2𝑦𝐴2+π‘₯.(3.1) Solutions of System (3.1) are(i)π‘₯=0, 𝑦=𝛼2/𝐴2βˆ’π›Ύ2 when 𝐴2>𝛾2, that is, 𝐸1=𝛼0,2𝐴2βˆ’π›Ύ2ξ‚Ά.(3.2)(ii) If π‘₯β‰ 0, then using System (3.1), we obtain 𝑦=𝛽1βˆ’π΅1π‘₯,0=π‘₯2𝐡1βˆ’π‘₯𝐡1𝛾2βˆ’π΄2ξ€Έ+𝛽1ξ€Έβˆ’ξ€·π›½1𝐴2βˆ’π›Ύ2ξ€Έβˆ’π›Ό2ξ€Έ.(3.3)

Solutions of System (3.3) are π‘₯3,2=𝐡1𝛾2βˆ’π΄2ξ€Έ+𝛽1±√𝐷02𝐡1,𝑦2,3=𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽1±√𝐷02,(3.4) where 𝐷0=(𝐡1(𝛾2βˆ’π΄2)βˆ’π›½1)2βˆ’4𝐡1𝛼2, which gives a pair of the equilibrium points 𝐸2=(π‘₯2,𝑦2) and 𝐸3=(π‘₯3,𝑦3).

Geometrically, the equilibrium points are the intersections of two equilibrium curves: 𝐢1∢π‘₯=0βˆͺ𝑦=βˆ’π΅1π‘₯+𝛽1 and 𝐢2βˆΆπ‘¦=𝛼2/(𝐴2βˆ’π›Ύ2+π‘₯). Depending on the values of parameters, 𝐢2 may have between 0 and 3 intersection points with two lines which constitutes 𝐢1.

The algebraic criteria for the existence of the equilibrium points are summarized in Table 1.

tab1
Table 1

Where 𝑒1=βŽ›βŽœβŽœβŽœβŽπ›½1+𝛽21βˆ’4𝐡1𝛼22𝐡1,𝛽1βˆ’ξ”π›½21βˆ’4𝐡1𝛼22⎞⎟⎟⎟⎠,𝑒2=βŽ›βŽœβŽœβŽœβŽπ›½1βˆ’ξ”π›½21βˆ’4𝐡1𝛼22𝐡1,𝛽1+𝛽21βˆ’4𝐡1𝛼22⎞⎟⎟⎟⎠.(3.5)

Remark 3.1. Observe the following: If the system (1.1) has two or three equilibrium points 𝐸1, 𝐸2, and 𝐸3 then, 𝐸1βͺ―𝐸2βͺ―𝐸3. Indeed, consider the critical curve 𝐢2βˆΆπ‘¦=𝛼2/(𝐴2βˆ’π›Ύ2+π‘₯). Observe that 𝑦(0)=𝛼2/(𝐴2βˆ’π›Ύ2), 𝑦(π‘₯2)=𝑦2, and 𝑦(π‘₯3)=𝑦3. It is obvious that the following holds 0≀π‘₯2≀π‘₯3. Since, the critical curve 𝐢2 decreases, we have 𝑦(0)β‰₯𝑦(π‘₯2)β‰₯𝑦(π‘₯3), that is, 𝐸1βͺ―𝐸2βͺ―𝐸3.

Lemma 3.2. Assume that π‘₯0=0, 𝑦0βˆˆβ„+=(0,∞). Then the following statements are true for solutions of the system (1.1). (i)If 𝐴2>𝛾2, then π‘₯𝑛=0, for all π‘›βˆˆβ„•, and 𝑦𝑛→𝛼2/(𝐴2βˆ’π›Ύ2), π‘›β†’βˆž.(ii)If 𝐴2<𝛾2, then π‘₯𝑛=0, for all π‘›βˆˆβ„•, and π‘¦π‘›β†’βˆž, π‘›β†’βˆž.(iii) If 𝐴2=𝛾2, then 𝑦𝑛=𝑦0+(𝛼2/𝛾2)𝑛, and π‘₯𝑛=0, for all π‘›βˆˆβ„•, π‘¦π‘›β†’βˆž, π‘›β†’βˆž.
Assume that π‘₯0β‰ 0 and (π‘₯0,𝑦0)βˆˆβ„+2. Then, the following statements are true for all 𝑛=1,2,…:(iv)π‘₯𝑛≀𝛽1/𝐡1. (v)𝑦𝑛≀𝑐(𝛾2/𝐴2)𝑛+(𝛼2/(𝐴2βˆ’π›Ύ2)) and (a)𝑦𝑛β‰₯𝐡1𝛼2/(𝐡1(𝐴2βˆ’π›Ύ2)+𝛽1)+πœ–1, 𝐴2>𝛾2, where πœ–1 is arbitrarily small positive number. (b)𝑦𝑛≀𝛼2/(𝐴2βˆ’π›Ύ2)+πœ€2, πœ€>0, 𝐴2>𝛾2, where πœ–2 is arbitrarily small positive number.

