Research Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane

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1 Hindawi Publishing Corporation Advances in Difference Equations Volume 009 Article ID pages doi:101155/009/1380 Research Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane S Kalabušić 1 M R S Kulenović and E Pilav 1 1 Department of Mathematics University of Sarajevo Sarajevo Bosnia and Herzegovina Department of Mathematics University of Rhode Island Kingston RI USA Correspondence should be addressed to M R S Kulenović kulenm@mathuriedu Received 6 August 009; Accepted 8 December 009 Recommended by Panayiotis Siafarikas We investigate global dynamics of the following systems of difference equations x n 1 α 1 x n /y n y n 1 α γ y n / A x n n 0 1 where the parameters α 1 α γ and A are positive numbers and initial conditions x 0 and y 0 are arbitrary nonnegative numbers such that y 0 > 0 We show that this system has rich dynamics which depend on the part of parametric space 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 of nonhyperbolic equilibrium points Copyright q 009 S Kalabušić 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 1 Introduction and Preliminaries In this paper we study the global dynamics of the following rational system of difference equations: x n 1 α 1 x n y n y n 1 α γ y n A x n n where the parameters α 1 α γ and A are positive numbers and initial conditions x 0 0andy 0 > 0 are arbitrary numbers System 11 was mentioned in 1 as a part of Open Problem 3 which asked for a description of global dynamics of three specific competitive systems According to the labeling in 1 system 11 is called 1 9 In this paper we provide the precise description of global dynamics of system 11 We show that system

2 Advances in Difference Equations 11 has a variety of dynamics that depend on the value of parameters We show that system 11 may have between zero and two equilibrium points which may have different local character If system 11 has one equilibrium point then this point is either locally saddle point or non-hyperbolic If system 11 has two equilibrium points then the pair of points is the pair of a saddle point and a sink The major problem is determining the basins of attraction of different equilibrium points System 11 gives an example of semistable non-hyperbolic equilibrium point The typical results are Theorems 41 and 45 below System 11 is a competitive system and our results are based on recent results developed for competitive systems in the plane; see 3 In the next section we present some general results about competitive systems in the plane The third section deals with some basic facts such as the non-existence of period-two solution of system 11 The fourth section analyzes local stability which is fairly complicated for this system Finally the fifth section gives global dynamics for all values of parameters Let I and J be intervals of real numbers Consider a first-order system of difference equations of the form x n 1 f ) x n y n y n 1 g ) n x n y n where f : I J Ig: I J J and x 0 y 0 I J When the function f x y is increasing in x and decreasing in y and the function g x y is decreasing in x and increasing in y the system 1 is called competitive When the function f x y is increasing in x and increasing in y and the function g x y is increasing in x and increasing in y the system 1 is called cooperative A map T that corresponds to the system 1 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 coordinate-wise strictly monotone If v u v R we denote with Q l v l {1 3 4} the four quadrants in R relative to v thatisq 1 v { x y R : x u y v} Q v { 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 the North-East partial order ne on R 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 : inf { x y :y A}ByintA 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: ) ) ) x 1 x x 1 x ) se F se F 13 y 1 y y 1 y Competitive systems were studied by many authors; see 4 19 and others All known results with the exception of deal with hyperbolic dynamics The results presented here are results that hold in both the hyperbolic and the non-hyperbolic cases We now state three results for competitive maps in the plane The following definition is from 18

