Global Behavior of Four Competitive Rational Systems of Difference Equations in the Plane

M. Garić-Demirović,1 M. R. S. Kulenović,2 and M. Nurkanović1 1 Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia And Herzegovina 2 Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA Correspondence should be addressed to M. R. S. Kulenović, kulenm@math.uri.edu Received 29 August 2009; Accepted 27 October 2009 Recommended by Guang Zhang We investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. 2009 . Copyright q 2009 M. Garić-Demirović 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.


Introduction and Preliminaries
We consider the following open problem see 1, Open Problem 2 .

1.6
The typical results are the following theorems.The first theorem is a combination of Theorems 2.3 and 2.5 and the second theorem is Theorem 3.3.Theorem 1.1.Consider system (14,21) and assume that γ 2 A 1 / α 2 .If β 1 > A 1 , then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, x n , y n 0, ∞ for every x 0 , y 0 ∈ W − , x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

1.8
Assume that β 1 ≤ A 1 .Every solution { x n , y n } of system (14,21), with x 0 > 0, y 0 ≥ 0, satisfies 1.9 Theorem 1.2.Consider system (21, 21).There exists a set C ⊂ R which is invariant and a subset of the basin of attraction of the unique equilibrium E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, x n , y n 0, ∞ for every x 0 , y 0 ∈ W − , x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