Proof. Since (i)–(iv) are immediate consequences of the system (1.1), we will prove only (v).
Take π‘₯0=0 and 𝑦0βˆˆβ„+. Then, we have π‘₯𝑛=0 for all π‘›βˆˆβ„•, and 𝑦𝑛+1=𝛼2𝐴2+𝛾2𝐴2𝑦𝑛.(3.6) Solution of (3.6), when 𝐴2≠𝛾2 is 𝑦𝑛𝛾=𝑐2𝐴2𝑛+𝛼2𝐴2βˆ’π›Ύ2(3.7) which immediately implies (i) and (ii). Statement (iii) follows from (3.6).
Equation π‘₯𝑛+1=𝛽1π‘₯𝑛𝐡1π‘₯𝑛+𝑦𝑛(3.8) implies that π‘₯𝑛≀𝛽1𝐡1.(3.9) Using the last inequality, we have 𝑦𝑛+1=𝛼2+𝛾2𝑦𝑛𝐴2+π‘₯𝑛β‰₯𝛼2+𝛾2𝑦𝑛𝐴2+𝛽1/𝐡1ξ€Έ=𝐡1𝛼2𝐴2𝐡1+𝛽1+𝛾2𝐡1𝐡1𝐴2+𝛽1𝑦𝑛,(3.10) which by difference inequality theorem [18] implies the following 𝑦𝑛𝛾β‰₯𝑐2𝐡1𝐴2𝐡1+𝛽1𝑛+𝐡1𝛼2𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽1β‰₯𝐡1𝛼2𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽1+πœ–1,π‘›βŸΆβˆž.(3.11) Furthermore, second equation in (1.1) implies that 𝑦𝑛+1≀𝛼2𝐴2+𝛾2𝐴2𝑦𝑛(3.12) which, by the difference inequalities argument, see [18], implies that 𝑦𝑛≀𝑒𝑛, where 𝑒𝑛 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 𝛽𝑇(π‘₯,𝑦)=1π‘₯𝐡1,𝛼π‘₯+𝑦2+𝛾2𝑦𝐴2ξ‚Ά+π‘₯.(3.13)

Lemma 3.3. System (1.1) has no minimal period-two solution.

