Bifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Terms

and Applied Analysis 3 can not encounter chaos in such systems as has been shown by Smith; see [16]. If k = (u, V) ∈ R, we denote with Ql(k), l ∈ {1, 2, 3, 4}, the four quadrants inR relative to k, that is,Q1(k) = {(x, y) ∈ R : x ≥ u, y ≥ V}, Q2(k) = {(x, y) ∈ R : x ≤ u, y ≥ V}, and so on. Define the South-East partial order ⪯se on R by (x, y) ⪯se (s, t) if and only if x ≤ s and y ≥ t. Similarly, we define theNorth-East partial order ⪯ne onR by (x, y) ⪯ne (s, t) if and only if x ≤ s and y ≤ t. For A ⊂ R and x ∈ R, define the distance from x to A as dist(x,A) fl inf{‖x − y‖ : y ∈ A}. By intA we denote the interior of a set A. It is easy to show that a map F is competitive if it is nondecreasing with respect to the South-East partial order, that is, if the following holds: (x1 y1)⪯se (x2 y2) 󳨐⇒ F(x1 y1)⪯se F(x2 y2) . (8) For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions see [10]. We now state three results for competitive maps in the plane. The following definition is from [16]. Definition 1. LetS be a nonempty subset ofR. A competitive mapT : S → S is said to satisfy condition (O+) if for every x, y in S, T(x) ⪯ne T(y) implies x⪯ne y, and T is said to satisfy condition (O−) if for every x, y in S, T(x) ⪯ne T(y) implies y⪯ne x. The following theorem was proved by de MottoniSchiaffino [17] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [13, 14]. Theorem 2. Let S be a nonempty subset of R. If T is a competitive map for which (O+) holds then for all x ∈ S, {Tn(x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T. If instead (O−) holds, then for all x ∈ S, {T2n(x)} is eventually componentwise monotone. If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point. The following result is from [16], with the domain of the map specialized to be the Cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O−). Theorem 3. Let R ⊂ R be the Cartesian product of two intervals in R. Let T : R → R be a C1 competitive map. If T is injective and det JT(x) > 0 for all x ∈ R then T satisfies (O+). If T is injective and det JT(x) < 0 for all x ∈ R then T satisfies (O−). The following result is a direct consequence of the TrichotomyTheoremofDancer andHess (see [18]) and is helpful for determining the basins of attraction of the equilibrium points. Corollary 4. If the nonnegative cone of ⪯ is a generalized quadrant in R, 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. Next result is well known global attractivity result which holds in partially ordered Banach spaces as well; see [18]. Theorem 5. Let T be a monotone map on a closed and bounded rectangular region R ⊂ R. Suppose that T has a unique fixed point e inR. Then e is a global attractor of T on R. The following theorems were proved by Kulenović and Merino [1] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or nonhyperbolic) is by absolute value smaller than 1 while the other has an arbitrary value. These results are useful for determining basins of attraction of fixed points of competitive maps. Theorem 6. Let T be a competitive map on a rectangular region R ⊂ R. Let x ∈ R be a fixed point of T such that Δ fl R∩ int(Q1(x)∪Q3(x)) is nonempty (i.e., x is not the NW or SE vertex ofR), and T is strongly competitive on Δ. Suppose that the following statements are true: (a) The map T has a C1 extension to a neighborhood of x. (b) The Jacobian JT(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 axis. Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace 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 ofR are either fixed points or minimal period-two points. In the latter case, the set of endpoints ofC is a minimal period-two orbit of T. The situation where the endpoints of C are boundary points ofR is of interest.The following result gives a sufficient condition for this case. Theorem7. For the curveC ofTheorem 6 to have endpoints in ∂R, it is sufficient that at least one of the following conditions is satisfied (i) The map T has no fixed points or periodic points of minimal period-two in Δ. (ii) ThemapT has no fixed points inΔ, det JT(x) > 0, and T(x) = x has no solutions x ∈ Δ. (iii) The map T has no points of minimal period-two in Δ, det JT(x) < 0, and T(x) = x has no solutions x ∈ Δ. 4 Abstract and Applied Analysis The next result is useful for