1.11
All considered systems are competitive systems, which we discuss next.
A first-order system of difference equations x n 1 f x n , y n y n 1 g x n , y n n 0, 1, 2, . . ., x −1 , x 0 ∈ R, 1.12 where R ⊂ R 2 , f, g : R → R, f, g are continuous functions, is competitive if f x, y is nondecreasing in x and nonincreasing in y, and g x, y is nonincreasing in x and nondecreasing in y.If both f and g are nondecreasing in x and y, the system 1.12 is cooperative.A map T that corresponds to the system 1.12 is defined as T x, y f x, y , g x, y .Competitive and cooperative maps, which are called monotone maps, are defined similarly.Strongly competitive systems of difference equations or maps are those for which the functions f and g are coordinatewise strictly monotone.If v u, v ∈ R 2 , we denote with Q v , ∈ {1, 2, 3, 4}, the four quadrants in R 2 relative to v, that is, x ≤ u, y ≥ v}, and so on.Define the South-East partial order se on R 2 by x, y se s, t if and only if x ≤ s and y ≥ t.Similarly, we define the North-East partial order ne on R 2 by x, y ne s, t if and only if x ≤ s and y ≤ t.For A ⊂ R 2 and x ∈ R 2 , define the distance from x to A as dist x, A : inf { x − y : y ∈ A}.By int A we denote the interior 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: Competitive systems were studied by many authors; see 2-17 , and others.All known results, with the exception of 2, 3, 18 , deal with hyperbolic dynamics.The results presented here are results that hold in both the hyperbolic and the nonhyperbolic case.
We now state three results for competitive maps in the plane.The following definition is from 17 .
Definition 1.3.Let S be a nonempty subset of R 2 .A competitive map T : 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 DeMottoni-Schiaffino for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations.Smith generalized the proof to competitive and cooperative maps 14, 15 .
Theorem 1.4.Let S be a nonempty subset of R 2 .If T is a competitive map for which (O ) holds then for all x ∈ S, {T n x } is eventually componentwise monotone.If the orbit of x has compact closure, then it converges to a fixed point of T .If instead (O−) holds, then for all x ∈ S, {T 2n } is eventually componentwise monotone.If the orbit of x has compact closure in S, then its omega limit set is either a period-two orbit or a fixed point.
The following result is from 17 , 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 1.5.Let R ⊂ R 2 be the cartesian product of two intervals in R. Let T : R → R be a C 1 (continuously differentiable) competitive map.If T is injective and det J T x > 0 for all x ∈ R, then T satisfies (O ).If T is injective and det J T x < 0 for all x ∈ R, then T satisfies (O−).
The next results are the modifications of 8, Theorem 4 .See 18 .
Theorem 1.6.Let T be a monotone map on a closed and bounded rectangular region R ⊂ R 2 .Suppose that T has a unique fixed point e in R. Then e is a global attractor T on R.
The following four results were proved by Kulenović and Merino 18 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.
Our first result gives conditions for the existence of a global invariant curve through a fixed point hyperbolic or not of a competitive map that is differentiable in a neighborhood of the fixed point, when at least one of two nonzero eigenvalues of the Jacobian matrix of the map at the fixed point has absolute value less than one.A region R ⊂ R 2 is rectangular if it is the cartesian product of two intervals in R.
Theorem 1.7.Let T be a competitive map on a rectangular region R ⊂ R 2 .Let x ∈ R be a fixed point of T such that Δ : R ∩ int Q 1 x ∪ Q 3 x is nonempty (i.e., x is not the NW or SE vertex of R , and T is strongly competitive on Δ. Suppose that the following statements are true.
a The map T has a C 1 extension to a neighborhood of x.
b The Jacobian matrix of T at x has real eigenvalues λ, μ such that 0 < |λ| < μ, where |λ| < 1, and the eigenspace E λ associated with λ is not a coordinate axis.
Then there exists a curve C ⊂ R through x that is invariant and a subset of the basin of attraction of x, such that C is tangential to the eigenspace E λ at x, and C is the graph of a strictly increasing continuous function of the first coordinate on an interval.Any endpoints of C in the interior of R are either fixed points or minimal period-two points.In the latter case, the set of endpoints of C is a minimal period-two orbit of T .
Corollary 1.8.If T has no fixed point nor periodic points of minimal period-two in Δ, then the endpoints of C belong to ∂R.
For maps that are strongly competitive near the fixed point, hypothesis b of Theorem 1.7 reduces just to |λ| < 1.This follows from a change of variables 17 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 axis.
The following result gives a description of the global stable and unstable manifolds of a saddle point of a competitive map.The result is a modification of 8, Theorem 5 .Theorem 1.9.In addition to the hypotheses of Theorem 1.7, suppose that μ > 1 and that the eigenspace E μ associated with μ is not a coordinate axis.If the curve C of Theorem 1.7 has endpoints in ∂R, then C is the global stable manifold W s x of x, and the global unstable manifold W u x is a curve in R that is tangential to E μ at x and such that it is the graph of a strictly decreasing function of the first coordinate on an interval.Any endpoints of W u x in R are fixed points of T .
The next result is useful for determining basins of attraction of fixed points of competitive maps.i W − is invariant, and dist T n x , Q 2 x → 0 as n → ∞ for every x ∈ W − .
ii W is invariant, and dist T n x , Q 4 x → 0 as n → ∞ for every x ∈ W .
If, in addition, x is an interior point of R and T is C 2 and strongly competitive in a neighborhood of x, then T has no periodic points in the boundary of Q 1 x ∪ Q 3 x except for x, and the following statements are true.
i For every x ∈ W − there exists n 0 ∈ N such that T n x ∈ int Q 2 x for n ≥ n 0 .
ii For every x ∈ W there exists n 0 ∈ N such that T n x ∈ int Q 4 x for n ≥ n 0 .
In this paper we study the global dynamics of four rational systems of difference equations mentioned earlier, where all parameters are positive numbers and initial conditions x 0 and y 0 are arbitrary nonnegative numbers.Two of these systems have a nonhyperbolic semistable equilibrium point.In general all four systems share the common feature that the global stable manifolds of either saddle points or nonhyperbolic equilibrium points serve as boundaries of basins of attraction of different local attractors or points at infinities.The techniques used here can be applied to treat number of competitive systems which appear in applications, such as Leslie-Gower competition model, see 19 , or Leslie-Gower competition model with stocking, see 20 , or genetic model, see 13 .An important new feature of our techniques is that they are applicable to nonhyperbolic case as well, which was shown for the first time in 18 where we have completed analysis of basic Leslie-Gower competition model from 19 .Furthermore, system 21, 38 can be considered as a variant of Leslie-Gower competition model, where the first equation has been replaced by another equation, which does not allow extinction of both species.In fact, all four considered competitive systems share common feature that they do not allow the extinction of both species.

System (14,21)
Now we consider the following system of difference equations: where the parameters A 1 , β 1 , α 2 , and γ 2 are positive numbers and initial conditions x 0 > 0, y 0 ≥ 0. System 2.1 was considered in 1, Example 1 , where it was shown that the associated map T x, y β 1 x/ A 1 y , α 2 γ 2 y /x is injective and Therefore, in view of Theorems 1.4 and 1.5 every solution of system 2.1 is eventually componentwise monotonic.If γ 2 A 1 < α 2 , then det J T x, y < 0, and four subsequences of every solution { x n , y n } of system 2.1 are eventually monotonic.Thus, if γ 2 A 1 / α 2 , the Jacobian matrix of T in x, y is invertible.
The Jacobian matrix of the corresponding map T x, y is of the form