Proof. We have 𝛽𝑇(𝑇(π‘₯,𝑦))=𝑇1π‘₯𝐡1,𝛼π‘₯+𝑦2+𝛾2𝑦𝐴2ξ‚Ά=𝛽+π‘₯21π‘₯𝐴2ξ€Έ+π‘₯𝐡1𝛽1π‘₯𝐴2ξ€Έ+𝛼+π‘₯2+𝛾2𝑦𝐡1ξ€Έ,𝐡π‘₯+𝑦1𝛼π‘₯+𝑦2𝐴2ξ€Έ+π‘₯+𝛾2𝛼2+𝛾2𝑦𝐴2𝐴+π‘₯ξ€Έξ€·2𝐡1ξ€Έπ‘₯+𝑦+𝛽1π‘₯ξ€Έξƒͺ.(3.14) Period-two solution satisfies 𝛽21π‘₯𝐴2ξ€Έ+π‘₯𝐡1𝛽1π‘₯𝐴2ξ€Έ+𝛼+π‘₯2+𝛾2𝑦𝐡1𝐡π‘₯+π‘¦βˆ’π‘₯=0,(3.15)1𝛼π‘₯+𝑦2𝐴2ξ€Έ+π‘₯+𝛾2𝛼2+𝛾2𝑦𝐴2𝐴+π‘₯ξ€Έξ€·2𝐡1ξ€Έπ‘₯+𝑦+𝛽1π‘₯ξ€Έβˆ’π‘¦=0.(3.16) We show that this system has no other positive solutions except the equilibrium points.
Equation (3.15) is equivalent to π‘₯𝑦𝛼2+π‘₯2𝐡1𝛽1βˆ’π΄2𝛽21ξ€·+π‘₯βˆ’π›½21+𝐡1𝛼2+𝐴2𝛽1ξ€Έξ€Έ+𝑦2𝛾2+π‘₯𝑦𝐡1𝛾2ξ€Έ=0,(3.17)βˆ’π‘₯𝑦2𝐴2+π‘₯2𝐡1𝛼2+π‘₯2π‘¦ξ€·βˆ’π΄2𝐡1βˆ’π›½1𝐴+𝑦2𝛼2+𝛼2𝛾2𝐴+π‘₯2𝐡1𝛼2+𝐡1𝛼2𝛾2ξ€Έ+𝑦2ξ€·βˆ’π΄22+𝛾22ξ€Έξ€·+π‘₯π‘¦βˆ’π΄22𝐡1+𝛼2βˆ’π΄2𝛽1+𝐡1𝛾22ξ€Έ=0.(3.18)
If π‘₯=0 then we obtain the fixed point 𝐸1. So assume that π‘₯β‰ 0. Then, using (3.17), we have 𝑦𝛼2+π‘₯2𝐡1𝛽1βˆ’π΄2𝛽21ξ€·+π‘₯βˆ’π›½21+𝐡1𝛼2+𝐴2𝛽1ξ€Έξ€Έ+𝑦2𝛾2+π‘₯𝑦𝐡1𝛾2=0.(3.19) Equation (3.19) implies that 𝑦2=βˆ’π‘¦π›Ό2βˆ’π‘₯2𝐡1𝛽1+𝐴2𝛽21ξ€·βˆ’π‘₯βˆ’π›½21+𝐡1𝛼2+𝐴2𝛽1ξ€Έξ€Έβˆ’π‘₯𝑦𝐡1𝛾2𝛾2.(3.20) Substituting (3.20) into (3.18), we have π‘₯+𝐴2=0(3.21) or 𝐴2𝑦+π‘₯𝐡1𝛼2+ξ€·π‘₯+𝐴2ξ€Έξ€·π‘₯𝐡1βˆ’π›½1𝛽1ξ€Έ+𝑦+π‘₯𝐡1𝛼2βˆ’π‘₯𝑦𝛽1𝛾2+𝛽1ξ€·βˆ’π‘₯𝐡1+𝛽1𝛾22=0.(3.22) Equation (3.21) implies thatπ‘₯=βˆ’π΄2, and (3.23) implies that 𝐴𝑦=2ξ€·ξ€·π‘₯+𝐴2𝛽21βˆ’π‘₯𝐡1𝛼2+ξ€·π‘₯+𝐴2𝛽1ξ€Έξ€Έβˆ’π‘₯𝐡1𝛼2𝛾2+ξ€·π‘₯𝐡1βˆ’π›½1𝛽1𝛾22𝐴2𝛼2+𝛼2βˆ’π‘₯𝛽1𝛾2.(3.23) Replacing (3.23) into (3.19), we get βˆ’π‘₯2𝐡1βˆ’π›Ό2+𝛽1𝐴2βˆ’π›Ύ2ξ€Έξ€·+π‘₯βˆ’π΄2𝐡1+𝛽1+𝐡1𝛾2ξ€Έ=0,(3.24)βˆ’ξ‚€π›½1𝛼2+𝛽1𝐴2βˆ’π›Ύ2𝐴2+𝛾2ξ€Έ2𝛼+π‘₯2+𝛽1𝐴2βˆ’π›Ύ2𝐴2+𝛾2𝐴2𝐡1βˆ’π›½1ξ€Έ+𝐡1𝛾2ξ€Έ+π‘₯2𝛽1𝐴22𝐡1+𝐴2𝐡1𝛾2+𝛽1𝛾2ξ€Έ=0.(3.25)
Solutions of (3.24) are the equilibrium points.
Consider (3.25). Discriminant of this equation is given by ξ‚€Ξ”=4𝛽21𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2ξ€Έ+𝛼2+𝛽1𝐴2βˆ’π›Ύ2𝛽1βˆ’π΅1𝐴2𝛾2ξ€Έξ€Έ2×𝛼2+𝛽1𝐴2βˆ’π›Ύ2.ξ€Έξ€Έ(3.26) Now, Ξ”β‰₯0 implies that π‘₯1𝐴=βˆ’2+𝛾2ξ€Έξ‚€Ξ”1+βˆšΞ”ξ‚2𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2ξ€Έ,π‘₯2𝐴=βˆ’2+𝛾2ξ€Έξ‚€Ξ”1βˆ’βˆšΞ”ξ‚2𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2ξ€Έ,(3.27) where Ξ”1=𝛼2+𝛽1𝐴2βˆ’π›Ύ2ξ€Έξ€Έξ€·βˆ’π›½1+𝐡1𝐴2+𝛾2ξ€Έξ€Έ.(3.28) Using (3.23), we obtain 𝑦1Ξ”=βˆ’2βˆ’βˆšΞ”2𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2ξ€Έ,𝑦2Ξ”=βˆ’2+βˆšΞ”2𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2ξ€Έ,(3.29) where Ξ”2=𝐴22𝐡1𝛽1+𝛼2𝛽1βˆ’π΅1𝛾2ξ€Έ+𝛽1𝛾2𝛽1+𝐡1𝛾2ξ€Έ+𝐴2𝛽21βˆ’π΅1𝛼2βˆ’2𝛽1𝛾2.ξ€Έξ€Έ(3.30) We prove the following claims.