determining basins of attraction of fixed points of competitive maps. Theorem 8. (A) Assume the hypotheses of Theorem 6, and let C be the curve whose existence is guaranteed by Theorem 6. If the endpoints ofC belong to ∂R, thenC separatesR into two connected components, namely, W− fl {x ∈ R \C : ∃y ∈ C with x ⪯se y} , W+ fl {x ∈ R \C : ∃y ∈ C with y⪯se x} , (9) 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 every x ∈ W− there exists n0 ∈ N such thatTn(x) ∈ intQ2(x) for n ≥ n0. (iv) For every x ∈ W+ there exists n0 ∈ N such thatTn(x) ∈ intQ4(x) for n ≥ n0. If T is a map on a setR and if x is a fixed point of T, the stable set Ws(x) of x is the set {x ∈ R : Tn(x) → x} and unstable set Wu(x) of x is the set {x ∈ R : there exists {xn}0n=−∞ ⊂ R s.t. T (xn) = xn+1, x0 = x, lim n→−∞xn = x} . (10) WhenT is noninvertible, the setWs(x)maynot be connected andmade up of infinitelymany curves, orWu(x)maynot be a manifold.The following result gives a description of the stable and unstable sets of a saddle point of a competitivemap. If the map is a diffeomorphismonR, the setsWs(x) andWu(x) are the stable and unstable manifolds of x. Theorem 9. In addition to the hypotheses of part (B) of Theorem 8, suppose that μ > 1 and that the eigenspace Eμ associated with μ is not a coordinate axis. If the curve C of Theorem 6 has endpoints in ∂R, thenC is the stable setWs(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 ofWu(x) inR are fixed points of T. 3. Number of Equilibria In this section we give some basic facts which are used later. Let T be the map associated with system (1) given by T (x, y) = (f (x, y) , g (x, y)) = ( x A1 + B1x + C1y, y2 A2 + B2x + C2y2) . (11) Let R = R2+. The equilibrium points (x, y) of system (1) satisfy equations x A1 + B1x + C1y = x, y2 A2 + B2x + C2y2 = y. (12) For x = 0 we have y = y2 − A2y − C2y3 (13) from which we obtain three equilibrium points E1 = (0, 0) , E2 = (0, 1 − √Δ 1 2C2 ) , E3 = (0, 1 + √Δ 1 2C2 ) , (14) where Δ 1 = 1 − 4A2C2. Assume that x ̸ = 0. Then, from the first equation of system (12) we have y = −A1 + B1x − 1 C1 . (15) By substituting this into the second equation we obtain A1 + B1x − 1 = 0 (16) or g (x) ≡ B2 1C2x2 + x (B1 (2 (A1 − 1)C2 + C1) + B2C2 1) + (A1 − 1)2 C2 + (A1 − 1)C1 + A2C2 1 = 0, (17) Abstract and Applied Analysis 5and Applied Analysis 5 from which we obtain the other three equilibrium points E4 = (1 − A1 B1 , 0) , E5 = (−2A1B1C2 − B2C2 1 − B1C1 + 2B1C2 + C1√Δ 2 2B2 1C2 , B1 + B2C1 − √Δ 2 2B1C2 ) , E6 = (−2A1B1C2 − B2C2 1 − B1C1 + 2B1C2 − C1√Δ 2 2B2 1C2 , B1 + B2C1 + √Δ 2 2B1C2 ) , (18) where Δ 2 = (B1 + B2C1)2 − 4B1 (A2B1 − (A1 − 1) B2) C2. (19) Lemma 10. The following hold: (i) The equilibrium points E2 and E3 exist if and only if Δ 1 ≥ 0 and E2 = E3 if and only if Δ 1 = 0. (ii) The equilibrium point E4 exists if and only if A1 ≤ 1 and E4 = E1 if and only if A1 = 1. (iii) Assume that Δ 2 ≥ 0.The equilibrium point E5 exists if and only if A1 < 1 and C1 ≤ (1 − A1) B1 (1 − A1) B2 + 2A2B1 , C2 ≤ (B2C1 + B1)2 4B1 (A2B1 + (1 − A1) B2) (20) or C1 > (1 − A1) B1 (1 − A1) B2 + 2A2B1 , C2 ≤ C1 (1 − A1 − A2C1) (1 − A1)2 . (21) (iv) Assume that Δ 2 ≥ 0.The equilibrium point E6 exists if and only if A1 < 1 and C1 ≤ (1 − A1) B1 (1 − A1) B2 + 2A2B1 , C1 (1 − A1 − A2C1) (1 − A1)2 ≤ C2 ≤ (B2C1 + B1)2 4B1 (A2B1 + (1 − A1) B2) . (22) Proof. Theproof of the statements (i) and (ii) is trivial and we skip it. Nowwe prove the statement (iii). In view of Descartes’ rule of signs we obtain that (17) has no positive solutions if A1 ≥ 1. Now, we suppose that A1 < 1. One can see that y5 > 0 for all values of parameters. We consider two cases: (1) Assume that −2A1B1C2 − B2C2 1 − B1C1 + 2B1C2 ≥ 0, (23) which is equivalent to C2 ≥ C1 (B2C1 + B1) 2 (1 − A1) B1 . (24) Since Δ 2 ≥ 0 ⇐⇒ C2 ≤ (B2C1 + B1)2 4B1 (A2B1 + (1 − A1) B2) (25) we have that x5 ≥ 0 if and only if C1 (B2C1 + B1) 2 (1 − A1) B1 ≤ C2 ≤ (B2C1 + B1)2 4B1 (A2B1 + (1 − A1) B2) , (26) (B2C1 + B1)2 4B1 (A2B1 − (A1 − 1) B2) − C1 (B2C1 + B1) 2 (1 − A1) B1 = −(B2C1 + B1) ((1 − A1) B2C1 + B1 (2A2C1 + A1 − 1)) 4 (1 − A1) B1 (A2B1 + (1 − A1) B2) ≥ 0 (27) which is equivalent to C1 ≤ (1 − A1) B1 2A2B1 + (1 − A1) B2 . (28) From (27) and (28) it follows x5 ≥ 0 if and only if C1 ≤ (1 − A1) B1 2A2B1 + (1 − A1) B2 , C1 (B2C1 + B1) 2 (1 − A1) B1 ≤ C2 ≤ (B2C1 + B1)2 4B1 (A2B1 + (1 − A1) B2) . (29) (2) A