Linearized Stability Analysis
The equilibrium points x, y of system 2.1 are solutions of the system of equations from which we obtain which is a saddle point.
ii If β 1 ≤ A 1 , then system 2.1 has no equilibrium points.
Proof.By 2.6 and 2.4 the Jacobian matrix evaluated at the equilibrium point E has the form The corresponding characteristic equation evaluated at the equilibrium point E is where

2.10
Notice that in view of 2.6 y/β 1 1 − A 1 /β 1 and so Since p > 0 and 1 q > 0, we need to show Indeed, which is satisfied because β 1 > 0 and y > 0 .Furthermore which is satisfied.
Proof.System 2.1 can be reduced to the following second-order difference equation: or to the following second-order difference equation: Now it is sufficient to prove that both of the difference equations 2.12 and 2.13 have no prime period-two solutions.Assume that this is not true for 2.12 , that is, that φ, ψ, φ, ψ, . . ., φ / ψ 2.14 is a prime period-two solution of 2.12 .Then we have This implies By subtraction, we obtain and this implies that φ ψ, which is a contradiction.Now assume that is a prime period-two solution of 2.13 .Then we have and this implies that χ ϕ, which is a contradiction.
Theorem 2.3.Consider system 2.1 and assume that β 1 > A 1 and γ 2 A 1 / α 2 .Then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

2.23
Proof.Clearly, system 2.1 is strongly competitive on 0, ∞ × 0, ∞ .In view of Theorem 2.2 we see that all conditions of Theorems 1.7, 1.9, and 1.10 and Corollary 1.8 are satisfied with R 0, ∞ × 0, ∞ and so the conclusion follows.
Remark 2.4 see 1 .If γ 2 A 1 α 2 , then system 2.1 can be decoupled as follows: and every solution of this system depending of the choice of the initial condition x 0 , y 0 is either bounded and converges to an equilibrium point or increases monotonically to infinity.

Case β
In this case system 2.1 has no equilibrium points.Now we have the following.
Theorem 2.5.Assume that On the other hand, if x n , and we obtain that the sequence {x n } ∞ n 0 is strictly decreasing.Because x n > 0 for all n, we see that {x n } ∞ n 0 is convergent and lim n → ∞ x n 0, since otherwise, that is, lim n → ∞ x n a > 0, the first equation of system 2.1 implies lim n → ∞ y n β 1 − A 1 0 or the second equation of system 2.1 implies lim n → ∞ y n α 2 / a−γ 2 / 0, which is a contradiction, since otherwise system 2.1 would have an equilibrium point in the first quadrant.
We see that if But then the denominator in is, for all large n, strictly less than a constant η < γ 2 , which in turn implies

2.28
Iterating this inequality we obtain and this forces y n to infinity.
The obtained results lead to the following characterization of the boundedness of solutions of system 2.1 .
Corollary 2.6.Consider system 2.1 subject to the condition α 2 / A 1 γ 2 .If β 1 > A 1 , then all bounded solutions converge to the unique equilibrium with the corresponding initial conditions belonging to the graph of a continuous increasing function C in the plane of initial conditions.All solutions that start in the complement of C are asymptotic to either ∞, 0 or 0, ∞ .If β 1 ≤ A 1 , then all solutions are unbounded in the sense that {x n } is bounded and {y n } approaches ∞.

System (21,21)
Now we consider the following system of difference equations: where the parameters α 1 , β 1 , α 2 , and γ 2 are positive numbers and initial conditions x 0 > 0, y 0 > 0. System 3.1 was considered in 1, Example 3 , where it was shown that the associated map T is injective and that is, the Jacobian matrix of T in x, y is invertible.Therefore, in view of Theorems 1.4 and 1.5, four subsequences of every solution { x n , y n } of system 3.1 are eventually monotonic.

Linearized Stability Analysis
Equilibrium points of system 3.1 are solutions of the system Since x / 0 and y / 0, we have where Since y − < 0 and y > 0, system 3.1 has a unique positive equilibrium E x , y , where where Lemma 3.1.System 3.1 has a unique positive equilibrium point: which is a saddle point.
Proof.The Jacobian matrix of the corresponding map T x, y β 1 x α 1 /y, α 2 γ 2 y /x is of the form 3.9 By using 3.4 we obtain The corresponding characteristic equation evaluated at the equilibrium point E of system 3.1 is where

3.12
Notice that Since p > 0 and 1 q > 0, we need to show Now, we get By using 3.4 , 3.5 , and 3.7 we obtain which is satisfied.