Claim 3.4. For all values of parameters𝑦2<0.Proof. If Ξ”2>0, then 𝑦2<0. Now, we assume that Ξ”2≀0. Then, 𝑦2<0βŸΊΞ”2+βˆšΞ”>0βŸΊΞ”βˆ’Ξ”22>0,(3.31)Ξ”βˆ’Ξ”22=βˆ’4𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2𝐴2𝐡1ξ€·βˆ’π›Ό2+𝛽1𝛾2ξ€Έ+𝛾2𝛽21+𝐡1ξ€·βˆ’π›Ό2+𝛽1𝛾2.ξ€Έξ€Έξ€Έ(3.32) Equation (3.32) implies that Ξ”βˆ’Ξ”22>0βŸΊπ›Ό2>𝐴2𝐡1𝛽1𝛾2+𝛽21𝛾2+𝐡1𝛽1𝛾22𝐡1𝐴2+𝛾2ξ€Έ.(3.33) Since Ξ”2≀0 if and only if 𝐡1>𝛽1𝐴2+𝛾2,𝛼2β‰₯𝐴22𝐡1𝛽1+𝐴2𝛽21+2𝐴2π΅βˆ’1𝛽1𝛾2+𝛽21𝛾2+𝐡1𝛽1𝛾22𝐴2𝐡1βˆ’π›½1+𝐡1𝛾2,𝐴22𝐡1𝛽1+𝐴2𝛽21+2𝐴2π΅βˆ’1𝛽1𝛾2+𝛽21𝛾2+𝐡1𝛽1𝛾22𝐴2𝐡1βˆ’π›½1+𝐡1𝛾2βˆ’π΄2𝐡1𝛽1𝛾2+𝛽21𝛾2+𝐡1𝛽1𝛾22𝐡1𝐴2+𝛾2ξ€Έ=𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2𝛽1+𝐡1𝐴2+𝛾2𝐡1𝐴2+𝛾2ξ€Έξ€·βˆ’π›½1+𝐡1𝐴2+𝛾2ξ€Έξ€Έ>0.(3.34) This implies that (3.33) is true. That is 𝑦2<0.Claim 3.5. Assume that π‘₯1β‰₯0. Then that 𝑦1<0.Proof. Assume that π‘₯1>0. This is equivalent to 𝐴2<𝛾2,𝐡1+𝛽1𝐴2+3𝛾2𝐴22βˆ’π›Ύ22𝛼>0,(3.35)2ξƒ©π›½β‰€βˆ’1𝐴2𝐴2𝐡1+𝛽1ξ€Έ+3𝛽1𝛾2βˆ’π΅1𝛾22𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξƒͺξ€Έξ€Έ.(3.36) Now 𝑦1<0βŸΊΞ”2β‰₯0,Ξ”βˆ’Ξ”22<0,(3.37) which is equivalent to Ξ”2β‰₯0⟺𝐡1<𝛽1𝐴2+𝛾2,𝛼2ξƒ©π›½β‰€βˆ’1𝐴2+𝛾2𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξ€Έξƒͺ(3.38) or 𝐡1β‰₯𝛽1𝐴2+𝛾2.(3.39)
Also, Ξ”βˆ’Ξ”22<0βŸΊπ›Ό2<𝛽1𝛾2𝛽1+𝐡1𝐴2+𝛾2𝐡1𝐴2+𝛾2ξ€Έ.(3.40)
Since, 𝛽1𝐴2𝐴2𝐡1+𝛽1ξ€Έ+3𝛽1𝛾2βˆ’π΅1𝛾22𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2+𝛽1𝛾2𝛽1+𝐡1𝐴2+𝛾2𝐡1𝐴2+𝛾2ξ€Έ=𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2𝛽1+𝐡1𝐴2+𝛾2ξ€Έξ€Έ2𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξ€Έξ€Έ2>0.(3.41)
This inequality and (3.36) imply (3.40).
Since 𝛽1𝐴2𝐴2𝐡1+𝛽1ξ€Έ+3𝛽1𝛾2βˆ’π΅1𝛾22𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξ€Έξ€Έ2βˆ’ξƒ©π›½1𝐴2+𝛾2𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξ€Έξƒͺ=2𝛽1𝐴22𝐡1+𝐴2𝐡1+𝛽1𝛾2𝛽1+𝐡1𝐴2+𝛾2𝛽1βˆ’π΅1𝐴2+𝛾2ξ€Έξ€Έ2>0.(3.42)
Last inequality, (3.36) and (3.39) imply that Ξ”2β‰₯0. So we prove, if π‘₯1>0, then 𝑦1<0.
Assume that π‘₯1=0.
We have π‘₯1=0βŸΊΞ”1+βˆšΞ”=0.(3.43)
Since, Ξ”21βˆ’Ξ”=4𝛽21𝛼2+𝛽1𝐴2βˆ’π›Ύ2𝐴2+𝛾2ξ€Έ2𝐡1𝐴22+𝐡1𝛾2𝐴2+𝛽1𝛾2ξ€Έ,(3.44) we have that π‘₯1=0 if and only if 𝐴2<𝛾2,𝛼2+𝐴2𝛽1=𝛽1𝛾2,Ξ”1≀0,(3.45) which is true, because Ξ”1ξ€·π›ΌβˆΆ=2+𝛽1𝐴2βˆ’π›Ύ2ξ€Έξ€Έξ€·βˆ’π›½1+𝐡1𝐴2+𝛾2ξ€Έξ€Έ.(3.46)
Replacing 𝛼2 with 𝛽1(𝛾2βˆ’π΄2) in the formula for 𝑦1, we obtain that 𝑦1=βˆ’π›½1.
Hence, there does not exist period-two solution.