Introduction
In this paper we study the global dynamics of the following rational system of difference equations: where the parameters  1 ,  2 ,  1 ,  2 ,  1 , and  2 are positive numbers and initial conditions  0 and  0 are arbitrary nonnegative numbers.System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane; see [1].System (1) can be used as a mathematical model for competition in population dynamics.System (1) is related to Leslie-Gower competition model where the parameters  1 ,  2 ,  1 ,  2 ,  1 , and  2 are positive numbers and initial conditions  0 and  0 are arbitrary nonnegative numbers, considered in [2].System (2) globally exhibits three dynamic scenarios in five parametric regions which are competitive exclusion, competitive coexistence, and existence of an infinite number of equilibrium solutions; see [1][2][3].System (2) does not exhibit the Allee effect, which is desirable from modeling point of view.The simplest variation of system (2) which exhibits the Allee effect is probably system where the parameters  1 ,  2 ,  1 ,  2 ,  1 , and  2 are positive numbers and initial conditions  0 and  0 are arbitrary nonnegative numbers, considered in [4].System (3) has between 1 and 9 equilibrium points and exhibits nine dynamics scenarios part of each is the Allee effect.In the case of the dynamic which exhibits nine dynamic scenarios and whose dynamics is very similar to the corresponding system without quadratic terms considered in [7].
In general, it seems that an introduction of quadratic terms in equations of the Leslie-Gower model (2) generates the Allee effect.We will test this hypothesis in this paper by introducing the quadratic terms only in the second equation.System (1) can be considered as the competitive version of the decoupled system where the parameters  1 ,  2 ,  1 , and  2 are positive numbers and initial conditions  0 and  0 are arbitrary nonnegative numbers, whose dynamics can be directly obtained from two separate equations.Unlike system (2) which has five regions of parameters with distinct local behavior system (1) has eighteen regions of parameters with distinct local behavior, which is caused by the geometry of the problem, that is, by the geometry of equilibrium curves.More precisely, the equilibrium curves of system (2) are lines while the equilibrium curves of system (1) are a line and a parabola.In the case when  1 > 1, all equilibrium points are hyperbolic and all solutions are attracted to the three equilibrium points on the -axis and we can describe this situation as competitive exclusion case.When  1 = 1, the equilibrium point  1 is nonhyperbolic and dynamics is analogous to the case when  1 > 1.In both cases the Allee effect is present.When  1 < 1, there exist 11 regions of parameters with different global dynamics.In nine of these regions the global dynamics is in competitive exclusion case, which means that all solutions converge to one of the equilibrium points on the axes and in only two situations we have competitive coexistence case, which means that the interior equilibrium points have substantial basin of attraction.In all 11 cases, the zero equilibrium has some basin of attraction which is a part of -axis so we can say that in these cases system (1) exhibits weak Allee's effect.Figure 3 gives the bifurcation diagram showing the transition from different global dynamics situations when  1 < 1, since the cases  1 ≥ 1 are simple and do not need graphical interpretation.
The paper is organized as follows.Section 2 contains some necessary results on competitive systems in the plane.Section 3 provides some basic information about the number of equilibrium points.Section 4 contains local stability analysis of all equilibrium solutions.Section 5 contains some global results on injectivity of the map associated with system (1).Section 6 gives global dynamics of system (1) in all regions of the parameters.