Global Results
Theorem 3.2.System 3.1 has no prime period-two solutions.
Proof.System 3.1 can be reduced to the following second-order difference equation: or to the following second-order difference equation: Now it is sufficient to prove that both of the difference equations 3.15 and 3.16 have no prime period-two solutions.Assume that this is not true for 3.15 , that is, that φ, ψ, φ, ψ, . . ., φ / ψ 3.17 is a prime period-two solution of 3.15 .Then we have that is, and this implies that φ ψ, which is a contradiction.Now assume that χ, ϕ, χ, ϕ, . . ., χ / ϕ 3.21 is a prime period-two solution of 3.16 .Then we have and this implies that χ ϕ, which is a contradiction.The global behavior system 3.1 is described by the following result.x n , y n 0, ∞ for every x 0 , y 0 ∈ W − , x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

3.25
Proof.In view of Theorem 3.2 and the injectivity of the map T we see that all conditions of Theorems 1.7, 1.9, and 1.10 and Corollary 1.8 are satisfied with R 0, ∞ × 0, ∞ and so the conclusion follows.
The obtained result leads to the following characterization of the boundedness of solutions of system 3.1 .
Corollary 3.4.All bounded solutions of system 3.1 converge to the unique equilibrium with the corresponding initial conditions which belong to the graph of a continuous increasing function C in the plane of initial conditions.All solutions that start in the complement of C are asymptotic to either ∞, 0 or 0, ∞ .

System (15,21)
Now we consider the following system of difference equations: where the parameters β 1 , B 1 , α 2 , and γ 2 are positive numbers and initial conditions x 0 > 0, y 0 ≥ 0. The Jacobian matrix of the corresponding map T x, y β 1 x/ B 1 x y , α 2 γ 2 y /x is of the form System 4.1 was considered in 1, Example 2 , where it was shown that the corresponding map T is injective and that is, the Jacobian matrix of T in x, y is invertible.Therefore, in view of Theorems 1.4 and 1.5, four subsequences {x 2n }, {x 2n 1 }, y 2n , y 2n 1 4.4 of every solution { x n , y n } of system 4.1 are eventually monotonic.

Linearized Stability Analysis
Equilibrium points of system 4.1 are solutions of the system Since x / 0, we obtain where This implies that we have the following three cases for the equilibrium points.
i If then there exist two equilibrium points of system 4.1 :

4.7
ii If β 1 − B 1 γ 2 2 B 1 α 2 , then system 4.1 has a unique equilibrium point: Next, by using 4.5 we have The corresponding characteristic equation evaluated at the equilibrium point E x, y is where
, then the equilibrium point E of system 4.1 is locally asymptotically stable and the equilibrium point E − is a saddle point. If For the equilibrium point E we need to prove that p < 1 q < 2, 4.12 or equivalently because p > 0 : Indeed, I we have

4.13
which is true.Furthermore II we have 4.14 which is true.

4.16
Now I we have 4.17 which is true.Similarly II we have We need to prove that 1 q p , 4.19 that is because p > 0 and, 1 q > 0 , 1 q p. 4.20 We have

Global Results
Theorem 4.2.System 4.1 has no prime period-two solutions.
Proof.The second iterate of map T is

4.22
Period-two solution satisfies

4.23
From this system we have In cases i and ii solutions x, y are equilibrium points E and E − , and in case v solution x, y is not in the first quadrant in the plane.It is sufficient to prove that solutions x, y in cases iii and iv are not in the first quadrant in the plane.Namely, if Δ 1 < 0, x and y are not real.Supose that By analogous reasoning we have that the same conclusion for case iv holds.
Our linearized stability analysis indicates that there are three cases with different asymptotic behavior, depending on the values of parameters β 1 , B 1 , α 2 , and γ 2 .

Global Results-Case 1
for every x 0 , y 0 ∈ W

4.25
Proof.Clearly, system 4.1 is strongly competitive on R 0, ∞ × 0, ∞ .In view of injectivity of T , invertibility of J T , and Theorem 4.2, we see that all conditions of Theorems 1.7, 1.9, and 1.10 and Corollary 1.8 are satisfied and the conclusion of the theorem follows.

Global Results-Case 2
Theorem 4.4.Consider system 4.1 and assume that β 1 − B 1 γ 2 2 B 1 α 2 .Then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, for every x 0 , y 0 ∈ W .