4. Linearized Stability Analysis

The Jacobian matrix of the map 𝑇, given by (3.13), has the form𝐽𝑇=βŽ›βŽœβŽœβŽœβŽœβŽœβŽπ›½1𝑦𝐡1ξ€Έπ‘₯+𝑦2βˆ’π›½1π‘₯𝐡1ξ€Έπ‘₯+𝑦2βˆ’π›Ό2+𝛾2𝑦𝐴2ξ€Έ+π‘₯2𝛾2𝐴2⎞⎟⎟⎟⎟⎟⎠+π‘₯.(4.1) The determinant of (4.1) is given by det𝐽𝑇𝛽(π‘₯,𝑦)=1𝑦𝐴2𝛾2βˆ’π‘₯𝛼2ξ€Έξ€·π‘₯+𝐴2ξ€Έ2𝑦+π‘₯𝐡1ξ€Έ2.(4.2) The value of the Jacobian matrix of 𝑇 at the equilibrium point 𝐸=(π‘₯,𝑦), π‘₯β‰ 0 is𝐽𝑇π‘₯,𝑦=βŽ›βŽœβŽœβŽœβŽπ‘¦π΅1π‘₯+π‘¦βˆ’π‘₯𝐡1π‘₯+π‘¦βˆ’π‘¦π΄2+π‘₯𝛾2𝐴2+π‘₯⎞⎟⎟⎟⎠.(4.3) The determinant of (4.3) is given by det𝐽𝑇π‘₯,𝑦=𝑦𝛾2βˆ’π‘₯𝐴2+π‘₯𝐡1π‘₯+𝑦,(4.4) and the trace of (4.3) isTr𝐽𝑇π‘₯,𝑦=𝑦𝐡1π‘₯+𝑦+𝛾2𝐴2+π‘₯.(4.5) The characteristic equation has the form πœ†2ξ‚΅βˆ’πœ†π‘¦π΅1π‘₯+𝑦+𝛾2𝐴2+π‘₯ξ‚Ά+𝑦𝛾2βˆ’π‘₯𝐴2+π‘₯𝐡1π‘₯+𝑦=0.(4.6)

Theorem 4.1. Assume that 𝐴2>𝛾2. Then there exists the equilibrium point 𝐸1 and (i)𝐸1(0,𝛼2/(𝐴2βˆ’π›Ύ2)) is locally asymptotically stable if 𝛽1<𝛼2/(𝐴2βˆ’π›Ύ2),(ii)𝐸1 is a saddle point if 𝛽1>𝛼2/(𝐴2βˆ’π›Ύ2). The corresponding eigenvalues are πœ†1=𝛾2𝐴2∈(0,1),πœ†2=𝛽1𝐴2βˆ’π›Ύ2𝛼2∈(1,+∞),(4.7)(iii)𝐸1 is nonhyperbolic if 𝛽1=𝛼2/(𝐴2βˆ’π›Ύ2). The corresponding eigenvalues are πœ†1=𝛾2𝐴2∈(0,1),πœ†2=1,(4.8) and the corresponding eigenvectors are (0,1) and (βˆ’1/𝛼2,1), respectively.