Preliminaries
A first-order system of difference equations  +1 =  (  ,   ) ,  +1 =  (  ,   ) ,  = 0, 1, . . ., (7) where S ⊂ R 2 , (, ) : S → S, ,  are continuous functions is competitive if (, ) is nondecreasing in  and nonincreasing in , and (, ) is nonincreasing in  and nondecreasing in .If both  and  are nondecreasing in  and , system (7) is cooperative.Competitive and cooperative maps are defined similarly.Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions  and  are coordinate-wise strictly monotone.
Competitive and cooperative systems have been investigated by many authors; see [1][2][3][7][8][9][10][11][12][13][14][15][16].Special attention to discrete competitive and cooperative systems in the plane was given in [1-3, 16, 17].One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species.Another reason is that the theory of two-dimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three-dimensional and higher systems.Part of the reason for this situation is de Mottoni-Schiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems.However, this does not mean that one can not encounter chaos in such systems as has been shown by Smith; see [16].
and so on.Define the South-East partial order ⪯ se on R 2 by (, ) ⪯ se (, ) if and only if  ≤  and  ≥ .Similarly, we define the North-East partial order ⪯ ne on R 2 by (, ) ⪯ ne (, ) if and only if  ≤  and  ≤ .For A ⊂ R 2 and  ∈ R 2 , define the distance from  to A as dist(, A) fl inf{‖ − ‖ :  ∈ A}.By int A we denote the interior of a set A.
It is easy to show that a map  is competitive if it is nondecreasing with respect to the South-East partial order, that is, if the following holds: For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions see [10].
We now state three results for competitive maps in the plane.The following definition is from [16].Definition 1.Let S be a nonempty subset of R 2 .A competitive map  : S → S is said to satisfy condition (+) if for every ,  in S, () ⪯ ne () implies  ⪯ ne , and  is said to satisfy condition (−) if for every ,  in S, () ⪯ ne () implies  ⪯ ne .
The following theorem was proved by de Mottoni-Schiaffino [17] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations.Smith generalized the proof to competitive and cooperative maps [13,14].Theorem 2. Let S be a nonempty subset of R 2 .If  is a competitive map for which (+) holds then for all  ∈ S, {  ()} is eventually componentwise monotone.If the orbit of  has compact closure, then it converges to a fixed point of .If instead (−) holds, then for all  ∈ S, { 2 ()} is eventually componentwise monotone.If the orbit of  has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point.
The following result is from [16], with the domain of the map specialized to be the Cartesian product of intervals of real numbers.It gives a sufficient condition for conditions (+) and (−).Theorem 3. Let R ⊂ R 2 be the Cartesian product of two intervals in R. Let  : R → R be a  1 competitive map.If  is injective and det   () > 0 for all  ∈ R then  satisfies (+).If  is injective and det   () < 0 for all  ∈ R then  satisfies (−).
The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess (see [18]) and is helpful for determining the basins of attraction of the equilibrium points.