3 Advances in Difference Equations 3 Definition 11 Let R be a nonempty subset of R A competitive map T : R R is said to satisfy condition O if for every x y in R T x ne T y implies x ne yandt is said to satisfy condition O if for every x y in R T x ne T y implies y ne x The following theorem was proved by de Mottoni and Schiaffino 0 for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations Smith generalized the proof to competitive and cooperative maps Theorem 1 Let R be a nonempty subset of R IfT is a competitive map for which O ) holds then for all x R {T n x } is eventually componentwise monotone If the orbit of x has compact closure then it converges to a fixed point of T IfinsteadO ) holds then for all x R {T n } is eventually componentwise monotone If the orbit of x has compact closure in R then its omega limit set is either a period-two orbit or a fixed point The following result is from 18 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 13 Smith 18 Let R R be the Cartesian product of two intervals in R LetT : R R be a C 1 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 ) Theorem 14 Let T be a monotone map on a closed and bounded rectangular region R R Suppose that T has a unique fixed point e in R Then e is a global attractor of T on R The following theorems were proved by Kulenović and Merino 3 for competitive systems in the plane when one of the eigenvalues of the linearized system at an equilibrium hyperbolic or non-hyperbolic 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 is rectangular if it is the Cartesian product of two intervals in R Theorem 15 Let T be a competitive map on a rectangular region R R Letx Rbe a fixed point of T such that Δ : R int Q 1 x Q 3 x is nonempty ie 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 <μwhere λ < 1 and the eigenspace E λ associated with λ is not a coordinate axis Then there exists a curve C Rthrough 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

4 4 Advances in Difference Equations Corollary 16 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 15 reduces just to λ < 1 This follows from a change of variables 18 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 quadrants 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 the modification of Theorem 17 from 1 Theorem 17 In addition to the hypotheses of Theorem 15 suppose that μ > 1 and that the eigenspace E μ associated with μ is not a coordinate axis If the curve C of Theorem 15 has endpoints in R thenc 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 Theorem 18 Assume the hypotheses of Theorem 15 and let C be the curve whose existence is guaranteed by Theorem 15 If the endpoints of C belong to RthenC separates R into two connected components namely W : {x R\C: y Cwith x se y} W : {x R\C: y Cwith y se x} 14 such that the following statements are true i W is invariant and dist T n x Q 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 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 iii For every x W there exists n 0 N such that T n x int Q x for n n 0 iv For every x W there exists n 0 N such that T n x int Q 4 x for n n 0 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 and O condition

5 Advances in Difference Equations 5 1 Equilibrium Points The equilibrium points x y of system 11 satisfy x α 1 x y y α γ y A x 1 First equation of System 1 gives y α 1 x Second equation of System 1 gives y A x α γ y 3 Now using weobtain α1 x ) A x x α 1 x α γ 4 x This implies α1 x ) A x α x γ α 1 γ x 5 which is equivalent to x x [ α 1 α A γ )] α1 A γ ) 0 6 Solutions of 6 are x 1 [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ x [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ 7

6 6 Advances in Difference Equations Table 1 E 1 A <γ E 1 E A >γ α 1 α A γ 4α 1 A γ > 0 A γ <α α 1 E 1 E A >γ α 1 α A γ 4α 1 A γ 0 A γ <α α 1 E A γ α >α 1 No equilibrium A >γ α 1 α A γ 4α 1 A γ < 0 No equilibrium A >γ α 1 α A γ 4α 1 A γ 0 A γ >α α 1 No equilibrium A γ α α 1 Now gives y 1 α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ y α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ 8 The equilibrium points are: E 1 x 1 y 1 ) E 1 x y ) 9 where x 1 y 1 x y are given by the above relations Note that xy / γ 10 The discriminant of 6 is given by D [ α 1 α A γ )] 4α1 A γ ) 11 The criteria for the existence of equilibrium points are summarized in Table 1 where E α α 1 α α α 1 ) 1 Condition O and Period-Two Solution In this section we prove three lemmas Lemma 1 System 11 satisfies either O or O Consequently the second iterate of every solution is eventually monotone

7 Advances in Difference Equations 7 Proof The map T associated to system 11 is given by T x y ) α1 x α ) γ y 13 y A x Assume T x 1 y 1 ) ne T x y ) 14 then we have α 1 x 1 y 1 α 1 x y 15 α γ y 1 A x 1 α γ y A x 16 Equations 15 and 16 are equivalent respectively to α 1 y y 1 ) β1 x1 y x y 1 ) 0 α x x 1 A γ y1 y ) γ x y 1 x 1 y ) Now using 17 and 18 wehavethefollowing: If y y 1 x 1 y <x y 1 x 1 <x x 1 y 1 ) ne x y ) If y <y 1 x <x 1 and/or x y 1 x 1 y < 0 x <x 1 x y ) ne x1 y 1 ) 19 Lemma System 11 has no minimal period-two solution Proof Set T x y ) α1 x α ) γ y 0 y A x Then T T x y )) α1 x T α ) γ y y A x x A α 1 )) ) α1 x /y y α )) )) γ α yγ /x A α yγ ya α 1 x 1