4.27
Proof.In this case system 4.1 has a unique equilibrium point This implies and It is obvious that p > 0. We will show that p < 1. Indeed

4.31
It means that all conditions of Theorems 1.7 and 1.10 are satisfied with R 0, ∞ × 0, ∞ .
Assume that x 0 , y 0 ∈ W . Then x n , y n ∈ W for all n, and sequences {x 2n }, {x 2n 1 }, {y 2n }, and {y 2n 1 } are monotone and bounded since x n ≤ β 1 /B 1 .Thus these sequences are convergent, which in view of Theorem 4.2 shows that they converge to the equilibrium point.Since E is the unique equilibrium point in W the statement for W follows.The same conclusion is obtained by using Theorem 1.6.
If x 0 , y 0 is in W − , by Theorem 1.10 the orbit of x 0 , y 0 eventually enters Q 2 E .Assume without loss of generality that x 0 , y 0 ∈ int Q 2 E .An eigenvector associated with the nonhyperbolic eigenvalue λ 2 1 is v −1, B 1 .Choose a value of t small enough so that E tv ∈ Q 2 E and x 0 , y 0 E tv.Let us show that T E tv E tv.Indeed where the last equality follows from the condition Since T E tv E tv, it follows that {T n E tv } is a monotonically decreasing sequence in Q 2 E which is bounded above by E. Since {T n E tv } is coordinatewise monotone and it does not converge if it did it would have to converge to E, which is impossible , we have that T n E tv has second coordinate which is monotone and unbounded.But x n , y n : T n x 0 , y 0 T n E tv , which implies that y n → ∞.From 4.1 it follows that x n → 0.

Global Results-Case 3
Theorem 4.5.Consider system 4.1 and assume that

4.35
Proof.In this case system 4.1 has no equilibrium points.Consider now the following system satisfied by subsequences of the solution of system 4.1 :

4.36
We know that each of the four subsequences of every solution { x n , y n } of system 4.1 is eventually monotonic.The subsequences {x 2k } and {x 2k 1 } are bounded by β 1 /B 1 , which implies that they are convergent.Suppose that a lim k → ∞ x 2k x E and b lim k → ∞ x 2k 1 x O .For the other two subsequences the following four cases are possible: Case 1 and Case 3 imply 4.38 that is, system 4.1 has a period-two solution, which is a contradiction by Theorem 4.2.Case 1 and Case 4 imply which is a contradiction by Case 1. Case 2 and Case 3 imply which is a contradiction by Case 3. Case 2 and Case 4 imply The obtained results lead to the following characterization of the boundedness of solutions of system 3.1 .
Corollary 4.6.Consider system 4.1 and assume that β 1 − B 1 γ 2 ≥ 2 B 1 α 2 .All bounded solutions of system 4.1 converge to the unique equilibrium with the corresponding initial conditions which belong to region below and on the graph of a continuous increasing function C in the plane of initial conditions.All solutions that start above C are asymptotic to 0, ∞ .
Consider system 4.1 and assume that either Then every solution of 4.1 is asymptotic to 0, ∞ .
The Jacobian matrix of the corresponding map T x, y α 1 β 1 x /y, γ 2 y/ A 2 B 2 x y is of the form System 5.1 was considered in 1, Example 4 , where it was shown that the map T is injective.In addition, when Therefore, when 5.3 holds, the Jacobian matrix of T in x, y is invertible and in view of Theorems 1.4 and 1.5 every solution of system 5.1 is eventually componentwise monotonic.When we see that and the Jacobian matrix of T in x, y is invertible.Therefore, in view of Theorems 1.4 and 1.5, four subsequences {x 2n }, {x 2n 1 }, y 2n , y 2n 1 5.7 of every solution { x n , y n } of system 5.1 are eventually monotonic.

Linearized Stability Analysis
Equilibrium points of system 5.1 are solutions of the system Since y / 0, we have where

5.10
It is easy to prove that the following result holds.

5.11
ii If γ 2 − A 2 − β 1 2 α 1 B 2 , then system 5.1 has a unique equilibrium point: Proof.By using 5.8 we have The corresponding characteristic equation evaluated at the equilibrium point E x, y is λ 2 − pλ q 0, 5.14 where

5.15
For the equilibrium point E we need to prove that p > 1 q , p 2 − 4q > 0, 5.16 that is because p > 0 and 1 q > 0 , which is always true, and in view of 5.8 γ 2 − B 2 x y A 2 we obtain which is true because y < γ 2 .
Next, in view of 5.8 γ 2 − B 2 x y A 2 , I we have

5.19
which is true.
Similarly, II we have 5.20 which is true.
For the equilibrium point E − we need to prove that p < 1 q < 2, 5.21 or equivalently I p < 1 q, II q < 1.
Indeed, I we have

5.22
which is true, and in view of 5.8 β 1 /y 1 − α 1 /xy II we obtain Let us prove that 1 q p.We have 5.24 which is true.