Corollary 4.
If the nonnegative cone of ⪯ is a generalized quadrant in R  , and if  has no fixed points in ⟦ 1 ,  2 ⟧ other than  1 and  2 , then the interior of ⟦ 1 ,  2 ⟧ is either a subset of the basin of attraction of  1 or a subset of the basin of attraction of  2 .
Next result is well known global attractivity result which holds in partially ordered Banach spaces as well; see [18].Theorem 5. Let  be a monotone map on a closed and bounded rectangular region R ⊂ R 2 .Suppose that  has a unique fixed point e in R. Then e is a global attractor of  on R.
The following theorems were proved by Kulenović and Merino [1] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or nonhyperbolic) is by absolute value smaller than 1 while the other has an arbitrary value.These results are useful for determining basins of attraction of fixed points of competitive maps.

Theorem 6. Let 𝑇 be a competitive map on a rectangular
such that the following statements are true: (ii) W + is invariant, and dist(  (), Q 4 ()) → 0 as  → ∞ for every  ∈ W + .
(B) If, in addition to the hypotheses of part (A),  is an interior point of R and  is  2 and strongly competitive in a neighborhood of , then  has no periodic points in the boundary of Q 1 () ∪ Q 3 () except for , and the following statements are true: (iv) For every  ∈ W + there exists  0 ∈ N such that   () ∈ int Q 4 () for  ≥  0 .
If  is a map on a set R and if  is a fixed point of , the stable set W  () of  is the set { ∈ R :   () → } and unstable set W  () of  is the set When  is noninvertible, the set W  () may not be connected and made up of infinitely many curves, or W  () may not be a manifold.The following result gives a description of the stable and unstable sets of a saddle point of a competitive map.If the map is a diffeomorphism on R, the sets W  () and W  () are the stable and unstable manifolds of .Theorem 9.In addition to the hypotheses of part (B) of Theorem 8, suppose that  > 1 and that the eigenspace   associated with  is not a coordinate axis.If the curve C of Theorem 6 has endpoints in R, then C is the stable set W  () of , and the unstable set W  () of  is a curve in R that is tangential to   at  and such that it is the graph of a strictly decreasing function of the first coordinate on an interval.Any endpoints of W  () in R are fixed points of .