8 8 Advances in Difference Equations Period-two solution satisfies x A α 1 )) ) α1 x /y x 0 α yγ y α )) γ α yγ / x A ) y 0 3 ya α 1 x We show that this system has no other positive solutions except equilibrium points Equations and 3 are equivalent respectively to xyα 1 ya α 1 xyα xα 1 A α 1 x xa xy γ y α yγ ) 0 4 xy A y A xyα 1 ya α 1 xyα ya α x y xya yα γ y γ x A ya α 1 x ) 0 5 Equation 4 implies ya α 1 xy α 1 α A α 1 x x α1 A ) xy γ 0 6 Equation 5 implies xy A x y xy ) ) ) α 1 α A y A α 1 α α γ y A γ 0 7 Using 6 we have x ya ) α 1 xy α 1 α A α 1 x α1 A xy γ 8 Putting 8 into 7 we have ) )) ) y y x A α 1 y A x A xyγ γ α xy β1 x A γ 0 9 This is equivalent to x ya A yα1 α ) β1 γ α yγ ) yα 1 α y β1 ) y A yγ ) 30

9 Advances in Difference Equations 9 Putting 30 into 4 weobtain α yγ ) α y A γ ) y α1 α A γ )) ) 0 31 or y γ A γ ) β1 γ α1 A γ )) y A γ ) α1 α A γ )) 0 3 From 31 we obtain fixed points In the sequel we consider 3 Discriminant of 3 is given by Δ : A γ ) 4 γ α 1 A γ )) A γ ) α1 α A γ )) ) 33 Real solutions of 3 exist if and only if Δ 0 The solutions are given by ) )) A γ α1 α A γ Δ y 1 ) γ A γ y A γ ) α1 α A γ )) Δ γ A γ ) 34 Using 30 we have )) ) α1 α A γ A γ Δ x 1 γ )) ) α1 α A γ A γ Δ x γ 35 Claim Assume Δ 0 Then i for all values of parameters y 1 < 0; ii for all values of parameters x < 0 Proof 1 Assume α 1 α A γ > 0 Then it is obvious that the claim y 1 < 0istrue Now assume α 1 α A γ 0 Then y 1 < 0 if and only if Δ A γ ) α1 α A γ )) > 0 36 which is equivalent to 4 γ A γ ) α1 A γ )) > 0 37

10 10 Advances in Difference Equations This is true since α1 A γ )) < α1 α A γ )) 0 38 Assume α 1 α A γ < 0 Then it is obvious that x < 0 Now assume α1 α A γ )) 0 39 Then x < 0 if and only if Δ α 1 α A γ )) A γ ) > 0 40 This is equivalent to 4 γ A γ ) A α1 α A ) α γ γ ) > 0 41 Using 39 we have ) ) A α1 α A α γ γ A γ α γ γ α 1 γ > 0 4 which implies that the inequality 41 is true Now the proof of the Lemma follows from the Claim Lemma 3 The map T associated to System 11 satisfies the following: T x y ) x y ) only for x y ) x y ) 43 Proof By using 1 we have T x y ) x y ) α 1 x y α 1 x y α γ y A x α γ y A x 44 First equation implies α 1 y y ) β1 xy xy ) 0 45 Second equation implies α x x γ A y y ) γ xy yx ) 46

11 Advances in Difference Equations 11 Note the following xy xy x x y x y y ) 47 Using 47 Equations 45 and 46 respectively become y x x α 1 x ) y y ) 0 α γ y ) x x γ A x y y ) 0 48 Note that System 48 is linear homogeneous system in x x and y y The determinant of System 48 is given by y α 1 y α γ y γ A x 49 Using 1 the determinant of System 48 becomes y xy y A x γ A x y A x γ xy ) / 0 50 This implies that System 48 has only trivial solution that is x x y y 51 3 Linearized Stability Analysis The Jacobian matrix of the map T has the following form: J T y α 1 x y α γ y A x γ A x 31 The value of the Jacobian matrix of T at the equilibrium point is ) y J T x y y A x x y γ A x 3