Global Results
Theorem 5.3.System 5.1 has no prime period-two solutions.

5.33
The solutions of first equation depending of the choice of the initial condition x 0 are either bounded and converge to a finite limit or increase monotonically to infinity.Using this and 5.32 we find the behavior of solutions of second equation.
Our linearized stability analysis indicates that there are three cases with different asymptotic behavior, depending on the values of parameters β 1 , B 1 , α 2 , and γ 2 :

Case γ
Theorem 5.5.Consider system 5.1 and assume that γ Then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E .The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

5.35
Proof.Clearly, system 5.1 is strongly competitive on R 0, ∞ × 0, ∞ .In view of the injectivity of T , the invertibility of J T and Theorem 5.3, we see that all conditions of Theorems 1.7, 1.9, and 1.10 and Corollary 1.8 are satisfied and the conclusion of the theorem follows.

Case γ
Theorem 5.6.Consider system 5.1 and assume that γ 2 − A 2 − β 1 2 α 1 B 2 and β 1 A 2 / α 1 B 2 .Then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely x n , y n ∞, 0 for every x 0 , y 0 ∈ W .

5.41
It means that all conditions of Theorems 1.7 and 1.10 are satisfied with R 0, ∞ × 0, ∞ .In view of the fact that y n ≤ γ 2 we obtain the conclusion of the theorem in the case x 0 , y 0 ∈ W − .The same conclusion is obtained by using Theorem 1.6.
Next, assume that x 0 , y 0 ∈ W .By Theorem 1.10 the orbit of x 0 , y 0 eventually enters Q 4 E .Assume without loss of generality that x 0 , y 0 ∈ int Q 4 E .An eigenvector associated with the nonhyperbolic eigenvalue λ 2 1 is v 1, −B 2 .Choose a value of t small enough so that E tv ∈ Q 4 E and E tv x 0 , y 0 .Let us show that E tv T E tv .Indeed where the last equality follows from the condition γ 2 − A 2 − β 1 2 α 1 B 2 .
Since E tv T E tv , it follows that {T n E tv } is a monotonically increasing sequence in Q 4 E which is bounded below by E. Since {T n E tv } is coordinatewise monotone and it does not converge if it did it would have to converge to E, which is impossible , we have that T n E tv has a first coordinate which is monotone and unbounded.But T n E tv x n , y n : T n x 0 , y 0 , which implies that x n → ∞.From 5.1 it follows that y n → 0.

Case γ
In this case system 5.1 has no equilibrium points.Proof. 1 Assume that β 1 A 2 > α 1 B 2 .Then every solution of system 5.1 is eventually componentwise monotonic.The sequence {y n } is bounded by γ 2 , which implies that it converges, that is, lim n → ∞ y n Y .For the sequence {x n } the following two cases are possible: a lim n → ∞ x n X, b lim n → ∞ x n ∞.
W − : x ∈ R \ C : ∃y ∈ C with x se y , W : x ∈ R \ C : ∃y ∈ C with y se x , \ C : ∃y ∈ C with x se y , W : x ∈ R \ C : ∃y ∈ C with y se x .

Theorem 1 . 10 .
Assume the hypotheses of Theorem 1.7, and let C be the curve whose existence is guaranteed by Theorem 1.7.If the endpoints of C belong to ∂R, then C separates R into two connected components, namely, W − : x ∈ R? \ C : ∃y ∈ C with x se y , W : x ∈ R? \ C : ∃y ∈ C with y se x , 1.14 such that the following statements are true.

Theorem 3 . 3 .
Consider system 3.1 .There exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E. The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, W − : x ∈ R \ C : ∃y ∈ C with x se y , W : x ∈ R \ C : ∃y ∈ C with y se x ,

Theorem 4 . 3 .
Consider system 4.1 and assume that β 1 − B 1 γ 2 > 2 B 1 α 2 .Then there exists a set C ⊂ R which is invariant and a subset of the basin of attraction of E − .The set C is a graph of a strictly increasing continuous function of the first variable on an interval (and so is a manifold) and separates R into two connected and invariant components, namely, W − : x ∈ R \ C : ∃y ∈ C with x se y W : x ∈ R \ C : ∃y ∈ C with y se x ,