Number of Equilibria
In this section we give some basic facts which are used later.Let  be the map associated with system (1) given by  (, ) = ( (, ) ,  (, )) The equilibrium points (, ) of system (1) satisfy equations For  = 0 we have from which we obtain three equilibrium points where Assume that  ̸ = 0.Then, from the first equation of system (12) we have By substituting this into the second equation we obtain or from which we obtain the other three equilibrium points where Lemma 10.The following hold: or (iv) Assume that Δ 2 ≥ 0. The equilibrium point  6 exists if and only if  1 < 1 and Proof.The proof of the statements (i) and (ii) is trivial and we skip it.Now we prove the statement (iii).In view of Descartes' rule of signs we obtain that (17) has no positive solutions if  1 ≥ 1.Now, we suppose that  1 < 1.One can see that  5 > 0 for all values of parameters.We consider two cases: (1) Assume that which is equivalent to Since we have that  5 ≥ 0 if and only if which is equivalent to From ( 27) and (28) it follows  5 ≥ 0 if and only if (2) Assume that which is equivalent to 2 and the curves C 32 and C 31 are defined as part of the parabola Then  5 ≥ 0 if and only if which is equivalent to Since then from (33) and Δ 2 ≥ 0 we have Since we have that ( 31) and ( 36) are equivalent to or Now, the proof of the statement (iii) follows from (28), (38), and (39).The proof of the statement (iv) is similar and we skip it.
We now introduce the following notation for regions in parameter space ( 1 ,  2 ) (see Figure 1): Figure 1 gives a graphical representation of above sets.The following result gives a complete classification for the number of equilibrium solutions of system (1).
and  2 be positive real numbers.Then, the number of positive equilibrium solutions of system (1) with parameters  1 ,  2 ,  1 ,  2 ,  1 , and  2 can be from 1 to 6.The different cases are given in Table 1.
Proof.The proof follows from Lemma 10.

Linearized Stability Analysis
The Jacobian matrix of the map  has the form 2 ) . (41) or The determinant of (41) at the equilibrium point is given by and the trace of (41) at the equilibrium point is given by tr   (, ) =  1 +  1 Abstract and Applied Analysis 9 The characteristic equation has the form Lemma 12.The following statements hold: Proof.We have that, for the equilibrium point  1 , tr   ( 1 ) = 1/ 1 and det   ( 1 ) = 0.The characteristic equation of (50) at  1 has the form  2 − (1/ 1 ) = 0, from which the proof follows.
The following lemma describes the local stability of the equilibrium points  5 and  6 .
(ii) If Δ 2 > 0 and  5 exists then it is a saddle point.
Assume that Δ 2 = 0. Then  5 =  6 is zero of f() of multiplicity two.In view of Lemmas 6 and 7 from [19] we have that x ( 5 ) = 0.The rest of the proof follows from the proof of Lemma 15.
From the proof of the statements (a) and (b) if we obtain This implies tr   ( 2 ) − 2 = (1 − 4 2  2 )/ (√1 − 4 2  2 + 1) > 0. The rest of the proof follows from the fact that if Δ 1 = 0 then and if (57) holds then Lemma 18. Assume that Δ 1 ≥ 0. The following statements are true: (a)  3 is locally asymptotically stable if Δ 1 > 0, and If Δ 1 = 0 then the eigenvalues of   ( 3 ) are given by with corresponding eigenvectors . (71) If (57) holds then the eigenvalues of   ( 3 ) are given by with corresponding eigenvectors Proof.Since the proof of this lemma is similar to the proof of Lemma 17, it is omitted.
We summarize results about local stability in the following theorem.

Theorem 19.
Let  1 ,  2 ,  1 ,  2 ,  1 , and  2 be positive real numbers.Then, local stability of the equilibrium points for different parameter regions is given by Table 2.
Proof.The proof follows from Theorem 8 and Lemmas 17 and 18.
Figure 2 illustrates visually local stability of all equilibrium points of system (1).

Injectivity and Convergence to Equilibrium Points
In this section we prove some global properties of the map  such as injectivity and (+) property and give global behavior on the coordinate axes.
Lemma 20.The map  is injective.
The global behavior of  on the coordinate axes is described with the following result.