12 1 Advances in Difference Equations The determinant of 3 is given by det J T x y ) γ xy y A x 33 The trace of 3 is Tr J T x y ) y γ A x 34 The characteristic equation has the form ) λ β1 λ y γ γ xy 0 A x y A x 35 Theorem 31 Assume that A <γ Then there exists a unique positive equilibrium E 1 which is a saddle point and the following statements hold a If γ A <α 1 α then λ 1 1 and λ 1 0 b If γ A >α 1 α A <α < γ and α 1 γ α < γ α A then λ 1 1 and λ 0 1 c If γ A >α 1 α α > γ and α 1 α γ > γ α A then λ 1 1 and λ 0 1 d If γ A >α 1 α α > γ and α 1 α γ < γ α A then λ 1 1 and λ 1 0 Proof The equilibrium is a saddle point if and only if the following conditions are satisfied: Tr J T x y ) > 1 det J T x y ) Tr J T x y ) 4detJT x y ) > 0 36 The first condition is equivalent to y γ A x > 1 γ xy y A x 37 This implies the following: A x γ y>y A x γ xy A x y ) γ y β1 ) > xy 38 y ) A γ x ) < xy

13 Advances in Difference Equations 13 Notice the following: y 1 α ) 1 α A γ α 1 α A γ ) 4α 1 A γ ) A γ α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ ) A γ x A γ 39 That is y 1 x A γ 310 Similarly A γ x 1 A γ [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ β ) [ α1 )] ) 1 A γ α1 α α A γ 4α1 A γ β ) [ α1 )] ) 1 A γ α1 α α A γ 4α1 A γ α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ A γ y A γ 311 Now we have y1 ) A γ x 1 ) < x1 y 1 x A γ y A γ < x 1 y 1 31

14 14 Advances in Difference Equations This is equivalent to x y < x 1 y The last condition is equivalent to x y x 1 y 1 < 0 y x 1 x x 1 y1 y )) < which is true since x 1 > x and y 1 > y The second condition is equivalent to ) β1 y γ γ 4 A x y A x 4 xy y A x > This is equivalent to ) β1 y γ xy 4 A x y A x > establishing the proof of Theorem 31 Since the map T is strongly competitive the Jacobian matrix 3 has two real and distinct eigenvalues with the larger one in absolute value being positive From 35 at E 1 we have λ 1 λ γ y 1 A x 1 λ 1 λ γ x 1 y 1 y 1 A x The first equation implies that either both eigenvalues are positive or the smaller one is negative Consider the numerator of the right-hand side of the second equation We have γ x 1 y 1 γ α ) ) 1 α A γ D α 1 α A γ D ) A γ D α1 α A γ ) γ β ) 1 γ A α1 α D 318 where D α 1 α A γ 4α 1 A γ

15 Advances in Difference Equations 15 a If γ A <α 1 α then the smaller root is negative that is λ 1 0 If γ A > α 1 α then γ A ) α1 α > D β 1 γ A γ A ) α1 α α 1 α > α 1 α α 1 α ) ) A γ β ) 1 A γ 4α1 A γ ) ) γ β1 A α α1 α γ > From the last inequality statements b c and d follow We now perform a similar analysis for the other cases in Table 1 Theorem 3 Assume A >γ A γ ) <α α 1 [ α1 α A γ )] 4α1 A γ ) > 0 30 Then E 1 E exist E 1 is a saddle point; E is a sink For the eigenvalues of E 1 λ 1 E 1 1 the following holds a If γ <α < A then λ 0 1 b If α > A and α 1 α γ < γ α A then λ 1 0 c If α < γ and γ A α >α 1 γ α then λ 0 1 Proof Note that if A α < 0andα γ < 0 then α > A and α < γ which implies A <γ which is a contradiction The equilibrium is a sink if the following condition is satisfied: Tr JT x y ) < 1 det JT x y ) < 31 The condition Tr J T x y < 1 det J T x y is equivalent to y γ A y < 1 γ y A x xy y A x 3 This implies A x γ y<y A x γ xy A x y ) γ y β1 ) < xy 33 y ) A γ x ) > xy Now we prove that E is a sink