Proof. The Jacobian matrix (4.1) at the equilibrium point 𝐸1(0,𝛼2/(𝐴2βˆ’π›Ύ2)), 𝐽𝑇𝐸1ξ€Έ=βŽ›βŽœβŽœβŽœβŽπ›½1𝑦0βˆ’π›Ό2+𝛾𝑦𝐴22𝛾2𝐴2⎞⎟⎟⎟⎠.(4.9)
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 det𝐽𝑇π‘₯,𝑦=𝛽1𝑦𝛾2𝐴2=𝛽1𝛾2𝐴2βˆ’π›Ύ2𝛼2𝐴2.(4.10) Note that, under the hypothesis of Theorem, the determinant is greater than zero.
The trace of (4.9) is Tr𝐽𝑇π‘₯,𝑦=𝛽1𝑦+𝛾2𝐴2=𝛽1𝐴2βˆ’π›Ύ2𝛼2+𝛾2𝐴2.(4.11)
An equilibrium point is locally asymptotically stable if the following conditions are satisfied ||Tr𝐽𝑇π‘₯,𝑦||<1+det𝐽𝑇π‘₯,𝑦<2.(4.12)
Now, these two conditions become 𝛽1𝐴2βˆ’π›Ύ2𝛼2+𝛾2𝐴2𝛽<1+1𝛾2𝐴2βˆ’π›Ύ2𝛼2𝐴2<2.(4.13)
Condition 𝛽1𝐴2βˆ’π›Ύ2𝛼2+𝛾2𝐴2𝛽<1+1𝛾2𝐴2βˆ’π›Ύ2𝛼2𝐴2(4.14) implies that 𝐴2βˆ’π›Ύ2𝛽1<𝛼2.(4.15)
If 𝛽1<𝛼2/(𝐴2βˆ’π›Ύ2), then this condition is satisfied.
Condition 1+(𝛽1𝛾2(𝐴2βˆ’π›Ύ2))/𝛼2𝐴2<2 is equivalent to 𝛽1𝛾2𝐴2βˆ’π›Ύ2𝛼2𝐴2<1.(4.16) It is easy to see that the condition is satisfied if 𝛽1<𝛼2/(𝐴2βˆ’π›Ύ2).
Next, we prove (ii).
An equilibrium point is a saddle if and only if the following conditions are satisfied ||Tr𝐽𝑇π‘₯,𝑦||>||1+det𝐽𝑇π‘₯,𝑦||,Tr2𝐽𝑇π‘₯,π‘¦ξ€Έβˆ’4det𝐽𝑇π‘₯,𝑦>0.(4.17)
The first condition is equivalent to 𝐴2βˆ’π›Ύ2𝛽1>𝛼2,(4.18) which is satisfied if 𝛽1>𝛼2/(𝐴2βˆ’π›Ύ2). The second condition is equivalent to 𝛽1𝐴2βˆ’π›Ύ2𝛼2βˆ’π›Ύ2𝐴2ξƒͺ2>0.(4.19)
Finally, we prove (iii).
An equilibrium point is nonhyperbolic if the following conditions are satisfied ||Tr𝐽𝑇π‘₯,𝑦||=||1+det𝐽𝑇π‘₯,𝑦||,ξ€·det𝐽𝑇π‘₯,𝑦||=1orπ‘‡π‘Ÿπ½π‘‡ξ€·π‘₯,𝑦||ξ€Έ.≀2(4.20)
The first condition is equivalent to 𝐴2βˆ’π›Ύ2𝛽1=𝛼2,(4.21) which is satisfied if 𝛽1=𝛼2/𝐴2βˆ’π›Ύ2.
The second condition becomes 𝛽1𝐴2βˆ’π›Ύ2𝛼2+𝛾2𝐴2𝛾=1+2𝐴2<2,(4.22) establishing part (iii).

We now perform a similar analysis for the other cases in Table 1.

Theorem 4.2. Assume that 𝐴2>𝛾2,𝐡1𝛾2βˆ’π΄2ξ€Έ+√4𝐡1𝛼2<𝛽1<𝛼2𝐴2βˆ’π›Ύ2,𝛼2>𝐡1𝐴2βˆ’π›Ύ2ξ€Έ2.(4.23) Then 𝐸1, 𝐸2, and 𝐸3 exist and(i)the equilibrium point 𝐸1 is locally asymptotically stable,(ii)the equilibrium point 𝐸2 is a saddle point. Furthermore, if 𝐡1(𝛾2βˆ’π΄2√)+4𝐡2𝛼2<𝛽1<𝛼2/(𝐴2+𝐡1𝛾2) and 𝛼2>𝐡1𝐴22, then the smaller eigenvalue belongs to the interval (βˆ’1,0), and the larger eigenvalue belongs to (1,+∞). In all other cases, the smaller eigenvalues is in (0,1).That is |πœ†1|<1 is given by πœ†1=𝛽1𝛾2+𝐴2+π‘₯2𝑦2βˆ’ξ”ξ€·π›½1𝛾2+𝐴2+π‘₯2𝑦2ξ€Έ2+4𝛽1𝐴2+π‘₯2ξ€Έξ€·π‘₯2βˆ’π›Ύ2𝑦22𝛽1𝐴2+π‘₯2ξ€Έ,(4.24) and the corresponding eigenvector is 𝑣1=𝛽1𝛾2βˆ’ξ€·π΄2+π‘₯2𝑦2+𝛽1𝛾2+𝐴2+π‘₯2𝑦2ξ€Έ2+4𝛽1𝐴2+π‘₯2ξ€Έξ€·π‘₯2βˆ’π›Ύ2𝑦2,2𝛽1𝑦2ξ‚Ά.(4.25) Eigenvalue πœ†2, where |πœ†2|>1, is given by πœ†2=𝛽1𝛾2+𝐴2+π‘₯2𝑦2+𝛽1𝛾2+𝐴2+π‘₯2𝑦2ξ€Έ2+4𝛽1𝐴2+π‘₯2ξ€Έξ€·π‘₯2βˆ’π›Ύ2𝑦22𝛽1𝐴2+π‘₯2ξ€Έ.(4.26)(iii) The equilibrium point 𝐸3 is locally asymptotically stable.