Abstract Applied Analysis
Proof.
Theorem 23.Every solution of system (1) converges to an equilibrium point.
Remark 24.In view of Theorem 23 the main objective in determining the global dynamics of system ( 1) is to characterize the basins of attractions of all equilibrium points.As we will see in Theorem 25 the boundaries of these basins of attractions will be the global stable manifolds of the saddle or nonhyperbolic equilibrium points, whose existence is guaranteed by Theorems 7, 8, and 9.

Global Behavior
In this section we give results which precisely describe global dynamics of system (1) including precise characterization of basins of attraction of different equilibrium points.The main result of this paper is the following.
Theorem 25.The global behavior of system ( 1) is given by Table 3. See Figure 3 for visual illustration of dynamic scenarios.
Proof.We will prove statements (i)-(x) listed in the second column of Table 3 in the given order.The proof of other statements is similar.Let R = [0, ∞) × [0, ∞).
(xii)  1 = 1, Δ 1 > 0 There exist three equilibrium points  1 = There exist two equilibrium points  1 =  4 and  2 =  3 which are nonhyperbolic.There exists a continuous increasing curve C with endpoint at  2 =  3 which is a subset of the basin of attraction of  2 =  3 .All solutions with initial point above C converge to  2 =  3 , while solutions with initial point below C converge to  1 =  4 . (xiv) There exists one equilibrium point  1 =  4 which is nonhyperbolic and global attractor.The basin of attraction of  1 is There exist three equilibrium points There exist two equilibrium points  1 and  2 =  3 , where  1 is locally asymptotically stable and  2 =  3 is nonhyperbolic.There exists a continuous increasing curve C with endpoint at  2 =  3 which is a subset of the basin of attraction of  2 =  3 .All solutions with initial point above C converge to  2 =  3 , while solutions with initial point below C converge to  1 .Take ( 0 ,  0 ) ∈ W + ∩ int( 1 (0, 0)).By Theorem 8 and Lemma 21 we have that there exists   0 > 0 such that   ( 0 ,  0 ) ∈ int( Since, by Theorem 23, every solution of system (1) converges to an equilibrium point, we have that B( 4 ) = (0, ∞) × [0, ∞).
(v) Suppose (  (vi) The proof is similar to the proof of case (i) and we skip it.
(vii) The proof is the same as the proof of case (viii) and we skip it.

Figure 3 :
Figure 3: Parameter regions in terms of parameters  1 and  2 and corresponding dynamic scenarios for system (1) if  1 < 1.
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.
is not the NW or SE vertex of R), and  is strongly competitive on Δ. Suppose that the following statements are true:(a) The map  has a  1 extension to a neighborhood of .(b)The Jacobian   () of  at  has real eigenvalues ,  such that 0 < || < , where || < 1, and the eigenspace   associated with  is not a coordinate axis.Then there exists a curve C ⊂ R through  that is invariant and a subset of the basin of attraction of , such that C is tangential to the eigenspace   at , 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 .
The equilibrium points  2 and  3 exist if and only if Δ 1 ≥ 0 and  2 =  3 if and only if Δ 1 = 0. (ii) The equilibrium point  4 exists if and only if  1 ≤ 1 and  4 =  1 if and only if  1 = 1.

Table 2 :
The local stability of the equilibrium points.
Equilibria in different parameter regions for the number of equilibria of system (1) when  1 ,  2 ,  1 , and  2 are fixed positive real numbers, as given by Proposition 11.Each circle in the parameters ( 1 ,  2 )-plane indicates the existence of an isolated equilibrium point of system (1) in the nonnegative quadrant of the -plane.Local stability character of equilibria as given in Theorem 19 is indicated as follows: •, locally asymptotically stable equilibrium; ⊖, saddle; ⃝ , repelling equilibrium point: two-colored circle, semistable nonhyperbolic equilibrium.