16 16 Advances in Difference Equations We have to prove that y ) A γ x ) > x y 34 Notice the following: y α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ ) A γ [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ ) A γ [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ ) A γ A γ x 1 A γ 35 Similarly A γ x A γ [ )] [ α1 )] ) α 1 α A γ α A γ 4α1 A γ α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ α ) [ α1 )] ) 1 α A γ α A γ 4α1 A γ A γ ) A γ y 1 A γ 36 Now condition y ) A γ x ) > x y 37

17 Advances in Difference Equations 17 becomes x 1 A γ y 1 A γ > x y 38 that is x 1 y 1 > x y 39 which is true see Theorem 31 Condition 1 det JT x y ) < 330 is equivalent to γ y A x xy y A x < This implies γ xy <y A x γ 1 ya < xy 33 We have to prove that γ y A < x y 333 Using we have ) ) α1 α1 γ A < x 334 x x This is equivalent to γ A ) < α1 x α 1A x 335 which is always true since A >γ and the left side is always negative while the right side is always positive

18 18 Advances in Difference Equations Notice that conditions x 1 y 1 > x y ) β1 y γ xy 4 A x y A x > imply that E 1 is a saddle point From 35 at E 1 we have λ 1 λ γ y 1 A x 1 λ 1 λ γ x 1 y 1 y 1 A x The first equation implies that either both eigenvalues are positive or the smaller one is negative Consider the numerator of the right-hand side of the second equation We have γ x 1 y 1 γ α ) ) 1 α A γ D α 1 α A γ D ) A γ D α1 α A γ ) γ β ) 1 γ A α1 α D 338 We have γ xy >0 β ) 1 γ A α1 α D > Inequality γ A ) α1 α D>0 340 is equivalent to γ β1 A α ) α1 α γ ) > which is obvious if γ <α < A Then inequality 341 holds This confirms a The other cases follow from 341

19 Advances in Difference Equations 19 Theorem 33 Assume A >γ A γ ) <α α 1 [ α1 α A γ )] 4α1 A γ ) 0 34 Then there exists a unique positive equilibrium point ) α α 1 A γ E 1 E β ) ) 1 A γ α1 α ) 343 A γ which is non-hyperbolic The following holds a If A γ >α 1 α then λ 1 1 and λ 0 1 b If A γ <α 1 α then λ 1 1 and λ 1 0 Proof Evaluating the Jacobian matrix 3 at equilibrium E 1 E α α 1 A γ / A γ α 1 α / A γ we have ) J T x y )) ) β1 A γ α1 α ) / α α 1 A γ /β1 ) ) A γ β1 A γ α1 α ) / ) A γ ) β1 A γ α1 α ) / ) A γ A γ )) ) α α 1 A γ /β1 A )) ) α α 1 A γ /β1 344 The characteristic equation of J T x y is det ) ) J T x y λi )) ) β1 A γ α1 α ) / α α 1 A γ /β1 ) λ ) A γ β1 A γ α1 α ) / ) A γ ) β1 A γ α1 α ) / ) A γ A γ )) ) α α 1 A γ /β1 A )) ) λ α α 1 A γ /β which is simplified to ) λ λ ) β1 A γ α1 α ) / γ ) A γ A )) ) α α 1 A γ /β1 γ )) ) ) α α 1 A γ /β1 β1 A γ α1 α ) / )) A γ ) β1 A γ α1 α ) / )) A γ A )) )) 0 α α 1 A γ /β1 346