Proof. By Theorem 4.1, (i) holds.
Evaluating the Jacobian matrix (4.3) at the equilibrium point 𝐸2, we obtain 𝐽𝑇=βŽ›βŽœβŽœβŽœβŽπ‘¦π›½1βˆ’π‘₯𝛽1βˆ’π‘¦π΄2+π‘₯𝛾2𝐴2+π‘₯⎞⎟⎟⎟⎠.(4.27) Note that the Jacobian matrix (4.27) implies that the map 𝑇 is strongly competitive.
The determinant of (4.27) is given by det=𝑦𝛾2𝛽1𝐴2+π‘₯ξ€Έβˆ’π‘₯𝑦𝛽1𝐴2+π‘₯ξ€Έ,(4.28) and the trace of (4.27) is given by Tr𝐽𝑇π‘₯,𝑦=𝑦𝛽1+𝛾2𝐴2+π‘₯.(4.29) The equilibrium point 𝐸2 is a saddle if and only if (4.17) is satisfied. The first condition is equivalent to 𝑦𝛽1+𝛾2𝐴2+π‘₯>||||1+𝑦𝛾𝛽1𝐴2+π‘₯ξ€Έβˆ’π‘₯𝑦𝛽1𝐴2+π‘₯ξ€Έ||||(4.30) which is equivalent to 𝑦𝐴2+π‘₯ξ€Έ+𝛽1𝛾2>𝛽1𝐴2+π‘₯ξ€Έ+𝑦𝛾2βˆ’π‘₯ξ€ΈβŸΉπ‘¦ξ€·π΄2+π‘₯ξ€Έ+𝛽1𝐴π‘₯βˆ’2βˆ’π›Ύ2βˆ’ξ€Έξ€Έπ‘¦ξ€·π›Ύ2βˆ’π‘₯ξ€ΈβŸΉ>0𝑦𝐴2+π‘₯βˆ’π›Ύ2ξ€Έβˆ’π›½1𝐴2βˆ’π›Ύ2+π‘₯ξ€Έ>βˆ’π‘₯π‘¦βŸΉξ€·π΄2βˆ’π›Ύ2+π‘₯ξ€Έξ€·π‘¦βˆ’π›½1ξ€Έ>βˆ’π‘₯π‘¦βŸΉξ€·π›½1βˆ’π‘¦π΄ξ€Έξ€·2βˆ’π›Ύ2+π‘₯ξ€Έ<π‘₯𝑦.(4.31)
In light of (3.3) 𝛽1βˆ’π‘¦2=𝐡1π‘₯2, and by using (3.4), 𝐴2βˆ’π›Ύ2+π‘₯2=𝑦3/𝐡1. Now, we have 𝐡1π‘₯2𝑦3𝐡1<π‘₯2𝑦2.(4.32) This implies that 𝑦3<𝑦2 which is true.
Condition Tr2𝐽𝑇π‘₯,π‘¦ξ€Έβˆ’4det𝐽𝑇π‘₯,𝑦>0(4.33) is equivalent to 𝑦2𝛽1βˆ’π›Ύ2𝛽1𝐴2+π‘₯2ξ€Έξƒͺ2+π‘₯2𝑦2𝛽2𝐴2+π‘₯2ξ€Έ>0,(4.34) 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 πœ†1+πœ†2=𝑦𝛽1+𝑦𝛾2𝐴2+π‘₯𝛽1,πœ†1πœ†2=𝑦𝛾2βˆ’π‘₯𝐴2+π‘₯𝛽1.(4.35)
Since the map 𝑇 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 that 𝛾2βˆ’π‘₯2<0.
We have 𝛾2βˆ’π‘₯2=𝛾2βˆ’π΅1𝛾2βˆ’π΄2ξ€Έ+𝛽1βˆ’ξ”ξ€·π΅1𝛾2βˆ’π΄2ξ€Έβˆ’π›½1ξ€Έ2βˆ’4𝐡1𝛼22𝐡1=𝐡1𝐴2+𝛾2ξ€Έβˆ’π›½1+𝐡1𝛾2βˆ’π΄2ξ€Έβˆ’π›½1ξ€Έ2βˆ’4𝐡1𝛼22𝐡1.(4.36) Now 𝛾2βˆ’π‘₯2<0 if and only if 𝐡1𝐴2+𝛾2ξ€Έβˆ’π›½1+𝐡1𝛾2βˆ’π΄2ξ€Έβˆ’π›½1ξ€Έ2βˆ’4𝐡1𝛼2<0,(4.37) which holds if 𝛽1>𝐡1𝐴2+𝛾2ξ€Έ,𝐡1𝐴2+𝛾2ξ€Έβˆ’π›½1ξ€Έ2βˆ’ξ€·π΅1𝛾2βˆ’π΄2ξ€Έβˆ’π›½1ξ€Έ2+4𝐡1𝛼2𝐡>0,1𝐴2+𝛾2ξ€Έ<𝛽1<𝛼2𝐴2+𝐡1𝛾2,𝛼2𝐴2+𝐡1𝛾2βˆ’ξ€·π΄2𝐡1+𝛾2𝐡1ξ€Έ=𝛼2βˆ’π΄22𝐡1𝐴2>0⟺𝐡1<𝛼2𝐴22.(4.38)
Also, we have 𝛼2𝐴2+𝐡1𝛾2βˆ’ξ‚€π΅1𝛾2βˆ’π΄2ξ€Έβˆš+2𝐡1𝛼2β‰₯0.(4.39)
In all other cases 𝛾2βˆ’π‘₯2>0.
This proves that the smaller eigenvalue is negative. Since the equilibrium point is a saddle point, it has to belong to (βˆ’1,0). The larger one belongs to (1,+∞). The proof of second statement is similar.
Now, we prove that 𝐸3 is locally asymptotically stable.
Notice that ||Tr𝐽𝑇π‘₯,𝑦||<1+det𝐽𝑇π‘₯,𝑦<2(4.40) implies that 𝑦2>𝑦3 which is true.