20 0 Advances in Difference Equations Solutions of 346 are λ 1 1 α 1 λ α 1α α A A β 1 γ 3β 1 γ α 1 α 1α α A α 1 A α A γ 347 Note that λ can be written in the following form: λ α 1 α β 1 A γ ) β 1 γ ) A γ α 1 α A α α 1 β1 A γ ) 348 Note that λ < 1 The corresponding eigenvectors respectively are A ) γ 1 ) ) ) ) A γ α1 α A γ α1 α A γ ) 1 α1 α A γ 349 Note that the denominator of 348 is always positive Consider numerator of 348 α 1 α β 1 ) A γ β 1 γ ) A γ 350 From [ α1 α A γ )] 4α1 A γ ) we have α 1 α 4α 1 A γ ) β1 α 1 α A γ ) β 1 A γ ) 35 Substituting α 1 α from 35 in 350 weobtain ) 4α 1 A γ β1 α 1 α ) ) A γ β A γ β 1 γ ) A γ A γ ) β1 A γ ) α1 α ) 353 Now 348 becomes establishing the proof of the theorem λ β ) 1 A γ β1 A α α 1 α ) α 1 α A α α 1 β1 A γ ) 354

21 Advances in Difference Equations 1 Now we consider the special case of System 11 when A γ In this case system 11 becomes x n 1 α 1 x n y n y n 1 α A x n A y n n Equilibrium points are solutions of the following system: x α 1 x y y α A y A x 356 The second equation implies x α α 1 α >α Now the first equation implies The map T associated to System 355 is given by y α α α 1 α >α T x y ) α1 x α ) A y 359 y A x The Jacobian matrix of the map T has the following form: J T y α 1 x y α A y A x γ A x 360 The value of the Jacobian matrix of T at the equilibrium point is ) y J T x y y A x x y A A x 361

22 Advances in Difference Equations The determinant of 361 is given by det J T x y ) A xy y A x 36 The trace of 361 is Tr J T x y ) y A A x 363 Theorem 34 Assume A γ α >α Then there exists a unique positive equilibrium point E α α 1 α α α 1 ) 365 of system 11 which is a saddle point The following statements hold a If A α > 0 then λ 1 1 and λ 0 1 b If A α < 0 then λ 1 1 and λ 1 0 Proof We prove that E is a saddle point We check the conditions Tr JT x y ) > 1 det JT x y ) Tr J T x y ) 4detJT x y ) > Condition Tr J T x y > 1 det J T x y is equivalent to y A A x > 1 A A x y x A x 367 This implies A x A y>ya xy A xy x>0 368 Condition Tr J T x y ) 4detJT x y ) > 0 369

23 Advances in Difference Equations 3 is equivalent to ) β1 y γ xy 4 A x y A x > Hence E is a saddle point Now λ 1 λ y λ 1 λ A α y A x 371 The first equation implies that either both eigenvalues are positive or the smaller one is less then zero The second equation implies that If A >α then λ 1 1 λ 0 1 ; If A <α then λ 1 1 λ establishing the proof of theorem 4 Global Behavior Theorem 41 Assume A <γ 41 Then system 11 has a unique equilibrium point E 1 which is a saddle point Furthermore there exists the global stable manifold W s E 1 that separates the positive quadrant so that all orbits below this manifold are asymptotic to 0 and all orbits above this manifold are asymptotic to 0 All orbits that start on W s E 1 are attracted to E 1 The global unstable manifold W u E 1 is the graph of a continuous unbounded strictly decreasing function Proof The existence of the global stable manifold W s E 1 with the stated properties follows from Theorems 15 17and18 and Lemmas 1 and Theorem 4 Assume A >γ A γ ) <α α 1 [ α1 α A γ )] 4α1 A γ ) > 0 4 Then system 11 has two equilibrium points: E 1 which is a saddle point and E which is a sink Furthermore there exists the global stable manifold W s E 1 that separates the positive quadrant so that all orbits below this manifold are asymptotic to 0 and all orbits above this manifold are attracted to equilibrium E All orbits that start on W s E 1 are attracted to E 1 The global unstable

24 4 Advances in Difference Equations manifold W u E 1 is the graph of a continuous unbounded strictly decreasing function with end point E Proof The existence of the global stable manifold W s E with the stated properties follows from Theorems 15 17and18 and Lemmas 1 and Theorem 43 Assume A >γ A γ ) <α α 1 [ α1 α A γ )] 4α1 A γ ) 0 43 Then system 11 has a unique equilibrium E E 3 which is non-hyperbolic The sequences {x n } {x n 1 } {y n } and {y n 1 } are eventually monotonic Every solution that starts in Q 4 E is asymptotic to 0 and every solution that starts in Q E is asymptotic to the equilibrium E Furthermore there exists the global stable manifold W s E that separates the positive quadrant into three invariant regions so that all orbits below this manifold are asymptotic to 0 and all orbits that start above this manifold are attracted to the equilibrium E All orbits that start on W s E are attracted to E Proof The existence of the global stable manifold W s E with the stated properties follows from Theorems 15 17and18 and Lemmas 1 and First we prove that for all points M x x α 1 /x x/ 0 the following holds: M x se T M x 44 Observe that M x is actually an arbitrary point on the curve x α 1 x /y which represents one of two equilibrium curves for system 11 Indeed T ) x α 1 /x x α )) γ α1 /x A x x α x γ α1 x ) ) A x x 45 Now we have x α ) 1 x se x α x γ α1 x ) ) x A x x x α 1 x x α x γ α1 x ) x A x 46 The last inequality is equivalent to α1 x ) A x α x γ α 1 γ x 47

25 Advances in Difference Equations 5 This is equivalent to x x α 1 α A γ )) α1 A γ ) 0 48 which always holds since the discriminant of the quadratic polynomial on the left-hand side is zero Note that M x se E and M x E for x α α 1 A γ / Monotonicity of the map T implies T n M x se T n 1 M x 49 Set T n M x { x n y n } Then the sequence {x n } is increasing and bounded by x-coordinate of the equilibrium and the sequence {y n } is decreasing and bounded by y-coordinate of the equilibrium This implies that { x n y n } converges to the equilibrium as n Now take any point A x y Q E Then there exists point M x such that M x se A x y se E By using monotonicity of the map T we obtain T n M x se T n A x y )) se E 410 Letting n in 410 we have lim n T n A x y )) E 411 Now we consider Q 4 E By choosing M x such that E se M x wenotethat E se M x se T M x 41 By using monotonicity of the map T we have T n M x se T n 1 M x 413 Set T n M x { x n y n } Then the sequence {x n } is increasing and the sequence {y n } is decreasing and bounded by y-coordinate of equilibrium and has to converge If {x n } converges then { x n y n } has to converge to the equilibrium which is impossible This implies that x n n Since y n 1 α γ y n / A x n then lim n y n 0 Now take any point B x y in Q 4 E Then there is point M x such that E se M x se B x y Using monotonicity of the map T we have E se T n M x se T n B x y )) 414 Since T n M x is asymptotic to 0 then lim n T n B x y 0

26 6 Advances in Difference Equations Theorem 44 Assume A γ α >α Then system 11 has a unique equilibrium E which is a saddle point Furthermore there exists the global stable manifold W s E that separates the positive quadrant so that all orbits below this manifold are asymptotic to 0 and all orbits above this manifold are asymptotic to 0 All orbits that start on W s E are attracted to E The global stable manifold W u E is the graph of a continuous unbounded strictly increasing function Proof The existence of the global stable manifold W s E with the stated properties follows from Theorems 15 17and18 and Lemmas 1 and Theorem 45 Assume A >γ [ α1 α A γ )] 4α1 A γ ) < A >γ [ α1 α A γ )] 4α1 A γ ) 0 β1 A γ ) >α α or A γ α α Then system 11 does not possess an equilibrium point Its global behavior is described as follows: x n y n 0 n 419 Proof If the conditions of this theorem are satisfied then 6 implies that there is no real if the first condition of this theorem is satisfied or positive equilibrium points if the second condition of this theorem is satisfied Consider the second equation of system 11 Thatis y n 1 α γ y n A x n 40 Note the following y n 1 α A γ A y n 41 Now consider equation u n 1 γ A u n α A 4