Theorem 4.3. Assume that 𝐴2>𝛾2,𝛽1>𝐴2βˆ’π›Ύ2𝐡1,𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽1ξ€Έ2βˆ’4𝛼2𝐡1=0.(4.41) Then 𝐸1,𝐸2=𝐸3 exist and:(i)The equilibrium point 𝐸1 is locally asymptotically stable. (ii)The equilibrium point 𝐸2=𝐸3=(𝐡1(𝛾2βˆ’π΄2)+𝛽1/2𝐡1,𝐡1(𝐴2βˆ’π›Ύ2)+𝛽1/2) is nonhyperbolic. The eigenvalues of the Jacobian matrix evaluated at 𝐸2 are πœ†1=1,πœ†2=𝐴22𝐡21βˆ’π›½21+2𝐡1𝛽1𝛾2βˆ’π΅21𝛾222𝛽1𝐴2𝐡1+𝛽1+𝐡1𝛾2ξ€Έ,(4.42) and the corresponding eigenvectors, respectively, are ξ‚΅βˆ’1𝐡1ξ‚Ά,,1βˆ’π΄2𝐡1+𝛽1+𝐡1𝛾2𝐴2𝐡1+𝛽1+𝐡1𝛾2ξ€Έ2𝐡1𝛽1𝐴2𝐡1+𝛽1βˆ’π΅1𝛾2ξ€Έξƒͺ,1.(4.43) Furthermore, if 𝛽1>𝐡1(𝐴2+𝛾2), then πœ†2∈(βˆ’1,0). If 𝐡1(𝐴2βˆ’π›Ύ2)<𝛽1<𝐡1(𝐴2+𝛾2), then πœ†2∈(0,1).

Proof. By Theorem 4.1, 𝐸1 is a locally asymptotically stable.
Now, we prove that 𝐸2 is nonhyperbolic.
The Jacobian matrix (4.3) at the equilibrium point 𝐸2=𝐸3=(𝐡1(𝛾2βˆ’π΄2)+𝛽1/2𝐡1,𝐡1(𝐴2βˆ’π›Ύ2)+𝛽1/2) is 𝐽𝑇𝐸2ξ€Έ=βŽ›βŽœβŽœβŽœβŽπ‘¦π›½1βˆ’π‘₯𝛽1βˆ’π‘¦π΄2+π‘₯𝛾2𝐴2+π‘₯⎞⎟⎟⎟⎠.(4.44)
The eigenvalues of (4.44) satisfy 𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽12𝛽1βˆ’πœ†ξƒͺ2𝐡1𝛾22𝐡1𝐴2+𝐡1𝛾2βˆ’π΄2ξ€Έ+𝛽1ξƒͺβˆ’π΅βˆ’πœ†1𝛾2βˆ’π΄2ξ€Έ+𝛽12𝛽1𝐡1𝐡1𝐡1𝐴2βˆ’π›Ύ2ξ€Έ+𝛽1ξ€Έ