27 Advances in Difference Equations 7 Its solution is given by ) n γ α u n c A A γ 43 Since A >γ then letting n we obtain that u n 0 Now 41 implies y n α A γ ε n 44 This means that sequence {y n } is bounded for A >γ In order to prove the global behavior in this case we decompose System 11 into the system of even-indexed and odd-indexed terms as x n 1 α 1 x n y n x n α 1 x n 1 y n 1 y n 1 α γ y n A x n 45 y n α γ y n 1 A x n 1 for n 1 Lemma 1 implies that subsequences {x n 1 } {x n } {y n 1 } and {y n } are eventually monotone Since sequence {y n } is bounded then the subsequences {y n 1 } and {y n } must converge If the sequences {x n 1 } and {x n } would converge to finite numbers then the solution of 11 would converge to the period-two solution which is impossible by Lemma Thus at least one of the subsequences {x n 1 } and {x n } tends to n Assume that x n as n In view of third equation of 45 y n 1 0 and in view of first equation of 45 x n 1 which by fourth equation of 45 implies that y n 1 0asn Now we prove the case when A γ and α α 1 In this case System 11 becomes x n 1 α 1 x n y n y n 1 α 1 A y n A x n 46 The map T associated to System 46 is given by T x y ) α1 x α ) 1 A y 47 y A x

28 8 Advances in Difference Equations Equilibrium curves C 1 and C can be given explicitly as the following functions of x : C 1 : y 1 α 1 x C : y α 1 x 48 It is obvious that these two curves do not intersect which means that System 46 does not possess an equilibrium point Similarly as in the proof of Theorem 43 for all points C 1 x x α 1 /x x/ 0 the following holds: C 1 x T C 1 x 49 Indeed α1 T x α ) α 1 x β 1 A 1 x x A x ) x α 1x A α1 x ) ) A x x 430 Now we have x α ) 1 x se x α 1x A α1 x ) ) A x x x x and α 1 x α 1x A α1 x ) A x x 431 The last inequality is equivalent to α1 x ) A x α 1 x A α 1 A x 43 which always holds Monotonicity of T implies T n C 1 x se T n 1 C 1 x 433 Set T n C 1 x { x n y n } Then the sequence {x n } is increasing and the sequence {y n } is decreasing Since {y n } is decreasing and y n > 0 n 1 then it has to converge If {x n } converges then { x n y n } has to converge to the equilibrium which is impossible This implies that x n n The second equation of System 46 implies that 0 n y n

29 Advances in Difference Equations 9 Now take any point x y R Then there exists point C 1 x x α 1 /x such that C 1 x se x y ) 434 Monotonicity of T implies T n C 1 x se T n x y ) 435 Set T n x y ) x n y n ) T n C 1 x x ny n) 436 Then we have x n x n y n y n 437 Since x n y n 0 n 438 we conclude using the inequalities 437 that x n y n 0 n 439 Similarly we can prove the case A γ α <α 1 References 1 ECamouzisMRSKulenović G Ladas and O Merino Rational systems in the plane Journal of Difference Equations and Applications vol 15 no 3 pp M R S Kulenović and O Merino Global bifurcation for discrete competitive systems in the plane Discrete and Continuous Dynamical Systems Series B vol 1 no 1 pp M R S Kulenović and O Merino Invariant manifolds for competitive discrete systems in the plane to appear in International Journal of Bifurcation and Chaos 4 Dž Burgić S KalabušićandMRSKulenović Nonhyperbolic dynamics for competitive systems in the plane and global period-doubling bifurcations Advances in Dynamical Systems and Applications vol 3 no pp Dž Burgić M R S Kulenović and M Nurkanović Global dynamics of a rational system of difference equations in the plane Communications on Applied Nonlinear Analysis vol 15 no 1 pp D Clark and M R S Kulenović A coupled system of rational difference equations Computers & Mathematics with Applications vol 43 no 6-7 pp D Clark M R S Kulenović and J F Selgrade Global asymptotic behavior of a two-dimensional difference equation modelling competition Nonlinear Analysis Theory Methods & Applications vol 5 no 7 pp

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